Accepted Manuscript Title: Size, shape and temperature dependent surface energy of binary alloy nanoparticles Author: Mohammad Amin Jabbareh PII: DOI: Reference:
S0169-4332(17)32242-0 http://dx.doi.org/doi:10.1016/j.apsusc.2017.07.242 APSUSC 36776
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APSUSC
Received date: Revised date: Accepted date:
10-4-2017 21-7-2017 26-7-2017
Please cite this article as: Mohammad Amin Jabbareh, Size, shape and temperature dependent surface energy of binary alloy nanoparticles, Applied Surface Sciencehttp://dx.doi.org/10.1016/j.apsusc.2017.07.242 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Size, shape and temperature dependent surface energy of binary alloy nanoparticles Mohammad Amin Jabbareh* Department of materials engineering, Faculty of engineering, Hakim Sabzevari University, Sabzevar, Iran.
*E-mail address:
[email protected] Tel: +(98) 51 44012776; fax: +(98) 51 44012773 Address: Hakim Sabzevari university, Towhid shahr, Sabzevar, Iran, P.O.Box: 9617976487.
Highlights
Based on the liquid drop model and Butler's equation, a new model is proposed for prediction of the alloy nanoparticles surface energy.
The model incorporates the effects of size, concentration, temperature and the shape of nanoparticles to calculation of the surface energy.
The results show that the value of surface energy decreases with decreasing particle size, increasing temperature and increasing shape factor of the particle.
Abstract Surface energy has an important role in determining the properties of nanoparticles. Even though, extensive research has been done on the surface energy of pure nanoparticles, the surface energy of alloy nanoparticles has not been considered enough. In this work, based on the liquid drop model for surface energy of pure nanoparticles and Butler's equation, a model for size dependent surface energy of alloy nanoparticles has been developed. In addition to size and concentration, the model can describe the effects of shape and temperature on surface energy of alloy nanoparticles. Cu – Ag and Cu – Au systems have been studied as two examples and the results have been compared with other theoretical models and available simulated data. Reasonable agreements between the results were observed. The results show that the decreasing particle size decreases surface energy of alloy nanoparticles but decreasing temperature and shape factor increases the value of surface energy. Keywords: surface energy; alloy nanoparticle; size effect; shape effect.
1. Introduction In recent years, alloy nanoparticles have attracted the interest of scientists due to their potential for useing as catalysts [1–3], nanosolders [4,5], sensors [6,7] and their medical applications [8]. It has been shown that the physical, chemical and mechanical properties such as melting and Curie temperatures, chemical reactivity, Thermal conductivity and Young’s modulus of nanoparticles are quite different from those of corresponding bulk materials [9]. The reasons for special characteristic of nanoparticles lie in increased surface to volume ratio, so the surface characteristic of nanoparticles plays an important role in determining the properties of these materials. One of the most important physical quantities, which can explain the characteristics of the surfaces, is the surface energy. So to specify the properties of nanoparticles, determination of the surface energy is essential. Because of the small size of nanoparticles, experimental measurement of the surface energy is very difficult and therefore theoretical models are commonly used to study the surface energy of nanoparticles [10]. Due to its importance, extensive studies have been done on the surface energy of pure nanoparticles and various models have been proposed to calculate the surface energy [11]; However, rare studies focused on the surface energy of the alloy nanoparticles [12,13]. Based on Butler's equation, Liu et al. [12] proposed a model for calculation of the surface energy of liquid binary alloy nano – droplets and studied the effects of size and composition on the surface energy of Bi – Sn binary alloy. Recently Takrori and Ayyad [13] developed a model for calculation of the surface energy of alloy metallic nanoparticles. The model describes the effect of particle size and composition on the surface energy of binary alloy nanoparticles but ignores the effects of nanoparticle shapes and temperature, whereas previous studies showed that the temperature[14,15] and also particle shapes [16–18] could affect the surface energy of solid nanoparticles. To the author's knowledge, this model is the only available model for prediction of size dependent surface energy of nanoparticles. So in this work a simple thermodynamic procedure is developed for calculation of binary alloy nanoparticles surface energy which can describe the effects of temperature and particle shapes as well as size and composition on the surface energy. The model is based on the previously developed liquid drop model for the surface energy of
pure nanoparticles [17] and Butler's equation [19]. Cu – Au and Cu – Ag binary systems have been chosen as examples.
