# Effects of Number of Atoms, Shell Thickness, and Temperature on the Structure of Fe Nanoparticles Amorphous by Molecular Dynamics Method

Authors: Dung Nguyen Trong; Van Cao Long
Journal: Advances in Civil Engineering
 (2021)

---

## Abstract

This study aims to study the effect of several structural factors, such as number of atoms (N), shell thickness (d), and temperature (T), on the structure of amorphous iron nanoparticle (amorphous nano-Fe) by using the molecular dynamics (MD) method with Sutton–Chen (SC) dip interaction and free boundary conditions. The structural parameters of amorphous nano-Fe include their size (D), energy (E), radial distribution function (RDF), coordination number (CN), and coordination number density (CNd). The results show that the glass temperature (Tg) and the first peak position (r) of radial distribution function (RDF) have the values of Tg = 900 K and r = 2.55 Å, respectively. Furthermore, the values of parameters D and E are always proportional to N−1/3 and N−1, respectively. Regarding the effect of number of atoms, shell thickness, and the temperature on the structure of amorphous nano-Fe, we found that the increase in atoms number leads to decrease in the RDF height and increase in the coordination number (CN). However, increasing temperature leads to decreasing the shell thickness of amorphous nano-Fe.


---

## Body

## 1. Introduction

As ferromagnetic material, iron nanoparticles (nano-Fe) exhibit a transition temperature (Tm) in range from Tm = 1043 K to Tm = 1881 K [1]. It was reported that reducing the size (D) below D = 20 nm, nano-Fe moved from ferromagnetic state to super-para-magnetic state without any residual magnetism. This phenomenon provides a various promising application of nano-Fe [2], including the design of high-speed storage [3], magnetic resonance cameras [4], catalysts [5], nonlinear optics [6], sensors [7], and biomedicine (separation of biological molecules [8, 9], drug transmission, target drug transport, marking of cells, and so on [10]). As reported, turning the size of nano-Fe might enhance their heat capacity, magnetism, and biomedical compatibility [11]. In this regard, various many methods have been used to study the effect of influencing factors, such as number of atoms, shell thickness, and temperature, on the structure and phase transition of iron nanoparticle, such as theory methods, experiment methods, and simulation [12, 13] methods such as Green function method [14, 15] and Monte Carlo (MC) method [16] and Bethe model [17]. Among these methods, the molecular dynamics simulation (MD) method has been considered a powerful tool in both theoretical and experimental studies. Based on the MD method for Fe bulk, several authors found that its first peak position (r) of radial distribution function (RDF) was r = 2.55 Å [18–20] whereas it was r = 2.54, 2.618, and 2.570 Å, by using the experiment method, Neutrons, and X-rays, respectively [21, 22]. Similarly, its coordinate number (CN) was CN = 13.23, 12.92, and 12.10 by using MD, Neutrons, and X-rays methods, respectively. These obtained data confirmed their tight relation to the Icosahedron structure (ICO) or the Frank–Kasper polyhedron structure of the bulk material [23]. In this direction, in 2009, Hoang [24] determined the relationship between crystalline temperature (Tg) and the size (D) of nano-Fe. He found that, with D values D = 3.0, 4.0, and 5.0 nm, Tg values are Tg = 860, 878, and 909 K, respectively. The author also argued that their energy (E) and number of atoms (N) had a certain relationship with each other. In 2015, Van [25] identified the structure and crystallization process of nano-Fe at a temperature (T) of T = 300 K corresponding to annealing time (t), through 2.5 × 107 steps of MD simulation. The authors indicated that after the annealing time at T = 300 K, nano-Fe was in an amorphous state. In the case of Ni nanoparticles, several works identified their transition temperature (Tm), which was always proportional to their number of atoms (N), N−1/3 [26–31]. These authors reported that increasing atoms number (N) (from N = 336 to N = 8007 atoms) increased their phase transition temperature (Tm), from Tm = 980 K to Tm = 1380 K [26]. Similarly, Wen et al. confirmed that glass temperature (Tg) of Ni nanowires was always proportional to their D−1 value [27]. In recent years, we used the MD method for studying various metals and alloys, such as Fe [32–35], Ni [26–32, 36, 37], Ni alloy [38–42], and Al [43]. We found that the first peak position (r) radial distribution function (RDF) of Fe metal is r = 2.55 Å, which is consistent with both the experimental results [21, 22] and other previous simulations [18, 19, 21, 22, 25]. In the case of biomedical applications, to enhance their biological compatibility, the core/shell hybrid nanostructure has been developed using nonmagnetic materials (gold, silver, and polymer) as the shells. In this case, a method for simulation of core/shell model has been used to identify structural characteristic quantities based on the formation of separate layers [44, 45]. At present, there was a lack of the literature related to the factors influencing on structure of amorphous nano-Fe. To shed more light on this issue, in this study, we use the MD method with Sutton–Chen (SC) embedded interaction potential, the Verlet [46] algorithm, and free boundary conditions to research. The obtained results will serve as the basis for biomedical magnetic research in the future.


