Effects of temperature on surface clusters by molecular dynamics simulation

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Physica B 334 (2003) 369–374

Effects of temperature on surface clusters by molecular

dynamics simulation

Sheng-Rui Jian

a

, Te-Hua Fang

b,

*, Der-San Chuu

a a

Institute and Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan b

Department of Mechanical Engineering, Southern Taiwan University of Technology, Tainan 710, Taiwan Received 26 December 2002; received in revised form 10 March 2003; accepted 10 March 2003

Abstract

This article discusses the physical mechanisms for the evaporation phenomena of argon clusters on surfaces under various temperatures with the aid of molecular dynamics analysis by means of the Stoddard–Ford potential. Our simulated results indicate that the evaporation rate of the argon clusters increased drastically when the temperature was increased but the contact angle decreased. Furthermore, the thermal stability of the argon clusters is also discussed here. The evaporation mechanisms of argon clusters are clearly shown with the aid of molecular dynamics.

PACS: 68.03.Fg; 61.46.+w; 61.25.Bi

Keywords: Molecular dynamics; Stoddard–Ford potential; Clusters; Evaporation

1. Introduction

Nanoclusters are currently attracting consider-able attentions for their use in lithography, deposition, electronic devices, etc. because of their unique characterizations[1–3]. Better investigation of the mechanisms associated with nanometer-scale clusters has become increasingly important in the ﬁeld for both theoretical and experimental research[4,5].

The technique is rapidly developing, yet it is still quite difﬁcult to describe the phenomena on an atomic scale. Accordingly, molecular dynamics provides useful information for understanding

atomic behaviors that are difﬁcult to obtain experimentally. All physical phenomena, surface tension, evaporation, condensation and so on come from the intermolecular interaction.

It is well known that molecular dynamics simulations, by virtue of their high temporal and spatial resolutions, offer novel insights into atomistic mechanisms. Recently, a number of researchers have used atomistic simulations to investigate the physical mechanisms of clusters[6– 10], but there still are many inﬁnitesimal details that are not clearly understood. Therefore, the emphasis of this study is focused on the evolution of nanometer-size argon surface clusters in the course of the evaporation mechanisms at various temperatures via molecular dynamics simulation technique.

*Corresponding author. Fax: +886-62-42-5092. E-mail address:[email protected] (T.-H. Fang).

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During the simulation processes, it was found that the ‘‘vacancy’’ event and the inﬂuence of temperature played an important role in this system.

2. Molecular dynamics methodology

The evaporation of argon clusters on surfaces was investigated using molecular dynamics simu-lations. Periodic boundary conditions (PBC) [11]

are used in the transverse directions (x- and y-axis), and the bottom three layers of the Pt atoms of the substrate are ﬁxed in space[12]. The PBC is also used for the liquid argon nanoclusters. The force acting on an individual atom is obtained by summing the forces contributed by the surround-ing atoms; meansurround-ing that, the atoms are described to interact via the Stoddard–Ford potential[13], as follows: fðrijÞ ¼ 4e s rij 12 s rij 6 " # þ r rcut 2 ( 6 s rcut 12 3 s rcut 6 " # 7 s rcut 12 4 s rcut 6 " #) ; ð1Þ

where fðrijÞ is a pair energy function, rcut(cut-off

distance) is set at 3:5s; s ¼ 3:40 1010m and e ¼ 1:67 1021J: The Stoddard–Ford potential has been selected for these simulations and it has been used previously in another study [14].

The force on the ith atom resulting from the interaction of all the other atoms can be derived from Eq. (1) such that

Fi¼ XN j¼1;jai rifðrijÞ ¼ Nmi d2riðtÞ dt2 ; ð2Þ

where Fi; mi; riand N are denoted as: the resultant

force on the ith atom, the mass of ith atom, the position of the ith atom and the total number of atoms, respectively.

The initial conﬁguration of the substrate was a face-centered cubic (FCC) lattice. Initial velocities were assigned from the Maxwell distribution [11]

and the magnitudes were adjusted with the purpose of keeping the temperature in the system constant according to vnew_i ¼ vold_i NfkBT0N 2 XN i¼1 mi voldi 2 2 " #1 8 < : 9 = ; 1=2 ; ð3Þ

where viis the velocity of ith atom, T0is a speciﬁed

temperature, kB¼ 1:381 1023J K1 is the

Boltzmann’s constant and Nf is the freedom of

the system. The initial displacement and velocity were values determined independently and the time integration of motion was performed by Gear’s ﬁfth predictor-corrector method [11] with time step of 1 fs.

The initial atomic configuration of the simulated system is illustrated in Fig. 1. The eight-layers of the FCC, each of which contained about 640 atoms, were set up as the substrate. An argon cluster of 3 nm in size (with a cluster consisting of 1372 atoms) was placed on the substrate. This was not claimed to be a truly minimum array but was rather an empirically derived practical array. For this simulation, the NVT model [15] was used to control the number of atoms N; volume V and temperature T; in addition, the substrate was initially assumed to have a well-defined atomic surface and the atomic array model of the surface was constructed at a specific constant temperature. In addition, the velocities of the substrate atoms at a specific constant temperature were satisfied with the Maxwell velocity distribution.

Fig. 1. Initial conﬁguration of the simulated system using the molecular dynamics model.

