#### Abstract

The influence of the number of atoms, *N* = 3000, 5000, 7000, and 9000 atoms, at temperature *T* = 300 K and temperatures *T* = 300, 500, 700, 900, 1100, 1300, and 1500 K at *N* = 9000 atoms, on microscopic structure, phase transition temperature, and mechanical property of bulk aluminium in an amorphous state is studied by the molecular dynamics method with the Sutton–Chen embedded interaction potential and the periodic boundary condition. Structural results are analyzed through the radial distribution function, the total energy of the system, the size, and the common neighbor analysis. The phase transition temperature is determined by the relationship between the total energy of the system and temperature. The mechanical property is derived from the deformation along the Z-axis. It can be noted that when the number of atoms increases, the first peak’s position for radial distribution function changes, the first peak’s height decreases, the number of FCC and HCP structural units decreases, the number of Amor structural units increases, and the total energy of system increases. It can be seen that when temperature increases, the first peak’s position changes, the first peak’s height decreases, the number of FCC and HCP structural units decreases, the number of Amor structural units increases, and the total energy of the system decreases. The obtained results are very useful for experimental studies in the future.

#### 1. Introduction

Aluminium (Al) is a nontaste, nontoxic, and nonmagnetic metal material with high conductivity. It has a density of 2.7 g/cm^{3} and exists on Earth. Aluminium has a low specific weight and high strength, and, therefore, it is applied in planes, rockets, cars, etc. In a pure state, aluminium easily reacts with air to formulate a surface oxidation layer with high stability and anticorrosion. The structure of aluminium has many forms such as icosahedral structure and decahedral structure. Properties of aluminium with nanometer-size were studied by the scanning tunneling microscope (STM) [1, 2], the atomic force microscope (AFM) [3–5], the molecular dynamics (MD) method [6], etc.

In recent decades, several methods have been proposed and developed to produce nanomaterials that have controlled morphologies such as particle size, surface, and geometry, which give them specific characteristics and applications in different fields [7].

Porous materials obtained from aluminium have many different characteristics compared with origin materials. For example, in origin materials, the stress is concentrated at a point, but in porous materials, the stress is propagated on the whole structure [8].

In practical applications, aluminium is applied majorly in the form of multicrystal, but in simulations and experiments, aluminium normally is in the form of a monocrystal. At normal pressure, aluminium has a face-centered cubic (FCC) structure [8].

Theoretical and experimental studies at the nanoscale still are a new field of research. In recent years, some researchers have been considered the phase transition from a liquid state to crystalline and amorphous states in the action of many factors such as the heating rate, the temperature, the number of atoms, and the pressure on the structure of materials. Experimental methods on these studies are very difficult and have low stability.

The molecular dynamics method is considered as a very effective tool in studying the influence of temperature, size, etc. on the stress-strain relationship [9], the microscopic structures, the phase transitions and the dynamic crystallization in Ni nanoparticles [10], and the magnetic properties of the iron nanoparticles [11].

Many types of equipment and techniques such as the microscopic electronic mechanical system (MEMS) and the microscopic optical-electronic mechanical system (MOEMS) [12] were developed to provide favorable conditions in analyzing nanodevices. Features of nanomaterials such as testing the tensile strength in nanoscale by experiments are very difficult [13]. The MD simulation studies the behavior of many monocrystal bulk metals with the FCC structure such as Al, Cu, and Ni, and body-centered cubic (BCC) structure such as Fe, Cr, and W under single-axis weight for materials of a large mass [14].

Parinello and Rahman [15] introduce a new Lagrangian to carry out MD simulations on the system stressed externally. Using this method, they apply single-axis stress with a FCC bulk grid on the system under external stress. Using this method, they applied the uniaxial tension of Ni with the periodic boundary conditions at *T* = 350 K. Selinger et al. [16] proposed a transformation in crystalline structure from FCC structure to HCP structure. Crystals then are expanded, continuously stabilizing as far as they are broken. At high temperatures, the direct fail of the system because of melting is caused by stress [17, 18].