## 2. Calculation Methods

Initially, we randomly sowed these atoms Fe in spherical blocks with size (D), as follows:(1)D=23N4πρ,where density (ρ = 7.0 g cm−3) and atoms numbers (N) are obtained by the molecular dynamics (MD) method. The embedded interaction potential Sutten–Chen (SC) and free boundary conditions [47–50] can be calculated by the following equations:(2)E=∑i=1N12∑j=1,j≠1NΦrij+Fρi,Φrij=εarijn,Fρi=−εC∑i=1Nρi,ρi=∑j=1,jiNρrij,where ε = 0.017306 eV, a = 3.471392 Å, C = 24.939, m = 4.7877, n = 8.137381, and rc = 3.35 Å [26, 51].Initially, we run the recovery statistics 1 × 104 steps NVT (N: number of atoms, volume: V, and temperature: T at a constant value of T = 7000 K) to nano-Fe existing in a liquid state, the atoms do not stick together. Thereafter, we decrease the temperature (T) from T = 7000 K to T = 300 K to change the state of the material from liquid to amorphous ones with heating rate ∼1 × 106 K/s, and the time in each step of MD simulation is 0.46 fs.The effect of various factors, such as number of atoms (N, N = 3000 atoms: Fe3000, N = 4000 atoms: Fe4000, N = 5000 atoms: Fe5000, N = 10000 atoms: Fe10000, N = 15000 atoms: Fe15000, N = 20000 atoms: Fe20000), temperature (T, T = 300 K), and pressure (P, P = 0.0 GPa), on the structure of amorphous nano-Fe have been evaluated. Besides, to study the effect of shell thickness (d = 1 Å to d = 2, 3, 4 Å) with Fe10000 at T = 300 K on the structure of amorphous nano-Fe, we run 1 × 105 steps NVE. On the studying structure of amorphous nano-Fe, we use the Verlet algorithm with the heating process (followed the laws of Nosé [52] and Hoover [53]). To determine the radial distribution function (RDF) of the amorphous nano-Fe, the X-ray diffraction method has been used for determining the structural factors. Then, the Fourier function has been used to calculate their structural factors [54–58]. Based on these data, their g(r), radial distribution function (RDF) [59–64], is evaluated by using the statistical method as follows:(3)gr=nr4πr2drρ0.Also, the coordination number (CN) [65] can be calculated by using the following equation:(4)CN=4πρ∫0Rgrr2dr,where r is the cut radius (or the minimum position right after the first vertex of RDF). To check the accuracy of results, we use tools as the centrosymmetric parameter [29, 66], the bond angle analysis [67], the bond order analysis [68], and the AFM analysis method to find the morphology of surface [69] and fractal and multifractal geometric structures determined directly from the database [70].


