# Molecular Dynamics Simulation of the Coalescence and Melting Process of Cu and Ag Nanoparticles

Authors: Hui Guo; LinFu Zhang; Qiang Zhu; ChuanJie Wang; Gang Chen; Peng Zhang
Journal: Advances in Condensed Matter Physics
 (2021)

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## Abstract

The coalescence and melting process of different sizes and arrangements of Ag and Cu nanoparticles is studied through the molecular dynamics (MD) method. The results show that the twin boundary or stacking fault formation and atomic diffusion of the nanoparticles play an important role in the different stages of the heating process. At the beginning of the simulation, Cu and Ag nanoparticles will contact to each other in a very short time. As the temperature goes up, Cu and Ag nanoparticles may generate stacking fault or twin boundary to stabilize the interface structure. When the temperature reaches a critical value, the atoms gain a strong ability to diffuse and eventually melt into one liquid sphere. The coalescence point and melting temperature increase as cluster diameter increases. Moreover, the arrangement of Cu and Ag nanoparticles has a certain effect on the stability of the initial joint interface, which will affect subsequent coalescence and melting behavior.


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## Body

## 1. Introduction

Nanoparticles have attracted much attention because of their unique chemical and physical properties [1, 2]. Also, they are already used in many fields, such as microelectronics, biomedicine, catalytic chemistry, and sensors [3–8]. In recent years, more and more researchers from a variety of engineering and academic fields are especially interested in bimetallic nanoparticles [9, 10]. The interaction between different metal nanoparticles makes them possess more unique physical and chemical properties that single metal does not have, and these properties are not those of a simple combination of two metals[11]. As for a Cu-Ag nanoparticle, many research studies have been conducted on its synthesis method [12–14]. In addition, its practical application was studied by lots of scholars [8, 15–20]. For example, Zhang et al. [8] found that copper and silver nanoparticles could improve the electrical conductivity and tensile strength of the silver-plate-based conductive adhesive. Vengatesan et al. [15] found that the storage space might be improved by Cu-Ag-nanoparticle- (NP-) decorated graphene composites. Aditya et al. [16] proved that Cu-Ag nanoparticles have relative high activity in protein denaturation inhibition, antidiabetic, antioxidative, and anticancer. Medina [17] reported that Cu-Ag nanoparticles could provide strong bactericidal effect. Liu et al. [18, 19] developed Cu-Ag nanoparticles paste to solve the problems of oxidation and high bonding temperature of Cu nanoparticles. Moreover, it was found that the polypropylene-based composite with Cu-Ag nanoparticles plays a great role in catalytic reduction of 4-nitrophenol [20].During the application of nanoparticles, the coalescence and melting of nanoparticles may be an unavoidable problem. For the coalesce and melting process, it is difficult to observe them directly at the atomic scale through experiments [21]. Therefore, many scholars [22–24] have used the MD method to simulate the melting process of pure metals and the coalescence and melting process of bimetallic nanoparticles. In this paper, the MD method is used to explore the coalescence and melting process of Cu and Ag nanoparticles at the atomic scale.


## 2. Method of Simulation

### 2.1. Interaction Potential

The embedded atomic method (EAM) potential has been widely used to describe the interactions of atoms with the FCC structure, such as Cu and Ag [25, 26]. Therefore, the EAM potential developed by Zhou et al. [27] is used in this paper. Also, the total energy (Etotal) in the EAM potential can be described as(1)Etotal=∑iFiρi+12∑i,j,j≠iQijrij,where Qij is the pair energy between atoms i and j at a distance rij. Also, Fi is the embedding energy to embed an atom i in an electron density ρi.The electron densityρi can be expressed as(2)ρi=∑j,j≠ipjrij.pj(rij) is the electron density of an atom i. Also, it can be calculated by the following equation:(3)pr=pe⋅expβ1−r/rer/re−λ20+1.The function of pair potentials is written as(4)Qr=A⋅expα1−r/rer/re−κ20+1−B⋅expβ1−r/rer/re−λ20+1.In eq. (4), re is the equilibrium distance of the nearest neighboring atoms. Also, α, β, κ, λ, and A, B are the parameters used to adjust the formula.As for a bimetallic system, the abovementioned formula can be written as(5)Qabr=12pbrparQaar+parpbrQbbr.The embedding energy functions should be divided as eq. (6) so that the embedding energy can vary smoothly.(6)Fρ=∑i=03Fniρρn−1i,ρ<ρnρn=0.85ρe,∑i=03Fiρρe−1i,ρn≤ρ<ρ0ρ0=1.15ρe,Fe1−lnρρsη⋅ρρsη,ρ0≤ρ.All the parameters used in the abovementioned equations are listed in Table1.Table 1
Parameters for EAM potential.
CuAgCuAgre2.5561622.891814Fn0−2.170269−1.729364pe1.5544851.106232Fn1−0.263788−0.255882ρe21.17587114.604100Fn21.0888780.912050ρs21.17539514.604144Fn3−0.817603−0.561432α8.1276209.516052Fe−2.186568−1.748423β4.3347315.075228F0−2.19−1.75κ0.3087820.356570F100λ0.7565150.748798F20.5618300.744561A0.3966200.229762F3−2.100595−1.150650B0.5480850.356666η0.3104900.783924


