In this paper, the interatomic interaction in Pt–Pd–Cu–Au quaternary alloy NPs is described by the quantum corrected Sutton–Chen (Q-SC) many-body potentials. According to the Q-SC force field, the total potential energy of the system is expressed as

Table 1.

Table 1.

Table 1. Potential parameters for Pt–Pd–Cu–Au NPs.[31] . Element n m ɛ/meV c a/Å Pt 11 7 9.7894 71.336 3.9163 Pd 12 6 3.2864 148.205 3.8813 Cu 10 5 5.7921 84.843 3.6030 Au 11 8 7.8052 53.581 4.0651

Table 1. Potential parameters for Pt–Pd–Cu–Au NPs.[31] .

For describing the interatomic interaction between different metals, the geometric mean is used to obtain the energy parameter ɛ, while the arithmetic mean is used for the remaining parameters.^[5,27]

To conduct the atomistic simulation, tetrahexahedral (THH) NPs,^[32] covered by high-index facets, are chosen as the initial atomic configuration. Firstly, a large face-centered cubic (fcc) single crystal is created. Subsequently, a square pyramid is used to cap this cube on each face along the direction of the crystal. Finally, THH NPs bounded by 24 {210} facets are obtained. Furthermore, Pt and other metallic atoms (Pd, Cu, and Au) are randomly distributed in the THH NPs by computer-produced random seeds. Figure 1 shows the illustration of THH NPs.

Fig. 1.

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	Fig. 1. Schematic illustration of THH NPs.

Generally, the most stable structure of NPs corresponds to the lowest-energy structure. Therefore, we can transfer the structural optimization of NPs into searching for their lowest-energy structures.

For Pt–Pd–Cu–Au quaternary alloy NP consisting of N atoms, let w₀, w₁, w₂, and w₃ denote the proportions of the four types of atoms (element 1, element 2, element 3, and element 4, respectively) and w₀+w₁+w₂+w₃ = 1. Subsequently, arabic numerals are used to code different types of atoms, where 0, 1, 2, and 3 represent element 1, element 2, element 3, and element 4, respectively. Therefore, we can define a combinational optimization model for Pt–Pd–Cu–Au NPs as follows:

The earliest idea of SA algorithm was proposed by Metropolis et al. in 1953.^[33] In 1983, Kirkpatrick and co-authors successfully introduced the annealing thought to the combinatorial optimization field.^[34] Annealing thought is a kind of stochastic optimization algorithm based on the Monte–Carlo iterative solution strategy. The starting point for annealing thought is based on the similarity between the annealing processes of a material in physics and the general combinatorial optimization problem. The SA algorithm starts from a high initial temperature. As the temperature decreases, it may search for the objective function’s global optimum in solution space. So far, the SA algorithm has achieved great success in combinatorial optimization, production scheduling, machine learning, circuit design, neural network, and other fields.

The structural optimization of NPs belongs to a discrete problem. Therefore, we use an integer coding method. For an NP with N atoms, let w₁, w₂, w₃, w₄ denote the proportions of four types of atoms, respectively, and w₁+w₂+w₃+w₄=1. Use numbers 0, 1, 2, and 3 to represent the Pt, Pd, Cu, and Au atoms, respectively. All the coordinates of atoms in the NP are numbered from 1 to N. The coding scheme is defined as follows:

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The corresponding numbers of the four types of atoms are p₀ = N × w₀, p₁ = N × w₁, p₂ = N × w₂, and p₃ = N × w₃. Each sequence of 0, 2, 2, and 3 represents a solution of the problem in the solution space, and it also represents a structure of the Pt–Pd–Cu–Au quaternary NP. To construct an initial feasible solution, various atoms are randomly distributed in the N coordinate space subject to the above constraints.

Through continuous searching iteration, a neighborhood searching makes further adjustments to the feasible solution so as to approach an optimal solution. In this paper, we use the SA algorithm for neighborhood construction. With a feasible solution with N atoms, we randomly exchange two atoms’ positions to generate new solutions and calculate the corresponding fitness. To avoid useless searches, two exchanged atoms must be different types. As shown below, atom 0 and atom 2 exchange will result in a new explanation, and this exchange is effective. However, when atom 3 exchanges with another atom 3, there is no change to the current solution, so this exchange is invalid.

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There are ρ (ρ = ρ₀ ρ₁ ρ₂ ρ₃) solutions for each state. As the particle size increases, the neighborhood solutions grow exponentially. However, we change only a pair of atoms each time. During the late stage of the search, the optimization speed is slow. Therefore, for the larger size NP, we make the following improvements.

