Thermodynamic model for prediction of binary alloy nanoparticle phase diagram including size dependent surface tension effect

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 58 (2017) 1–5

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Thermodynamic model for prediction of binary alloy nanoparticle phase diagram including size dependent surface tension effect

MARK

⁎

Fatemeh Monji, Mohammad Amin Jabbareh

Department of Materials Engineering, Faculty of Engineering, Hakim Sabzevari University, Sabzevar, Iran

A R T I C L E I N F O

A BS T RAC T

Keywords: Nano-phase diagram Nanoalloy Nanoparticle Surface tension

Considering the size effect of nanoparticles on surface tension, a new CALPHAD type thermodynamic model was developed to predict phase diagram of binary alloy nanoparticle systems. In contrast to conventional model, the new model can be applied to the nanoparticles smaller than the critical size (5 NM in radius). For an example, the model applied to Ag – Au binary system and the results were compared with experimental data as well as conventional CALPHAD model and molecular dynamics simulation results.

1. Introduction Because of their unique properties, alloy nanoparticles have attracted the interest of scientists and industries, especially for use as catalysts [1–5] or nanosolders [6–8]. Due to these applications, the thermal stability of nanoparticles has great importance. Because of the increased surface to volume ration of nanoparticles, the phase stability of nanoparticles is significantly different from those of the bulk materials, for example melting point of the nanoparticles decreased by decreasing particle radius [9,10]. To predict the effect of size on phase stability of alloy nanoparticles, two general approaches are used. The first approach lies on atomistic modeling, including density functional theory (DFT), Monte Carlo (MC) and molecular dynamics (MD) simulations [11,12]. In all cases the knowledge of atomic interactions is required for the simulations. The methods are extremely computer intensive and they are limited to very small nanoparticles (e.g. MD simulations are limited to the particles smaller than approximately 10 nm [13]). The second approaches are based on CALPHAD (Computer Calculation of Phase Diagrams) framework. By adding the surface Gibbs free energy contributions to the total Gibbs energy of the system, Tanaka et al. [14] extended the CALPHAD method for the prediction of the phase diagram of nanoparticles. Later Park and Li [15] suggested a method to calculating phase diagram of nanoparticles with commercial software products. Garzel et al. [13] modified this model further and added the effect of particle shapes in calculations. The model also improved for systems containing intermetallic compounds [16–18] and extended for ternary alloy systems [7]. In comparison with atomistic modeling, the CALPHAD method is much easier to implement but cannot be applied to nanoparticles generally

⁎

smaller than 5 nm in radius [19]. It should be mentioned that the well known macroscopic thermodynamic laws such as the Second law of thermodynamics may break down for very small systems [20]. Moreover previous studies have found that some thermodynamic properties such as heat capacity and latent heat of phase transformations are size dependent and so changed in nano-scale systems [9,11]. Furthermore it was showed that in very small solid particles lattice parameter [21] and surface tension [22–24] are also changed. Surface energy (or surface tension) is the most important physical quantity, which can explain the identical properties of nanoparticles [9] and also the equilibrium conditions depend strongly on the curvature of the surfaces [25–27]. So, in this work, by considering the size effect of nanoparticles on surface tension, a new CALPHAD type thermodynamic model is developed which can be applied to nanoparticles smaller than critical size (5 nm in radius). The model applied to binary Ag – Au system as an example, and the results compared with the conventional CALPHAD model as well as experimental and MD simulation data from the literature. 2. Thermodynamic model For an A–B binary alloy system, the total Gibbs energy of a nanoparticle is expressed as [14]: ϕ

Gnano = ϕGBulk + ϕGSurf

where ϕ denotes the phase of the system, GBulk is the molar Gibbs energy of the bulk binary alloy and ϕGSurf is the surface free energy contribution.

Correspondence to: Hakim Sabzevari University, Towhid Shahr, P.O.Box: 9617976487, Sabzevar, Iran. E-mail address: [email protected] (M.A. Jabbareh).

http://dx.doi.org/10.1016/j.calphad.2017.04.003 Received 19 January 2017; Received in revised form 9 April 2017; Accepted 12 April 2017 0364-5916/ © 2017 Elsevier Ltd. All rights reserved.

