Available online at www.HighTechJournal.org  

HighTech and Innovation  
Journal 

Vol. 2, No. 3, September, 2021 

 

 

158 

 

 

Study of Effect of Size on Iron Nanoparticle by Molecular 

Dynamics Simulation 

Pham Huu Kien a , Yiachu Khamphone a, Giap Thi Thuy Trang a* 
a Department of Physics, Thainguyen University of Education, No. 28 Luong Ngoc Quyen, Thainguyen, Vietnam. 

Received 05 January 2021; Revised 13 April 2020; Accepted 12 May 2021; Published 01 September 2021 

Abstract 

We use molecular dynamics simulation to study iron Nanoparticles (NPs) consisting of 4000, 5000, and 6000 atoms at 

temperatures of 300 and 900 K. The crystallization and microstructure were analyzed through the pair radial distribution 

function (PRDF), the potential energy per atom, the distribution of atom types and dynamical local structure parameters 

<fx>, where x is the bcc, ico or 14. The simulation indicated that amorphous NP contains a large number of ico-type 

atoms that play a role in preventing crystallization. Amorphous NP is crystallized through transformations of f14 > 0 and 

fbcc = 0 type to bcc-type atoms when it is annealed at 900 K upon 40 ns. The growth of crystal clusters happens in parallel 

with the changing of their microstructure. The behavior of the crystal cluster resembles the nucleation process described 

by classical nucleation theory. Furthermore, we found that the amorphous NP has two parts: the core has a structure 

similar to that of amorphous bulk, while the surface structure is more porous and amorphous. Unlike amorphous NP, 

crystalline NP also has three parts: the core is the bcc, the next part is the distorted bcc and the surface is amorphous. 

Amorphous and crystalline NPs have part of a core which has a structure that does not depend on size. 

Keywords: Nanoparticle; Crystallization; Molecular Dynamics; Amorphous Iron; Effect Size. 

 

1. Introduction 

Iron nanoparticle has many attractive properties and finds important applications in different areas of industry. So, 

the interest in this type of material has continued to grow in recent years [1-3]. NP can be produced in either 

amorphous or crystalline states. The crystallization process, which plays an important role in modern science and 

technology [1, 2], has been investigated by both experiment and simulation [4-6]. Computational simulations have 

been successfully conducted to study the amorphous solid-crystal transitions at atomic levels because simulation has 

advanced in probing those transitions since it allows calculating the trajectory of individual atoms [7-9].  

These studies supplement the commonly used classical nucleation theory (CNT) [10]. For instance, large-scale 

molecular dynamics (MD) simulations [9] captured the spontaneous nucleation and subsequent grain growth from the 

atomistic viewpoint. The temperature dependence of nucleation rate and incubation time as obtained from the 

simulation shows a characteristic shape with nose at critical temperature. The simulation method also has the 

capability to distinguish structures of different phases using geometric units or cells [11, 12]. Most of these works 

indicated that CNT can be applied to the crystallization process observed in the simulation, while other works 

suggested that CNT does not properly describe all aspects of the nucleation process. It was shown that nuclei could be 

formed through complex pathways and exhibit different structures, shapes, and surface morphology from those 

assumed in the CNT.  

                                                           
* Corresponding author: tranggtt@tnue.edu.vn 

 
http://dx.doi.org/10.28991/HIJ-2021-02-03-01 

 This is an open access article under the CC-BY license (https://creativecommons.org/licenses/by/4.0/). 

© Authors retain all copyrights. 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0003-1002-9678


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In particular, the calculation results of Voronoi polyhedra indicate that the liquid-crystal transition for pure iron 

comprises the enlargement of the coordination number and a transformation of local symmetry from five-fold to four-

fold [13]. In accordance with the simulation [14, 15], the crystallization proceeds by forming a small cluster with a 

BCC structure, then the core of this cluster transforms into an FCC structure, but the surface has a BCC structure. 