2. Model According to the liquid drop model, the size dependent surface energy of a solid nanoparticle is given by Eq. (1) [17] 𝜎 = 𝜎 ∞ (1 − 2𝑐𝛽/𝑟)
(1)
where 𝜎 ∞ is the bulk surface energy, 𝑟 is the particle radius, 𝑐 is the shape factor which is equal to unity for spherical nanoparticles [20] and 𝛽 is a material constant related to atomic volume [17] as 6𝑣
𝜎
𝛽 = 0.022944 (𝑇 ∞) 𝑚
(2)
Where 𝜎 is the bulk surface energy at melting temperature, 𝑇𝑚∞ , is the melting temperature of bulk material and 𝑣, is the atomic volume. To calculate the surface energy of a binary alloy nanoparticle, Eq. (1) can be rewritten as follows ∞ 𝜎𝐴𝐵 = 𝜎𝐴𝐵 (1 − 2𝑐𝛽𝐴𝐵 /𝑟)
(3)
∞ where 𝜎𝐴𝐵 is the surface energy of bulk alloy and 𝛽𝐴𝐵 is the alloy constant .
Considering that the value of 𝜎/𝑇𝑚∞ for most fcc metals is approximately equal to unity [21], the problem of 𝛽𝐴𝐵 calculation reduced to calculation of atomic volume of A – B alloy. According to the Zen's law [22] if two components of the binary alloy have similar crystal structure, atomic volume of the alloy can be simply derived from Eq. (4). 𝑣𝐴𝐵 = (1 − 𝑋𝐵 )𝑣𝐴 + 𝑋𝐵 𝑣𝐵
(4)
where 𝑋𝐵 , is the mole fraction of B component. So, 𝛽𝐴𝐵 can be calculated as 𝛽𝐴𝐵 = (1 − 𝑋𝐵 )𝛽𝐴 + 𝑋𝐵 𝛽𝐵
(5)
It should be mentioned that the Zen's law not often perfectly obeyed, however it is commonly uses in practice to obtain rough estimates when experimental data are not available [23]. For example, according to our calculations in the case of Ag – Cu, the errors are between 0.0% and 15 % at different compositions. The surface energy of liquid alloys can be successfully calculated from Butler’s equation [19]. If the differences in shape and surface strain with respect to composition are ignorable, Butler’s equation also can use for prediction of solid alloys surface energy [24]. Butler’s equation for an A – B binary alloy is expressed as Eq. (6)
𝑅𝑇 𝐴
𝑅𝑇
𝑋
𝑆𝑢𝑟𝑓
𝑋𝐴
∞ 𝜎𝐴𝐵 = 𝜎𝐴∞ + 𝑆 ln (
𝑋𝐴𝐵𝑢𝑙𝑘
𝑆𝑢𝑟𝑓
1
1
𝐸𝑥,𝑆𝑢𝑟𝑓
) + 𝑆 [𝐺𝐴 𝐴
𝐸𝑥,𝑆𝑢𝑟𝑓
𝜎𝐵∞ + 𝑆 ln (𝑋𝐵𝐵𝑢𝑙𝑘 ) + 𝑆 [𝐺𝐵 𝐵
𝐵
𝐵
− 𝐺𝐴𝐸𝑥,𝐵𝑢𝑙𝑘 ] =
− 𝐺𝐵𝐸𝑥,𝐵𝑢𝑙𝑘 ]
(6)
where 𝜎𝐴∞ and 𝜎𝐵∞ are the bulk surface energy and 𝑆𝐴 and 𝑆𝐵 are the molar surface areas of 𝑆𝑢𝑟𝑓
the components A and B respectively. 𝑋𝐴
𝑆𝑢𝑟𝑓
and 𝑋𝐵
are concentrations of the
component A and B in the surface, and 𝑋𝐴𝐵𝑢𝑙𝑘 and 𝑋𝐵𝐵𝑢𝑙𝑘 are concentrations of the 𝐸𝑥,𝑆𝑢𝑟𝑓
component A and B in the bulk. 𝐺𝐴
𝐸𝑥,𝑆𝑢𝑟𝑓
(𝐺𝐵