(1) ϕ

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 58 (2017) 1–5

F. Monji, M.A. Jabbareh

For a non – ideal solution, the molar Gibbs energy of the bulk could be written as:

Table 1 Thermodynamic and physical properties used in the calculation of Ag – Au nanoparticle phase diagram. L denotes liquid and S denotes solid phases. Atomic diameters are taken from Ref [36] and other data from Ref [15].

GBulk = XA ϕGA0 + XB ϕGB0 + RT (XA ln XA + XB ln XB ) + ϕG Ex, Bulk

ϕ

ϕ

GA0

Variables

Equations

Surface tension

L ∞ σAg (Nm−1)=1.207

Molar volume

L

L

GB0

and are the standard Gibbs energies of the components where A and B in phase ϕ , XA and XB are the mole fractions of the components A and B; and R and T are the universal gas constant and absolute temperature, respectively. ϕG Ex, Bulk is the excess free energy of the bulk in phase ϕ which is usually given by means of the Redlich–Kister polynomials as [28]:

− 2.28 × 10−4T

L ∞ σAu (Nm−1)=1.33 − 1.4 × 10−4T S ∞ σAg (Nm−1)=1.675 − 0.47 × 10−3T S ∞ σAu (Nm−1)=1.947

− 0.43 × 10−3T

ϕ

G Ex, Bulk = XA XB ∑ L v (XA − XB )v (v = 0, 1, 2, …)

V Ag (m3 /mol )=1.0198 × 10−5+1.1368 × 10−9T

V Au

(m3 /mol )=1.02582

×

10−5+7.797

(2)

ϕ

×

10−10T

(3)

where L v is the temperature dependent bulk interaction parameter:

S

V Ag (m3 /mol )=1.12066 × 10−5

L v = a v + bv T + cv T ln T + …

S

V Au (m3 /mol )=1.07109 × 10−5

Excess Gibbs energy

L Ex, Bulk GAg, Au (J /mol )

Where av, bv and cv are constants. For isotropic spherical particles the surface Gibbs energy is expressed by Eq. (5) [19].

= XAg XAu [−16402 + 1.14T ]

S (fcc) Ex, Bulk GAg, Au (J /mol )

= XAg XAu [−15599]

ϕ

Atomic diameter

(4)

∞ϕ GSurf = 2CϕσAB V AB / r

hAg (nm ) = 0.32 hAu (nm ) = 0.27

(5)

where r is the particle radius, C is the shape factor that is introduced to account the effects of shape, surface strain and the uncertainty of the

Fig. 1. Concentration dependent surface tension of Ag-Au alloy for solid (a), and liquid (b) phases. Dotted line indicates the surface tension of bulk Ag-Au calculated from Eq. (7).

Fig. 2. Calculated surface Gibbs free energy of Ag-Au alloy for nanoparticles with different radius at 1100 K for solid (a), and liquid (b) phases. Solid lines: this work, Eq. (12), Dotted lines: conventional model, Eq. (5).

2

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 58 (2017) 1–5

F. Monji, M.A. Jabbareh

Table 2 Calculated liquidus temperature for three different Ag – Au alloy nanoparticles with various particle sizes. ΔTL indicates difference between liquidus temperatures calculated from two models. r (nm)

XAu

TL (K) (This work)

TL (K) (Conventional model)

ΔTL (K)

10

0.2 0.5 0.8

1219 1258 1283

1217 1256 1281

2 2 2

5

0.2 0.5 0.8

1149 1193 1224

1141 1185 1216

8 8 8

2

0.2 0.5 0.8

846 908 962

746 811 871

100 97 91

1

0.2 0.5 0.8

270 382 490

No Data No Data 303

_ _ 187

where ϕV A and ϕV B are the molar volumes of the components A and B, respectively. In the conventional CALPHAD type model for calculation of the nanoparticles phase diagram, it is assumed that the surface tension of pure components (ϕσA∞ and ϕσB∞) are only a function of temperature which is calculated from theoretical methods such as DFT [21] or derived from the experiments carried out on the bulk materials [24]. These functions are used to predict the surface tension of binary alloys ∞ (ϕσAB ) via Butler's equation [31]: ϕ ∞ σAB