### 2.2. Simulation Models

The model used in this work is that one copper nanoparticle plus one silver nanoparticle inside the center of simulation box. The dimension of the cube simulation box is 30 nm × 30 nm × 30 nm. Cu and Ag nanoparticles were constructed in FCC crystal, and the lattice parameters of Cu and Ag are 0.3615 nm and 0.409 nm, respectively. In the simulation box, the number of atoms in the Cu nanoparticle is equal to that of Ag. Also, the simulation box is big enough to prevent the self-interaction of nanoparticles. The geometric center of Cu and Ag nanoparticles was placed along theX-axis, and there was a space approximately 0.4 nm along the X-axis between the Cu and Ag nanoparticle. The specific parameters of nanoparticles are described in Table 2.Table 2
Parameters of models for simulation.
ModelThe number of atoms in the Ag nanoparticleCrystal orientation along theX-axis of the simulation boxCuAg-2 nm-001442<001>Cu//<001>AgCuAg-3 nm-0011072<001>Cu//<001>AgCuAg-4 nm-0012101<001>Cu//<001>AgCuAg-5 nm-0014586<001>Cu//<001>AgCuAg-6 nm-0018584<001>Cu//<001>AgCuAg-8 nm-00116736<001>Cu//<001>AgCuAg-4 nm-0112101<011>Cu//<011>AgCuAg-4 nm-0122101<012>Cu//<012>AgCuAg-4 nm-0132101<013>Cu//<013>AgCuAg-4 nm-1112101<111>Cu//<111>AgCuAg-4 nm-1122101<112>Cu//<112>AgCuAg-4 nm-1132101<113>Cu//<113>AgCuAg-4 nm-1222101<122>Cu//<122>AgCuAg-4 nm-1232101<123>Cu//<123>AgCuAg-4 nm-1332101<133>Cu//<133>Ag


### 2.3. MD Procedure

In this study, Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code [28] is used to carry out all the simulations. Also, all the simulations were performed under an NVT (constant number of atoms, constant volume, and constant temperature) ensemble. In addition, verlet leapfrog algorithm with a time-step of 0.4 fs was selected to integrate the motion equation.The initial models were heated from 0.1 K to 900 K with a temperature increment of 100 K, and all the systems were equilibrated 1 million trajectory steps every 100 K. Then, the models were equilibrated every 20 K in the region of 900–1200 K in order to promise the accuracy of the melting point. Moreover, the whole heating process rate was 0.25 K/ps.


## 2.1. Interaction Potential

The embedded atomic method (EAM) potential has been widely used to describe the interactions of atoms with the FCC structure, such as Cu and Ag [25, 26]. Therefore, the EAM potential developed by Zhou et al. [27] is used in this paper. Also, the total energy (Etotal) in the EAM potential can be described as(1)Etotal=∑iFiρi+12∑i,j,j≠iQijrij,where Qij is the pair energy between atoms i and j at a distance rij. Also, Fi is the embedding energy to embed an atom i in an electron density ρi.The electron densityρi can be expressed as(2)ρi=∑j,j≠ipjrij.pj(rij) is the electron density of an atom i. Also, it can be calculated by the following equation:(3)pr=pe⋅expβ1−r/rer/re−λ20+1.The function of pair potentials is written as(4)Qr=A⋅expα1−r/rer/re−κ20+1−B⋅expβ1−r/rer/re−λ20+1.In eq. (4), re is the equilibrium distance of the nearest neighboring atoms. Also, α, β, κ, λ, and A, B are the parameters used to adjust the formula.As for a bimetallic system, the abovementioned formula can be written as(5)Qabr=12pbrparQaar+parpbrQbbr.The embedding energy functions should be divided as eq. (6) so that the embedding energy can vary smoothly.(6)Fρ=∑i=03Fniρρn−1i,ρ<ρnρn=0.85ρe,∑i=03Fiρρe−1i,ρn≤ρ<ρ0ρ0=1.15ρe,Fe1−lnρρsη⋅ρρsη,ρ0≤ρ.All the parameters used in the abovementioned equations are listed in Table1.Table 1
Parameters for EAM potential.
CuAgCuAgre2.5561622.891814Fn0−2.170269−1.729364pe1.5544851.106232Fn1−0.263788−0.255882ρe21.17587114.604100Fn21.0888780.912050ρs21.17539514.604144Fn3−0.817603−0.561432α8.1276209.516052Fe−2.186568−1.748423β4.3347315.075228F0−2.19−1.75κ0.3087820.356570F100λ0.7565150.748798F20.5618300.744561A0.3966200.229762F3−2.100595−1.150650B0.5480850.356666η0.3104900.783924