Other simulations on the binary Ni50Al50 system showed that the crystallization proceeds with the formation of initial 

non-equilibrium long range order regions (LROR) and a subsequent transition to equilibrium LROR [16]. Furthermore, 

it was revealed that there were multiple intermediate states between disordered and crystalline phases. For the Cu-Ni 

system [17, 18], the bcc structures act as unstable intermediate states, which are dominated by initial nuclei and 

eventually transformed into FCC and HCP structures. Such intermediate structures have demonstrated the validity of 

the step rule of Ostwald. 

Although the nanoparticle (NP) has been intensively investigated by simulation and experiment [4-6, 19], the 

crystallization as well as the microstructure of Fe NP remain poorly understood. Namely, the structure heterogeneity 

and effect of size on NP are still unclear yet. In the present paper, we develop previous simulations for mechanisms 

underlying the nucleation and growth of crystals in Fe NP and study the effect of size on iron nanoparticles. The 

micro-structural evolution is directly traced on the basis of newly proposed dynamical local structure parameters, 

analyzed in terms of the common neighbor method, transition of different x-types, and angle distribution. We also 

perform a systematic analysis of the temporal structure of crystalline-like clusters in order to identify intermediate 

states. The local microstructure of amorphous and crystalline NP will also be discussed.  

2. Calculation Procedure 

 We have prepared a NP model at 300 K containing 5000 Fe atoms under free boundary conditions. All atoms are 

placed in a sphere with a radius of 28 Å. Pak-Doyma potential was used to calculate atomic interactions [20]. We also 

prepared another model at 900 K. For the convenience of discussion, we call those models as 300- and 900-sample, 

respectively. The 300-sample was constructed by statistic relaxation and MD simulation [21].  

Figure 1. Flowchart of molecular dynamics simulation method 

Start 

- Inputting the characteristic coefficients: T, n, ,dt…. 

- Select the initial coordinates and velocities for the atoms 

k=1 

- Calculation of force  

- Analyzing Newton's equations of motion. 

- Let the atoms move freely under the effect of force. 

- Determine the new coordinates and velocity of each atom after each time step. 

k<kmax? 

Show results 

Determine the average value of the quantity 

End 

True 

False 

k=k+1 



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Higher temperature sample is constructed by heating up the 300-sample to desired temperature. Then the obtained 
sample is isothermally annealed for long time by using NVT ensemble (constant temperature and volume). The 
annealing time is about 40 ns. The structure characteristics were collected for different configurations during the 
annealing process. The pair radial distribution function (PRDF) is calculated by the procedure reported in Kodama et 
al. (2006) [22]. The preparedness for 4000-atoms, 6000-atoms sample at 300, 900K is similar to that for 5000-atoms 
sample. 

We determine the atomic identity (x-type) through the coordination cell which consists of a central atom and 
neighbors. Here the distance between central atom and neighbor is less than cutoff distance rcutoff = 3.35 Å. The cutoff 
distance is determined from first minimum of PRDF. The atom having 14 neighbor is denoted as 14-type atom. Other 
x-type atoms are identified by CNA method [23].  

For instance, the central atom of bcc cell satisfies two conditions: (i) it has 14 neighbors; (ii) six among its 

neighbors possesses “[4, 4, 4]” and other eight ones have “[6, 6, 6]” numbers where the characteristic numbers 

indicate respectively the number of common neighbors, number of total bonds between these common neighbors and 

number of bonds in largest chain of bonds connecting common neighbors. We also determine the x-type identity such 

as bcc-type and ico-type which correspond respectively to the bcc and ico cell. Note that the bcc-type also belongs to 

14-type. Due to the atomic rearrangement the identity of every atom can change from one to another x-type during 

annealing process. To characterize the local environment around particular atom we propose dynamical local structure 

parameters given as:  

 

 
x obs

x

obs

n t
f

n t
                         (1)  

Where n(tobs) is the total number of MD steps during the time tobs; nx(tobs) is the number of steps when the identity of 

given atom is the x-type. For instance, fbcc = 0.9 means that the given atom possesses the bcc-type at about 90% of MD 

steps within the time tobs [24, 25].  

