Size, Dimensionality, and Constituent Stoichiometry Dependence of

Article pubs.acs.org/JPCC

Size, Dimensionality, and Constituent Stoichiometry Dependence of Physicochemical Properties in Nanosized Binary Alloys Chun Cheng Yang* and Yiu-Wing Mai Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering J07, The University of Sydney, Sydney, NSW 2006, Australia ABSTRACT: The physicochemical properties of nanosized binary alloys are investigated with respect to the effects of size, dimensionality, and composition using a nanothermodynamics model. The results indicate that the cohesive energy, melting temperatures of solid solution and eutectic alloys, Debye temperature, and order−disorder transition temperature decrease with decreasing material size for isolated nanoalloys, which is attributed to the severe bond dangling associated with increased surface/volume ratio. The model predictions are consistent with available test data and other theoretical results. This study has provided new insights into the basic understanding of the physicochemical properties in nanoalloys and also their potential applications in nanodevices.

1. INTRODUCTION Over the past decade, low-dimensional bimetallic nanoalloys AxB1−x, with x being the molar fraction of element A, have attracted intensive interest owing to their potential catalytic, optoelectronic, magnetic, and medical applications.1−3 For example, bimetallic nanoparticles, such as equi-atomic FePt, CoPt, or FePd with L10-type ordered structure, are promising candidates for ultrahigh-density magnetic-storage applications due to their large magnetocrystalline anisotropies.4 When the size of the bimetallic alloys is decreased to the nanometer scale, the electronic, magnetic, optic, catalytic, biomedical, and thermodynamic properties differ substantially from their bulk properties. In general, these physicochemical properties are intrinsically dominated by the size, shape, dimensionality, crystallinity, structure, composition, etc. With rapid progress in nanoscience and nanotechnology, the desired properties of nanoalloys can be achieved by tuning one or more of the above parameters. However, understanding the nature behind the above parameter-dependent properties in nanoalloys is still one of the most important research topics within the nanomaterials community. Although single-component metallic nanocrystals have already been extensively investigated, it is only recently that many studies are focused on bimetallic nanoalloys. For example, the cohesive energy Ec,5 melting temperatures of solid solution Tm6−9 and eutectic alloys Teu,10−12 Debye temperature ΘD,13,14 and order−disorder transition temperature To15−20 of binary nanoalloys have been examined experimentally and theoretically due to their scientific and industrial importance. It is recognized that the thermal stability of nanomaterials is increasingly becoming one of the major concerns in upcoming technologies. Alloying has emerged as an effective approach to improve the thermal stability of nanocrystalline structures, © 2013 American Chemical Society

which can be characterized by the size-dependent cohesive energy.3,21 The understanding of melting temperatures in nanosized solid solution and eutectic alloys is beneficial for the construction of size-dependent binary phase diagrams,8−14 which are critical for industrial applications.22 The Debye temperature is a most important physical parameter to characterize many material properties since it is directly related to the thermal vibration of atoms and atomic bonding.23 Moreover, from an industrial and fundamental point of view, it is essential to study the quantitative relationship between the ordered state and the size of magnetic nanoalloys.18 Through experiments or simulation results, it was found that Ec, Tm, Teu, ΘD, and To of isolated nanoalloys are progressively reduced with decreasing crystal size. Several analytical models have also been developed to study the size dependence of the above properties.5,10,11,19 Both the computer simulations and theoretical models developed from various perspectives could describe the size dependence of these properties in some aspects. However, (a) the computer simulation process is complex and time-consuming; (b) current analytical models are not in good agreement with experimental data in the size range r < 5 nm,10,11 where r is either the radius of nanoparticles and nanowires or the half-thickness of thin films; and (c) the origin underpinning the size effect on the above properties remains unclear. Further, a systematic investigation of the size-, dimensionality-, and composition-dependent physicochemical properties in nanoalloys is necessary since this is essential for designing possible processing routes to synthesize metallic nanoalloys with desired properties. Received: December 4, 2012 Revised: December 31, 2012 Published: January 16, 2013 2421

dx.doi.org/10.1021/jp311893d | J. Phys. Chem. C 2013, 117, 2421−2426

The Journal of Physical Chemistry C

Article

By utilizing the Ec(r, d) function reported in the literature33 and considering the composition effect, a universal relation can be obtained below