## 3. Results and Discussion

### 3.1. Effect of Number of Atoms

Figure1 and Table 1 present the morphology and structure of amorphous nano-Fe with the different atomic numbers. As can be seen in Figure 1 and Table 1, the obtained results indicated that amorphous nano-Fe, with number of atoms N = 3000 atoms (Fe3000) at T = 300 K, has the spherical shape. These Fe atoms are uniformly distributed, as seen as the red color in Figure 1(a), with size (D = 4.68 nm) and energy (E = -2.208 eV).Figure 1
Shape (a) and radial distribution function RDF (b) of amorphous nano-Fe with number of atomsN = 3000 atoms.
(a)(b)Table 1
The size (D), energy (E), link length (r), and height of first peak RDF g (r) of Fe nanoparticles amorphous with various atoms number N.
N (atoms)300040005000100001500020000D (nm)4.684.985.306.607.778.39E (eV)−2.208−2.218−2.234−2.256−2.276−2.289r (Å)2.552.552.552.552.552.55g (r)4.2424.1414.1624.1704.2414.275As shown in Figure1(b), the first peak height of RDF, g (r), has the maximal value of g(r) = 4.242 with the first peak position (r) at r = 2.55 Å. When we increase atoms number (N) from N = 3000 atoms to N = 4000, 5000, 10000, 15000, and 20000 atoms, size (D) increases from D = 4.68 nm to D = 4.98, 5.30, 6.60, 7.77, and 8.39 nm and energy (E) decreases from E = −2.208 eV to E = −2.218, −2.234, −2.256, −2.276, and −2.289 eV whereas r has a constant value of r = 2.55 Å and g (r) value varies from g(r) = 4.242 to g(r) = 4.141, 4.162, 4.170, 4.241, and 4.275 (Table 1), which does not follow the rule indicating far interaction of amorphous nano-Fe. This almost near interaction is consistent with reported results from experimental method (r = 2.54 Å), Neutron method (r = 2.62 Å), X-ray method (r = 2.57 Å) [2, 21, 22], and simulation method (r = 2.55 Å) [18–20]. Figure 2 shows the relationship between size (D) and energy (E) with the number of atoms (N) of amorphous nano-Fe.Figure 2
Relationship between size (D) with number of atoms (N) of amorphous nano-Fe (a) and relationship between energy (E) with number of atoms (N) of amorphous nano-Fe (b).
(a)(b)As can be seen in Figure2, the increase in number of atoms N leads to increasing size D and decreasing energy E. Based on these data, the relationship between D and E with N can be satisfied with the following formula. In case that D = 12.99–127.5 N−1/3 (nm) (Figure 2(a)), E = 2.209 + 0.004 N−1 (eV) (Figure 2(b)). Thus, in all cases, D and E are proportional to N−1/3 and N−1, respectively. These results match well with the published works, such as the phase transition temperature Tm was proportional to atom number N−1/3 [26, 28, 71, 72] and to size D−1 [27]. To confirm our findings, we select the Fe10000 nanoparticles amorphous at temperature T = 300 K to match the experimental data for cubic Fe at temperature T = 300 K with r = 2.52 Å [21]. Figure 3 presents the simulation result.Figure 3
Coherence between the radial distribution function of Fe10000 nanoparticles amorphous by the simulation method and the experimental data of cubic Fe at temperature T = 300 K.As shown in Figure3, the first peak position RDF of Fe10000 nanoparticles amorphous at temperature T = 300 K obtained by the MD method has r = 2.55 Å consistent with the experimental data for Fe bulk material at T = 300 K with r = 2.52 Å [21]. To further verify these results, we combine the visualized methods (Figure 4 and Table 2).Figure 4
The radial distribution function (a) (a1, a2, … a6) and shape (b) (b1, b2, … b6) of amorphous Fe nanoparticles with various atoms numbers.
(a)(b)Table 2
Coordination number (CN) and coordination number density (CNd) of amorphous Fe nanoparticles with different number of atoms.
Atoms numbersN (atoms)300040005000100001500020000Coordination number CN131313131313Coordination number density CNd (%)29.230.231.432.232.432.5As seen, when increase in atoms numbersN (from N = 3000 atoms to N = 4000, 5000, 10000, 15000, 20000 atoms) leads to r has the constant values is r = 2.55 Å and g(r) increasing from g(r) = 4.242 (Figure 4a1) to g(r) = 4.141 (Figure 4a2), 4.162 (Figure 4a3), 4.170 (Figure 4a4), 4.241 (Figure 4a5), 4.246 (Figure 4a6), corresponding to shape of nanoparticle Fe amorphous (Figure 4b1, 4b2,.. 4b6) and coordination number (CN) has to the constant valuesis CN = 13. Also, the density coordination number (CNd) increases from CNd = 29.2% to 30, 2%, 31.4%, 32.2%, 32.4%, and 32.5% (Table 2). These data confirmed that the increase in N leads to increasing the density of atoms, thus affecting the structure of amorphous nano-Fe, and the cause of this phenomenon is the size effect. The results obtained serve as the basis for experimental studies in future biomedical applications. We choose amorphous Fe10000 nanoparticles to study the influence of factors in the next section.


