**1. Introduction**

Today, the alloys PtAu [1], PdAu [2], NiAu [3] are receiving great attention from theoretical and experimental scientists [4,5] because they have many special properties compared to pure materials [6,7]. In particular, NiAu alloy is synthesized by two metals, Ni and Au, and applied in many fields of science, technology, and life such as magnetism [8–10], photocatalyst [11,12], DNA markers [13,14], or cancer treatment [15] as the agent in cell separation [16,17], and biological processing [18,19] which increase contrast and biological agents [20]. The properties of alloys such as ionization, optics, and magnetism [21] depend on the shape and the concentration of doping [22].

There are a lot of methods to research and manufacture NiAu alloy, such as experimental, theoretical, and simulated. The experimental method includes mechanical grinding [23], electric arc [24], deposition [25,26], electrochemical [27], hydrothermal [28], Sol-Gel [29], mechanics [30], micro-emulsion [31], and colloidal solution [32].

These methods can change the size and shape of the alloy in normal conditions and do not require an environment of pressure (P) and high temperature (T) [33]. Theoretical methods include initial principles, Ab initio model [34], and methods of Molecular Dynamics (MD) simulation [35–37] combined with different interaction potentials such as Finnis–Sinclair (FS) [38], and Sutton–Chen (SC) [39,40]. In particular, the method of MD simulation is considered as the most preeminent method today with low research costs, capable of researching at the atomic level and providing a huge amount of information o the structure and explaining relevant physical mechanisms [41,42].

The result of research of Ni, Au metal and NiAu alloy in the liquid state, crystalline state, an amorphous state [43–45] shows that at the temperature (T), T = 300 K, pure Ni, Au metals do not change the structure transition process when being combined to form NiAl alloy; their electronic mobility is in ranges from 5% to 99% depending on the impurity concentration [46] that leads to the crystallize processes, and the structural transitions occurring quickly [47]. They use NiAu alloys as catalysts for clean water [48–50] by using Au atoms in combination with Ni atoms to ionize water atoms. However, to meet below phase diagram, the study and synthesis of NiAu alloy [51] are being performed by electrolysis, the results showing an irregularly distributed shape and a medium particle size, which is 25 nm [52].

Recently, researchers have proposed a method using low temperature [53,54] and reduction as a way to synthesize NiAu alloy [55]. Vasquez et al. [56] also used this method to synthesize Au3Fe, Au3Co, and Au3Ni alloys. With this one, the shape and size of NiAu alloy are better controlled, and it has carried out more research in recent years [57–61]. With these obtained results [3], at high pressure [62,63] and Morse potential interaction, we can measure accurately the elastic modulus of AuNi. Lecadre et al. [64] studied the scattering and diffusion mechanism in Au-Ni alloys with Au3Ni and Au3Ni2 ratios. Berendsen et al. [65] have identified the transition temperature (Tm) of NiAu ranges from Tm = 1100 K to Tm = 1300 K with Au impurity concentration of 58% [66]. Combined with our recent results as Al [67], FeNi [68], AlNi [69], Ni1−<sup>x</sup> Fex [70], Ni1−xCux [71], Ni [72,73], these results show that the transition temperature (Tm) of Ni material; Tm is always proportional with atom number (N), N−1/3 [74,75], and the electronic structure of AuCu [76] and AgAu [77]. The phase transition of Ni material can be determined by stress or temperature [78–81], and the bonding length of Ni-Ni determined by the experimental method is r = 2.43 Å [82], while the simulation method of Dung, N.T is r = 2.45 Å [73], and P.H. Kien is r = 2.52 Å [72]. Meanwhile, Ni and Au both have significant differences in atomic radius (R) sizes such as: Ni is R = 1.245 Å, Au is R = 1.44 Å, and surface energy (E) of Ni is E = 149 meVÅ−2, Au is E = 96.8 meVÅ−<sup>2</sup> [83], which lead to the diffusion of Au atoms in the crust and Ni atoms in the core layer [84]. So, what processes were happened to NiAu alloy when there was a change in heating rate, atomic number, and temperature? To answer this question, we focus on studying the factors that affect the structure and crystallization process of NiAu alloys.

