- Email: [email protected]
The size and material consideration on single bond characteristics and their application in elasticity of nanomaterials
Solid State Communications 299 (2019) 113655
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
The size and material consideration on single bond characteristics and their application in elasticity of nanomaterials
T
H. Li*, Y.Y. Shen, H.N. Du, H.X. Zhang College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan, 030024, China
ARTICLE INFO
ABSTRACT
Communicated by A. Nikita
A model for the size-dependent Young's modulus Y(D) of nanomaterials is established in this work through introducing their single bond characteristics, with D being size. The size effect on single bond energy ε(D) is evident, while bond length h(D) is directly related with both size and materials index (m) of nanomaterials, resulting in single bond modulus y(D) always not smaller than that in bulk. Based on y(D) and coordination number consideration, Y(D) ≥ Y0 is found for some metallic materials with m = 1, and Y(D) ≤ Y0 is observed for some other materials when their m values are respectively 2.56, 4, and 4.88. Moreover, a good agreement between model predictions and other experimental or simulation results is obtained.
Keywords: Size effect Thermodynamic Elasticity Modeling Nanomaterial
1. Introduction For nanostructures and nanometer-scale patterning, they are still attractive for scientific research since their many novel yet unexplored properties with the increasing demand for further miniaturization of nanodevices [1–3]. As one knows, the properties of nanomaterials have great difference from the corresponding bulk ones, and moreover many of these properties are not invariable, having great dependence on size [4–6]. This is mainly due to the surface/volume ratio of nanomaterials, where the broken bonds existing on surface atoms lead to their difference from that of bulk interior [7,8]. Except the change of many physical or chemical properties of nanomaterials, broken bonds of surface atoms will lead to the variation of the single bond characteristics in a nanosystem, since the structural relaxation inevitably occurs and this influence becomes more and more obvious when size is gradually decreased. One should note, both single bond energy ε and bond length h are two key parameters in determining interatomic potential V(h) in a crystal. In the process of structural relaxation, ε and h will be changed and behave as size-related parameters [9]. In the past, the lattice contraction of many metallic nanomaterials has been widely manifested [10]. And the single bond energy of Ag clusters is also reported to have an increase trend as size decreases [6]. Moreover, their size effects will be determined by material type to a large extent. As reported in Sun et al.‘s work [9], if introducing a parameter c which represents the bond length contraction coefficient in a nanomaterial, the bond length h(D) can be read as h(D) = ch0, where h0 means the single bond length in bulk interior. Then, the
*
corresponding bond energy has the form of ε(D) = c-mε0 with ε0 being bulk single bond energy and here m is material index, representing material type. As one knows, the parameter ′c′ is not a constant, depending on the coordination number of the atom and it's site in a nanomaterial [9], however, if considering h(D) being the average value of bond length in a nanomaterial, it's size dependence also possesses an similar relationship with the corresponding average value of ε(D). As a result, there is [9].
h (D)/ h 0 = [ (D )/ 0]
1/m
(1)
Even though bond contraction has been reported in some previous works, Eq. (1) provides another possible way to derive bond length in a nanomaterial. Experience have revealed that m = 1 is suitable for elemental metals, m ≈ 4 for alloys or compounds, and m has been optimized to be 2.56 and 4.88 for carbon and silicon, respectively [11]. As the important parameters in determining the interatomic potential in a nanomaterial, both h(D) and ε(D) must be known, because they directly determine atomic structure and materials' properties at atomic level. For example, if considering the elasticity of a material, the elasticity of a single bond y(D) could provide some useful information. This is because y(D) is closely related with both h(D) and ε(D), and so the variation of bond arrangement in different structures may change the local interatomic interaction and then change it's elasticity [12]. Obviously, as one part of a nanomaterial, y(D) inevitably influences the Young's modulus Y(D) of the whole nanomaterial [13]. However, how the relation between Y(D) and y(D) is, and how does Y(D) change with size, is still not completely clear for researchers [14]. Although to some extent, this is
Corresponding author. E-mail address: [email protected] (H. Li).