## 2.2. Simulation Models

The model used in this work is that one copper nanoparticle plus one silver nanoparticle inside the center of simulation box. The dimension of the cube simulation box is 30 nm × 30 nm × 30 nm. Cu and Ag nanoparticles were constructed in FCC crystal, and the lattice parameters of Cu and Ag are 0.3615 nm and 0.409 nm, respectively. In the simulation box, the number of atoms in the Cu nanoparticle is equal to that of Ag. Also, the simulation box is big enough to prevent the self-interaction of nanoparticles. The geometric center of Cu and Ag nanoparticles was placed along theX-axis, and there was a space approximately 0.4 nm along the X-axis between the Cu and Ag nanoparticle. The specific parameters of nanoparticles are described in Table 2.Table 2
Parameters of models for simulation.
ModelThe number of atoms in the Ag nanoparticleCrystal orientation along theX-axis of the simulation boxCuAg-2 nm-001442<001>Cu//<001>AgCuAg-3 nm-0011072<001>Cu//<001>AgCuAg-4 nm-0012101<001>Cu//<001>AgCuAg-5 nm-0014586<001>Cu//<001>AgCuAg-6 nm-0018584<001>Cu//<001>AgCuAg-8 nm-00116736<001>Cu//<001>AgCuAg-4 nm-0112101<011>Cu//<011>AgCuAg-4 nm-0122101<012>Cu//<012>AgCuAg-4 nm-0132101<013>Cu//<013>AgCuAg-4 nm-1112101<111>Cu//<111>AgCuAg-4 nm-1122101<112>Cu//<112>AgCuAg-4 nm-1132101<113>Cu//<113>AgCuAg-4 nm-1222101<122>Cu//<122>AgCuAg-4 nm-1232101<123>Cu//<123>AgCuAg-4 nm-1332101<133>Cu//<133>Ag


## 2.3. MD Procedure

In this study, Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code [28] is used to carry out all the simulations. Also, all the simulations were performed under an NVT (constant number of atoms, constant volume, and constant temperature) ensemble. In addition, verlet leapfrog algorithm with a time-step of 0.4 fs was selected to integrate the motion equation.The initial models were heated from 0.1 K to 900 K with a temperature increment of 100 K, and all the systems were equilibrated 1 million trajectory steps every 100 K. Then, the models were equilibrated every 20 K in the region of 900–1200 K in order to promise the accuracy of the melting point. Moreover, the whole heating process rate was 0.25 K/ps.


