Molecular Dynamics Simulation of Bulk Cu Material under Various Factors

: In this paper, the molecular dynamics (MD) method was used to study the inﬂuence of factors of bulk Cu material, such as the effect of the number of atoms (N) at temperature (T), T = 300 K, temperature T, and annealing time (t) with Cu 5324 on the structure properties, phase transition, and glass temperature T g of the bulk Cu material. The obtained results showed that the glass transition temperature (T g ) of the bulk Cu material was T g = 652 K; the length of the link for Cu-Cu had a negligible change; r = 2.475 Å; and four types of structures, FCC, HCP, BCC, Amor, always existed. With increasing the temperature the FCC, HCP, and BCC decrease, and Amorphous (Amor) increases. With an increasing number of atoms and annealing time, the FCC, HCP, and BCC increased, and Amor decreased. The simulated results showed that there was a great inﬂuence of factors on the structure found the gradient change, phase transition, and successful determination of the glass temperature point above T g of the bulk Cu material. On the basis of these results, essential support will be provided for future studies on mechanical, optical, and electronic properties.


Introduction
Metals are the most used materials, and they are readily available in nature. Among them, copper (Cu) is a metal with an atomic number of Z = 29 with outstanding properties, such as high ductility and good thermal and electrical conductivity [1][2][3][4][5][6][7], and it is a soft, ductile, orange-red metal. Therefore, Cu is widely used in human life and science and technology; for example, one can produce materials that effectively conduct heat and electricity [8], Cu thin films indispensable in electronics [9][10][11], phosphorescent materials [12], etc. As a consequence, for a long time, Cu has been a center of interest for many researchers, both in theoretical and experimental terms [13][14][15].
Experimental methods include direct MW irradiation in the presence of triethylenetetramine (TETA) [16], sliding friction plate strain [17], and the reduction method [18][19][20]. Instruments such as scanning electron microscopes (SEMs) and transmission electron microscopes (TEMs) are called electron microscopes (SEM/TEM) and are used to observe the surface details of materials, while the melting and crystallization temperatures are directly measured by differential scanning calorimetry (DSC), X-ray diffraction, optical microscopy, and electron microscopy [16]. Li-Cheng Jheng et al. [16] successfully studied the melting and crystallization process and determined the phase transition temperature (T m ) to be T m = 1358 K. In addition, it can be noted that Cu with a size less than 100 nm can be MW irradiated (microwave) with or without polyols [21,22].
Theoretically, results showed that for 5 nm materials, a significant decrease in the phase transition temperature T m to several hundred degrees for metals was produced. Furthermore, if the Cu material has a size of less than 3 nm, the T m decreases sharply (T m < 373 K) because of the size effect (surface effect) [23]. Recently, Xin Zhou et al. [24] method for fixing the glass transition temperature using the procedure they discovered earlier in [58,59], which led to the result of 794 K. This is a higher than the value of 773 K obtained from the well-known Wen-Abraham empirical criterion given in [60]. These values are also different from the result of 652 K obtained in this paper by our simulation method, which, in our opinion, is more in agreement with reality because it is a result of the fast convergent MD simulation procedure presented below.
Therefore, the question arises of whether there is any influence on the structure, phase transition, and the phase transition temperature of the bulk Cu material. To answer that question, in the content of this paper, we studied the influence of factors such as atoms number, temperature, and annealing time on the structural characteristics and phase transition of bulk Cu material using the MD simulation method. The obtained results will serve as the basis for the future study of the structural features of the experiment.