https://doi.org/10.1016/j.ssc.2019.113655 Received 16 February 2019; Received in revised form 27 April 2019; Accepted 10 June 2019 Available online 11 June 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 299 (2019) 113655
H. Li, et al.
due to technological challenge. Therefore, in this work, with the help of the size-dependent cohesive energy in nanomaterials and their coordination number, the models for size-dependent h(D) and ε(D) are well established, with some simulation and experimental results confirming their validity. Furthermore, we apply both h(D) and ε(D) into the filed of the elasticity of single bond and nanomaterials, providing the quantitative determination for both y(D) and Y(D). It is interesting to find the successful application of h(D) and ε(D) in predicting Y(D) of some nanomaterials, and so the model predictions are well consistent with the related theoretical or simulation results. Moreover one finds Y(D) ≥ Y0 in whole size range for Cu nanoparticles, while Y(D) ≤ Y0 is estimated for diamond nanoparticles, NaCl and SiC films. 2. Theoretical model Considering the surrounding environment of an atom, it's single bond energy is one part of it's cohesive energy, with coordination number (CN) connecting them. So, for a nanomaterials, it's cohesive energy Ec(D) can be expressed as Ec(D) = ε(D) × Ba, where Ba is the bond number concerned in the nanomaterial. Similarly, the corresponding bulk material will have Ec0 = ε0 × Bt, with Ec0 showing bulk cohesive energy and Bt the total bond number without any broken bond in a system. In this way, the size dependence of ε(D) can be resolved.
(D)/
0
(2)
= [Ec (D)/ Ec 0]/[Ba /Bt ]
The size dependence of cohesive energy is widely reported in previous work [6], and moreover a reasonable model for Ec(D) has been presented as the function of bond number, that is. (3)
Ec (D )/ Ec0 = [(Ba /Bt )1/2 + Ba / Bt ]/2 1/2
Another valid model for Ec(D) is Ec(D)/Ec0 = (Ba/Bt) , however it overestimates attractive interaction between atoms and is suitable in predicting the simulation results. Then, by using Eqs (2) and (3), the size dependence of ε(D) and even h(D) can be expressed.
(D)/
0
= [(Ba / Bt )
1/2
h (D)/ h 0 = {[(Ba /Bt )
(4.1)
+ 1]/2
1/2
Fig. 1. The function of (a) Ec(N), (b) ε(N) and (c) h(N) in terms of Eqs (4) and (6), respectively. The symbols ′■′, ′●′ denote the simulation results for Ag nanoparticles [6].
+ 1]/2}
1/ m
(4.2)
From math point, Ba ranges from zero to infinite, and so Ba/Bt → 0 as D or N → 0, while Ba/Bt → 1 as D or N → ∞. It is evident that Eqs (4)–(6) present the model predictions for ε(D), h(D), y(D) and Y(D) even in full size range.
Arriving here, it is clear that one just needs the bond numbers of Ba and Bt in order to obtain Ec(D), ε(D) and h(D). In fact, it is not difficult to get Ba and Bt if the nanomaterials’ structures are known, this is because there are Ba = [Zs(D) × Ns + Zb × (N-Ns)]/2 and Bt = [Zb × N]/ 2. Here, Zs(D) is the average CN of surface atoms in a nanomaterial, Zb is CN of bulk interior atoms, Ns is the surface atoms number, and N is the total atom number in a nanomaterial and has the similar meaning with size D. For example, for the nanoparticles with cubooctahedral structure, there is Ba/Bt = (10n3+6n2+6n)/(10n3+15n2+11n+3) [7]. Considering the interatomic potential V(h) in a ideal bulk crystal, it is easy to derive the Young's modulus y0 of the bulk single bond. Simply, it can be written as y0 = Cε0/h03 with ′C′ being a coefficient. If taking ′C′ as size independent, the single bond of the nanomaterial will have the similar form, that is y(D) = Cε(D)/h(D)3. So the size dependence of y(D) has y(D)/y0 = [ε(D)/ε0]/[h(D)/h0]3. Based on Eq. (4), y (D)/y0 can be expressed as the following.