#### **2. Method of Calculation**

Initially, the ratio between NiAu alloy and Ni:Au is 1:1, as in 2048 NiAu atoms, there are 1024 Ni atoms, 1024 Au atoms (NiAu2048), 2916 atoms (NiAu2916), 4000 atoms (NiAu4000), 5324 atoms (NiAu5324), 6912 atoms (NiAu6912); all samples are studied by molecular dynamics (MD) simulation method [85–95] with embedded Sutton–Chen (SC) interaction [39,96–99] and boundary conditions recirculating with the Equation (1):

$$\mathbf{E}\_{\text{tot}} = \sum\_{\mathbf{i}=1}^{N} \frac{1}{2} \sum\_{\mathbf{j}=1,\mathbf{j}\neq\mathbf{i}}^{N} \Phi\left(\mathbf{r}\_{\overline{\mathbf{i}}}\right) + \mathbf{F}(\rho\_{\overline{\mathbf{i}}}),\\\Phi\left(\mathbf{r}\_{\overline{\mathbf{i}}}\right) = \varepsilon \left(\frac{\mathbf{a}}{\mathbf{r}\_{\overline{\mathbf{i}}}}\right)^{n},\\\mathbf{F}(\rho\_{\mathbf{i}}) = -\varepsilon \mathbf{C} \sum\_{\mathbf{i}=1}^{N} \sqrt{\mathbf{p}\_{\mathbf{i}}},\\\ \rho\_{\mathbf{i}} = \sum\_{\mathbf{j}=1,\mathbf{j}\neq\mathbf{i}}^{N} \rho\left(\mathbf{r}\_{\overline{\mathbf{i}}}\right),\\\ \rho\left(\mathbf{r}\_{\overline{\mathbf{i}}}\right) = \left(\frac{\mathbf{a}}{\mathbf{r}\_{\overline{\mathbf{i}}}}\right)^{n} \qquad \text{(1)}$$

The parameters of the NiAu alloy (Table 1) are shown below.

**Table 1.** Parameters of NiAu alloy.


The parameters of the alloy are determined by the mathematical Formula (2):

$$\varepsilon\_{\text{NiAu}} = \sqrt{\varepsilon\_{\text{Ni}} \cdot \varepsilon\_{\text{Au}}}; \text{ } \mathfrak{a}\_{\text{NiAu}} = \frac{(\mathfrak{a}\_{\text{Ni}} + \mathfrak{a}\_{\text{Au}})}{2}; \text{ } \mathfrak{m}\_{\text{NiAu}} = \frac{(\mathfrak{n}\_{\text{Ni}} + \mathfrak{n}\_{\text{Au}})}{2}; \text{ } \mathfrak{m}\_{\text{NiAu}} = \frac{(\mathfrak{m}\_{\text{Ni}} + \mathfrak{m}\_{\text{Au}})}{2}; \text{ } \mathbb{C}\_{\text{NiAu}} = \sqrt{\mathbb{C}\_{\text{Ni}} \cdot \mathbb{C}\_{\text{Au}}} \tag{2}$$

At all samples, there is an increase in temperature (T) from T = 0.0 K to T = 2000 K to NiAu alloy at the liquid state. From the liquid state, the temperature of the samples was reduced from T = 2000 K to T = 300 K to change from a liquid state to a crystalline one. After getting NiAu alloy, NiAu6912 alloys are run MD with a heating speed of 4 × 1011 K/s, <sup>4</sup> × <sup>10</sup><sup>12</sup> K/s, 4 × 1013 K/s, 4 × 1014 K/s at (T), T = 300 K. After determining the heating speed of 4 × 1012 K/s to be appropriate, the effects of NiAu2048, NiAu2916, NiAu4000, NiAu5324, NiAu6912 at T = 300 K; NiAu6912 at T = 300 K, 400 K, 500 K, 600 K, 700 K, 900 K, 1100 K are studied. All given samples are structurally studied through shape, size (l) as (3),

$$\rho = \frac{\mathbf{N}}{\mathbf{V}} \to 1 = \sqrt[3]{\frac{\mathbf{N}}{\rho}} = \sqrt[3]{\frac{(\mathbf{m}\_{\text{Ni}} \cdot \mathbf{n}\_{\text{Ni}} + \mathbf{m}\_{\text{Au}} \cdot \mathbf{n}\_{\text{Au}})}{\rho}} \tag{3}$$

radial distribution function (RDF) as (4):

$$\log(\mathbf{r}) = \frac{\mathbf{V}}{\mathbf{N}^2} \left\langle \frac{\sum\_{i} \mathbf{n}\_i(\mathbf{r})}{4\pi \mathbf{r}^2 \Delta \mathbf{r}} \right\rangle \tag{4}$$

In it: 1, ρ, r, N, ni(r), V, g(r) is the size, density, radial distance, the number of atoms, the coordinates, the volume, the probability of finding an atom in the distance from r to r + Δr. To determine the number of structural units, are applied the Common Neighborhood Analysis (CNA) method [100–103]. The crystallizing process is carried out based on the laws of Nosé el [104] and Hoover el [105] and uses the techniques of particle size analysis, atomic composition, and configuration [106].