Method of Calculation
To study the structural characteristics of bulk Cu material, we applied the following procedure: Originally, Cu atoms were randomly sown into a cube with the size l, determined by Formula (1): and the molecular dynamics (MD) simulation method with the embedded interaction potential of Sutton-Chen (SC) (2) [61,74,75], and periodic boundary conditions were applied. As emphasized in [73], this choice of potential is the most appropriate one.
The corresponding parameters of bulk Cu [61,74,75] are given in Table 1. Table 1. Parameters of the cubic Cu material model [76].  The program code LAMMPS [74,75] based on Embedded Atom Method (EAM) potential [49] was applied to describe the interaction between Cu atoms.
Initially, to make bulk Cu material, the material was left to run 5 × 10 4 MD steps of recovery statistics with a heating rate of 4 × 10 11 K/s and a heating step of 1 fs at temperature (T) = 4500 K. Then, the temperature decreased from T = 4500 K to T = 1000, 900, 800, 700, 600, 500, 400, and 300 K to increase the crystallization state of the material. To determine the effect number of atoms, models with the corresponding number of atoms (N) were applied: N = 2916 atoms Cu (Cu 2916 ), 4000 atoms Cu (Cu 4000 ), 5324 atoms Cu (Cu 5324 ), and 6912 atoms Cu (Cu 6912 ). Then, the material Cu 5324 was selected as the standard to study the effect of temperature, and the glass transition temperature of the material and the annealing time (t) = 200 ps at T = 652 K were determined to increase the crystallization of the material. To study the structural features, radial distribution function (RDF), size (l), and the total energy of the system (E tot ) [77] were determined through the values of the shape quantities. All samples are conducted through the phase transition with the laws of Nosé [78], Hoover [79], and the Common Neighbors Analysis (CNA) [80,81] was applied on the basis of the glass transition temperature (T g ) by the E tot and T. For the number of atoms distribution in the material structure, the radial distribution function (RDF) was applied [82] to determine g(r): where g(r), n(r), ρ 0 , and r are functions used to determine the probability of finding an atom i in the space, the atom number, the atomic density, and the distance from atom i to another atom.
To determine the diffusion mechanism of the atoms, mean squared distance (MSD) was applied using the following expression [83]: where N is the number of atoms in the system, and r i (0) → r i (t) is the position of atom i at time t. For visualization, the software OVITO [55,56] was used. Finally, to check the accuracy of the results, Dual Energy X-ray Absorptiometry (DXA) was used [84]. These programs were edited and applied on the computer system of the Institute of Physics, University of Zielona Góra, Poland.

Structural Properties of Bulk Cu Materials
The structural properties were characterized by the number of structural units, as shown in Figure 1.
The obtained results show that the material of Cu 2916 metal was identified by a cube-shaped structure, made up of Cu metal atoms, and identified in purple (Figure 1a). To determine the interaction between atoms (which is characterized by the number of structural units) the Common Neighbors Analysis (CNA) was applied. The obtained results show that Cu 2916 material is determined by four structural units: face-centered cubic (FCC) is red, hexagonal close-packed (HCP) is blue, body-centered cubic (BCC) is black, and amorphous (Amor) is yellow ( Figure 1b); the number of structural units was shown by 183 FCC, 1216 HCP, 356 BCC, and 1258 Amor ( Figure 1c); the length of links was r Cu-Cu = 2.475 Å; the height of the first peak radial distribution function was g(r) = 4.86 ( Figure 1d). The obtained results are completely consistent with the experimental result r Cu-Cu = 2.48 Å [28] and the simulation result r Cu-Cu = 2.46 Å [43].
In addition, the material had a size (l) corresponding to l = 3.25 nm, E tot = −9931 eV. Through the obtained results, it is shown that the structural characteristic quantities that have been adopted by the Cu 2916 material model are consistent with the results of [28,43] and create a premise for the process of studying the influencing factors. The obtained results show that the material of Cu2916 metal was identified by a cube-shaped structure, made up of Cu metal atoms, and identified in purple ( Figure 1a). To determine the interaction between atoms (which is characterized by the number of structural units) the Common Neighbors Analysis (CNA) was applied. The obtained results show that Cu2916 material is determined by four structural units: face-centered cubic (FCC) is red, hexagonal close-packed (HCP) is blue, body-centered cubic (BCC) is black, and amorphous (Amor) is yellow ( Figure 1b); the number of structural units was shown by 183 FCC, 1216 HCP, 356 BCC, and 1258 Amor ( Figure 1c); the length of links was rCu-Cu = 2.475 Å ; the height of the first peak radial distribution function was g(r) = 4.86 ( Figure 1d). The obtained results are completely consistent with the experimental result rCu-Cu = 2.48 Å [28] and the simulation result rCu-Cu = 2.46 Å [43].
In addition, the material had a size (l) corresponding to l = 3.25 nm, Etot = −9931 eV. Through the obtained results, it is shown that the structural characteristic quantities that have been adopted by the Cu2916 material model are consistent with the results of [28,43] and create a premise for the process of studying the influencing factors.