An extension of the classical thermodynamics theory to nanometer scale has generated a new interdisciplinary theory, “nanothermodynamics”, which is an indispensable tool for the investigation of size-dependent physicochemical properties in nanocrystals. In this work, we have demonstrated that Ec determines a number of physicochemical properties of nanoalloys. On the basis of this understanding, a unified model has been developed to calculate Ec, Tm, Teu, ΘD, and To of binary nanoalloys. The accuracy of this model is assessed through comparisons with experimental data and other theoretical results available in the published literature.

Ec(x , r , d)/Ec(∞) = Tm(x , r , d)/Tm(∞) = Teu(x , r , d)/Teu(∞) = ΘD2(x , r , d)/ΘD2(∞) = To 2(x , r , d)/To 2(∞) ⎫ ⎧ 1 ⎬ = ⎨1 − 12r /[(3 − d)h(x)] − 1 ⎭ ⎩ ⎫ ⎧ 2S (x) 1 ⎬ exp⎨− b 3R 12r /[(3 − d)h(x)] − 1 ⎭ ⎩

2. METHODOLOGY The cohesive energy Ec is defined as the difference between the average energy of the atoms in a solid and the isolated atoms. It is the sum of bond strength ε over all the coordinates of the specific atom with coordination z, and Ec = zNaε/2, where Na is the Avogadro constant.23 It has been shown that the variation of the potential profile for nanocrystals, which is related to the crystallographic structures and corresponding transition functions, can be determined from the size (r)- and dimensionality (d)-dependent Ec(r, d).24−29 Note that d is 0, 1, and 2 for nanoparticles, nanowires, and thin films, respectively. Hence, Ec(r, d) dominates the size effect on a number of physicochemical properties in low-dimensional materials. Since most current investigations are focused on single-component nanocrystals, there are very few systematic studies on nanoalloys. The composition (x)-, size-, and dimensionality-dependent physicochemical properties in nanoalloys could be predicted if the Ec(x, r, d) function is available. On the basis of Lindemann’s criterion for melting,30 the correlations between Tm(∞), ΘD(∞), and Ec(∞) could be obtained with ∞ denoting the bulk crystal. This criterion has been verified experimentally and is valid for both bulk and nanocrystals.23,26 Combining with Einstein’s explanation for the low-temperature specific heats of crystals, ΘD(∞) can be written as ΘD(∞) = c[Tm(∞)/(MVm2/3)]1/2, where c is a constant, Tm(∞) bulk melting temperature, M molar atomic or molecular weight, and Vm molar volume of crystals.31 Thus, ΘD2(∞) ∝ Tm(∞). Moreover, from Lindemann’s criterion, Tm(∞) ∝ Ec(∞) can be obtained for elemental crystals.24 Since (a) Tm(∞) and Ec(∞) are parameters to describe the bond strength and (b) the nature of chemical bonding in alloys does not change during the transition, it is understood that Tm(∞) ∝ Ec(∞) for alloys. Note that Tm(∞) here includes both melting temperatures of solid solution and eutectic alloys. Hence, ΘD2(∞) ∝ Tm(∞) ∝ Teu(∞) ∝ Ec(∞). It is assumed that these relations are also valid at the nanometer scale as a first-order approximation. So, we have: ΘD2(x, r, d)/ΘD2(∞) = Tm(x, r, d)/Tm(∞) = Teu(x, r, d)/Teu(∞) = Ec(x, r, d)/Ec(∞). A theoretical study has revealed that the possible cause for stabilization of disordered phases was lattice softening with decreasing material size,32 which could result in the depression of T0. By calculating the temperature-dependent difference of Gibbs free energy between disordered and ordered phases of Au−Cu nanoalloys, it is found that T0 decreases with decreasing ΘD, and a relation between To and ΘD can be described approximately by: To(x, r, d)/To(∞) = ΘD(x, r, d)/ ΘD(∞). Considering that the chemical ordering of Co−Pt and Fe−Pt alloys is similar to the ordering in the Au−Cu alloy, this relationship should also apply to Co−Pt and Fe−Pt nanoalloys. Hence, we have To2(x, r, d)/To2(∞) = ΘD2(x, r, d)/ΘD2(∞) = Ec(x, r, d)/Ec(∞).