In the SA search process, according to the search generation and cluster size, we determine a sequence position parameter for each search. Early in the search, to accelerate the optimization progress, each of the variable parameters is assigned a larger progressively decreasing constant K. To obtain the optimal result, iterations are terminated when K reaches 1. In this paper, the setting of K is as follows:

In the SA algorithm, we choose the Q-SC potential function as the fitness function. When the algorithm runs, the current solution is updated if we find a lower energy than the energy of the current solution. If the mutated energy is higher than the current solution’s energy, the new solution is accepted based on a probability calculated as follows:

Fig. 2.

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	Fig. 2. Flowchart of SA.

In order to verify the efficiency of the improved SA algorithm, the stable structures of THH Pt–Pd–Cu–Au quaternary alloy NPs with 443 atoms and 1417 atoms are studied. Figures 3(a) and 3(b) show the lowest-energy configurations of Pt–Pd–Cu–Au quaternary alloy NPs obtained by the SA algorithm. The Pt–Pd–Cu–Au quaternary atomic ratios of 0.25:0.25:0.25:0.25 are selectively illustrated in this figure to simplify the discussion. Meanwhile, the experimental results illustrate that Au and Cu tend to distribute on the surface, Pt tends to occupy the middle layers, and Pd tends to occupy the inner layers.

This stable structure of Pt–Pd–Cu–Au NPs is caused by the balance of different energies. Table 2 shows the properties of Pt, Pd, Cu, and Au. Owing to the difference of cohesive energy and surface energy, Au and Cu tend to segregate on the surface, Pt tends to distribute in the middle layers, and Pd preferentially distributes in the inner layers. This segregation will produce a lower total energy.

Table 2.

Table 2.

Table 2. Materials’ properties calculated with the Q-SC potentials.[31] . Element Atomic radius/Å Lattice constant/Å Cohesive energy/eV Surface energy of [111]/J·m−2 Pt 1.39 3.916 5.84 1.404 Pd 1.37 3.881 3.89 1.229 Cu 1.28 3.603 3.49 1.495 Au 1.44 4.065 3.81 0.670

Table 2. Materials’ properties calculated with the Q-SC potentials.[31] .

Furthermore, we calculate the pair correlation function of Pt–Pd–Cu–Au quaternary alloy NPs, N(r). The pair correlation function N(r) is defined as the counted atom number at distance r from the center of mass. For two atom sizes (443 atoms and 1417 atoms), the atomic distribution tendencies are almost the same. Figures 3(c) and 3(d) illustrate that Au and Cu preferentially segregate on the surface, Pt tends to occupy the middle layers, while Pd tends to occupy the inner layers.

Fig. 3.

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Fig. 3. Atomic configurations of Pt–Pd–Cu–Au quaternary alloy NPs with (a) 443 atoms and (b) 1417 atoms. For each pair of columns, the first and the second columns represent the whole and the cross-section atomic arrangements of the NPs, respectively. Pt atoms are in orange, Pd atoms in red, Cu atoms in gray, and Au atoms in yellow. Panels (c) and (d) show the corresponding pair correlation functions.

The interior distribution of the Pt–Pd–Cu–Au quaternary alloy NPs with atomic ratio of 0.25:0.25:0.25:0.25 is also explored. As shown in Fig. 4, the alloy NPs are divided into several layers (a layer is also called a shell) according to the distance from the core to the atom. Evidently, there are five, seven, and nine layers in the NPs with 443 atoms, 1417 atoms, and 3285 atoms, respectively. These layers (shells) are numbered according to their distances to the center. Figure 4 shows that Au and Cu atoms tend to distribute at the outermost layers, Pt atoms tend to occupy the middle layers, and Pd atoms preferentially locate in the inner layers.

Fig. 4.

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	Fig. 4. Calculated atoms’ ratios in each shell: (a) 443 atoms, (b) 1417 atoms, (c) 3285 atoms.

Noble metal NPs have broad applications in chemical and petroleum industries due to their excellent catalytic performances. Generally, chemical reactions take place on the surface, and the distribution of the atoms on the surface of multi-alloy NPs is always related to the chemical properties of the component elements. Therefore, it is vital to study Pt–Pd–Cu–Au quaternary alloy NPs’ surface atomic distributions.^[35]

According to Figs. 3 and 4, we have learned that Cu and Au atoms preferentially segregate to the surface. To further analyze the segregation phenomena, two cases are investigated: (i) common segregation, and (ii) competitive segregation. At the same time, we selected two NPs of 443 atoms and 1417 atoms for comparison.