Effect of Atoms Number
The results showed the influence of the number of atoms on the structural properties, as shown in Figure 2.
The obtained results show that for the material of Cu 2916 metal, the structure shape is given in Figure 2(a1); the number of structural units was shown by 183 FCC, 1216 HCP, 356 BCC, and 1258 Amor (Figure 2(a2)); the length of the links were r Cu-Cu = 2.475 Å, g(r) = 4.86, l = 3.25 nm, and E tot = −9931 eV (Figure 2(a3)). When the number of atoms increased from N = 2916 (Cu 2916 ) to N = 4000 (Cu 4000 ), 5324 (Cu 5324 ), and 6912 (Cu 6912 ) atoms, the structural shape of the material changed with the FCC increase from 183 FCC to 458 FCC, HCP increased from 1216 HCP to 2863 HCP, BCC increased from 356 BCC to 855 BCC, Amor increased from 1258 Amor to 1606, 2162, and 2736 Amor (Figure 2(a2,b2,c2,d2)). r had no change, with r Cu-Cu = 2.475 Å; g(r) changed from g(r) = 4.86 to g(r) = 4.80. l increased from l = 3.25 nm to l = 4.33 nm. E tot decreased from E tot = −9931 eV to E tot = −23525 eV ( Figure 2(a3,b3,c3,d3)). The obtained results showed that there was a close relationship between the atom number and the size and total energy of the system (Figure 3).

Effect of Atoms Number
The results showed the influence of the number of atoms on the structural properties, as shown in Figure 2.  to 855 BCC, Amor increased from 1258 Amor to 1606, 2162, and 2736 Amor ( Figure  2(a2,b2,c2,d2)). r had no change, with rCu-Cu = 2.475 Å ; g(r) changed from g(r) = 4.86 to g(r) = 4.80. l increased from l = 3.25 nm to l = 4.33 nm. Etot decreased from Etot = −9931 eV to Etot = −23525 eV (Figure 2(a3,b3,c3,d3)). The obtained results showed that there was a close relationship between the atom number and the size and total energy of the system (Figure 3). The results showed that there was a linear relationship between size (l) and the number of atoms (N), described by the expression l = 7.52-61.38N −1/3 . Similarly, the total energy of the system (Etot) and number of atoms (N) is described by the expression Etot = −13.22-3.402N (Figure 3). The obtained results showed that there was an influence between the values, similar to the results for metal [42,43,62] and alloy [44][45][46][47]. In addition, the size increase of the material was confirmed by the increases in enthalpy and entropy [43]. When increasing the atom number, the size of the matter increased by the ratio l~N −1/3 , and the total energy of the system decreased by the ratio Etot~N; this phenomenon is caused by the size effect and the surface effect. The results obtained will be the basis for experimentation methods to deploy future applications. To ensure the calculation speed as well as the stability of the structural features, we choose the Cu5324 material model as the standard material to investigate the influencing factors in the following sections.

Effect of Temperature
The effect of temperature on the structural characteristics and phase transition of bulk Cu5324 material are shown in Table 2. The results showed that there was a linear relationship between size (l) and the number of atoms (N), described by the expression l = 7.52-61.38N −1/3 . Similarly, the total energy of the system (E tot ) and number of atoms (N) is described by the expression E tot = −13.22-3.402N (Figure 3). The obtained results showed that there was an influence between the values, similar to the results for metal [42,43,62] and alloy [44][45][46][47]. In addition, the size increase of the material was confirmed by the increases in enthalpy and entropy [43]. When increasing the atom number, the size of the matter increased by the ratio l~N −1/3 , and the total energy of the system decreased by the ratio E tot~N ; this phenomenon is caused by the size effect and the surface effect. The results obtained will be the basis for experimentation methods to deploy future applications. To ensure the calculation speed as well as the stability of the structural features, we choose the Cu 5324 material model as the standard material to investigate the influencing factors in the following sections.