(1)

where Sb(x) is the composition-dependent bulk evaporation entropy of the crystals, R ideal gas constant, and h(x) composition-dependent nearest atomic distance. For first-order approximation, Sb(x), and h(x) can be obtained by the Fox equation as follows34 1/S b(x) = (1 − x)/S b(0) + x /S b(1)

(2)

1/h(x) = (1 − x)/h(0) + x /h(1)

(3)

where Sb(0), Sb(1), h(0), and h(1) denote the corresponding values with x = 0 or x = 1.

3. RESULTS AND DISCUSSION Figure 1 plots the predicted results according to eq 1 and a semiempirical model result5,35 of Ec(x, r, d) for disc-shaped

Figure 1. Ec(x, r, d) of disc-shaped Au0.3Cu0.7 nanoalloys versus size r. The solid curve denotes model predictions from eq 1, and the symbol Δ is the result obtained from a semiempirical model.5,35

Au0.3Cu0.7 nanoalloys. Relevant parameters used in the modeling are listed in Table 1 (same for other figures). For the bulk Au0.3Cu0.7 alloy, Ec(∞) = 3.593 eV is calculated from: 1/Ec(∞) = 0.7/EcCu(∞) + 0.3/EcAu(∞), with EcCu(∞) = 3.505 eV and EcAu(∞) = 3.816 eV, respectively.36 Note that the absolute value of Ec is used here. Moreover, the disc-shaped nanocrystal is assumed to have a quasi-dimensionality of d = 1 since its surface/volume ratio is between those of a particle and a film.37 It is evident from this figure that the calculated results of Ec(x, r, d) decrease with decreasing material size, indicating the instability of nanoalloys compared to their corresponding bulk counterparts. This is in good agreement with the semiempirical model result denoted by the symbol Δ.5,35 The surface/volume ratio increases with reducing nanoalloy size, resulting in severe bond dangling and a higher energetic state of surface atoms, thus depressing the cohesive energy. 2422



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Table 1. Parameters Used in Model Predictionsa AxB1−x

EbA

TbA

SbA

EbB

TbB

SbB

Sb(x)

hA

hB

h(x)

Au0.3Cu0.7 Cu0.8Ni0.2 Cu0.5Ni0.5 Cu0.25Ni0.75 Cu0.2Ni0.8 Pd0.5Pt0.5 Ag0.045Pb0.955 Pb0.261Sn0.739 Bi0.43Sn0.57 In0.517Sn0.483 Zr0.9Al0.1 Zr0.84Al0.16 Co0.5Pt0.5 Fe0.5Pt0.5

368 338 338 338 338 377 285 195 207 243 605 605 426 415

3129 3200 3200 3200 3200 3236 2435 2022 1837 2345 4682 4682 3200 3134

117.61 105.63 105.63 105.63 105.63 116.5 117.04 96.44 112.68 103.62 129.22 129.22 133.13 132.42

338 431 431 431 431 565 195 302 302 302 326 326 565 565

3200 3186 3186 3186 3186 4098 2022 2875 2875 2875 2792 2792 4098 4098

105.63 135.28 135.28 135.28 135.28 137.87 96.44 105.04 105.04 105.04 116.76 116.76 137.87 137.87

109.22 110.47 118.63 126.41 128.09 126.29 97.21 102.65 108.19 104.3 127.86 127.05 135.46 135.09

0.2884 0.2556 0.2556 0.2556 0.2556 0.2751 0.2889 0.3501 0.3071 0.3252 0.3232 0.3232 0.2497 0.2027

0.2556 0.2492 0.2492 0.2492 0.2492 0.2775 0.3501 0.3022 0.3022 0.3022 0.2863 0.2863 0.2775 0.2775

0.2654 0.2543 0.2524 0.2508 0.2505 0.2763 0.3468 0.3134 0.3043 0.3137 0.3191 0.3167 0.2629 0.2343