## 3. Results and Discussion

The first-order phase transition temperature is the temperature that the step change of enthalpy occurs due to the latent heat [29]. Potential energy curve is a useful tool to determine the phase transition temperature. Therefore, potential energy curves of different models shown in Figure 1 are used to determine the melting point. From these curves, one can find that potential energy curves of small-size models will drop at the beginning of the simulation obviously. The reason why this phenomenon happens is that the contacting of Cu and Ag nanoparticles leads to the disappearance of part free surface and the formation of the Cu/Ag interface. Also, the potential energy dropping caused by this process is bigger than the elevating caused by temperature rising for small size models. Then, after the curves go up when temperature increases almost linearly, the rate of potential energy elevation decreases due to the formation of Cu-Ag bonds in the coalescence process. At the temperature range from 900 K to 1100 K, there is a sharp increase in the curves, which means phase transition occurs. Also, the melting point of different models is marked by black arrow as shown in Figure 1. In addition, it can be found that the potential energy increased with the decrease of model size, which is because the nanoparticle size smaller is higher of the surface atom proportion.Figure 1
Potential energy per atom versus temperature.The coalescence process of nanoparticles is always accompanied by structure translation. Also, some scholars defined the coalescence as a temperature that the rod-like structure is formed [22, 30]. In fact, the coalescence of two particles just likes the metallurgical bonding process because a low-interfacial free energy interface will be formed between two nanoparticles. Generally, formation of twin boundary or stacking fault and atomic diffusion may occur [31] during the formation of this kind of interface. Therefore, the coalescence and melting process can be analysis though charactering the information about atomic migration. For example, the mean distance of atoms from the mass center (Dmass) of two nanoparticles is a very useful tool to analyze the coalescence temperature. Also, mean square displacement (MSD) can also be used to characterize the atomic diffusion [32]. It can be expressed as [33](7)MSD=1N∑i=1Nrit0+t−rit02,where N is the total number of particles, t is the time, and ri(t0+t)-ri(t0) is the vector distance traveled by an atom over a period of time.In the present study, the Cu and Ag nanoparticles are placed along theX-axis. Therefore, the parameter Dmass−X (mass center displacement at the X-axis) and MSD are used to characterize the coalescence and melting behavior of nanoparticles during the heating process. Figure 2 shows the Dmass−X and MSD curves of CuAg-3 nm-001. One can find that Dmass−X curves of Cu as well as Ag gradually approach the curve of Cu-Ag with simulation time going on, which means that the melting process is going on as temperature rises. In addition, the motion magnitude of the Cu nanoparticle is bigger than the Ag nanoparticle because the surface energy of Cu is higher than that of Ag [34, 35]. Moreover, when the temperature arrives at a specific value labeled by the dashed line in Figure 2, the curves of MSD and Dmass−X will suddenly jump due to the enhanced diffusion ability of atoms. Actually, this specific value is consistent with the melting point obtained by Figure 1.Figure 2
Temperature dependence ofDmass−X and MSD of CuAg-3 nm.Additionally, several “step-like” regions emerge in theDmass−X and MSD curves before temperature arriving at the melting point. The cross section of snap shots and the corresponding displacement cloud pictures of the CuAg-3 nm-001 model are used to explore the reasons of the generation of “step-like” regions more intuitively, as shown in Figure 3. Displacement vector of each atom should be calculated before getting the displacement cloud picture. The way to obtain the displacement vector is to calculate the displacement of each atom between the simulation time A, B, and C and A’, B’, and C’, which means the structure at simulation time A, B, and C will be a reference structure. From Figure 3(a), it can be found that the rotation presents between two nanoparticles, which caused the orientation of two nanoparticles to tend to the same direction. The diffusion phenomenon is not significant at ∼300 K, which can also be verified by the low value of the MSD curve. Therefore, the jump in the Dmass−X curve at point A should be stemmed from lots of atoms in nanoparticle migration commonly which presents as part of Cu or Ag slide. Figures 3(a)–3(c) illustrate stacking fault and twin boundary formed in two nanoparticles which may promote the reduction of the Cu/Ag interface mismatch. Besides the presentation of stacking fault and twin boundary in BB’ and CC’ “step-like” regions, as shown in Figures 3(b) and 3(c), the atom diffusion process can also be found in these regions. However, the process of forming stacking fault and twin boundary is still the main reason for the sudden jump of Dmass−X value in B’B due to only few atoms with the long-range diffusion. Figure 3(c) shows that there are several atoms with big movement distance on the surface of particle, which leads to the disappearance of the neck part between two nanoparticles. It indicates that the atom diffusion promotes the coalescence process at ∼900 K. One can find that there is a stable interface region between two nanoparticles from Figure 3(c) and there is no more significant shaking in Dmass−X and MSD curves after the formation of a stable interface region. Therefore, the temperature of the “step-like” corresponding to the elimination of the neck part by atom diffusion can be set as the coalescence point. Following the third “step,” the Dmass−X value changes monotonously which results from the atoms diffusion. Due to the melting temperature of silver lower than that of copper, silver atoms will wrap the Cu nanoparticle in the initial stage of the melting process. Therefore, it can be concluded that the shakings of Dmass−X and MSD curves are caused by relative sliding of nanoparticles and diffusion of atoms.Figure 3