### 3.2. Effect of Crust Thickness

Besides the effect of atoms number, we evaluate also the effect of thickness factor on the structure of amorphous nano-Fe in its core/shell structure. Figure5 shows the result for Fe10000 nanoparticles amorphous.Figure 5
Shape (a) and local structure (b) of core/shell nano-Fe10000 amorphous.
(a)(b)As can be seen in Figure5(a), the core/shell shape of amorphous nano-Fe has its core (marked in red), shell (marked in blue), and the thickness (d) of the shell layer. We assume that its atoms in the core are distributed evenly. Regarding the core radius (R), when R < 28 Å (Figure 5(b)), the local structural density ρ(r) has an almost constant value of 0.0825 atoms/Å3 (distributing evenly from the core layer to the shell). However, when R > 28 Å, the local structural density ρ (r) reduces strongly (Figure 5(b)). In this case, when the thickness of the shell (d) is 5 Å (Figure 5(b)), the core exhibits a dense phase, while the shell presents porous structure. To confirm this result, we combine the shape, radial distribution function RDF, coordination number CN, and the visualized method as shown in Figure 6 and Table 3.Figure 6
Shape (a) (a1, b1, c1, and d1) and radial distribution function RDF (b) (a2, b2, c2, and d2) of amorphous Fe nanoparticles with various crust thicknesses.
(a)(b)(c)(d)Table 3
Coordination number and coordination number density of amorphous Fe nanoparticles with various crust thicknesses.
Thicknessesd (Å)1234Coordination number CN13131313Coordination number density CNd (%)32.2432.3032.4032.50As shown in Figure6, with the thickness of shell (d), d = 1 Å, the respective values ofr, g(r), CN, and CNd are 2.55 Å, 3.907, 13, and 32.24%. .Then, the increase in d values (from d = 1 Å to d = 2, 3, 4 Å) leads to changing the shape of the Fe nanoparticles amorphous, the coordination number CN has value constant CN = 13, the coordination number density CNd increases from 32.4% to 32.5% and increases g (r) (from 4.19 to 4.23, 4.32, 4.46, and 4.51). However, by increasing d, the constant values of r and CN are observed at 2.55 Å and values, respectively, while g(r) and CNd increasing. An interesting result is that the coordination number density of the shell is smaller than that of the core (the core is in a dense structure).


### 3.3. Influence of Temperature

Along with the influence of both atom numbe0r and shell thickness, temperature also affects the structure of amorphous nano-Fe. Figure7 and Table 4 present the obtained data for Fe10000 nanoparticles amorphous at various temperatures (T) from T = 300 K to T = 500, 700, 800, 900, 1000, and 1100 K.Figure 7
Shape of amorphous Fe10000 nanoparticles at temperatures T = 300 K (a) and radial distribution function RDF (b) of amorphous Fe10000 nanoparticles at different temperatures (from T = 300 K to T = 1100 K).
(a)(b)Table 4
Coordination number and coordination number density of amorphous Fe nanoparticles with different temperatures.
TemperaturesT (K)30050070080090010001100EnergyE (eV)−2.256−2.192−2.121−2.083−2.065−2.018−1.991Coordination number CN13131313131212Coordination number density CNd (%)32.2030.2028.2525.4326.7425.8524.94As shown in Figure7, at temperatures T = 300 K, the results show that Fe10000 nanoparticles amorphous have spherical shape with size D = 6.6 nm, energy E = −2.256 eV, link length r = 2.55 Å, height of first peak RDF g(r) = 4.18, coordination number CN = 13, and coordination number density CNd = 30.72%. However, the increase in temperatures T from 300 K to 500, 700, 800, 900, 1000, and 1100 K leads to increasing E (from −2.256 eV to −2.192, −2.121, −2.083, −2.065, −2.018, and −1.991 eV), decreasing r (from 2.55 Å to 2.55, 2.55, 2.55, 2.55, 2.5, and 2.5 Å), decreasing g (r) (from 4.18 to and 2.26), decreasing CN (from 13 to 13, 13, 13, 13, 12, and 12), and decreasing CNd from 32.2% to 30.2, 28.25, 25.43, 26.74, 25.85, and 24.94% (Table 4).In the case ofD value, it keeps a constant value of D = 6.6 nm when increasing T. Figure 8 presents the relationship between energy (E) and temperature (T). As shown in Figure 8, the increase in T value (from 300 K to 500, 700, 800, 900, 1000, and 1100 K) leads to increasing energy E (−2.256 eV to −1.991 eV). Regarding the temperature range from T = 800 K to T = 1000 K, the energy of Fe10000 nanoparticles amorphous significant changes, corresponding to the crystallization phenomenon (Tg) at T = 900 K (glass temperature, Tg = 900 K, corresponding with E = −2.065 eV). The result obtained that Tg = 900 K is completely consistent with the Curie phase transition temperature (Tc) [34] and glass transition temperature Tg [71], which shows that, with amorphous Fe nanoparticles, Tg = Tc and there was no sudden increase in energy at the crystallization phase transition. The reason is that, with very small heating rate, the Fe nanoparticles gradually transit from the amorphous state to the liquid state and vice versa. These results are quite interesting. The results obtained are basis for future experimental studies to use amorphous Fe nanoparticles in magnetic applications.Figure 8
Relationship between energy and temperature of amorphous Fe10000 nanoparticles.