The units for Eb, Tb, Sb, and h are kJ·mol−1, K, J·mol−1·K−1, and nm, respectively. SbA and SbB are calculated by Sb = Eb/Tb where values of Eb and Tb are taken from ref 36. The values of hA and hB are also from ref 36. Sb(x) and h(x) are determined by eqs 2 and 3, respectively. a

Figures 2 and 3 compare the theoretical curves based on eq 1 and the molecular dynamics simulation results of Tm(x, r, d) for

Figure 3. Tm(x, r, d) of Pd0.5Pt0.5 nanowires versus size r. The solid curve denotes model predictions from eq 1, and the symbols Δ are molecular dynamics simulation results.8

the simulation results of Tm(x, r, d) for CuxNi1−x nanoparticles. In Figure 3, although the simulation results of Tm(x, r, d) for Pd0.5Pt0.5 nanowires do not match perfectly our predicted curve, they show the same trend within measurement errors. It is believed that such a difference is caused by the solid−solid structural phase transformation from fcc to hcp, which occurs prior to the melting of Pd0.5Pt0.5 nanowires.8 Figure 4 plots the theoretical results from eq 1 and the experimental data of T eu (x, r, d) for Ag 0.045 Pb 0.955 , 10 Pb0.261Sn0.739,11 Bi0.43Sn0.57,11 and In0.517Sn0.48311 nanoparticles, where the Teu(∞) values are 577 K,39 456 K,40 412 K,41 and 393 K,42 respectively. It is clear that Teu(x, r, d) decreases with decreasing r due to the increased surface/volume ratio, and such a size effect is pronounced when r < 5 nm. Moreover, the size effects on Teu(x, r, d) of these eutectic nanoalloys are different, especially for r < 5 nm, which is caused by the corresponding Sb(x) and h(x) at eutectic composition. Our predicted results agree well with recent test data for different nanoscale eutectic systems,10,11 which were measured using in situ transmission electron microscopy. This confirms that: (a) the developed model is accurate and (b) Teu(x, r, d)/Teu(∞) = Ec(x, r, d)/Ec(∞) applies to nanosized eutectic alloys, which is similar to the size-dependent melting temperature of pure elements. As a result, Lindamann’s melting criterion, which is valid for both bulk materials and elemental nanocrystals, can also be used to study the eutectic transition of nanoalloys. This

Figure 2. Tm(x, r, d) of (a) Cu0.8Ni0.2, (b) Cu0.5Ni0.5, (c) Cu0.25Ni0.75, and (d) Cu0.2Ni0.8 nanoparticles versus size r. The solid curves denote model predictions from eq 1, and the symbols Δ6 and ○9 are molecular dynamics simulation results.

Cu0.8Ni0.2, Cu0.5Ni0.5, Cu0.25Ni0.75, and Cu0.2Ni0.8 nanoparticles6,9 as well as Pd0.5Pt0.5 nanowires.8 For the bulk CuxNi1−x alloys, the Tm(∞) values are 1461, 1588, 1666, and 1680 K for Cu0.8Ni0.2, Cu0.5Ni0.5, Cu0.25Ni0.75, and Cu0.2Ni0.8, respectively, which are taken from ref 9. For bulk Pd0.5Pt0.5, Tm(∞) = 2100 K.8 From these figures, we see that Tm(x, r, d)/Tm(∞) < 1 for isolated nanoalloys owing to the higher energetic state of the surface atoms than that of the interior atoms. This is caused by the enhanced atomic vibrational amplitude with decreasing r associated with the increase of surface/volume ratio.23,26,38 As shown in Figure 2, our model predictions agree quite well with 2423



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nanocrystals consist of only some tens to hundreds of atoms, and the statistical meaning is lost. The crystallographic structure also becomes unstable due to the bond deficit so that the increase of broken bonds causes localized structural distortion or variation and the solid has cluster characteristics. In this case, the long-range ordering of the crystalline solids no longer exists, resulting in bonding structures different from their bulk counterparts. This is thus beyond the scope and validity of our developed model. Figure 5 plots the theoretical results from eq 1 and experimental data of ΘD(x, r, d) for Zr0.9Al0.1 and Zr0.84Al0.16