Rentsch and Inasaki [19] carried out the MD simulation for single-axis stress of silicon. Their simulated Young modulus and the surface energy of silicon respectively are 171 GPa and 393 Jm^{−2}. A concern on the strong plastic deformation (SPD) is raised greatly in recent times because many special properties of materials can be got by processing SPD [20–22].

A series of recent studies show that the ECAP can lead to an original combination of strength and plasticity in the case of aluminium alloy [23–25]. That is a requirement in manufacturing advanced materials. These features are combined with the making of some concrete structures and related to the concrete processing regime. Therefore, the main difficulties in the ECAP are to consistently design and manufacture working tools used in the process of strong plastic deformation, and simulation methods are applied to evaluate exactly technical parameters. From the experimental data, at *T* = 300 K, pure aluminium materials have the Young modulus *E* = 71.9 GPa, the rigidity modulus = 27.2 GPa, and the strain factor *α* = 0.34 for deformation along the Z-axis. In recent time, researchers considered the influence of the heating rate, the temperature of the number of atoms, and the time of heat annealing on the structure and the mechanical property of crystalline nickel. According to the obtained results, when temperature and heating rate increase, the stress decreases, and when the number of atoms and the time of heat annealing increase, the stress increases [26]. Up to now, there is no full explanation for the action of temperature and the number of atoms on the structure, and the mechanical property of amorphous aluminium.

In this study, the influence of temperature on the structure and the mechanical property of amorphous materials is studied through the radial distribution function (RDF), the coordination number, the energy, the size, the common neighbor analysis (CNA), and the strain along the Z-axis. However, the change of microscopic structure and mechanical property from an amorphous state to a liquid state, as well as the density of atoms, is unclear, and recently, some papers were published concerning the structure of aluminium materials [27–29]. These results show that, in crystalline aluminium, the position of the first peak for the radial distribution function changes from 2.8 Å to 2.85 Å [27, 29], in amorphous aluminium, this position decreases to 2.75 Å [28, 30], and the experimental data is 3.24 Å [31]. Besides, our understanding of the mechanism of phase transition from the amorphous state to the liquid state still has many limits. Therefore, studying the influence of some concrete factors on the microscopic structure, the phase transition temperature, and the mechanical property of amorphous aluminium will have an important contribution to materials research in a liquid state, amorphous state, and crystalline state in the future [32].

Section 1 is a brief review of the microscopic structure, the phase transition temperature, and the mechanical property of amorphous aluminium. Section 2 is about the method of calculation. Section 3 is about the simulation results and discussion, and Section 4 refers to conclusions.

#### 2. Method of Calculation

Initially, amorphous aluminum (Al) atoms are thrown randomly into a cube with a density of *ρ* = 2.9 g/cm^{3}, a heating rate of 4.10^{13} K/s, and time of each step of 2 fs to set up the amorphous Al by the MD method with the Sutton–Chen embedded interaction potential [33, 34] and the periodic boundary condition [33, 34]:

The main parameters of the material are given in Table 1. In this table, *r*_{ij} is the distance between the *i*th atom and the *j*th atom, *a* is a parameter with the length dimension, *ρ*_{i} is the density of *i*th atoms, *E*_{tot} is the total energy of the system, Φ (*r*_{ij}) is the energy between the *i*th atom and the *j*th atom, *F* (*ρ*_{i}) is the interaction force on the *i*th atom, *r*_{c} is the disconnect radius, *ε* is the energy, and *C*, *m*, *n*, and *N* are constants.