y (D)/ y0 = {[(Ba / Bt )
1/2
+ 1]/2}1 + 3/m
3. Results and discussion Taking nanoparticles with cubooctahedral structure [7] as example, Fig. 1 shows the model predictions for the size dependence of Ec(N), ε(N) and h(N), associated with some simulation results of Ag nanoparticles to compare [6]. As one sees, when size increases, Ec(N) has an increase trend, and it is well consistent with many other reports [15,16]. However, the opposite trend exists for ε(N) with ε(N) ≥ ε0 being found even in full size range. This means the single bond strength is enhanced when size is dropped, which can be expected from the surface atoms relaxation in a nanomaterial. For both Ec(N) and ε(N), there are Ec(N) → Ec0 and ε(N) → ε0 when N → ∞ is satisfied. It should be noted that there have four theoretical curves in Fig. 1c for the change of bond length with size, even though all of their h(N) values behave as the increase trend with size. This is because the material index m is different. It is evident that there is a relatively large change of h(N) with size when m = 1, while it is smaller when m = 4.88. So, it seems that the larger m value will result in the larger in h(N) value. For example, when size is given, one can see the sequence of hm=1 < hm=2.56 < hm=4 < hm=4.88. To further confirm the model of Eq. (4.2), the comparison of model predictions and the corresponding experimental results for some metallic nanoparticles is made. As displayed in Fig. 2, their good consistence is obvious, where m = 1 is
(5)
Considering the interactions in metals and semiconductors are many body types, a premise that the elasticity of a whole nanomaterial is regarded as the summation of all single bonds is taken. So, for bulk Young's modulus Y0, there will be Y0 = Bt × y0, and correspondingly Young's modulus Y(D) of the nanomaterial will have Y(D) = Ba × y(D). In this way, the size-dependent Young's modulus can be read.
Y(D)/Y0 = (Ba / Bt ) × {[(Ba / Bt )
1/2
+ 1]/2}1 + 3/m
(6) 2
Solid State Communications 299 (2019) 113655
H. Li, et al.
Fig. 2. The change trends of h(D) with size as m = 1 (solid lines) respectively for (a) Cu, (b) Al, (c) Ag and (d) Pt nanoparticles. The symbols ′♦′ [17], ′◊′ [18], ′▼′ [19], ′★′[20] and ′▶′ [18] are the experimental results.
Fig. 4. The model predictions (solid lines) of Eq. (6) for (a) Cu nanoparticles, (b) diamond nanoparticles, (c) NaCl film and (d) Si(110) film. The symbols ′●′ [21], ′ ′ [22],′ ′ [23] and ′ ′ [24] are the theoretical or simulation results for comparison.
(4)–(6), the model predictions for Y(D) of nanomaterials are also made to compare the reported results. Considering material/bond type, m = 1 is used for Cu nanoparticles, m = 2.56 is applied for diamond nanoparticles, and m = 4, 4.88 are respectively used for NaCl for Si nanofilms. As displayed in Fig. 4, it is clearly found the stiffening of Cu nanoparticles, and the softening for diamond nanoparticles, NaCl and Si (110) films. The good agreement between model predictions of Eq. (6) and other theoretical or simulation results is obtained. All this means that, based on ε(D) and h(D), single bond modulus is one of the valid methods to resolve the stiffening or softening of nanomaterials, providing a possible way to analyze the mechanical behavior at atomic level. 4. Conclusion A reasonable model for Young's modulus Y(D) of nanomaterials is established in this work. It means the single bond characteristics including ε(D), h(D) and y(D) makes the clarification of Y(D) > Y0 or Y (D) < Y0 at atomic level become possible. Except size, material type is one of the important parameter in determining the elasticity of nanomaterials. The good agreement between model predictions for Y(D) and other researchers' results further confirms the reasonability of using single bond modulus to resolve Y(D).
Fig. 3. The model predictions for (a) y(N) and (b) Y(N) respectively when m = 1, 2.56, 4, and 4.88.