Figure 4. Teu(x, r, d) of Ag0.045Pb0.955, Pb0.261Sn0.739, Bi0.43Sn0.57, and In0.517Sn0.483 nanoparticles versus size r. The solid curves denote model predictions from eq 1, and the symbols ⧫,10 ○,11 Δ,11 and ∇11 are experimental data of Ag0.045Pb0.955, Pb0.261Sn0.739, Bi0.43Sn0.57, and In0.517Sn0.483 nanoparticles, respectively.

criterion is much simpler than other criteria since it does not consider complicated phase boundary and phase equilibrium. As noted above, while a quantitative expression of Teu(x, r, d) in binary nanoalloys has been determined, it is found that severe bond dangling is the predominant factor which determines the thermal stability of nanosized alloys. This reveals the origin underlying the size-dependent Teu of eutectic alloys in the nanometer scale. The nanothermodynamics approach proposed in this work can also be used to predict Tm(x, r, d) and Teu(x, r, d) of other nanomaterials if relevant thermodynamic parameters are available. This is critical to construct the size-dependent binary phase diagrams of solid solution and eutectic alloys, which will have potential industrial applications.7,22 As the phase equilibrium on the nanometer scale is metastable and is difficult to measure experimentally, theoretical estimates are an alternative. Our work provides a new method to study the thermal stability of nanoalloys, which will add to the basic understanding of the phase transition theory. In general, we have 1 − [1/{12r/[(3 − d)h(x)] − 1}] ≈ 1 − (3 − d)h(x)/(12r) and exp{−[(2Sb(x))/(3R)][1/{12r/[(3 − d)h(x)] − 1}]} ≈ 1 − [Sb(x)/(18R)][(3 − d)h(x)]/r as a firstorder approximation when r is sufficiently large (r > 10 nm). Thus, the Tm(x, r, d)/Tm(∞) and Teu(x, r, d)/Teu(∞) functions in eq 1 can be rewritten as

Figure 5. ΘD(x, r, d) of (a) Zr0.9Al0.1 and (b) Zr0.84Al0.16 nanoparticles versus size r. The solid curves denote model predictions from eq 1, and the symbols Δ are experimental data.13

nanoparticles, where the corresponding ΘD(∞) values are 290.32 and 286.7 K,13 respectively. It is found that ΘD(x, r, d) of Zr0.9Al0.1 and Zr0.84Al0.16 nanoparticles decreases with decreasing r, and the theoretical results show good agreement with experimental data. This verifies the accuracy of the developed model and shows that the Fox equation can be used to calculate the thermodynamics parameters of bimetallic crystals. For a compatible system, the Sb(x) value is within the range Sb(0) and Sb(1). It can be determined by eq 2. This is the same for h(x) with eq 3. A number of equations have been proposed to relate the composition-dependent parameters of miscible blends to the parameters of its constituents, for example, Vegard formula, Fox equation, and other empirical formulas, etc.33,44 It has been confirmed that the results calculated from the Fox equation are consistent with experimental data for blends of polymers,44 semiconductors,33 and metallic alloys14 in both bulk and nanosized materials. An obvious advantage of the Fox equation is that it has no adjustable parameters, thus substantially simplifying the calculation of Sb(x) and h(x) for bimetallic alloys. Figures 6 and 7 plot theoretical predictions obtained from eq 1, a test datum, and Monte Carlo simulation results of To(x, r, d) for Co0.5Pt0.518 and Fe0.5Pt0.515−17 nanoalloys whose To(∞) values are 109818 and 1573 K,16,17 respectively. It should be noted that there are no test data or simulation results available for d = 2 in Figure 6, and the dotted theoretical curve is shown only for comparison. Clearly, To(x, r, d) decreases with decreasing r, and our model predictions are consistent with the test datum and simulation results. This shows it is possible to induce order in nanoalloys at a lower critical temperature. Since atomic ordering by heat treatment is mainly a kinetics problem and cooling rate plays an important role during this process,4 these details will not be discussed here. Moreover, for the same

Tm(x , r , d)/Tm(∞) = Teu(x , r , d)/Teu(∞) ≈ 1 − [S b(x)/(18R ) + 1/12][(3 − d)h(x)]/r

