A new approach to construct bulk and size-dependent continuous binary solution phase diagrams of alloys

Abstract

The construction of bulk and size-dependent temperature–composition phase diagrams of alloys is critical for their industrial applications. However, the nano-phase diagrams are difficult to be determined accurately by experiments since the nano-phase equilibrium is metastable. In this work, a new approach was developed to construct both bulk and size-dependent continuous binary solution phase diagrams with three steps: (1) determining bulk atomic interaction energy by using ab initio molecular dynamics simulation; (2) calculating size-dependent melting enthalpy, melting temperature, and atomic interaction energy using a unified nanothermodynamics model; and (3) constructing phase diagrams with the above parameters, where a typical Au–Ag alloy was studied here as an example. It is found that (i) the simulated bulk atomic interaction energy is consistent with experimental data; (ii) the melting enthalpy, melting temperature, and atomic interaction energy decrease with decreasing material size for isolated nanocrystals; and (iii) the temperatures of the solidus and liquidus curves drop and the two-phase zone becomes small for the Au–Ag nanoalloy. The general approach developed here can be used to investigate other continuous binary alloy systems and can be extended to construct other phase diagrams, for example, the eutectic phase diagram.

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1. Introduction

Recently nanoalloys have been receiving great attention due to their scientific and industrial importance even though single-component metallic nanocrystals have been extensively investigated. A number of reports have demonstrated that nanoalloys show superior physicochemical properties comparable to or even better than their alloy components.^1–4 For example, a recent study indicated that Au–Ag alloy nanoparticles (NPs) have a higher catalytic activity than pure Au and Ag NPs for CO oxidation.¹ Moreover, alloying is also an efficient approach to enhance the thermal stability of nanoscale materials.³ As a result, the construct of bulk and size-dependent phase diagrams of alloys is of fundamental importance for their fabrications and industrial applications.^5–7 However, how to accurately determine the phase diagrams in experiments is a difficult task, especially at the nanometer scale, due to metastable nature of the nano-phase equilibrium.⁸ As a result, theoretical modeling has become an attractive alternative approach in recent years. For example, several theoretical studies have been implemented to establish solid solution phase diagrams of nanosized binary alloys, which have larger surface/volume ratio than that of bulk alloys.^5,6,9–12 Wautelet et al.⁵ developed a model to construct phase diagrams of Ge–Si nanoparticles by considering the size-dependent melting temperature T_m(r) of pure elements, where r is the radius of nanoparticles and nanowires or half-thickness of thin films. It is found that both solidus and liquidus curves shift to lower temperatures compared to the bulk phase diagram. But, the size dependence of the melting enthalpy H_m(r) was neglected in this model. Tanaka et al.⁶ calculated binary phase diagrams of nanoparticles by considering the composition- and temperature-dependent excess Gibbs energies and surface tensions of the solid and liquid phases with some rough approximations. Liang et al.¹⁰ derived a model to establish phase diagrams of solid solution binary systems of metals Cu–Ni, semiconductors Ge–Si, ceramics Al₂O₃–Cr₂O₃ and V₂O₃–Cr₂O₃, and organic crystals p-chlorobromobenzene–p-dibromobenzene by considering T_m(r), H_m(r) and size-dependent atomic interaction energy Ω(r). It is found that the two-phase field shrinks with decreasing r. More recently, Guisbiers et al. presented phase diagrams of Au–Cu¹¹ and Cu–Ni¹² nanoparticles with considerations of effects of size, shape and surface segregation. Moreover, both experimental and theoretical efforts^13–17 have also been implemented to understand the size-dependent eutectic phase diagrams, which are more complicated than those of the solid solution. These approaches have contributed greatly to the basic understanding of nano-phase equilibria of different alloy systems. However, to date related reports on this topic are still limited and many challenges remain even for solid solution phase diagrams. For example, the critical parameters of bulk atomic interaction energy Ω^S(∞) and Ω^L(∞) are not considered or obtained from the known bulk phase diagrams in the above theoretical methods,^5,6,9–12 where ∞ denotes the bulk, and the superscripts L and S denote the liquid and solid, respectively. This has limited applications of these approaches.

In this work, a new approach was developed to construct both bulk and size-dependent continuous binary solution phase diagrams by combining ab initio molecular dynamics (MD) simulation and nanothermodynamics modeling. A typical Au–Ag alloy system was studied here as an example due to its widespread applications in antibacterial materials,¹⁸ catalysts,¹⁹ sensors,²⁰ surface-enhanced Raman scattering,^21,22 etc.