Figure 5 shows Pt–Pd–Cu–Au quaternary alloy NPs’ surface segregation curves when both Cu and Au compositions change from 0 to 50%. The segregation degree is measured by the surface atomic fraction, which is defined as C = N_A/N_S (A is Cu or Au), where N_A is the number of surface A atoms and N_S is the number of surface atoms. In Figs. 5(a) and 5(b), the left black dotted lines represent the max segregation, while the others denote no segregation. In Fig. 5, we can see that for small amounts of Cu and Au, the Cu and Au atoms tend to segregate to the surface, and reach the max segregation. We define this segregation as the common segregation. When the amounts of Cu and Au grow to the surface-to-volume ratio, they maintain the max segregation. With the compositions further growing, the surface Cu fraction begins to drop. We call this behavior the competitive segregation. This phenomenon is related to the lower surface energy of Au. By comparing Fig. 5(a) with Fig. 5(b), we find that the segregation degree will increase with the atomic size increasing. Herein, the surface segregation degree of Au is stronger than that of Cu with the same amounts of Cu and Au atoms. This effect is also associated with the lower surface energy of Au.

Fig. 5.

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	Fig. 5. Surface atomic fractions of Cu and Au with the Cu (Au) composition changing from 0 to 50% at particle sizes of (a) 443 atoms and (b) 1417 atoms.

Furthermore, we investigate the preferred sites of different metal atoms. We select Pt–Pd–Cu–Au quaternary alloy NPs of 1417 atoms and analyze the preferred sites of different metal atoms (Pt, Pd, Cu and Au) by the distribution of atoms at different positions. Firstly, we change the atomic ratio of one metal; secondly, the atomic ratios of the other three metals change with the first one, and these three metals have the same atomic ratio. Figure 6 shows the concentrations of Pt and Pd atoms at different positions (different-coordinated sites), and figure 7 shows the concentrations of Cu and Au atoms at different positions (different-coordinated sites).

Fig. 6.

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	Fig. 6. Pt, Pd, Cu, and Au concentrations at different coordination sites as a function of composition of (a)–(d) Pt or (e)–(h) Pd.

Fig. 7.

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	Fig. 7. Pt, Pd, Cu, and Au concentrations at different coordination sites as a function of composition of (a)–(d) Cu or (e)–(h) Au.

As shown in Figs. 6(a)–6(d), the atomic ratio of Pt increases and the atomic ratios of the other metals decrease with the increasing atomic number of Pt. It is found that the number of Pd atoms decreases most rapidly in the high coordination sites, the number of Au atoms decreases most rapidly in the middle coordination sites, whereas the number of Cu atoms decreases most rapidly in the low coordination sites.

As shown in Figs. 6(e)–6(h), the atomic ratio of Pd increases and the ratios of the other metals decrease with the increasing atomic number of Pd. It is found that, the number of Pt atoms decreases most rapidly in the high coordination sites, the number of Au atoms decreases most rapidly in the middle coordination sites, whereas the number of Cu atoms decreases most rapidly in the low coordination sites.

As shown in Figs. 7(a)–7(d), the atomic ratio of Cu increases and the atomic ratios of the other metals decrease with the increasing atomic number of Cu. It is found that, the number of Pt atoms decreases most rapidly in the high coordination sites, the number of Au atoms decreases most rapidly in the middle coordination sites, whereas the number of Pd atoms decreases most rapidly in the low coordination sites.

As shown in Figs. 7(e)–7(h), the atomic ratio of Au increases and the ratios of the other metals decrease with the increasing atomic number of Au. It is found that, the number of Pt atoms decreases most rapidly in the high coordination sites, the number of Pd atoms decreases most rapidly in the middle coordination sites, whereas the number of Cu atoms decreases most rapidly in the low coordination sites.

By comparing Fig. 6 with Fig. 7, we can find some interesting phenomena. The number of Pt atoms with the coordination numbers of 9 and 10 decreases slower than that of Pt atoms with the coordination numbers of 11 and 12. By comparison of the results in Figs. 3(c) and 3(d), we can find that the Pd atoms preferentially occupy high coordination sites (i.e., 11-fold or 12-fold coordination sites). The Pt atoms prefer to occupy the low coordination sites (9-fold or 10-fold coordination sites), the Au atoms tend to locate at middle coordination positions (6-fold or 8-fold coordination sites), and the Cu atoms tend to locate at the lowest coordination sites (3-fold or 5-fold coordination sites). The analysis of coordination number indicates that Au and Cu prefer to occupy the surface, Pt prefers to occupy the middle layers, while Pd tends to distribute in the inner layers.