) and 𝐺𝐴𝐸𝑥,𝐵𝑢𝑙𝑘 (𝐺𝐵𝐸𝑥,𝐵𝑢𝑙𝑘 ) are the partial
excess Gibbs energy of the component A (B) in the surface and the bulk phases. The molar surface area of each component can be determined by Eq. (7) [25] 𝑆𝑖 = 1.091 𝑁0 (𝑉𝑖 )2/3 (𝑖 = 𝐴 𝑜𝑟 𝐵)
(7)
where 𝑁0 is the Avogadro’s number and 𝑉𝑖 is the molar volume of the component 𝑖. According to Yeum’s model [26] the surface excess Gibbs energy is related to that of the bulk phase as follows 𝐸𝑥,𝑆𝑢𝑟𝑓
𝐺𝑖
= 𝛼𝐺𝑖𝐸𝑥,𝐵𝑢𝑙𝑘 (𝑖 = 𝐴 𝑜𝑟 𝐵)
(8)
Where 𝛼 is a parameter corresponding to the ratio of the coordination number in the surface to that in the bulk. If only the ratio of the coordination number is considered, calculations will be shown that α=0.75 for all closed packed crystal structures. However, it has been shown that due to the surface relaxation and surface atomic rearrangement, the parameter α could be changed [27]. Although the value of α slightly differ from element to
element (e.g. value of α for solid Ag, Au and Cu elements are 0.806, 0.835 and 0.841 respectively [28]), but commonly one single value is used for all similar components which is determined from the relation between the surface energy and the heat for solid – gas transformation (∆𝐻𝑆𝐺 ) of various elements as follows[27,28] 𝜎𝑖 𝑆𝑖 = (1 − 𝛼)∆𝐻𝑆𝐺
(9)
From the relation between the surface energy and the heat for solid – gas transformation of 22 solid metals, Park and Lee [28] estimated that 𝛼 = 0.84, which will be used in this study. 𝐺𝑖𝐸𝑥,𝐵𝑢𝑙𝑘 can be expressed by Eq. (10) 𝐸𝑥 𝐸𝑥 𝐺𝑖𝐸𝑥,𝐵𝑢𝑙𝑘 = 𝐺𝐴𝐵 + (1 − 𝑋𝑖 )d𝐺𝐴𝐵 /d𝑋𝑖
(𝑖 = 𝐴 𝑜𝑟 𝐵)
(10)
𝐸𝑥 where 𝐺𝐴𝐵 is the excess Gibbs free energy of the bulk alloy, which is usually given by means
of the Redlich–Kister polynomials [29] 𝐸𝑥 𝐺𝐴𝐵 = 𝑋𝐴 𝑋𝐵 ∑ 𝐿𝑣 (𝑋𝐴 − 𝑋𝐵 )𝑣
(𝑣 = 0, 1, 2, … )
(11)
where 𝐿𝑣 is the bulk interaction parameter which is a function of temperature 𝐿𝑣 = 𝑎𝑣 + 𝑏𝑣 𝑇
(12)
where 𝑎𝑣 and 𝑏𝑣 are empirical constants. ∞ Substituting calculated values of 𝛽𝐴𝐵 and 𝜎𝐴𝐵 from Eq. (5) and Eq. (6) to Eq. (3), gives the
surface energy of binary alloy nanoparticles as a function of concentration and temperature as well as the shape and size of nanoparticles. The model used to predict the surface energy of Cu – Ag and Cu – Au alloy nanoparticles and the results compared with available data from the literature. Thermodynamic and physical property data used in the present calculations are provided in Table 1.