Cross section of sample at AA’, BB’, CC’ and D steps in Figure2. (Green atoms, red atoms, white atoms are FCC structure, HCP structure, disorder structure, respectively.)
(a)(b)(c)(d)For small-size models, such as CuAg-2 nm-001, CuAg-3 nm-001, and CuAg-4 nm-001, the obvious shaking phenomenon can also be found inDmass−X and MSD curves. However, the steps in the Dmass−X and MSD curves of CuAg-6 nm-001 are insignificant at lower temperature, as shown in Figure 4. When the temperature reaches to 800 K, some stack fault in the nanoparticles and few atom diffusions on the particle surface can be found, which suggests that it reduces the interface energy mainly by forming stack fault or twin boundary (Figure 5(a)). With temperature increase, as shown in Figure 5, there are lots of Ag atom diffusions on the surface of the structure similar to that shown in Figure 3(c). At the same time, the neck part of two nanoparticles disappeared. Therefore, the coalecense point of CuAg-6 nm-001 is about 1040 K estimated from the Dmass−X and MSD curves. In the melting process, silver atoms wrap the copper nanoparticle firstly (Figure 5(c)). Then, the copper particle will melt completely. At last, mutual diffusion of Cu and Ag atoms can be found in Figure 5(d).Figure 4
Temperature dependence ofDmass−X and MSD of CuAg-6 nm.Figure 5
Cross section of CuAg-6 nm. (Green atoms, red atoms, white atoms are FCC structure, HCP structure, disorder structure, respectively.)
(a)(b)(c)(d)In the temperature increase process, the stacking fault or twin boundary formation and atomic diffusion can promote formation of the stable interface structure [31]. For the small-size models, forming stacking fault or twin boundary plays a dominant role in this process. However, atomic diffusion is the main way to form the stable interface structure for big-size models. The relative high temperature is an important condition for atom diffusion, which is not necessary for forming stacking fault or twin boundary. In addition, a twin boundary will be formed after the coalescence for all different-size models, as shown in Figure 6. It can also be found that the melting point of nanoparticles increases with nanoparticle size increase, which is consistent with lots of thermodynamic models [29, 36–38]. In fact, melting point and coalescence temperature decrease linearly with N−1/3(N is the atomic number), as shown in Figure 6.Figure 6
Size effect of coalescence and melting temperatures.The coalescence and melting process of Cu and Ag nanoparticles with different arrangements (Table2) is also studied in the present study. Also, the stereographic projection of different arrangements is shown in Figure 7. The simulation results illustrate that the coalescence point and melting temperature of the two nanoparticle systems are 900 K and 1000 K, respectively. However, the coalescence process of these models can be roughly classified into three categories which are marked by different colors in Figure 7. In order to reflect the different contacting modes intuitively, the CNA analysis results, Dmass−X and MSD curves are shown in Figure 8. As shown in Figure 8(a), due to the initial crystal orientation along the X-axis of CuAg-4 nm-013 close to <001>, the coalescence process is similar to that of CuAg-4 nm-001, shown in Figure 2 and 3. Therefore, the coalescence of models with the initial crystal orientation of <001> or <013> is affected by forming stacking fault or twin boundary and the diffusing process commonly. For the models with initial crystal orientation including <011>, <012>, and <133>, a relative stable interface can form after two particles contacting at low temperature, as shown in Figures 8(b) and 8(c). However, in order to form a more stable interface as (111), stack fault or twin boundary may be formed near the interface of two nanoparticles (Figure 8(c)). Because the initial crystal orientation including <112>, <113>, <122>, and <123> approach the <111> direction, the interface as the (111) crystal face will be formed easily after two nanoparticles’ contact. Due to the stable interface being formed at lower temperature for the model with initial crystal orientation approach <011> or <111>, the coalescence will realize only by the diffusion process at high temperature, as shown in Figures 8(d)–8(f). Though the coalescence process is different for the three categories’ models, the phenomenon that lots of Ag atoms’ diffusion on the surface of Cu side will present after the coalescence for all models, which indicates that atom diffusion play a major role at high temperature. However, forming stacking fault or twin boundary can promote producing stable interface for model with big difference in orientation from the (111) face at lower temperature.Figure 7
Schematic plot of stereographic projection of different arrangements.Figure 8
TheDmass−X and MSD curves and cross section of snapshots, (a),(b), (c), (d), (e) and (f) are corresponding to CuAg-4 nm with arrangement of <013>, <011>, <012>, <111>, <113> and <123>, respectively.
(a)(b)(c)(d)(e)(f)


## 4. Conclusions

The coalescence and melting process of Cu and Ag nanoparticles indicate that the movement of atoms can be realized by forming stacking fault or twin boundary and the diffusion process. The stacking fault or twin boundary can promote production of the stable interface at lower temperature. With temperature increase, the atom diffusion process will play an important role for the coalescence and melting process. There is a phenomenon that nanoparticle size effect on the coalescence and melting temperature decreases linearly withN−1/3 (N is the atomic number). The different arrangements of Cu and Ag nanoparticles will decide different contacting modes of two nanoparticles, which will affect the subsequent coalescence process.



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*Source: 9945723-2021-04-29.xml*