Surf RT ⎛ XA ⎞ 1 ln ⎜⎜ Bulk ⎟⎟ + [GAEx, Surf −GAEx, Bulk ] AA ⎝ XA ⎠ AA RT ⎛ XBSurf ⎞ 1 ⎟ + [GBEx, Surf −GBEx, Bulk ] = ϕσB∞+ ln ⎜ AB ⎝ XBBulk ⎠ AB

= ϕσA∞+

(7)

where AA and AB are the molar surface areas of components A and B, when a close-packed monolayer is assumed. XASurf ( XBSurf ) and XABulk ( XBBulk ) are concentrations of the component A (B) in the surface and bulk phases, respectively. GAEx, Surf (GBEx, Surf ) and GAEx, Bulk (GBEx, Bulk ) are the partial excess Gibbs energy of the component A (B) in surface and the bulk phases. The molar surface area could be calculated from Eq. (8).

Ai = 1. 091N0 ( ϕV i )

2/3

(i = AorB )

(8)

where N0 is the Avogadro's number. Yeum et al. [32] suggested that the surface excess Gibbs energy is related to that of the bulk phase as follows:

GiEx, Surf = βGiEx, Bulk (i = AorB )

Where β is a parameter corresponding to the ratio of the coordination number in the surface to that in the bulk. Park and Lee [15] estimated that β = 0.85 and 0.84 for liquids and solids respectively. Although in conventional model, the surface tension of the pure components is assumed as a function of temperature only, Tolman [33] showed that the surface tension of the metallic nanoparticles is different from that of the corresponding bulk metals and it depends on the particle size as well as temperature. This is due to the increasing of edge and corner atoms contribution to the cohesive energy with the decreasing of the particle radius [23]. Since the effects of edges and vertices in the conventional model neglected, the model could not accurately estimate the surface tension when the particle size decreased below the critical size and therefore could not predict the transition temperatures correctly. In order to include the effects of the edges and vertices to the model, current description of surface tensions replaced by a size dependent surface tension model. Different expressions are proposed

Fig. 3. Calculated nano – phase diagram of Ag-Au alloy in this work (Black lines), in comparison with conventional CALPHAD model (red lines), MD simulations [39] (black symbols) and some experimental data [40–43] (white symbols). (a) Particle radius: 2−10 nm, (b) Particle radius: 1 nm. Blue lines: Bulk phase diagram. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

surface tension measurements. For the liquid phases and spherical ∞ is the surface solid particles it is usually assumed that C = 1 [29]. ϕσAB ϕ tension of A-B alloy in phase ϕ and V AB is the molar volume of the system. For metallic binary alloys ϕV AB is calculated as [30]. ϕ

V AB = XA ϕV A + XB ϕV B

(9)

(6) 3

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 58 (2017) 1–5

F. Monji, M.A. Jabbareh

Fig. 4. Comparison of calculated melting point of Ag (a), and Au (b) nanoparticles in this work with experimental data (white symbols) and MD simulation results (black symbols).

Fig. 3a shows the calculated phase diagram of Ag – Au alloy for nanoparticles with radius from 2 nm to 10 nm. Comparison of the calculated results from conventional and revised model shows that the predicted solidus and liquidus temperatures with revised model are greater than that of predicted with the conventional model. It is reasonable because considering size dependent surface tension effect, reduced calculated surface free energies in comparison with conventional model (Fig. 2). The differences between the results are increased by decreasing particle radius. For comparison, calculated liquidus temperatures for three different Ag – Au alloys are given in Table 2. As can be seen, for Ag – 50% Au nanoparticles with r =10 nm, for example, the difference between predicted liquidus temperatures, ∆TL , is 2 K, which is increased to 97 K for nanoparticles with r =2 nm. The values of ∆TL also show that, for very small nanoparticles (i.e. r =2 nm), the difference between calculated liquidus temperatures is concentration dependent. Lue and Hu [24] show that considering size effect on surface tension will decrease calculated melting point depression of pure nanoparticles due to introduction of factor (1−λ/r); in a way that increasing of λ (0.725 hi in this work) increases the calculated melting point, and so the difference between calculated melting point with and without considering size dependent surface tension effect will increase. Thus, given that the atomic diameter of silver (hAg) is greater than that of the gold (hAu), it is expected that the increasing Ag concentration increases the differences between calculated liquidus (and solidus) temperatures from two models as can be seen in the results. Increasing of particle radius decreases the effect of λ parameter. Calculated phase diagram of Ag – Au nanoparticle with r =1 nm is shown in Fig. 3b. For comparison, full freezing temperature (solidus temperature) of 1 nm radius Ag – Au nanoparticle, predicted by Yeo et al. [39] with MD simulation method and some experimental data on melting temperature of pure Ag [40,41] and Au [41–43] are plotted together. The results show that the calculated phase diagram with revised model is in reasonable agreements with experimental data. Also considering this point that the MD simulations overestimate the melting temperature of the system [44], agreements between calculated phase diagram and MD simulation results are acceptable too. While the results of conventional model show significant differences from other results, especially for silver rich part of the phase diagram. Due to the lack of sufficient empirical data in the literature on the phase diagram of Ag – Au nanoparticle systems, the calculated melting point of pure Au and Ag compared to the experimental and MD simulation results reported by others [22,40–43,45–48] (Fig. 4). Reasonable agreements between the results of this work and compared data can be seen. It is noteworthy that in comparison with calculated