Samples with *N* = 3000, 5000, 7000, and 9000 atoms at *T* = 300 K and the sample with *N* = 9000 atoms at *T* = 300, 500, 700, 900, 1100, 1300, and 1500 K are run with 4 × 10^{4} steps NVT (keeping *N*, , *T* = const) and are run together 2 × 10^{5} steps NVE (keeping *N*, , *E* = const) steps so that samples get the equilibrium state. The structures of obtained samples are studied through the RDF, the CAN, and the mechanical property is studied through the strain along the *Z-*axis as follows [26]: where , are the size of the sample at the initial time *t* = 0 and the time *t*, is the strain along the *Z-*axis, anpha is the strain factor (1/GPa), is the stress (GPa), *E* is the Young modulus (GPa), is the mass of the *i*th atom, is the velocity of the *i*th atom along the axis , is the velocity of the *i*th atom along the axis , is the interaction force between the *i*th atom and the *j*th atom, is the distance between the *i*th atom and the *j*th atom, is the volume of the *i*th atom, and is the rigidity modulus. To confirm the accuracy of this result, we use tools such as the centrosymmetric [35, 36], the bond angle analysis [37], and the bond order [38].

#### 3. Results and Discussion

##### 3.1. Influence of Number of Atoms

The radial distribution function (RDF) for bulk aluminum samples with *N* = 3000, 5000, 7000, and 9000 atoms at *T* = 300 K is shown in Table 2 and Figure 1.

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According to Figure 1 and Table 2, when the number of atoms increases from *N* = 3000 atoms to *N* = 5000, 7000, and 9000 atoms, the first peak of RDF dominates, and the position of this peak changes very little from *r* = 2.76 to *r* = 2.78 Å. Other simulation results are *r* = 2.8 Å, 2.85 Å [27, 29], 2.75 Å [28, 30], and the experimental data is *r* = 3.24 [31]. This result shows that the bulk aluminium sample does not exist in the long order but always exists in the near order, and the distance between atoms changes very little. Also, the first peak’s height for RDF changes strongly. When the number of atoms increases from *N* = 3000 to *N* = 5000, 7000, and 9000 atoms, initially, the height of the first peak decreases from (*r*) = 6.383 to (*r*) = 6.264 and then increases from (*r*) = 6.264 to (*r*) = 6.491. This shows that the sample of 3000 atoms has not got enough atoms to get bulk materials. Only when the number of atoms increases from *N* = 5000 atoms to *N* = 9000 atoms, the height of the first peak increases, and the second peak of RDF is split into two new peaks with approximately equal heights after annealing 2.10^{5} steps displacement. This result shows that the density of atoms increases, and in the sample, there always exist two structures of FCC and HCP when the number of atoms increases. Similarly, when increasing the number of atoms from *N* = 3000 to *N* = 5000, 7000, and 9000 atoms at *T* = 300 K, then the total energy of samples decreases very strongly from *E*_{tot} = –359.198 to *E*_{tot} = –598.618, −838.124, and −1077.596 eV, the size of samples increases from = 7.09 nm to = 8.31, 9.30, and 10.11 nm, and, therefore, the density of atoms increases. Next, the CNA is used to analyze the shape of the structure. The obtained results are presented in Figures 2–4 and Table 3.