3. Results and Discussion 

3.1. Effect of Size for Amorphous Nano 

PRDF for 4000-, 5000-, 6000-atoms sample at temperature of 300K is shown in Figure 2. It is clearly that PRDF 
for samples of different sizes are similar to each other. PRDF fit well the experimental result for amorphous iron: the 
second peak is split into two sub-peaks. The left sub-peak is higher than the right one. The splitting of second peak 
may be related to large amount of ico-type atoms which play a role in the preventing of the transformation from 
amorphous to crystalline phase [8]. 

For a more detailed understanding of the microstructure of iron amorphous nanoparticle, we consider distribution 
of atoms, ico-atoms and dynamical local structure parameters <f14>, <fico> for 4000, 5000, 6000 atoms sample 
through concentric spherical layers in Fig.3 and Fig.4. The characteristic lines of 4000, 5000, 6000 atoms samples are 
the same with r < 14 Å. So, amorphous NP has part core (r < 14 Å) which has the structure not depend on size and the 
surface has more porous structure. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2. PRDF for 4000, 5000, 6000 atoms sample at 300 K 

5 10 15 20

0

1

2

3

4

5

g
(r

)

r, Å

 g4000(r)

 g5000(r)

 g6000(r)



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Figure 3. Distribution of atoms, ico-atoms for 4000, 5000, 6000 atoms sample through concentric spherical layers at 300 K 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4. Dynamical local structure parameters <f14>, <fico> for 4000, 5000, 6000 atoms sample through concentric 

spherical layers at 300 K 

3.2. Effect of Size for Crystalline Nan 

In Figure 5 we plot the potential energy per atom Epot as a function of annealing time. In the case of 300-sample the 

curve is nearly straight line and Epot slightly fluctuates around -1.292, -1.300 and -1.305 eV of 4000, 5000 and 6000 

atoms sample, respectively. Unlike 300-sample, the 900-sample undergoes various structural transformations under 

annealing to a stable state. Here this process can be divided into three separate stages as demonstrated in Figure 5. Within 

the first stage (stage 1), the energy Epot moderately varies with time. This indicates that the system is in metastable 

state. During stage 2, the energy Epot decreases fast. This clearly demonstrates that unlike first stage, the system is in 

unstable states and transforms to bcc-crystalline phase. The last stage (stage 3) is characterized by that the energy Epot 

are slightly fluctuated around fix values. This indicates that the crystallization is completed and the system is in a 

stable state. Epot after crystallization process of 4000, 5000, 6000 atoms sample are decrease from -1.27÷ -1.29 eV.  

 

3,5 7,0 10,5 14,0 17,5 21,0 24,5 28,0 31,5

0

300

600

900

1200

1500

1800

T
h
e
 n

u
m

b
e
r 

o
f 
a
to

m
s

r, Å

 N4000

 N5000

 N6000

 Nico,4000

 Nico,5000

 Nico,6000

3,5 7,0 10,5 14,0 17,5 21,0 24,5 28,0 31,5

0,0

0,2

0,4

0,6

0,8

1,0

D
y
n

a
m

ic
a

l 
lo

c
a

l 
s
tr

u
c
tu

re
 p

a
ra

m
e

te
rs

 

r, Å

 f14,4000

 f14,5000

 f14,6000

 fico,4000

 fico,5000

 fico,6000



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The crystallization in NP can be seen from PRDF. As shown in Fig.6, many peaks appear at large distance r as the 

4000, 5000, 6000 atoms samples at temperature of 900K is annealed for a long time. Compared to PRDF obtained 

from the bcc lattice we observe that first (31/2a/2) and third (21/2a) peaks are coincided exactly with that from NP, 

where a is the lattice constant. However, the first and second peaks are not completely separated. Other peaks also 

reproduce well those of bcc lattice. It is clearly that PRDF for samples of different size are similar to each other.  

To determine the structural evolution we divide all atoms into three groups. G1 group includes atoms with f14 = 0. 

The atoms having f14 > 0 and fbcc = 0 belong to G2 group, while the G3 group consists of atoms with fbcc > 0. For 

simplicity these atoms are denoted as G1-, G2- and G3-atom, respectively. Obviously, the crystal cluster consists of 

G3-atoms.  