for the calculation of size dependent surface tension from various researchers [25,34–36]. We used the recently developed model by Xiong et al. [37]. According to this model the size dependent surface tension of a pure element is given by:

σir = σi∞ (1 − 0. 725

hi / r )

(10)

where σi∞ is the surface tension of bulk material, hi is the atomic diameter of the element i, and r is the particle radius. Substitution σir instead of σi∞ to Eq. (7) gives the size dependent surface tension of the A-B binary alloy as follows: ϕ r σ AB

Surf ⎛ 0. 725hA ⎞ RT ⎛ XA ⎞ 1 = ϕσA∞ ⎜1 − ln ⎜⎜ Bulk ⎟⎟ + [GAEx, Surf −GAEx, Bulk ] ⎟+ ⎠ AA ⎝ XA ⎠ AA ⎝ r RT ⎛ XBSurf ⎞ 1 ⎟ + [GBEx, Surf −GBEx, Bulk ] = ϕσB∞ (1 − 0. 725hB / r )+ ln ⎜ AB ⎝ XBBulk ⎠ AB

(11) (ϕσi∞)

By the replacement of the surface tension of bulk A-B alloy in Eq. (5) with calculated size dependent surface tension from Eq. (11), the surface Gibbs free energy of each phase could be rewritten as: ϕ

r ϕ GSurf = 2Cϕσ AB V AB / r

(12)

Using Eq. (12) instead of Eq. (5) can increase the accuracy of phase diagram calculation for nanoparticles with the radius of smaller than 5 nm. In the present study a simple binary alloy system, Ag – Au, was selected to assess the validity of the model. Gibbs energy terms of bulk pure elements are taken from the SGTE database [38]. Table 1 listed the thermodynamic and physical property data used in the present calculation. 3. Results and discussion Fig. 1 shows the calculated surface tension of Ag – Au binary system for nanoparticles with different radius in comparison with the surface tension of bulk Ag – Au at 1100 K. As can be seen, by decreasing the particle radius the values of surface tension decreased. Even though the differences between the surface tension of bulk Ag – Au and the particle with 10 nm radius are not considerable, but by decreasing the particle radius the differences become significant. This difference in the calculation of the surface tension, leads to the difference in calculated values of the surface Gibbs free energy as shown in Fig. 2. As can be seen in the Fig. 2, for nanoparticles with r =10 nm calculated surface Gibbs free energy from both models are almost the same, but by decreasing the particle radius the differences gradually increased. 4

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 58 (2017) 1–5

F. Monji, M.A. Jabbareh

and experimental data on Au melting point, in the case of Ag melting point the difference between calculated and experimental results relatively large. This is maybe due to the scarce experimental information on the surface tension of pure Ag which is lead to inaccurate description of the surface tension function for this element in solid and liquid phases.

[20]

[21]

[22]

4. Conclusion In this work a revised version of the thermodynamic model for calculating of binary alloy nanoparticles phase diagram is suggested. The model can incorporate the effects of edges and vertices to calculate of the surface Gibbs free energy via considering size dependent surface tension effect. So, the main limitation of conventional thermodynamic models, i.e. calculation of phase diagram for nanoparticles smaller than 5 nm in radius, is eliminated. The model applied to Ag – Au binary system and the results compared with the experimental and MD simulation data from the literature. Reasonable agreements between the results were observed.