(4)

Equation 4 obeys the thermodynamic law of low dimensional materials, in which the variation of a size-dependent quantity is related to the surface (interface)/volume ratio, or 1/r.23−29,38,43 This indicates that surface or interface engineering plays an important role in developing high-performance nanoalloys. Moreover, eq 4 can also be roughly used to compare (a) sizedependent melting temperatures of solid solutions or eutectic alloys in different systems or (b) size-dependent melting temperatures of nanoalloys and those of their constituent metal nanocrystals. Conversely, thermodynamics has a statistical mechanics origin, and our model is only suitable for crystalline solids based on the consideration of a continuous medium.23 When r < (3 − d)h(x), i.e., r < 0.5−1 nm in this study, the 2424



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4. CONCLUSIONS A unified thermodynamic quantitative model has been developed to calculate the physicochemical properties of binary nanoalloys based on the size-, dimensionality-, and composition-dependent cohesive energy. For isolated nanoalloys, the cohesive energy, melting temperatures of solid solution and eutectic alloys, Debye temperature, and order−disorder transition temperature all decrease with decreasing material size. The cohesive energy change, attributed to severe bond dangling associated with increased surface/volume ratio, is an intrinsic factor controlling the physicochemical properties of nanoalloys. Moreover, the effect of interaction energy between two constituents on the properties of binary nanoalloys is negligible when r < 5 nm. The predictions obtained from this new model agree well with experimental data and other theoretical results. It affords a basic understanding of the physics underpinning the size-dependent properties in nanoalloys, leading to potential applications, such as construction of nanoscale binary phase diagrams and design of nanoparticles as ultrahigh-density magnetic-storage media.

Figure 6. To(x, r, d) of Co0.5Pt0.5 nanoalloys versus size r for different dimensionality d = 0, 1, and 2. The solid, dashed, and dotted curves denote model predictions from eq 1 for Co0.5Pt0.5 nanoparticles, discshaped Co0.5Pt0.5 nanoalloys, and Co0.5Pt0.5 thin films, respectively. The symbols ⧫ and ○ are experimental datum and Monte Carlo simulation results, respectively, for Co0.5Pt0.5 nanoparticles.18 The symbol Δ is a Monte Carlo simulation result for a disc-shaped Co0.5Pt0.5 nanoalloy.18

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +61-2-93517145. Fax: +61-2-93517060. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We wish to thank the University of Sydney Postdoctoral Fellowship program (Project ID: 2010-02190) for financially supporting this project.

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Figure 7. To(x, r, d) of Fe0.5Pt0.5 nanoparticles versus size r. The solid curve denotes model predictions from eq 1, and the symbols ○,15 Δ,16 and ∇17 are Monte Carlo simulation results.

r, the sequence of size effects, from strong to weak, on To(x, r, d) determined by eq 1 is nanoparticles, nanowires (or discshaped nanocrystals), and thin films due to their different surface/volume ratios, of 3/r, 2/r, and 1/r, respectively, as is evident from eq 4 and Figure 6. Hence, our model can also be used to determine the dimensionality dependence of physicochemical properties of nanoalloys. It must be mentioned that the composition-, size-, and dimensionality-dependent interaction energy Ω(x, r, d) between two constituents can also affect the properties of binary nanoalloys, which is not considered in our model. On the basis of general quantum chemistry consideration,7,45 the Ω(x, r, d) function can be obtained from Ω(x , r , d)/Ω(∞) = 1 − 2(3 − d)h(x)/r

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(5)

As a result, Ω(x, r, d) decreases with decreasing r in nanoalloys, and Ω(x, r, d) = 0 when r = 2(3 − d)h(x). For example, for Fe0.5Pt0.5 nanoparticles, d = 0, h = 0.2343 nm, and Ω(x, r, d) = 0 when r = 1.4 nm. Thus, although Ω(x, r, d) is not included in our model, its influence on the accuracy of the predicted results when r < 5 nm is not significant. For larger nanoalloys, the Ω(x, r, d) effects on their physicochemical properties are stronger, which need further investigation. 2425



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Size, Dimensionality, and Constituent Stoichiometry Dependence of

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