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5. Potential energy per atom as a function of annealing time 4000, 5000, 6000 atoms sample at 300 and 900 K 

0 5 10 15 20

-1,310

-1,305

-1,300

-1,295

-1,290

(A)

300 K

P
o
te

n
ti
a
l 
e
n
e
rg

y
 p

e
r 

a
to

m
 E

p
o

t, 
e
V

 

Time, ns

 E4000

 E5000

 E6000

0 5 10 15 20 25 30 35 40

-1,30

-1,28

-1,26

-1,24

-1,22

-1,20

(stage 3)

(stage 2)

(stage 1)

(B)

900 K

P
o
te

n
ti
a
l 
e
n
e
rg

y
 p

e
r 

a
to

m
 E

p
o

t,
 e

V
 

Time, ns

 E4000

 E5000

 E6000



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Figure 6. PRDF for 4000, 5000, 6000 atoms sample at 900 K 

The characteristics of different type atoms are listed in Table 1. It is crystallized via can be seen that the number of 

G1-atoms changes slightly with time. In contrast, the G2-atoms convert fast into G3-atoms which mean that NP 

transformation from amorphous type to bcc-type atoms. 

Table 1. Characteristics of atoms of different groups for 5000 atoms sample at 900 K. Here nAt, nCl and sLC is the number of 

atoms, number of cluster and size of largest cluster, respectively 

t, ns 
G1 G2 G3 

nAt nCl sLC nAt nCl sLC nAt nCl sLC 

2.25 1225 47 1162 3697 1 3697 78 23 29 

3.25 1291 52 1220 3591 1 3591 118 19 73 

4.25 1334 48 1269 3506 1 3506 160 18 107 

5.25 1310 45 1244 3297 2 3296 393 25 351 

6.25 1253 30 1215 2921 12 2904 826 15 797 

7.25 1246 31 1208 2751 15 2722 1003 15 979 

8.25 1218 12 1203 1946 33 1888 1836 17 1802 

9.25 1187 4 1184 256 83 24 3557 1 3557 

10.25 1250 3 1248 185 91 14 3565 1 3565 

11.25 1252 2 1251 191 90 17 3557 1 3557 

35.00 1237 1 1237 183 75 15 3580 1 3580 

As shown in Table 1, at early time t, G3-atoms form a number of clusters, but most among them are small and 

consist of fewer atoms. There is a large cluster (main cluster) with size significantly larger than other clusters. As the 

time proceeds, the main cluster grows up, while small clusters either disappear or merge to the main cluster. It is worth 

to note that unlike amorphous NP, <fico> for crystalline NP is close to zero. This result confirms the role of ico-atoms 

for preventing the crystallization.  

Table 2 presents the distribution of G3-atoms through concentric spherical layers of NP. It is shown that at early 

time t the main cluster spread is concentrated in layers 14-16, 16-18 and 18-20. Then this cluster grows and is spread 

over other layers. The main cluster develops from one to other side of NP. At long time t the main cluster covers up a 

major part of NP core.  

5 10 15 20

0

1

2

3

4

5

g
(r

)

r, Å

 g4000(r)

 g5000(r)

 g6000(r)



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Table 2. Distribution of G3-atoms through different spherical layers for 5000 atoms sample at 900 K: a) all G3-atoms; b) the 

G3-atoms of main cluster 

lint - lout 

t = 2.25 ns t = 5.25 ns t = 7.25 ns t = 8.25 ns 

a b a b a b a b 

0 - 2 0 0 0 0 1 1 2 2 

2 - 4 0 0 2 0 9 9 19 19 

4 - 6 0 0 4 1 22 22 42 42 

6 - 8 0 0 10 8 50 49 73 73 

8 - 10 2 0 17 14 66 66 116 116 

10 - 12 4 0 37 31 96 93 135 133 

12 - 14 13 2 47 41 103 103 177 175 

14 - 16 12 4 54 49 130 127 238 230 

16 - 18 16 9 63 57 148 144 271 265 

18 - 20 19 10 76 72 159 151 342 333 

20 - 22 7 2 62 59 157 153 326 319 

22 - 24 5 2 20 18 62 61 93 93 

24 - 26 0 0 1 1 0 0 2 2 

 