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The sample of *N* = 3000 atoms at *T* = 300 K is of cube shape (Figure 2(a)), and only the FCC structure exists, where there are 13 common neighbors of FCC structure and 2987 common neighbors of Amor structure (Figure 3). When the number of atoms increases from *N* = 3000 to *N* = 5000, 7000, and 9000, common neighbors of FCC structure change from 13 to 26, 78, and 62, common neighbors of HCP structure change from 0 to 32, 58, and 130, common neighbors of FCC and HCP structures change from 13 to 58, 136, and 192, and common neighbors of Amor structure change from 2987 to 4942, 6864, and 8808. When the number of atoms increases from *N* = 3000 to *N* = 5000, 7000, and 9000 atoms, common neighbors of structure change from 0.43% to 2.17%, the FCC and HCP structures appear, and the density of atoms in samples increases (according to Table 3), and that agrees with simulation and experimental results. For amorphous material annealed at *T* = 300 K, the density of crystalline atoms increases very little, and, therefore, the amorphous state of material always exists in an equilibrium state. According to Figure 4 for the sample of *N* = 3000 atoms, the stress of the sample gets the maximum value. When the number of atoms increases from *N* = 3000 to *N* = 5000, 7000, and 9000 atoms, the stress of the sample decreases, and when the number of atoms is *N* = 9000 atoms, the stress of the sample increases again. The deformation of the sample is determined by that in a very small sphere. The linear deformation shows that the sample of *N* = 9000 atoms at *T* = 300 K has the strain factor anpha = 0.0202 and the Young modulus *E* = 49.56 GPa (see Table 4). The Young modulus of amorphous aluminium is smaller than that of crystalline aluminium, and this simulation result is in good agreement with experimental results of crystalline Ni and Al at *T* = 300 K with *E* = 71.9 GPa, = 27.2 GPa, *µ* = 0.34 [37]. When the number of atoms increases from *N* = 3000 to *N* = 5000, 7000, and 9000 atoms, elastic moduli *E*, increase, and the strain factor decreases. This shows that the material leads to the bulk material, and the influence of the number of atoms on the structure and the mechanical property of amorphous aluminium exists. We choose the sample of *N* = 9000 atoms to investigate the influence of temperature on the structure and the mechanical property of amorphous aluminium in the next section.

##### 3.2. Influence of Temperature on Structure, the Phase Transition Temperature, and the Mechanical Properties

###### 3.2.1. Influence of Temperature on the Structure

According to Table 5 and Figure 5 when temperature increases from *T* = 300 K to *T* = 500, 700, 900, 1100, 1300, and 1500 K, the first peak of RDF dominates (see Figure 5), and the position of this peak changes extremely little from *r* = 2.76 Å to *r* = 2.78 Å (Table 5). This result shows that the bulk aluminum sample does not exist in the long order but always exists in the near order and the distance between atoms changes extremely little. Also, the height of the first peak changes strongly. When temperature increases from *T* = 300 K to *T* = 500 K and from *T* = 700 K to *T* = 1100 K, the height of the first peak decreases, and the density of atoms increases. When temperature increases from *T* = 500 K to *T* = 700 K and from *T* = 1100 K to *T* = 1500 K, the height of the first peak increases, and the density of atoms decreases. Moreover, the second peak of RDF is split into two new peaks with approximately equal heights after annealing 2.10^{5} steps displacement. This result shows that the density of atoms increases, and in the sample, at least two structures exist when the temperature increases. The shape and characteristic quantities of the bulk aluminium sample with 9000 atoms at *T* = 300, 500, 700, 900, 1100, 1300, and 1500 K are given in Figure 6, and Table 6.

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According to Figure 6 and Table 6, atoms have the form of a cube. They are created by a kind of aluminium atom and distribute uniformly in the cube. When temperature increases from *T* = 300 K to *T* = 500, 700, 900, 1100, 1300, and 1500 K, the energy of samples increases from *E*_{tot} = −1077.59 to *E*_{tot} = −1072.98, −1056.39, −1047.66, −1037.88, −1028.97, and −1021.25 eV, the size of sample does not change, and, therefore, the density of atoms in the material increases. Next, the CNA is used to analyze the shape of the structure. The obtained results are presented in Figure 6.

According to Figure 6, for the sample of *N* = 9000 atoms at *T* = 300 K, the FCC and HCP structures exist, where there are 192 common neighbors of crystalline aluminium (62 common neighbors of FCC structure and 130 common neighbors of HCP structure) and 8808 common neighbors of amorphous aluminium. When temperature increases from *T* = 300 to *T* = 500, 700, 900, 1100, 1300, and 1500 K, common neighbors of FCC structure change from 62 to, 51, 58, 42, 58, 64, and 51, common neighbors of HCP structure change from 130 to 118, 123, 153, 95, 95, and 71, and common neighbors of Amor structure change from 8808 to 8831, 8819, 8805, 8847, 8841, and 8878. When temperature increases from *T* = 300 to *T* = 500, 700, 900, 1100, 1300, and 1500 K, common neighbors of structures decrease from 2.13% to 1.35%, and always the FCC and HCP structures exist. That agrees with simulation and experimental results. When temperature increases from *T* = 300 to *T* = 500, 700, 900, 1100, 1300, and 1500 K for the Amor structure of aluminium, the density of atoms decreases, and the amorphous state of material always exists in the equilibrium state.