2. Methodology

When a binary system is in a liquid–solid equilibrium state, the chemical potentials of component A (and B) in the both phases (solid and liquid phases) are equal. As a result, the solidus and liquidus curves of Au–Ag alloy system can be expressed by:¹⁰

H_mB(T_mB − T)/T_mB = Ω^S(1 − x^S_B)² − Ω^L(1 − x^L_B)² + RT ln(x^S_B/x^L_B) (1a)

H_mA(T_mA − T)/T_mA = Ω^S(x^S_B)² − Ω^L(x^L_B)² + RT ln[(1 − x^S_B)/(1 − x^L_B)] (1b)

where x is the mole fraction, T absolute temperature and R ideal gas constant. Eqn (1) can be utilized to (i) determine both x^L_B and x^S_B (or x^L_A and x^S_A with x_A + x_B = 1) in a bulk phase diagram at a certain T when T_m, H_m, Ω^S and Ω^L are known; and (ii) calculate Ω^S and Ω^L when T, T_m, H_m, x^L_B and x^S_B are available from the bulk phase diagram. For the latter, Ω^S and Ω^L are generally determined at T ≈ (T_mA + T_mB)/2 as a first-order approximation since the composition effects on Ω^S and Ω^L are weak for continuous solution alloys due to small electronegativity difference between their components.¹⁰ In this case, it is evident that a well-constructed bulk phase diagram measured in experiments is necessary if one wants to establish a size-dependent phase diagram. This has impeded the development of nanothermodynamics database. In this work, Ω^S(∞) and Ω^L(∞) will be obtained by ab initio MD simulations.

2.1 Ab initio MD simulations

Ω^S and Ω^L of regular solution are given by:¹⁰where Z is the coordination number, N_a the Avogadro constant, and ε the bond strength.¹⁰ ε can be determined by:where eqn (3a) and (3b) are for elemental crystals (Au and Ag) and alloys (Au₅₀Ag₅₀), respectively. Here, E_t is the total energy, E_a the single-atom energy, N the total number of atoms, A the number of Au atoms in Au₅₀Ag₅₀ solid solution. Note that eqn (3) is applicable for pure metals or alloys in both solid and liquid states.

For geometry optimization, first-principles density functional theory (DFT)^23,24 calculations on Au, Ag and Au₅₀Ag₅₀ solid solution are implemented by using CASTEP code.²⁵ The exchange–correlation interaction was treated within the generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) function.²⁶ All 4 atoms in the unit cells are fully relaxed with a convergence tolerance of 5 × 10⁻⁶ eV per atom of energy, 0.01 eV Å⁻¹ of maximum force, and 5 × 10⁻⁴ Å of maximum displacement. The 8 × 8 × 8 k-points²⁷ were used to sample the Brillouin zone and the energy cutoff of 360 eV. The ultrasoft pseudopotential²⁸ was used for all structures in the simulations. Based on the optimized systems, 3 × 3 × 3 supercells (Au, Ag and Au₅₀Ag₅₀ solid solution) with 108 atoms were built.

Then, the MD simulations were performed by using ab initio calculations. At first, the MD simulations were performed with NPT (dynamics with a thermostat to maintain a constant temperature and with a barostat to maintain a constant pressure). To characterize the liquid states, the constant temperatures are set as the melting points of Au, Ag and Au₅₀Ag₅₀, which are 1337, 1235 and 1306 K, respectively.²⁹ The simulated lattice parameters for liquid Au, Ag and Au₅₀Ag₅₀ are 4.397, 4.466 and 4.436 Å, respectively. The corresponding volume changes for the melting of Au, Ag and Au₅₀Ag₅₀ are 25.351%, 30.575% and 28.906%, respectively, which are much larger than the reported data of 1.2–6.8% for face-centered-cubic (FCC) structures.³⁰ Hence, these supercell structures cannot describe the liquid states accurately and the NPT method is invalid in this case. Alternatively, in this work, we conducted all MD simulations with NVT (dynamics at a fixed volume with a thermostat where a constant temperature is kept). The details will be given below.

The MD simulations with the NVT method were performed with the total simulation time and time step are 10.0 ps and 1.0 fs, respectively. The 2 × 2 × 2 k-points²⁷ were used to sample the Brillouin zone and the energy cutoff of 300 eV. The MD simulations were performed at T = 300 K, where the lattice parameters of a^S and a^L for the solid and liquid states, respectively, used in the simulations are listed in Table 1. For the solid states, the simulated results of E^S_t and E^S_a, and the calculation results of ε^S and Ω^S are also listed in Table 1. The corresponding solid structures of Au, Ag and Au₅₀Ag₅₀ solid solution after MD simulations are plotted in Fig. 1. For the liquid states, Fig. 2 shows the supercell structures of liquid Au, Ag and Au₅₀Ag₅₀ after simulations. The simulated E^L_t and E^L_a, and the calculated ε^L and Ω^L are also given in Table 1.