[23]

[24]

[25] [26] [27]

[28]

References [1] A. Wang, Y.P. Hsieh, Y.F. Chen, C.Y. Mou, Au-Ag alloy nanoparticle as catalyst for CO oxidation: effect of Si/Al ratio of mesoporous support, J. Catal. 237 (2006) 197–206. http://dx.doi.org/10.1016/j.jcat.2005.10.030. [2] S. Tokonami, N. Morita, K. Takasaki, N. Toshima, Novel synthesis, structure, and oxidation catalysis of Ag/Au bimetallic nanoparticles, J. Phys. Chem. C 114 (2010) 10336–10341. http://dx.doi.org/10.1021/jp9119149. [3] J. Krajczewski, K. Kolataj, A. Kudelski, Enhanced catalytic activity of solid and hollow platinum-cobalt nanoparticles towards reduction of 4-nitrophenol, Appl. Surf. Sci. 388 (2016) 624–630. http://dx.doi.org/10.1016/j.apsusc.2016.04.089. [4] L. Srisombat, J. Nonkumwong, K. Suwannarat, B. Kuntalue, S. Ananta, Simple preparation Au/Pd core/shell nanoparticles for 4-nitrophenol reduction, Colloids Surf. A Physicochem. Eng. Asp. 512 (2017) 17–25. http://dx.doi.org/10.1016/ j.colsurfa.2016.10.026. [5] C. Chu, Z. Su, Gold-decorated platinum nanoparticles in polyelectrolyte multilayers with enhanced catalytic activity for methanol oxidation, Appl. Catal. A Gen. 517 (2016) 67–72. http://dx.doi.org/10.1016/j.apcata.2016.03.002. [6] H. Jiang, K.S. Moon, C.P. Wong, Recent advances of nanolead-free solder material for low processing temperature interconnect applications, Microelectron. Reliab. 53 (2013) 1968–1978. http://dx.doi.org/10.1016/j.microrel.2013.04.005. [7] A. Roshanghias, J. Vrestal, A. Yakymovych, K.W. Richter, H. Ipser, Sn – Ag – Cu nanosolders: melting behavior and phase diagram prediction in the Sn-rich corner of the ternary system, Calphad 49 (2015) 101–109. http://dx.doi.org/10.1016/ j.calphad.2015.04.003. [8] Y. Shu, S. Gheybi Hashemabad, T. Ando, Z. Gu, Ultrasonic powder consolidation of Sn/In nanosolder particles and their application to low temperature Cu-Cu joining, Mater. Des. 111 (2016) 631–639. http://dx.doi.org/10.1016/ j.matdes.2016.09.013. [9] C.C. Yang, Y. Mai, Thermodynamics at the nanoscale: a new approach to the investigation of unique physicochemical properties of nanomaterials, Mater. Sci. Eng. R. 79 (2014) 1–40. http://dx.doi.org/10.1016/j.mser.2014.02.001. [10] G. Guenther, O. Guillon, Models of size-dependent nanoparticle melting tested on gold, J. Mater. Sci. 49 (2014) 7915–7932. http://dx.doi.org/10.1007/s10853-0148544-1. [11] Felorent Calvo, Thermodynamics of nanoalloys, Phys. Chem. Chem. Phys. 17 (2015) 27922–27939. http://dx.doi.org/10.1039/b000000x. [12] A.S. Barnard, Modelling of nanoparticles: approaches to morphology and evolution, Rep. Prog. Phys. 73 (2010) 86502–86554. http://dx.doi.org/10.1088/0034-4885/ 73/8/086502. [13] G. Garzel, J. Janczak-rusch, L. Zabdyr, Reassessment of the Ag – Cu phase diagram for nanosystems including particle size and shape effect, Calphad 36 (2012) 52–56. http://dx.doi.org/10.1016/j.calphad.2011.11.005. [14] T. Tanaka, S. Hara, Thermodynamic evaluation of binary phase diagrams of small particle systems, Z. Fur Met. 92 (2001) 467–472. [15] J. Park, J. Lee, Phase diagram reassessment of Ag–Au system including size effect, Calphad 32 (2008) 135–141. http://dx.doi.org/10.1016/j.calphad.2007.07.004. [16] S. Bajaj, M.G. Haverty, R. Arróyave, W.A. Goddard Iii Frsc, S. Shankar, Phase stability in nanoscale material systems: extension from bulk phase diagrams, Nanoscale 7 (2015) 9868–9877. http://dx.doi.org/10.1039/c5nr01535a. [17] K. Sim, J. Lee, Phase stability of Ag-Sn alloy nanoparticles, J. Alloy. Compd. 590 (2014) 140–146. http://dx.doi.org/10.1016/j.jallcom.2013.12.101. [18] M. Ghasemi, Z. Zanolli, S. Martin, J. Johansson, Size- and shape-dependent phase diagram of In-Sb nano-alloys, Nanoscale 7 (2015) 17387–17398. http:// dx.doi.org/10.1039/C5NR04014K. [19] J. Lee, T. Tanaka, J. Lee, H. Mori, Effect of substrates on the melting temperature