## 3.1. Effect of Number of Atoms

Figure1 and Table 1 present the morphology and structure of amorphous nano-Fe with the different atomic numbers. As can be seen in Figure 1 and Table 1, the obtained results indicated that amorphous nano-Fe, with number of atoms N = 3000 atoms (Fe3000) at T = 300 K, has the spherical shape. These Fe atoms are uniformly distributed, as seen as the red color in Figure 1(a), with size (D = 4.68 nm) and energy (E = -2.208 eV).Figure 1
Shape (a) and radial distribution function RDF (b) of amorphous nano-Fe with number of atomsN = 3000 atoms.
(a)(b)Table 1
The size (D), energy (E), link length (r), and height of first peak RDF g (r) of Fe nanoparticles amorphous with various atoms number N.
N (atoms)300040005000100001500020000D (nm)4.684.985.306.607.778.39E (eV)−2.208−2.218−2.234−2.256−2.276−2.289r (Å)2.552.552.552.552.552.55g (r)4.2424.1414.1624.1704.2414.275As shown in Figure1(b), the first peak height of RDF, g (r), has the maximal value of g(r) = 4.242 with the first peak position (r) at r = 2.55 Å. When we increase atoms number (N) from N = 3000 atoms to N = 4000, 5000, 10000, 15000, and 20000 atoms, size (D) increases from D = 4.68 nm to D = 4.98, 5.30, 6.60, 7.77, and 8.39 nm and energy (E) decreases from E = −2.208 eV to E = −2.218, −2.234, −2.256, −2.276, and −2.289 eV whereas r has a constant value of r = 2.55 Å and g (r) value varies from g(r) = 4.242 to g(r) = 4.141, 4.162, 4.170, 4.241, and 4.275 (Table 1), which does not follow the rule indicating far interaction of amorphous nano-Fe. This almost near interaction is consistent with reported results from experimental method (r = 2.54 Å), Neutron method (r = 2.62 Å), X-ray method (r = 2.57 Å) [2, 21, 22], and simulation method (r = 2.55 Å) [18–20]. Figure 2 shows the relationship between size (D) and energy (E) with the number of atoms (N) of amorphous nano-Fe.Figure 2
Relationship between size (D) with number of atoms (N) of amorphous nano-Fe (a) and relationship between energy (E) with number of atoms (N) of amorphous nano-Fe (b).
(a)(b)As can be seen in Figure2, the increase in number of atoms N leads to increasing size D and decreasing energy E. Based on these data, the relationship between D and E with N can be satisfied with the following formula. In case that D = 12.99–127.5 N−1/3 (nm) (Figure 2(a)), E = 2.209 + 0.004 N−1 (eV) (Figure 2(b)). Thus, in all cases, D and E are proportional to N−1/3 and N−1, respectively. These results match well with the published works, such as the phase transition temperature Tm was proportional to atom number N−1/3 [26, 28, 71, 72] and to size D−1 [27]. To confirm our findings, we select the Fe10000 nanoparticles amorphous at temperature T = 300 K to match the experimental data for cubic Fe at temperature T = 300 K with r = 2.52 Å [21]. Figure 3 presents the simulation result.Figure 3
Coherence between the radial distribution function of Fe10000 nanoparticles amorphous by the simulation method and the experimental data of cubic Fe at temperature T = 300 K.As shown in Figure3, the first peak position RDF of Fe10000 nanoparticles amorphous at temperature T = 300 K obtained by the MD method has r = 2.55 Å consistent with the experimental data for Fe bulk material at T = 300 K with r = 2.52 Å [21]. To further verify these results, we combine the visualized methods (Figure 4 and Table 2).Figure 4
The radial distribution function (a) (a1, a2, … a6) and shape (b) (b1, b2, … b6) of amorphous Fe nanoparticles with various atoms numbers.
(a)(b)Table 2
Coordination number (CN) and coordination number density (CNd) of amorphous Fe nanoparticles with different number of atoms.
Atoms numbersN (atoms)300040005000100001500020000Coordination number CN131313131313Coordination number density CNd (%)29.230.231.432.232.432.5As seen, when increase in atoms numbersN (from N = 3000 atoms to N = 4000, 5000, 10000, 15000, 20000 atoms) leads to r has the constant values is r = 2.55 Å and g(r) increasing from g(r) = 4.242 (Figure 4a1) to g(r) = 4.141 (Figure 4a2), 4.162 (Figure 4a3), 4.170 (Figure 4a4), 4.241 (Figure 4a5), 4.246 (Figure 4a6), corresponding to shape of nanoparticle Fe amorphous (Figure 4b1, 4b2,.. 4b6) and coordination number (CN) has to the constant valuesis CN = 13. Also, the density coordination number (CNd) increases from CNd = 29.2% to 30, 2%, 31.4%, 32.2%, 32.4%, and 32.5% (Table 2). These data confirmed that the increase in N leads to increasing the density of atoms, thus affecting the structure of amorphous nano-Fe, and the cause of this phenomenon is the size effect. The results obtained serve as the basis for experimental studies in future biomedical applications. We choose amorphous Fe10000 nanoparticles to study the influence of factors in the next section.