Table 1 Related parameters used in the simulations and modeling

	Au	Ag	Au50Ag50
a ε^S and ε^L are determined by eqn (3).b Ω^S(∞) and Ω^L(∞) are determined by eqn (2a) and (2b), respectively.c For the molar weight M and the liquid density ρ^L of Au50Ag50, the average values of those of Au and Ag are used.d Vmol = M/ρ^L.e The atomic volume can be obtained by Va = (Vmol × 10^24)/Na. The volume of unit cells V in FCC crystals is given by V = 4Va, and the lattice parameter in liquid state a^L = V^1/3.f h = (2^1/2/2)a for FCC crystals.g For Au and Ag, Sb = Eb/Tb. For Au50Ag50, the average value of those of Au and Ag is used.h The values of Tm(r), Hm(r) and Ω(r) are calculated from eqn (4).
a^S (Å)^31	4.078	4.086	4.076
E^St (eV)	−98678.488	−110991.869	−104844.038
E^Sa (eV)	−910.646	−1025.035
ε^S (eV)^a	−0.507	−0.445	−0.490
Ω^S(∞) (J mol^−1)^b		−15828.562
ρ^L (g cm^−3)	17.4 (ref. 32)	9.15 (ref. 32)	13.3^c
M (g mol^−1)	196.97 (ref. 33)	107.87 (ref. 33)	152.42^c
Vmol (cm^3 mol^−1)^d	11.320	11.789	11.460
a^L (Å)^e	4.221	4.279	4.239
E^Lt (eV)	−98684.644	−110989.958	−104845.234
E^La (eV)	−910.650	−1025.028
ε^L (eV)^a	−0.516	−0.443	−0.492
Ω^L(∞) (J mol^−1)^b		−14171.621
h (nm)^f	0.2884	0.2889	0.2882
Eb (kJ mol^−1)^33	368	285
Tb (K)^33	3129	2435
Sb (J mol^−1 K^−1)^g	117.609	117.043	117.326
Tm(r = 5 nm) (K)^h	1147.643	1060.509
Hm(∞) (kJ mol^−1)^33	12.5	11.3
Hm(r = 5 nm) (kJ mol^−1)^h	10.730	9.703
Ω^S(r = 5 nm) (J mol^−1)^h		−13592.761
Ω^L(r = 5 nm) (J mol^−1)^h		−12169.865

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Fig. 1 Solid supercell structures of (a) Au, (b) Ag and (c) Au₅₀Ag₅₀ after MD simulations with NVT at 300 K. The dark and light balls represent Au and Ag atoms, respectively, and a is the lattice parameter of each structure.

Fig. 2 Liquid supercell structures of (a) Au, (b) Ag and (c) Au₅₀Ag₅₀ after MD simulations with NVT at 300 K.

2.2 Nanothermodynamics modeling

In order to obtain the size-dependent phase diagrams of Au–Ag alloys, the size effects on these parameters in eqn (1), T_m, H_m and Ω, should be considered. Recently, it has been revealed that vacancy formation determined by the cohesive energy variation is one of the intrinsic factors that dominate the variation of the potential profile and thus size-dependent physicochemical properties of low-dimensional nanocrystals.^8,34 Therefore, the size-dependent cohesive energy function E_c(r) dominates the size dependence of a number of physicochemical properties of nanocrystals, including T_m(r), H_m(r) and Ω(r).

Combining the E_c(r) function reported in the literature,^8,34 a universal relation can be obtained for isolated nanoparticles,

where S_b = E_b/T_b is bulk solid–vapor transition entropy of a crystal as determined by bulk solid–vapor transition enthalpy E_b and solid–vapor transition temperature T_b, and h the nearest atomic distance. Substituting eqn (4) into eqn (1), we can construct size-dependent Au–Ag phase diagrams.