Table 1. Thermodynamic and physical property data used in the present calculations
Variables Surface energy
Equations ∞ (𝐽𝑚−2 ) 𝜎𝐶𝑢 = 2.158512 − 0.4 × 10−3 𝑇 ∞ (𝐽𝑚−2 ) 𝜎𝐴𝑔 = 1.675 − 0.47 × 10−3 𝑇 ∞ (𝐽𝑚−2 ) 𝜎𝐴𝑢 = 1.947 − 0.43 × 10−3 𝑇
Molar volume
𝑉𝐶𝑢 (𝑚3 𝑚𝑜𝑙 −1 ) = 0.709× 10−5 𝑉𝐴𝑔 (𝑚3 𝑚𝑜𝑙 −1 ) = 1.12066 × 10−5 𝑉𝐴𝑢 (𝑚3 𝑚𝑜𝑙 −1 ) = 1.07109 × 10−5
[31] [30] [28]
Excess Gibbs energy
Cu – Au alloy:
𝐿0 𝐿1 𝐿2 𝐿0 𝐿1
[32]
Cu – Ag alloy:
Material constant
= −27755 − 3.492 × 𝑇 = 3184.9 + 3.113 × 𝑇 = 1193.3 = 36772.58 − 11.02847 × 𝑇 = −4612.43 + 0.28869 × 𝑇
Reference [30] [28] [28]
[30]
𝛽𝐶𝑢 (𝑛𝑚) = 0.225
[17]
𝛽𝐴𝑔 (𝑛𝑚) = 0.2415
[17]
𝛽𝐴𝑢 (𝑛𝑚) = 0.280
[17]
3. Results and discussion Variation of the surface energy with respect to particle radius for Cu – Au alloys with different compositions is shown in figure 1(a). As can be seen, in all cases the surface energy decreases with decreasing particle radius. The total surface energy of a solid nanoparticle would be considered as the sum of the contributions from its faces, edges and vertices. The contributions of edges and vertices can be ignored when the particle radius is larger than approximately 5 nm [33]. By decreasing the particle size below this critical size, the contributions of edges and vertices greatly increase which is lead to sudden drop of the curves. The results also show that the increasing of Au content in the alloy decreases the surface energy of the nanoparticle. This is expected, because surface energy of Au is smaller than that of Cu; on the other hand previous studies show that Au tends to segregate on the surface [34,35], which lead to decrease of surface energy. Comparison of calculated value of the surface energy at mole fraction of 0.95 Cu – 0.05 Au for nanoparticles with r =20 nm (1.67 J/m2) for example, with the weighted average value of surface energies (1.70 J/m2) confirms partially segregation of Au atoms on the surface. Figure 1(b) shows the surface energy of Cu – Ag alloy nanoparticles. The general trend of decreasing surface energy with decreasing particle radius is the same as Cu – Au system. However, due to the larger
differences between the surface energy of pure Cu and Ag in comparison with Cu and Au, in the same mole fractions, the surface energy of copper – silver alloy is smaller than that of the copper – gold alloy. Another difference between the two systems is that the concentration dependency of surface energy in Cu – Au alloy is relatively linear (figure 1(c)) while in the case of Cu –Ag non – linear dependency is observed. As can be seen in figure 1(c), addition of only 0.05 mole fraction Ag to pure copper for a particle with r =20 nm decreases the value of the surface energy to 1.43 J/m2 which is smaller than that of the weighted average value of surface energy for this alloy (1.64 J/m2) and based on our calculations it is equal to the weighted average value of the surface energy for 0.5 Cu – 0.5 Au alloy. This means that the concentration of Ag on the surface is approximately 50%, whereas it is only 5% in the alloy, which indicates a strong tendency of silver for segregation to surface in Cu – Ag nanoparticles. Surface segregation of Ag in Cu – Ag nano – alloys is experimentally observed and reported by Lu et al. [36]. Deviation from linear relation decreases with decreasing particle radius in both Cu – Au and Cu – Ag systems (figure 1(c)), which demonstrates that the tendency to surface segregation decreases with decreasing particle size. It has been shown that the surface energy difference between the pure components is the main driving force in determining segregation behavior for most binary alloys [37]. As can be seen in figures 1 (a) and (b) by decreasing the particle size the differences between the surface energy of pure elements decrease and consequently the segregation driving force decreases. As a result, the tendency to surface segregation decreases with decreasing the particle radius. This trend agrees with the MD simulation results [38]. Liu et al. [12] reported similar phenomena for Bi – Sn liquid nano-droplets. The model developed by Takrori and Ayyad [13] also shows this phenomena in different binary systems.
Figure 1. Calculated surface energy of binary alloy nanoparticles at T=1000 K. (a) Cu – Au and (b) Cu – Ag alloys with different mole fractions. (c) Variation of the surface energy with respect to composition for nanoparticles with r=20 nm and r= 1 nm, dashed lines indicate weighted average value of surface energy.