Acknowledgement
taken for these metallic nanoparticles. Similarly, the elasticity of both single bond and nanomaterials is determined not only by size, but also by m value. As presented in Fig. 3a, firstly one can find y(N) has an increase trend with size decreasing, and also it is interesting to find y(N) ≥ y0 in whole size range. This is the comprehensive result of both the enhanced bond strength in ε(N) and the shortened bond length in h(N). However, as shown in Fig. 3b, this case does not always exist in the elasticity Y(N) of nanomaterials. If compared with Y0, one can see Y(N) ≥ Y0 in the case of m = 1, while Y(N) ≤ Y0 is found as m = 2.56, 4 or 4.88 is used. That is to say, the change trend of Y(D) with size is also related with material or bond type. For example, Y(D) decreases with size increasing when m = 1, while the opposite trend exists when m = 2.56, 4, and 4.88 respectively. Finally, an interesting phenomenon should be mentioned. That is, under a given size, a sequences of ym=1 > ym=2.56 > ym=4 > ym=4.88 and Ym=1 > Ym=2.56 > Ym=4 > Ym=4.88 is obtained, which is completely different from that of h(N). To further confirm the reasonability of the derivations of Eqs
The authors acknowledge the financial supports of the National Natural Science Foundation of China (grant No. 11404235 and 51701137), and also acknowledge the support by Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
3
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Solid State Communications 299 (2019) 113655
H. Li, et al. 2007, p. 18 (Chapter 1). [12] J. Schiøtz, F.D. Di Tolla, K.W. Jacobsen, Nature 391 (1998) 561–563. [13] Y. Chen, T. Burgess, X. An, Y.W. Mai, H.H. Tan, J. Zou, S. Ringer, C. Jagadish, X. Liao, Nano Lett. 16 (2016) 1911–1916. [14] M. Dunaevskiy, P. Geydt, E. Lähderanta, P. Alekseev, T. Haggrén, J.P. Kakko, H. Jiang, H. Lipsanen, Nano Lett. 17 (2017) 3441–3446. [15] M. Goyal, B.R.K. Gupta, Chin. J. Phys. 56 (2018) 282–291. [16] S. Bhatt, R. Kumar, M. Kumar, Mod. Phys. Lett. B 31 (2017) 1750011–1750013. [17] G. Apai, J.F. Hamilton, J. Stohr, A. Thompson, Phys. Rev. Lett. 43 (1979) 165.
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H.J. Wasserman, J.S. Vermaak, Surf. Sci. 32 (1972) 168–174. J. Woltersdorf, A.S. Nepijko, E. Pippel, Surf. Sci. 106 (1981) 64–69. H.J. Wasserman, J.S. Vermaak, Surf. Sci. 22 (1970) 164–172. R. Dingreville, J. Qu, C. Mohammed, J. Mech. Phys. Solids 53 (2005) 1827–1854. X. Jiang, J. Zhao, C. Zhuang, B. Wen, X. Jiang, Diam. Relat. Mater. 19 (2010) 21–25. [23] C.T. Sun, H. Zhang, J. Appl. Phys. 93 (2003) 1212. [24] H. Sadeghian, J.F.L. Goosen, A. Bossche, B.J. Thijsse, F. van Keulen, Thin Solid Films 520 (2011) 391–399.
4
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
The size and material consideration on single bond characteristics and their application in elasticity of nanomaterials
T
H. Li*, Y.Y. Shen, H.N. Du, H.X. Zhang College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan, 030024, China
ARTICLE INFO
ABSTRACT
Communicated by A. Nikita
A model for the size-dependent Young's modulus Y(D) of nanomaterials is established in this work through introducing their single bond characteristics, with D being size. The size effect on single bond energy ε(D) is evident, while bond length h(D) is directly related with both size and materials index (m) of nanomaterials, resulting in single bond modulus y(D) always not smaller than that in bulk. Based on y(D) and coordination number consideration, Y(D) ≥ Y0 is found for some metallic materials with m = 1, and Y(D) ≤ Y0 is observed for some other materials when their m values are respectively 2.56, 4, and 4.88. Moreover, a good agreement between model predictions and other experimental or simulation results is obtained.