###### 3.2.2. Influence of Temperature on the Phase Transition Temperature

To investigate the phase transition temperature of the model, the relationship between the energy and temperature in the equilibrium state was plotted. The obtained results are shown in Table 6 and Figure 7.

According to Table 6 and Figure 7, when temperature increases from *T* = 300 K to *T* = 1500 K, the energy increases linearly indicating that the phase transition temperature does not exist for amorphous aluminium material. The obtained results confirm the influence of the number of atoms and temperature on the structure and the mechanical property of amorphous aluminium material. Because of the size effect and the energy effect when the number of atoms increases, the size also increases. It can be seen that when temperature increases, the energy increases, the density of atoms increases, and the structure changes. Thus, when the number of atoms increases, the mechanical property increases, and when temperature increases, the mechanical property decreases.

###### 3.2.3. Influence of Temperature on the Mechanical Properties

According to Figure 8 when temperature increases from *T* = 300 K to *T* = 500, 700, 900, 1100, 1300, and 1500 K, the strain factor also increases from anpha = 0.01909 to anpha = 0.01913, 0.01931, 0.01945, 0.01965, 0.01974, and 0.01983, the Young modulus decreases from *E* = 52.36 GPa to *E* = 52.28, 51.78, 51.42, 50.89, 50.65, and 50.43 GPa, and the rigidity modulus also decreases from = 19.537 GPa to = 19.507, 19.321, 19.186, 18.988, 18.899, and 18.817 GPa (Figure 8). The Young modulus of amorphous aluminium is smaller than that of crystalline aluminium and these simulation results are in good agreement with experimental results of crystalline Ni and Al at *T* = 300 K with *E* = 71.9 GPa.

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#### 4. Conclusion

Through the process of researching and investigating the structure and the mechanical property of amorphous aluminium with *N* = 3000, 5000, 7000, and 9000 atoms at temperature *T* = 300 K and amorphous aluminium with *N* = 3000 atoms at temperature *T* = 300, 500, 700, 900, 1100, 1300, and 1500 K, we have the following conclusion: with the Sutton–Chen embedded interaction potential, the periodic boundary condition, and selected parameters, the obtained structural results are in good agreement with previous results [27–31]. The influence of the number of atoms and temperature on the structure and the mechanical property of amorphous aluminium is caused by the size effect. It can be noted that when temperature increases, the density of atoms decreases, the number of FCC and HCP structural units decreases, the number of Amor structural units increases, the energy of material increases, elastic moduli *E*, decrease, and the strain factor anpha increases. It can be seen that when the number of atoms increases, the size increases, the number of FCC and HCP structural units increases, the number of Amor structural units decreases, the energy of material decreases, elastic moduli *E*, increase, and the strain factor anpha decreases. The obtained results are in good agreement with the crystalline Ni results [26]. Our obtained results of the influence of temperature on the structure and the mechanical property of amorphous aluminium with *N* = 3000, 5000, 7000, and 9000 atoms at temperature *T* = 300 K and amorphous aluminium with *N* = 3000 atoms at temperature *T* = 300, 500, 700, 900, 1100, 1300, and 1500 K have important practical and scientific significance. These results are used as a basis for experimental research in the future for amorphous metal materials.

#### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

#### Disclosure

The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Authors’ Contributions

Tuan Tran Quoc performed validation, provided resources, and wrote the draft. Dung Nguyen Trong carried out conceptualization, methodology, investigation, resources, supervision, original draft preparation, and formal analysis. Ştefan Ţălu performed reviewing and editing.