Effect of Temperature
The effect of temperature on the structural characteristics and phase transition of bulk Cu 5324 material are shown in Table 2. The obtained results show that when T increased from T = 300 K to T = 1000 K, the E tot decreased from E tot = −18125 eV to E tot = −17491 eV, the size l increased trivially from l = 3.970 nm to l = 3.974 nm, r decreased from r Cu-Cu = 2.475 Å to r Cu-Cu = 2.472 Å, and the g(r) decreased from g(r) = 4.87 to g(r) = 3.60. Correspondingly, with the number of structural units FCC, HCP decreased from 297 FCC to 163 FCC and 2190 HCP to 1779 HCP, BCC increased from 675 BCC to 794 BCC, and Amor increased from 2162 Amor to 2558 Amor ( Table 2). The obtained results show that with an increase in the temperature, the length of the link of Cu-Cu decreased and was consistent with the results obtained earlier for metal [42,43,62] and alloy [44][45][46][47]. The relationship between the E tot and T is shown in The obtained results show that when T increased from T = 300 K to T = 100 Etot decreased from Etot = −18125 eV to Etot = −17491 eV, the size l increased triviall = 3.970 nm to l = 3.974 nm, r decreased from rCu-Cu = 2.475 Å to rCu-Cu = 2.472 Å , g(r) decreased from g(r) = 4.87 to g(r) = 3.60. Correspondingly, with the nu structural units FCC, HCP decreased from 297 FCC to 163 FCC and 2190 HCP HCP, BCC increased from 675 BCC to 794 BCC, and Amor increased from 2162 A 2558 Amor ( Table 2). The obtained results show that with an increase in the temp the length of the link of Cu-Cu decreased and was consistent with the results o earlier for metal [42,43,62] and alloy [44][45][46][47]. The relationship between the Etot a shown in Figure 4. The obtained results show that with increases in the temperature T, Etot alw creased in linearity and was proportionally linear with T from about T = 300 K to K and from T = 700 K to T = 1000 K. Increasing T led to l negligibly increasing creasing; the number of structural units FCC, HCP, and BCC decreasing; and A creasing, which indicates that approximately T = 600 K to T = 700 K is the glass ature of bulk Cu5324 material. According to the graph results, the intersection p tween the two temperature ranges is called the glass point or the glass temperat with the value Tg = 652 K ( Figure 4); the causes of this phenomenon are the siz and the surface effect, and the phase transition of bulk Cu5324 materials is a phas tion of type 1. To determine the crystallization process of bulk Cu5324 material at t transition temperature Tg = 652 K, we investigated the effect of annealing time; tent will be presented in detail in the following section. The obtained results show that with increases in the temperature T, E tot always increased in linearity and was proportionally linear with T from about T = 300 K to T = 600 K and from T = 700 K to T = 1000 K. Increasing T led to l negligibly increasing; E tot increasing; the number of structural units FCC, HCP, and BCC decreasing; and Amor increasing, which indicates that approximately T = 600 K to T = 700 K is the glass temperature of bulk Cu 5324 material. According to the graph results, the intersection point between the two temperature ranges is called the glass point or the glass temperature (T g ), with the value T g = 652 K ( Figure 4); the causes of this phenomenon are the size effect and the surface effect, and the phase transition of bulk Cu 5324 materials is a phase transition of type 1.
To determine the crystallization process of bulk Cu 5324 material at the glass transition temperature T g = 652 K, we investigated the effect of annealing time; the content will be presented in detail in the following section.
Fang et al. [85] first made a proper choice of the metal Cu as the subject of their research for analyzing the structural change from the liquid to the solid state. Instead of using the Sutto-Chen (SC) potential as the present work did, they performed an ab initio MD simulation that described the transition procedure to Cu glass and some structural features of metallic glasses. Although the number of Cu atoms was not large (150), the authors conducted a very interesting analysis that is similar to our consideration related to Figure 4. On the basis of the energy dependence of time (Figure 2 in [85]), they noticed the difference in simple deviations around the transition temperature to the glass. To determine this temperature, they took the intersection of these straight lines as 800 K, different from the one we established in the present work (see below) of 652 K, which is in good agreement with recent studies. In our opinion, this difference appeared because the number of Cu atoms of supercell considered in [85] was too small. Our simulations show that the simulation procedure using the SC potential stabilizes for numbers of atoms on the order of thousands. Meanwhile, our analysis results on the structural phase transition of bulk Cu materials were shown more in detail.