[29] [30]

[31] [32]

[33] [34]

[35]

[36] [37]

[38] [39]

[40]

[41]

[42] [43]

[44]

[45]

[46]

[47]

[48]

5

of gold nanoparticles, Calphad 31 (2007) 105–111. http://dx.doi.org/10.1016/ j.calphad.2006.10.001. E.M. Sevick, R. Prabhakar, S.R. Williams, D.J. Searles, Fluctuation Theorems, Annu. Rev. Phys. Chem. 59 (2008) 603–633. http://dx.doi.org/10.1146/annurev.physchem.58.032806.104555. B. Medasani, Y.H. Park, I. Vasiliev, Theoretical study of the surface energy, stress, and lattice contraction of silver nanoparticles, Phys. Rev. B - Condens. Matter Mater. Phys. 75 (2007) 235436–235441. http://dx.doi.org/10.1103/ PhysRevB.75.235436. I. Shyjumon, M. Gopinadhan, O. Ivanova, M. Quaas, H. Wulff, C.A. Helm, R. Hippler, Structural deformation, melting point and lattice parameter studies of size selected silver clusters, Eur. Phys. J. D 37 (2006) 409–415. http://dx.doi.org/ 10.1140/epjd/e2005-00319-x. S. Xiong, W. Qi, B. Huang, M. Wang, Y. Cheng, Y. Li, Size and shape dependent surface free energy of metallic nanoparticles, J. Comput. Theor. Nanosci. 8 (2011) 2477–2481. http://dx.doi.org/10.1166/jctn.2011.1982. W. Luo, W. Hu, Gibbs free energy , surface stress and melting point of nanoparticle, Phys. B Phys. Condens. Matter 425 (2013) 90–94. http://dx.doi.org/10.1016/ j.physb.2013.05.025. Y. Yao, Y. Wei, S. Chen, Size effect of the surface energy density of nanoparticles, Surf. Sci. 636 (2015) 19–24. http://dx.doi.org/10.1016/j.susc.2015.01.016. W.W. Mullins, R.F. Sekerka, On the thermodynamics of crystalline solids, J. Chem. Phys. 82 (1985) 5192–5202. http://dx.doi.org/10.1098/rspa.1955.0277. W.A. Jesser, R.Z. Shneck, W.W. Gile, Solid-liquid equilibria in nanoparticles of PbBi alloys, Phys. Rev. B. 69 (2004) 1–13. http://dx.doi.org/10.1103/ PhysRevB.69.144121. J. Lee, J. Park, T. Tanaka, Effects of interaction parameters and melting points of pure metals on the phase diagrams of the binary alloy nanoparticle systems: a classical approach based on the regular solution model, Calphad 33 (2009) 377–381. http://dx.doi.org/10.1016/j.calphad.2008.11.001. W.H. Qi, M.P. Wang, Q.H. Liu, Shape factor of nonspherical nanoparticles, J. Mater. Sci. 40 (2005) 2737–2739. http://dx.doi.org/10.1007/s10853-005-2119-0. J. Sopousek, J. Vrestal, J. Pinkas, P. Broz, J. Bursik, A. Styskalik, D. Skoda, O. Zobac, J. Lee, Cu – Ni nanoalloy phase diagram – Prediction and experiment, Calphad 45 (2014) 33–39. J.A.V. Butler, The thermodynamics of the surfaces of solutionsProc. R. Soc. A (1932), 1932, pp. 348–375. K. Yeum, R. Speiser, D. Poirier, Estimation of the surface tensions of binary liquid alloys, Metall. Trans. B 2013 (1989), 1989, pp. 693–703 〈http://link.springer.com/ article/10.1007/BF02655927〉. R.C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys. 17 (1949) 