## 3.2. Effect of Crust Thickness

Besides the effect of atoms number, we evaluate also the effect of thickness factor on the structure of amorphous nano-Fe in its core/shell structure. Figure5 shows the result for Fe10000 nanoparticles amorphous.Figure 5
Shape (a) and local structure (b) of core/shell nano-Fe10000 amorphous.
(a)(b)As can be seen in Figure5(a), the core/shell shape of amorphous nano-Fe has its core (marked in red), shell (marked in blue), and the thickness (d) of the shell layer. We assume that its atoms in the core are distributed evenly. Regarding the core radius (R), when R < 28 Å (Figure 5(b)), the local structural density ρ(r) has an almost constant value of 0.0825 atoms/Å3 (distributing evenly from the core layer to the shell). However, when R > 28 Å, the local structural density ρ (r) reduces strongly (Figure 5(b)). In this case, when the thickness of the shell (d) is 5 Å (Figure 5(b)), the core exhibits a dense phase, while the shell presents porous structure. To confirm this result, we combine the shape, radial distribution function RDF, coordination number CN, and the visualized method as shown in Figure 6 and Table 3.Figure 6
Shape (a) (a1, b1, c1, and d1) and radial distribution function RDF (b) (a2, b2, c2, and d2) of amorphous Fe nanoparticles with various crust thicknesses.
(a)(b)(c)(d)Table 3
Coordination number and coordination number density of amorphous Fe nanoparticles with various crust thicknesses.
Thicknessesd (Å)1234Coordination number CN13131313Coordination number density CNd (%)32.2432.3032.4032.50As shown in Figure6, with the thickness of shell (d), d = 1 Å, the respective values ofr, g(r), CN, and CNd are 2.55 Å, 3.907, 13, and 32.24%. .Then, the increase in d values (from d = 1 Å to d = 2, 3, 4 Å) leads to changing the shape of the Fe nanoparticles amorphous, the coordination number CN has value constant CN = 13, the coordination number density CNd increases from 32.4% to 32.5% and increases g (r) (from 4.19 to 4.23, 4.32, 4.46, and 4.51). However, by increasing d, the constant values of r and CN are observed at 2.55 Å and values, respectively, while g(r) and CNd increasing. An interesting result is that the coordination number density of the shell is smaller than that of the core (the core is in a dense structure).