Effect of Annealing Time
The obtained results show that when annealing the bulk Cu 5324 material at the glass temperature T g = 652 K, at t1 = 0 ps, E tot = −17791 eV. When the annealing time was increased from t1 = 0 ps to t2 = 40 ps, t3 = 100 ps, t4 = 150 ps, and t5 = 200 ps, E tot decreased from E tot = −17791 eV to E tot = −17821, −17832, −17837, and −17848 eV, which proves that when annealing at the temperature point T g in glass, E tot decreases very quickly ( Figure 5). This obtained result is completely consistent with the simulation results of previously studied metals Al [63], Fe [64], and Ag [65].
Through the obtained results, it was shown that when metal was annealed temperature Tg, metallic materials always tend to increase crystallization. Th problem: in the course of experimental research, there is always the question o point is the most standard glass temperature of a metal. To confirm the accurac obtained results, we continue to study the effect of the annealing time at glass t tures above Tg, and the results are shown in Figure 6.  Through the obtained results, it was shown that when metal was annealed at glass temperature T g , metallic materials always tend to increase crystallization. There is a problem: in the course of experimental research, there is always the question of which point is the most standard glass temperature of a metal. To confirm the accuracy of the obtained results, we continue to study the effect of the annealing time at glass temperatures above T g , and the results are shown in Figure 6.
As has been emphasized in [38,85], several kinds of crystalline structures, HCP, BCC, and FCC, can exist in simulations of different quenching processes (for different heating rates) and even of used potentials. It is well-known that in the imperfect crystallization of Cu, the FCC structure should be dominated, but in [85], the authors used the bond pair analysis technique to show that icosahedral, BCC, and tetrahedral SRO were dominant in the glass state for the quenching rate, with an average cooling rate of 2 × 10 14 K/s, and that during the rapid quenching process of Cu, there may exist an intermediate BCC phase between the fully amorphous phase and the FCC phase. By contrast, under an average cooling rate of 4.0 × 10 13 K/s, the liquid Cu crystallized into the FCC phase as it should. In [38], the authors concluded that at a cooling rate of 5 × 10 12 K/s and lower, the studied metals started to crystallize directly on cooling. High critical cooling rate values indicate the high instability of the supercooled liquid. The difference could be connected with the type of potentials used. In the analysis of Figure 6b in this paper, one can see that BCC and hexagonal close-packed (HCP) atomic arrangements are also found in Cu to approximately the same degree as FCC. Thus, the crystal morphology depends strongly on the quenching rate and simulation models used. In our paper, we see that when the annealing time increases, the structural shape of the material changes in relation to the FCC increases. Therefore, we predict that if we increased the annealing time more and simulate a larger number of Cu atoms as in [38], we would have domination of FCC structure. To prove this prediction is beyond the initial purpose of the current paper, and further work should be performed in a future publication. Nevertheless, the recently obtained results will serve as the basis for future experimental research.

Conclusions
The molecular kinetics method shows that there are factors affecting the structural characteristics of bulk Cu materials, such as the number of atoms, temperature, and annealing time. To determine these characteristics, we used the Sutton-Chen embedded force field to determine the interaction force between atoms, combined with periodic boundary conditions and the Velvet algorithm. At equilibrium, bulk Cu materials always have four types of structural units: FCC, HCP, BCC, and Amor. When increasing the number of atoms (N), leading to an increase in size (l), the total energy of the system (E tot ) decreased; while increasing the annealing time (t), l decreased and E tot decreased. As the temperature (T) increased, l increased, and E tot increased. In addition, if t was increased at T = 652 K, the number of FCC, HCP, BCC structural units increased sharply; Amor decreased the most; and the glass transition temperature of bulk Cu material was T g = 652 K. That determines the effect of N, T, and t on the structure and phase transition, crystallization of bulk Cu material. Correspondingly, with Cu-Cu bond lengths increasing in the range of r Cu-Cu = 2.473 Å to r Cu-Cu = 2.475 Å with increasing t and decreasing from r Cu-Cu = 2.475 Å to r Cu-Cu = 2.472 Å with increasing T. The obtained results are completely consistent with previous experimental and simulation results. In addition, as N increased, the l of the material was always proportional to N −1/3 , and the total E tot energy was always proportional to N. However, the obtained results showed that the bulk Cu 5324 material had a glass temperature (T g ) = 652 K. When the material was thermally annealed at T g = 652 K after time (t), t = 200 ps had an increase in crystallinity; as the number of structural units FCC, HCP, and BCC increased, Amor decreased, these results are completely consistent with previous studies. Finally, the phase transition and crystallization increase at the glass temperature T g = 652 K of the bulk Cu 5324 material are still unclear and need to be studied further in the future to explain and elucidate why this occurs in the upper and lower temperature ranges of the glass temperature. The obtained results serve as a basis for future research directions of new materials.

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

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that supports the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.