Keywords: Size effect Thermodynamic Elasticity Modeling Nanomaterial
1. Introduction For nanostructures and nanometer-scale patterning, they are still attractive for scientific research since their many novel yet unexplored properties with the increasing demand for further miniaturization of nanodevices [1–3]. As one knows, the properties of nanomaterials have great difference from the corresponding bulk ones, and moreover many of these properties are not invariable, having great dependence on size [4–6]. This is mainly due to the surface/volume ratio of nanomaterials, where the broken bonds existing on surface atoms lead to their difference from that of bulk interior [7,8]. Except the change of many physical or chemical properties of nanomaterials, broken bonds of surface atoms will lead to the variation of the single bond characteristics in a nanosystem, since the structural relaxation inevitably occurs and this influence becomes more and more obvious when size is gradually decreased. One should note, both single bond energy ε and bond length h are two key parameters in determining interatomic potential V(h) in a crystal. In the process of structural relaxation, ε and h will be changed and behave as size-related parameters [9]. In the past, the lattice contraction of many metallic nanomaterials has been widely manifested [10]. And the single bond energy of Ag clusters is also reported to have an increase trend as size decreases [6]. Moreover, their size effects will be determined by material type to a large extent. As reported in Sun et al.‘s work [9], if introducing a parameter c which represents the bond length contraction coefficient in a nanomaterial, the bond length h(D) can be read as h(D) = ch0, where h0 means the single bond length in bulk interior. Then, the
*
corresponding bond energy has the form of ε(D) = c-mε0 with ε0 being bulk single bond energy and here m is material index, representing material type. As one knows, the parameter ′c′ is not a constant, depending on the coordination number of the atom and it's site in a nanomaterial [9], however, if considering h(D) being the average value of bond length in a nanomaterial, it's size dependence also possesses an similar relationship with the corresponding average value of ε(D). As a result, there is [9].
h (D)/ h 0 = [ (D )/ 0]
1/m
(1)
Even though bond contraction has been reported in some previous works, Eq. (1) provides another possible way to derive bond length in a nanomaterial. Experience have revealed that m = 1 is suitable for elemental metals, m ≈ 4 for alloys or compounds, and m has been optimized to be 2.56 and 4.88 for carbon and silicon, respectively [11]. As the important parameters in determining the interatomic potential in a nanomaterial, both h(D) and ε(D) must be known, because they directly determine atomic structure and materials' properties at atomic level. For example, if considering the elasticity of a material, the elasticity of a single bond y(D) could provide some useful information. This is because y(D) is closely related with both h(D) and ε(D), and so the variation of bond arrangement in different structures may change the local interatomic interaction and then change it's elasticity [12]. Obviously, as one part of a nanomaterial, y(D) inevitably influences the Young's modulus Y(D) of the whole nanomaterial [13]. However, how the relation between Y(D) and y(D) is, and how does Y(D) change with size, is still not completely clear for researchers [14]. Although to some extent, this is
Corresponding author. E-mail address: [email protected] (H. Li).
https://doi.org/10.1016/j.ssc.2019.113655 Received 16 February 2019; Received in revised form 27 April 2019; Accepted 10 June 2019 Available online 11 June 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 299 (2019) 113655
H. Li, et al.
due to technological challenge. Therefore, in this work, with the help of the size-dependent cohesive energy in nanomaterials and their coordination number, the models for size-dependent h(D) and ε(D) are well established, with some simulation and experimental results confirming their validity. Furthermore, we apply both h(D) and ε(D) into the filed of the elasticity of single bond and nanomaterials, providing the quantitative determination for both y(D) and Y(D). It is interesting to find the successful application of h(D) and ε(D) in predicting Y(D) of some nanomaterials, and so the model predictions are well consistent with the related theoretical or simulation results. Moreover one finds Y(D) ≥ Y0 in whole size range for Cu nanoparticles, while Y(D) ≤ Y0 is estimated for diamond nanoparticles, NaCl and SiC films. 2. Theoretical model Considering the surrounding environment of an atom, it's single bond energy is one part of it's cohesive energy, with coordination number (CN) connecting them. So, for a nanomaterials, it's cohesive energy Ec(D) can be expressed as Ec(D) = ε(D) × Ba, where Ba is the bond number concerned in the nanomaterial. Similarly, the corresponding bulk material will have Ec0 = ε0 × Bt, with Ec0 showing bulk cohesive energy and Bt the total bond number without any broken bond in a system. In this way, the size dependence of ε(D) can be resolved.