## 3.3. Influence of Temperature

Along with the influence of both atom numbe0r and shell thickness, temperature also affects the structure of amorphous nano-Fe. Figure7 and Table 4 present the obtained data for Fe10000 nanoparticles amorphous at various temperatures (T) from T = 300 K to T = 500, 700, 800, 900, 1000, and 1100 K.Figure 7
Shape of amorphous Fe10000 nanoparticles at temperatures T = 300 K (a) and radial distribution function RDF (b) of amorphous Fe10000 nanoparticles at different temperatures (from T = 300 K to T = 1100 K).
(a)(b)Table 4
Coordination number and coordination number density of amorphous Fe nanoparticles with different temperatures.
TemperaturesT (K)30050070080090010001100EnergyE (eV)−2.256−2.192−2.121−2.083−2.065−2.018−1.991Coordination number CN13131313131212Coordination number density CNd (%)32.2030.2028.2525.4326.7425.8524.94As shown in Figure7, at temperatures T = 300 K, the results show that Fe10000 nanoparticles amorphous have spherical shape with size D = 6.6 nm, energy E = −2.256 eV, link length r = 2.55 Å, height of first peak RDF g(r) = 4.18, coordination number CN = 13, and coordination number density CNd = 30.72%. However, the increase in temperatures T from 300 K to 500, 700, 800, 900, 1000, and 1100 K leads to increasing E (from −2.256 eV to −2.192, −2.121, −2.083, −2.065, −2.018, and −1.991 eV), decreasing r (from 2.55 Å to 2.55, 2.55, 2.55, 2.55, 2.5, and 2.5 Å), decreasing g (r) (from 4.18 to and 2.26), decreasing CN (from 13 to 13, 13, 13, 13, 12, and 12), and decreasing CNd from 32.2% to 30.2, 28.25, 25.43, 26.74, 25.85, and 24.94% (Table 4).In the case ofD value, it keeps a constant value of D = 6.6 nm when increasing T. Figure 8 presents the relationship between energy (E) and temperature (T). As shown in Figure 8, the increase in T value (from 300 K to 500, 700, 800, 900, 1000, and 1100 K) leads to increasing energy E (−2.256 eV to −1.991 eV). Regarding the temperature range from T = 800 K to T = 1000 K, the energy of Fe10000 nanoparticles amorphous significant changes, corresponding to the crystallization phenomenon (Tg) at T = 900 K (glass temperature, Tg = 900 K, corresponding with E = −2.065 eV). The result obtained that Tg = 900 K is completely consistent with the Curie phase transition temperature (Tc) [34] and glass transition temperature Tg [71], which shows that, with amorphous Fe nanoparticles, Tg = Tc and there was no sudden increase in energy at the crystallization phase transition. The reason is that, with very small heating rate, the Fe nanoparticles gradually transit from the amorphous state to the liquid state and vice versa. These results are quite interesting. The results obtained are basis for future experimental studies to use amorphous Fe nanoparticles in magnetic applications.Figure 8
Relationship between energy and temperature of amorphous Fe10000 nanoparticles.


## 4. Conclusion

The molecular dynamics (MD) method successfully indicates that amorphous nano-Fe exhibits the potential embedded Sutton–Chen (SC) and free boundary conditions. The obtained results show that the first peak position (r) of the radial distribution function (RDF) has the value r = 2.55 Å, which is consistent with the published data from the experimental method, Neutron method, X-ray method, and another simulation method. The size (D) and energy (E) of amorphous nano-Fe are proportional to the atoms number N−1/3 and N−1, respectively. The obtained data for radial distribution function (RDF), coordination number (CN), and density coordination number (CNd) are all consistent with the published works by simulation and experimental studies. Amorphous nano-Fe at temperature (T) T = 300 K has the amorphous structure (with CN = 13), in which the increase in D value leads to decreasing energy (E). The molecular dynamics method for the core/shell Fe10000 nanoparticles amorphous indicates the core layer (with radius r < 28 Å) in dense structure, while the shell layer is in the porous structure. In addition, identified glass temperature (Tg), Tg = 900 K, result obtained completely consistent with the Curie phase transition temperature (Tc), which shows that, with amorphous Fe nanoparticles, Tg = Tc = 900 K, and this is quite interesting and interesting results basis for future experimental studies.



---

*Source: 9976633-2021-09-02.xml*