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

Trong, Dung Nguyen; Long, Van Cao

doi:https://doi.org/10.1155/2021/9976633

Advances in Civil Engineering

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9976633 | https://doi.org/10.1155/2021/9976633

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

Dung Nguyen Trong^1,2and Van Cao Long¹

Academic Editor: Shangtong Yang

Received31 Mar 2021

Revised03 Aug 2021

Accepted16 Aug 2021

Published02 Sept 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 (CN_d). The results show that the glass temperature () and the first peak position (r) of radial distribution function (RDF) have the values of = 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.

1. Introduction

As ferromagnetic material, iron nanoparticles (nano-Fe) exhibit a transition temperature (T_m) in range from T_m = 1043 K to T_m = 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 () and the size (D) of nano-Fe. He found that, with D values D = 3.0, 4.0, and 5.0 nm, values are = 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 (T_m), 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 (T_m), from T_m = 980 K to T_m = 1380 K [26]. Similarly, Wen et al. confirmed that glass temperature () 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: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:where ε = 0.017306 eV, a = 3.471392 Å, C = 24.939, m = 4.7877, n = 8.137381, and r_c = 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 (r), radial distribution function (RDF) [59–64], is evaluated by using the statistical method as follows:

Also, the coordination number (CN) [65] can be calculated by using the following equation: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

Figure 1 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).

(a)

(b)

As shown in Figure 1(b), the first peak height of RDF, (r), has the maximal value of (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 (r) value varies from (r) = 4.242 to (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.

(a)

(b)

As can be seen in Figure 2, 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 T_m 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.

As shown in Figure 3, 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).

(a)

(b)

As seen, when increase in atoms numbers N (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 (r) increasing from (r) = 4.242 (Figure 4a1) to (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 (CN_d) increases from CN_d = 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. Figure 5 shows the result for Fe₁₀₀₀₀ nanoparticles amorphous.

(a)

(b)

As can be seen in Figure 5(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.

(a)

(b)

(c)

(d)

As shown in Figure 6, with the thickness of shell (d), d = 1 Å, the respective values ofr, (r), CN, and CN_d 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 CN_d increases from 32.4% to 32.5% and increases (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 (r) and CN_d 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. Figure 7 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.

(a)

(b)

As shown in Figure 7, 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 (r) = 4.18, coordination number CN = 13, and coordination number density CN_d = 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 (r) (from 4.18 to and 2.26), decreasing CN (from 13 to 13, 13, 13, 13, 12, and 12), and decreasing CN_d from 32.2% to 30.2, 28.25, 25.43, 26.74, 25.85, and 24.94% (Table 4).

In the case of D 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 () at T = 900 K (glass temperature, = 900 K, corresponding with E = −2.065 eV). The result obtained that T_g = 900 K is completely consistent with the Curie phase transition temperature (T_c) [34] and glass transition temperature [71], which shows that, with amorphous Fe nanoparticles, = T_c 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.

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 (CN_d) 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 (), = 900 K, result obtained completely consistent with the Curie phase transition temperature (T_c), which shows that, with amorphous Fe nanoparticles, = T_c = 900 K, and this is quite interesting and interesting results basis for future experimental studies.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Dung Nguyen Trong and Van Cao Long. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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