333–337. http://dx.doi.org/10.1063/1.1747247. D.H. Rasmussen, Energetics of homogeneous nucleation - approach to a physical spinodal, J. Cryst. Growth 56 (1982) 45–55. http://dx.doi.org/10.1016/00220248(82)90011-2. S.C. Vanithakumari, K.K. Nanda, A universal relation for the cohesive energy of nanoparticles, Phys. Lett. Sect. A Gen. At. Solid State Phys. 372 (2008) 6930–6934. http://dx.doi.org/10.1016/j.physleta.2008.09.050. K.K. Nanda, Liquid-drop model for the surface energy of nanoparticles, Phys. Lett. A. 376 (2012) 1647–1649. http://dx.doi.org/10.1016/j.physleta.2012.03.055. S. Xiong, W. Qi, Y. Cheng, B. Huang, M. Wang, Y. Li, Modeling size effects on the surface free energy of metallic nanoparticles and nanocavities, Phys. Chem. Chem. Phys. 13 (2011) 10648–10651. http://dx.doi.org/10.1039/c0cp02102d. A.T. Dinsdale, SGTE Data for pure elements, Calphad 15 (1991) 317–425. S.C. Yeo, D.H. Kim, K. Shin, H.M. Lee, Phase diagram and structural evolution of Ag-Au bimetallic nanoparticles: molecular dynamics simulations, Phys. Chem. Chem. Phys. 14 (2012) 2791–2796. http://dx.doi.org/10.1039/c2cp23547a. N.H. Kim, J.-Y. Kim, K.J. Ihn, Preparation of silver nanoparticles having low melting temperature through a new synthetic process without solvent, J. Nanosci. Nanotechnol. 7 (2007) 3805–3809. http://dx.doi.org/10.1166/jnn.2007.044. T. Castro, R. Reifenberger, E. Choi, R.P. Andres, Size-dependent melting temperature of individual nanometer-sized metallic clusters, Phys. Rev. B. 42 (1990) 8548–8556. http://dx.doi.org/10.1103/PhysRevB.42.8548. P. Buffat, J.P. Borel, Size effect on the melting temperature of gold particles, Phys. Rev. A. 13 (1976) 2287–2298. http://dx.doi.org/10.1103/PhysRevA.13.2287. K. Dick, T. Dhanasekaran, Z. Zhang, D. Meisel, Size-dependent melting of silicaencapsulated gold nanoparticles, J. Am. Chem. Soc. 124 (2002) 2312–2317. http:// dx.doi.org/10.1021/ja017281a. G. Guisbiers, S. Mejia-rosales, S. Khanal, F. Ruiz-zepeda, R.L. Whetten, M. Jose, Gold − copper nano-alloy , “ Tumbaga ”, in the era of nano: phase diagram and segregation, Nano Lett. 14 (2014) 6718–6726. W. Luo, W. Hu, S. Xiao, Size effect on the thermodynamic properties of silver nanoparticles, J. Phys. Chem. C 112 (2008) 2359–2369. http://dx.doi.org/ 10.1021/jp0770155. L.J. Lewis, P. Jensen, J.-L. Barrat, Melting, freezing, and coalescence of gold nanoclusters, Phys. Rev. 56 (1997) 2248–2257. http://dx.doi.org/10.1103/ PhysRevB.56.2248. J.H. Shim, B.J. Lee, Y.W. Cho, Thermal stability of unsupported gold nanoparticle: a molecular dynamics study, Surf. Sci. 512 (2002) 262–268. http://dx.doi.org/ 10.1016/S0039-6028(02)01692-8. S.J. Zhao, S.Q. Wang, D.Y. Cheng, H.Q. Ye, Three dinstinctive melting mechanims in isolated nanoparticles, J. Phys. Chem. B 105 (2001) 12857–12860. http:// dx.doi.org/10.1021/jp012638i.