(D)/
0
(2)
= [Ec (D)/ Ec 0]/[Ba /Bt ]
The size dependence of cohesive energy is widely reported in previous work [6], and moreover a reasonable model for Ec(D) has been presented as the function of bond number, that is. (3)
Ec (D )/ Ec0 = [(Ba /Bt )1/2 + Ba / Bt ]/2 1/2
Another valid model for Ec(D) is Ec(D)/Ec0 = (Ba/Bt) , however it overestimates attractive interaction between atoms and is suitable in predicting the simulation results. Then, by using Eqs (2) and (3), the size dependence of ε(D) and even h(D) can be expressed.
(D)/
0
= [(Ba / Bt )
1/2
h (D)/ h 0 = {[(Ba /Bt )
(4.1)
+ 1]/2
1/2
Fig. 1. The function of (a) Ec(N), (b) ε(N) and (c) h(N) in terms of Eqs (4) and (6), respectively. The symbols ′■′, ′●′ denote the simulation results for Ag nanoparticles [6].
+ 1]/2}
1/ m
(4.2)
From math point, Ba ranges from zero to infinite, and so Ba/Bt → 0 as D or N → 0, while Ba/Bt → 1 as D or N → ∞. It is evident that Eqs (4)–(6) present the model predictions for ε(D), h(D), y(D) and Y(D) even in full size range.
Arriving here, it is clear that one just needs the bond numbers of Ba and Bt in order to obtain Ec(D), ε(D) and h(D). In fact, it is not difficult to get Ba and Bt if the nanomaterials’ structures are known, this is because there are Ba = [Zs(D) × Ns + Zb × (N-Ns)]/2 and Bt = [Zb × N]/ 2. Here, Zs(D) is the average CN of surface atoms in a nanomaterial, Zb is CN of bulk interior atoms, Ns is the surface atoms number, and N is the total atom number in a nanomaterial and has the similar meaning with size D. For example, for the nanoparticles with cubooctahedral structure, there is Ba/Bt = (10n3+6n2+6n)/(10n3+15n2+11n+3) [7]. Considering the interatomic potential V(h) in a ideal bulk crystal, it is easy to derive the Young's modulus y0 of the bulk single bond. Simply, it can be written as y0 = Cε0/h03 with ′C′ being a coefficient. If taking ′C′ as size independent, the single bond of the nanomaterial will have the similar form, that is y(D) = Cε(D)/h(D)3. So the size dependence of y(D) has y(D)/y0 = [ε(D)/ε0]/[h(D)/h0]3. Based on Eq. (4), y (D)/y0 can be expressed as the following.
y (D)/ y0 = {[(Ba / Bt )
1/2
+ 1]/2}1 + 3/m
3. Results and discussion Taking nanoparticles with cubooctahedral structure [7] as example, Fig. 1 shows the model predictions for the size dependence of Ec(N), ε(N) and h(N), associated with some simulation results of Ag nanoparticles to compare [6]. As one sees, when size increases, Ec(N) has an increase trend, and it is well consistent with many other reports [15,16]. However, the opposite trend exists for ε(N) with ε(N) ≥ ε0 being found even in full size range. This means the single bond strength is enhanced when size is dropped, which can be expected from the surface atoms relaxation in a nanomaterial. For both Ec(N) and ε(N), there are Ec(N) → Ec0 and ε(N) → ε0 when N → ∞ is satisfied. It should be noted that there have four theoretical curves in Fig. 1c for the change of bond length with size, even though all of their h(N) values behave as the increase trend with size. This is because the material index m is different. It is evident that there is a relatively large change of h(N) with size when m = 1, while it is smaller when m = 4.88. So, it seems that the larger m value will result in the larger in h(N) value. For example, when size is given, one can see the sequence of hm=1 < hm=2.56 < hm=4 < hm=4.88. To further confirm the model of Eq. (4.2), the comparison of model predictions and the corresponding experimental results for some metallic nanoparticles is made. As displayed in Fig. 2, their good consistence is obvious, where m = 1 is
(5)
Considering the interactions in metals and semiconductors are many body types, a premise that the elasticity of a whole nanomaterial is regarded as the summation of all single bonds is taken. So, for bulk Young's modulus Y0, there will be Y0 = Bt × y0, and correspondingly Young's modulus Y(D) of the nanomaterial will have Y(D) = Ba × y(D). In this way, the size-dependent Young's modulus can be read.