3. Results and discussions

As shown in Table 1, the simulation result of Ω^S(∞) = −15828.562 J mol⁻¹ agrees well with the result [Ω^S(∞) = −16 402 J mol⁻¹] obtained from experimental bulk Au–Ag phase diagram³⁵ with the deviation of only 3.5%. For Ω^L(∞), the corresponding values are −14171.621 and −15 599 J mol⁻¹,³⁵ respectively, and the deviation is 9.15%. Such a deviation may arise from (i) the assumption of composition independence of Ω (only Au₅₀Ag₅₀ was taken into account in this work); and (ii) the density value of liquid Au₅₀Ag₅₀ (average value of those of Au and Ag) used in the simulations. Nonetheless, the ab initio MD simulation results of Ω^S(∞) and Ω^L(∞) for Au–Ag alloys exhibit relative accuracy within 10% error. These data can thus be used to establish bulk Au–Ag phase diagram later.

Fig. 3 plots (1) bulk Au–Ag phase diagrams obtained in experiments;³⁵ (2) the calculation results of bulk and nanosized (r = 5 nm) Au–Ag phase diagrams from eqn (1), where T_m(r), H_m(r) and Ω(r) are calculated from eqn (4); and (3) experimental data of T_m(r) of Au nanoparticles (r = 5 nm)³⁶ denoted as the symbols ∇ [T_m(r) = 1169.4 ± 63.5 and 1180.4 ± 88.2 K] and a MD simulation result of T_m(r) of Ag nanoparticles (r = 5 nm)³⁷ denoted as the symbol Δ [T_m(r) = 1112.1 K]. As shown in the figure, the calculated bulk Au–Ag phase diagram from eqn (1), where the simulation results of Ω^S(∞) and Ω^L(∞) are used in the modeling, is consistent with that measured in experiments within 3% error. Thus, our method provides a simple and effective way to calculate the interaction energy of continuous binary alloys.

Fig. 3 Bulk and nanosized (r = 5 nm) Au–Ag phase diagrams. The solid lines denote predictions from eqn (1). The symbols ● and ○ are experimental data of the solidus and liquidus curves, respectively.³⁵ The symbols ∇ are experimental data of T_m(r) for Au nanoparticles (r = 5 nm)³⁶ and the symbol Δ is a MD simulation result of T_m(r) for Ag nanoparticles (r = 5 nm).³⁷

It is clear that T_m(r), H_m(r) and Ω(r) all decrease with decreasing r according to eqn (4), indicating the instability of isolated nanoparticles compared with their counterparts in bulk materials. From Fig. 3, we can see that the model predictions of T_m(r) are consistent with the experimental data of Au nanoparticles (r = 5 nm). Moreover, the derivation is only 5% between our calculation and the MD simulation results of T_m(r) for Ag nanoparticles (r = 5 nm). These demonstrate the accuracy of eqn (4). As r reducing, the surface/volume ratio increases, resulting in the formation of a large number of surface dangling bonds. Thus, the surface atoms are in a higher energetic state than those of the interior atoms, depressing E_c(r), T_m(r), H_m(r) and Ω(r).

As shown in Fig. 3, the decreased T_m(r) results in the drop of the solidus and liquidus curves, and the reduced Ω(r) causes the shrinkage of the two-phase zone in nanosized Au–Ag phase diagram compared with that of the bulk. The constructed size-dependent phase diagrams and these findings would be validated in future experimental studies. Our calculation results are critical for the basic understanding of the phase transition theory of nanoalloys and also for their scale-up industrial applications. If the necessary parameters are available, the general approach developed in this work could be used to establish other continuous binary alloy systems, and also be extended to construct other types of phase diagrams, for instance, the eutectic phase diagram. Moreover, it should be noted that only the size effect was focused in this work for a simplification. The dimensionality and shape effects could also be considered in our nanothermodynamics models,^8,34 which will be explored in future studies.

4. Conclusions

In summary, a simple and effective approach was developed to establish both bulk and size-dependent Au–Ag phase diagrams based on (1) bulk interaction energy Ω(∞) calculated by ab initio MD simulations; and (2) size-dependent melting temperature T_m(r), melting enthalpy H_m(r), and interaction energy Ω(r) using a unified nanothermodynamics model. It is found that the simulated Ω(∞) values are consistent with the results obtained from experimental bulk Au–Ag phase diagram. T_m(r), H_m(r) and Ω(r) all decrease with decreasing r for isolated Au and Ag nanoparticles, resulting in the drop of solidus and liquidus curves and shrinkage of the two-phase zone in nanosized Au–Ag phase diagram. The developed general approach could also be used to construct different phase diagrams in other alloy systems if the relevant parameters are available.

Acknowledgements

We wish to thank ChangBai Mountain Scholars Program and Jilin University Basic Research Grants Program for financially supporting this project and the computing resources of High Performance Computing Center of Jilin University and National Supercomputing Center in Jinan, China. We also thank G. Guisbiers for helpful discussions.

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