Due to the lack of sufficient experimental or computational data in the literature on the surface energy of alloy nanoparticles, calculated results for surface energy of pure Cu, Au and Ag have been compared with calculated results by other authors at T=586 K. The results are shown in figure 2. Reasonable agreements between the results of this work and the compared data are observed. As can be seen in figure 2 (a) and (b), for nanoparticles with r < 1 nm, calculated value of the surface energy in this work and that of calculated by atomistic models are very close and far from that calculated by other models, but by decreasing the particle radius below 1 nm, calculated value of the surface energy with atomistic models become more closer to calculations of Takrori and Ayyad [13]. In the case of silver (figure 2 (c)), it should be noted that some experimental and theoretical research has shown lattice contraction with reducing the size of free Ag nanoparticles, which is led to increase of the surface energy with decreasing of the particle radius [39,40]. Since lattice contraction is not included in this model (and also in the models proposed by Takrori and Ayyad[13] and Xiong et al. [41]) the differences between the calculated surface energy of Ag nanoparticles in this work and that of calculated by density functional theory method [42] is relatively large .
Figure 2. Calculated value for surface energy of pure nanoparticles from different models at T=586 K. Solid line: this work, dashed line: Takrori and Ayyad [13], black points Xiong et al. [41], white points: MD simulation [14,43] , Monte Carlo simulation [44] or density functional theory calculations [42]. (a) Pure Cu, (b) pure Au and (c) pure Ag.
Due to the lake of some neighbor atoms, an atom in the surface has a higher potential energy than inside the particle. This excess energy equals to the surface energy of the particle. Palmer [45] showed that this excess energy (i.e. surface energy) per unit area is equal to N.n.ε/4 where N is the number of atoms per unit area in the surface, n is the coordination number of interior atoms and ε is the energy required to overcome a single nearest neighbor interaction which is related to the temperature as ε=KT where K is the Boltzmann's constant. This is shows that in addition to the size and concentration, temperature is another factor that can affect the value of surface energy . Figure 3(a) and (b) shows the effect of temperature change on surface energy of Cu – 0.25 Au and Cu – 0.25 Ag nanoparticles respectively. As can be seen, in both cases increase of temperature, reduces the surface energy of nanoparticles. For a particular system, the degree of reduction in all sizes of nanoparticles is the same but it varies between the two systems. Calculated reduction value of the surface energy from 586 K to 1300 K is 14% for Cu – Au and 16% for Cu – Ag system. This is due to the differences in the temperature dependence of the surface energy in different components as shown in table 1. Since the surface energy of silver is more dependent on temperature in comparison with gold, surface energy of Cu – Ag system shows more reduction with increasing temperature. Calculated value of the surface energy as a function of temperature for pure Cu compared with the calculated results from literature in figure 3 (c). White points demonstrate the
molecular dynamics simulation results by Jia et al. [14] which is implemented on a system with approximately 2100 Cu atoms with cubic structure. Black points are from Monte Carlo simulations implemented on approximately 2200 Cu atoms in an orthorhombic shaped nanoparticle [15]. Based on the calculations presented by Qi et al. [20] a spherical nanoparticle with r = 1.7 nm is included about 2100 Cu atoms. So the calculated surface energy of pure Cu nanoparticle with r = 1.7 nm in this work (solid line in figure 3 (c)) compared with the results presented in [14] and [15]. Dashed lines indicate the results calculated by Takrori and Ayyad [13] for a spherical Cu nanoparticle with 2100 atoms. As demonstrated by figure 3 (c) except the model proposed in Ref [13], all other models predict the decreasing of surface energy with increasing temperature which is in consistent with experimental data on the bulk systems [25]. Considering that the variation in particle shapes changes the value of surface energy [16–18], the results are in good agreements. In particular, the rate of surface energy change with temperature is relatively the same in all cases (except results from [13]). As can be seen, the calculated temperature dependent surface energy in this work is more accurate than that of calculated by the proposed model in Ref. [13]. Figure 3(d) shows the concentration dependence of the surface energy at different temperatures. As can be seen, in both cases, deviation from weighted average value of the surface energy increases by decreasing temperature. Which means, that the decreasing temperature, increases tendency of surface segregation. Although the differences of the surface energy between the pure components slightly increase by increasing temperature in both cases, but due to the reduction of the differences between the partial excess Gibbs 𝐸𝑥,𝑆𝑢𝑟𝑓
energy of the pure components in the surface (𝐺𝑖
), tendency of surface segregation
decrease by increasing temperature. The results are consistent with previous studies on Cu – Ag [46] and Cu – Au [47] systems.