## **3. Results and Discussion**

#### *3.1. Effect of Heating Rate*

The factors that affect the heating rate 4 × 1012 K/s, 2 × <sup>10</sup><sup>13</sup> K/s, 4 × 1013 K/s, <sup>2</sup> × <sup>10</sup><sup>14</sup> K/s, and 4 × <sup>10</sup><sup>14</sup> K/s on the structural characteristics and crystallization process of NiAu5324 alloy at temperature (T), T = 300 K, are shown in Figure 1.

**Figure 1.** The shape (**a**), g(r) of NiAu5324 alloy (**b**) at T = 300 K, and heating rate 4 <sup>×</sup> 1012 K/s.

The result shows that when NiAu5324 alloy at T = 300 K with the heating rate of <sup>4</sup> × 1012 K/s, it has a cube shape, made by two atoms: Ni shown in black and Au in yellow (Figure 1a), and has structural features such as r of radial distribution function (RDF) = 2.47 Å; the height of RDF is g(r) = 6.51, size (l), l = 10.16 nm, Etot = −446.09 eV (Figure 1b). That increasing heating rate from 4 × 1012 K/s to 2 × <sup>10</sup><sup>13</sup> K/s, 4 × <sup>10</sup><sup>13</sup> K/s, <sup>2</sup> × 1014 K/s, and 4 × 1014 K/s leads to r decreases from r = 2.47 Å to r = 2.43 Å and g(r) decreases from g(r) = 6.51 to g(r) = 6.05, l negligibly changes from l = 9.71 nm to l = 10.76 nm, and Etot negligibly changes from Etot = −440.25 eV to Etot = −447.18 eV (Table 2). These results show that increasing the heating rate leads to NiAu5324 alloy change, the state from crystalline to amorphous. To study the process of structural transition, the CNA method was used and the results are shown in Figure 2.

**Table 2.** The structural features such as r, g(r) of the radial distribution function, l, and Etot with t different heating rates.


**Figure 2.** The structural unit number shape includes FCC structure (**a**), HCP structure (**b**), Amor structure (**c**) of NiAu alloy.

The result shows that NiAu5324 alloy at the heating rate of 4 × 1012 K/s has structural shapes (Figure 3a) corresponding with 03 links Ni-Ni, Au-Au: Ni- Ni is r = 2.47 Å, Ni-Au is r = 2.47 Å, Au-Au is r = 3.17 Å (Figure 3b), and expresses through structural unit number FCC (Figure 2a), HCP (Figure 2b), Amor (Figure 2c). The obtained results are consistent with the results of Ni-Ni by experimental methods r = 2.43 Å [82], and R = 1.245 Å, with the simulation method r = 2.45 Å [73], r = 2.52 Å [72], for Au-Au, only X-ray diffraction results in an atomic radius value R = 1.44 Å [83,84]. Increasing heating rate from 4 × 1012 K/s to <sup>2</sup> × <sup>10</sup><sup>13</sup> K/s, 4 × <sup>10</sup><sup>13</sup> K/s, 2 × <sup>10</sup><sup>14</sup> K/s, and 4 × <sup>10</sup><sup>14</sup> K/s leads to r of link Ni-Ni, Ni-Au, Au-Au change values. Besides, when Ni-Ni changes from r = 2.47 Å to r = 2.41 Å, Ni-Au decreases from r = 2.47 Å to r = 2.43 Å, Au-Au decreases from r = 3.17 Å to r = 3.09 Å, corresponding to the change of g(r) and structural unit number FCC, HCP, Amor; as FCC decreases from 802 to 0.0 FCC, HCP decreases from 811 HCP to 13 HCP, Amor increases from 3711 Amor to 5311 Amor (Table 3). That confirms that there is an increase in the heating rate when the crystallization process decreases.

**Figure 3.** The structural shape (**a**), radial distribution function (**b**) of NiAu5324 alloy at heating rates of 4 <sup>×</sup> <sup>10</sup><sup>12</sup> K/s.


**Table 3.** The structural features such as links Ni-Ni, Ni-Au, Au-Au include r and g(r) with a different atomic number (N).