Y(D)/Y0 = (Ba / Bt ) × {[(Ba / Bt )
1/2
+ 1]/2}1 + 3/m
(6) 2
Solid State Communications 299 (2019) 113655
H. Li, et al.
Fig. 2. The change trends of h(D) with size as m = 1 (solid lines) respectively for (a) Cu, (b) Al, (c) Ag and (d) Pt nanoparticles. The symbols ′♦′ [17], ′◊′ [18], ′▼′ [19], ′★′[20] and ′▶′ [18] are the experimental results.
Fig. 4. The model predictions (solid lines) of Eq. (6) for (a) Cu nanoparticles, (b) diamond nanoparticles, (c) NaCl film and (d) Si(110) film. The symbols ′●′ [21], ′ ′ [22],′ ′ [23] and ′ ′ [24] are the theoretical or simulation results for comparison.
(4)–(6), the model predictions for Y(D) of nanomaterials are also made to compare the reported results. Considering material/bond type, m = 1 is used for Cu nanoparticles, m = 2.56 is applied for diamond nanoparticles, and m = 4, 4.88 are respectively used for NaCl for Si nanofilms. As displayed in Fig. 4, it is clearly found the stiffening of Cu nanoparticles, and the softening for diamond nanoparticles, NaCl and Si (110) films. The good agreement between model predictions of Eq. (6) and other theoretical or simulation results is obtained. All this means that, based on ε(D) and h(D), single bond modulus is one of the valid methods to resolve the stiffening or softening of nanomaterials, providing a possible way to analyze the mechanical behavior at atomic level. 4. Conclusion A reasonable model for Young's modulus Y(D) of nanomaterials is established in this work. It means the single bond characteristics including ε(D), h(D) and y(D) makes the clarification of Y(D) > Y0 or Y (D) < Y0 at atomic level become possible. Except size, material type is one of the important parameter in determining the elasticity of nanomaterials. The good agreement between model predictions for Y(D) and other researchers' results further confirms the reasonability of using single bond modulus to resolve Y(D).
Fig. 3. The model predictions for (a) y(N) and (b) Y(N) respectively when m = 1, 2.56, 4, and 4.88.
Acknowledgement
taken for these metallic nanoparticles. Similarly, the elasticity of both single bond and nanomaterials is determined not only by size, but also by m value. As presented in Fig. 3a, firstly one can find y(N) has an increase trend with size decreasing, and also it is interesting to find y(N) ≥ y0 in whole size range. This is the comprehensive result of both the enhanced bond strength in ε(N) and the shortened bond length in h(N). However, as shown in Fig. 3b, this case does not always exist in the elasticity Y(N) of nanomaterials. If compared with Y0, one can see Y(N) ≥ Y0 in the case of m = 1, while Y(N) ≤ Y0 is found as m = 2.56, 4 or 4.88 is used. That is to say, the change trend of Y(D) with size is also related with material or bond type. For example, Y(D) decreases with size increasing when m = 1, while the opposite trend exists when m = 2.56, 4, and 4.88 respectively. Finally, an interesting phenomenon should be mentioned. That is, under a given size, a sequences of ym=1 > ym=2.56 > ym=4 > ym=4.88 and Ym=1 > Ym=2.56 > Ym=4 > Ym=4.88 is obtained, which is completely different from that of h(N). To further confirm the reasonability of the derivations of Eqs
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