Figure 3. (a) Calculated surface energy of Cu – 0.25 Au and (b) Cu – 0.25 Ag nanoparticles at different temperatures. (c) Comparison of calculated surface energy of pure Cu from this work with calculated value by others. Solid line: this work for nanoparticle with r=1.7 nm, dashed line: Takrori and Ayyad [13] for spherical nanoparticle with 2100 atoms, white points: Jia et al. [14] for cubic nanoparticles with approximately 2100 atom and black points: Frolov and Mishin [15] for
orthorhombic nanoparticles with 2200 atoms. (d) Concentration dependent surface energy of Cu – Au and Cu – Ag at different temperatures for nanoparticles with r= 20 nm.
As mentioned above it has been shown that the surface energy of the nanoparticles is shape dependent. From atomistic point of view, when a nanoparticle is deformed from spherical shape, some interior atoms move to the surface and the number of the surface atoms increases, as a result, the surface free energy of non-spherical nanoparticles can be calculated as 𝑛 𝜎𝑎𝑡𝑜𝑚 + 𝜎𝑠𝑝ℎ𝑒𝑟 [48], where 𝜎𝑠𝑝ℎ𝑒𝑟 is the surface energy of a spherical nanoparticle, 𝑛 is the number of atoms added to the surface and 𝜎𝑎𝑡𝑜𝑚 is the surface free energy per atom which is defined as the work required to increase the surface of a
nanoparticle by an area of 𝜋𝑅 2 where R is the atomic radius. Variation of the surface energy due to the shape change can be calculated by introducing the shape factor parameter. Typically the shape factor, c, is specified as the ratio of the surface area of the nanoparticle to that of the spherical nanoparticle, where both types of nanoparticles have an equal volume [30]. Calculated shape factors for different particle shapes are presented in Table 2. Figure 4 (a) and (b) represent the variation of surface energy with respect to shape factor for Cu – 0.25 Au and Cu – 0.25 Ag systems at T = 1000 K respectively. As can be seen, the increasing of the shape factor decreases the surface energy of the system. However, by increasing the particle size the effect of shape factor decreases. These results are consistent with the simulation results performed on pure nanoparticles by Xiong et al [49].
Table 2. The calculated shape factors for various shapes of nanoparticles.
Nanoparticle shape Shape factor (c) Reference Sphere
1
[20]
Icosahedron
1.06
This work
Dodecahedron
1.09
This work
Cuboctahedron
1.10
This work
Octahedron
1.18
[20]
Cube
1.24
[20]
Tetrahedron
1.49
[20]
Disk like
>1.15
[20]
Nanofilm
0.33
[30]
By decreasing the shape factor, the value of surface energy for small nanoparticles approach to the surface energy value of bulk material so, it can be concluded that the decreasing of the shape factor increases the thermal stability of the nanoparticles. Calculated phase diagrams of Ag – Cu [30], Au – Cu [34], and Ni – Cu [50] for different shapes of nanoparticles confirm this conclusion.
Figure 4. Effect of particle shape on surface energy of binary alloy nanoparticles. (a) Cu – 0.25 Au and (b) Cu – 0.25 Ag systems at T = 1000 K. It should be noted that in the case of the nanofilm (c=0.33), r is equal to the half of the nanofilm thickness (See Ref [29]).
4. Conclusion In summary, a model for size dependent surface energy of binary metallic nanoparticles has been developed based on liquid drop model for pure nanoparticles and Butler's equation. The model can take into account the effects of temperature and particle shape as well as size and concentration on surface energy of alloy nanoparticles. The model applied to Cu – Au and Cu – Ag binary systems. The results show that the value of the surface energy decreases with decreasing particle size, increasing temperature and increasing shape factor of the particle. Reasonable agreements between the model prediction and previous works are obtained. To the knowledge of the author this is the first model for prediction of alloy nanoparticles surface energy which can incorporate the effects of size, shape, composition and temperature in calculations. Due to the important role of surface energy on nanoparticle properties, the present model can be used to study the size dependent properties of alloy nanoparticles related to the particle surface, such as phase diagram calculations.
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