For a more detailed understanding of the microstructure of iron nanoparticle, we consider distribution of atoms, 

bcc-atoms and dynamical local structure parameters <f14>, <fbcc> for 4000, 5000, 6000 atoms sample through 

concentric spherical layers as shown in Fig.7 and Fig.8. The characteristic lines of 4000, 5000, 6000 atoms samples 

are the same with r < 9 Å. It can be seen that the size of crystalline NP’s core is r < 9 Å. It means that, distribution of 

atoms, bcc-atoms and dynamical local structure parameters in core region don’t depend on size of NP. 

The amorphous and crystallized NP can be divided into separate regions as schematically described in Figure 9. 

The amorphous sample consists of surface and core. Unlike the core, the surface has a porous amorphous structure. 

The crystallized sample comprises three regions. The region located in the surface having the amorphous structure, the 

next part is distorted and the last region is bcc structure.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7. Distribution of atoms, bcc-atoms for 4000, 5000, 6000 atoms sample through concentric spherical at 900 K 

3,5 7,0 10,5 14,0 17,5 21,0 24,5 28,0 31,5

0

300

600

900

1200

1500

1800

T
h
e
 n

u
m

b
e
r 

o
f 
a
to

m
s

r, Å

 N4000

 N5000

 N6000

 Nbcc,4000

 Nbcc,5000

 Nbcc,6000



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Figure 8. Dynamical local structure parameters <f14>, <fbcc> for 4000, 5000, 6000 atoms sample through concentric 

spherical at 900 K 

Figure 9. Schematics of separate regions with different microstructure 

Amorphous 300-sample 

Crystallized 900-sample 

Porous shell 

Dense 

Amorphous 

Core 

Amorphous surface 

 Distorted bcc shell 

bcc core 

3,5 7,0 10,5 14,0 17,5 21,0 24,5 28,0 31,5

0,0

0,2

0,4

0,6

0,8

1,0

D
y
n

a
m

ic
a

l 
lo

c
a

l 
s
tr

u
c
tu

re
 p

a
ra

m
e

te
rs

 

r, Å

 f14,4000

 f14,5000

 f14,6000

 fbcc,4000

 fbcc,5000

 fbcc,6000



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4. Conclusion 

In this study, the annealing of Fe NP consisted of 4000, 5000, 6000 atoms at temperatures of 300 and 900 K has 

been simulated. When the amorphous NP is annealed at 900 K, it possesses various meta-stable states which differ 

strongly in the fraction and spatial distribution of low- and high-coordination atoms and the structural transformations 

that happen. Unlike amorphous NP, <fico> for crystalline NP is close to zero. This result confirms the role of ico-type 

atoms in preventing the crystallization and indicates that NP tends to crystallize into a BCC-crystal structure. On the 

other hand, the specified atoms form a number of clusters, but most of these atoms belong to a large cluster, and small 

clusters contain only a few atoms. The main cluster spread is concentrated in the middle layers. In the annealing 

process, this cluster grows and covers up a major part of the NP core, while small clusters either disappear or merge 

into the main cluster. We have indicated that the structure of the amorphous and crystalline NPs is divided into some 

separate regions. The first region is located near the surface of NP. The remaining region is called the core, and its size 

is unchanged by the number of atoms of NP. Besides, we also found that the amorphous NP has two parts: the core has 

a structure similar to that of amorphous bulk with a smaller radius 14 Å, in while the surface structure is more porous 

and amorphous. The crystalline NP also has three parts: the core is the bcc with the smaller radius 9 Å, the next part is 

the distorted BCC, and the surface is amorphous. 

5. Declarations  

5.1. Data Availability Statement 

The data presented in this study are available in article. 

5.2. Funding 

This work was supported by the Research program of Science and Technology of Thai Nguyen University of 

Education in 2021 (Grant CS-2021). 

5.3. Declaration of Competing Interest 

The authors declare that they have no known competing financial interests or personal relationships that could have 

appeared to influence the work reported in this paper. 

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