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3. Results and discussions

Fig. 2 shows time step snapshots of the trajec-tory of argon clusters in which adsorption and evaporation phenomena can be observed. At the time steps of 120(100 fs) to 500(100 fs), it can be seen that the volume expansion of the vapor argon atoms occurs rapidly at three various temperatures. In addition to that, there are adsorption and evaporation phenomena taking

place during the simulation processes and these phenomena are strongly dependent upon the temperature of the system.

It is important to note that at the highest temperature of 450 K, the smaller argon clusters formed and moved away from the surface. As the evaporation process advanced, it becomes clear that a large amount of the vapor argon atoms have already moved far away from the surface and this is illustrated inFigs. 2(b), (c).

Fig. 2. The adsorption and evaporation phenomena at various time steps and under various temperatures: (a) 50 K, (b) 250 K, and (c) 450 K, respectively.

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The original cluster was separated into several smaller clusters because the system’s temperature was raised, i.e. the energy of the system was

increased to reach a sufﬁcient level to break the atomic bonds (van der Waals forces) and then the evaporation phenomena began. During the simu-lation process, another new phenomenon was observed this was a ‘‘vacancy’’ materialized at the temperature of 250 K and is displayed in

Fig. 3. The clusters showed violent variations because the energy being increased rapidly, which produced a ‘‘vacancy’’ or split into a number of smaller clusters and into weaker binding energy which allowed the atoms to escape from the surface.

Fig. 4 depicts the time step snapshots of the contact angle (y). It can be seen that when the temperature is raised, the contact angle of the temperature at 250 K is smaller than the contact angle of the temperature at 150 K. The contact angle experiments of Rowan et al. [16]

and Hua et al. [17] provide further evidence of the evaporating phenomena of microscale clusters and the variation of the size of the clusters. The simulated results for the nanometer-scale clus-ters in this study were similar to those in previous studies. A linear relationship of the constant angle and the time steps were observed in angles between 50 _{and 140} _{and may be}

Fig. 3. Snapshots of the evaporating argon clusters conﬁgura-tion at a temperature of 250 K during the 120–240( 100 fs) time steps.

Fig. 4. The contact angle at various time steps for 150 and 250 K, respectively.

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expressed as

y ¼ Ay þ B; ð4Þ

where y is the contact angle, y is the time step, where A and B are constant values obtained by curve ﬁtting.

Fig. 5shows the ratio of Ne=Nclusterincreased as

the time steps were increased. Simpliﬁcation parameters representing the ratio of cluster eva-poration were deﬁned as Ne=Ncluster; where Ne is

the number of evaporating atoms of the cluster and Ncluster is the total number of atoms of the

cluster. The evaporating atoms increased rapidly at higher temperatures. The values of Ne=Ncluster(%) ranged from 0.001% to 27.33%

for temperatures of 50– 450 K. This is because the higher temperature of 450 K can cause the tegration of the clusters. A more detailed disin-tegration of the clusters can also be seen inFig. 2. The evaporation ratio, i.e. Ne=Ncluster for

for-mation of a vaporization region between (200ð100 fsÞ) and (400ð100 fsÞ) time steps, at the temperatures of 250, 350 and 450 K ranged 3.72–5.03%, 15.67–18.22% and 17.20–26.97%, respectively. The simulated results deduced are in agreement with the evaporation behavior of other

similar macroscopic experiments [18]. In addition to that, the expected rate of evaporation was 3:33 1024_{; 0:73 10}27_{; 2:49 10}27_{; 4:45 10}27

and 6:35 1027_{ð1 m}2_s1_{Þ for the temperatures of}

50, 150, 250, 350 and 450 K, respectively. The evaporation rate is deﬁned as the number of atoms evaporating per unit area of surface per unit time. These were based on previous studies

[18]and are commonly used to calculate evapora-tion rate. In comparing the expected evaporaevapora-tion rates of 50, 150 and 250 K to the calculated rates, it was discovered that the expected rates were higher than the calculated rates of 2:58 1023_{ð1 m}2_s1_Þ;

5:15 1023_{ð1 m}2_s1_{Þ and 2:25 10}27_{ð1 m}2_s1_Þ;

respectively. At 350 and 450 K, the expected rates were lower than the calculated rates of 7:08 1027_{ð1 m}2_s1_{Þ and 9:66 10}27_{ð1 m}2_s1_Þ:

The calculated potential energy of the system at different time steps is shown in Fig. 6 and are plotted in exponential-like decay curves. InFig. 6, it can be seen that the curve began to decrease abruptly at around 50ð100 fsÞ: The general decaying behavior of the energy was caused by the attractive interatomic forces of the clusters.

From the kinetic theory of gases [19], the temperature was directly related to the kinetic

Fig. 5. The evaporation ratios at different time steps and temperatures.

Fig. 6. The potential energy at various time steps and temperatures.

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energy. Thus, as the temperature was increased, the number of evaporating atoms increased and raised the potential energy of the system as expected. After complete evaporation, the system stabilized and the estimated energy in the higher temperature of 450 K (about –12.87 eV) was larger than lower temperature of 50 K (about –14.86 eV).

4. Conclusion

This study adopted the method of molecular dynamics to investigate the evaporation phenom-ena of nanometer-scale argon clusters on surfaces under various temperatures. A higher-temperature state manifests the formation of small clusters and the ‘‘vacancy’’ event. The results indicated that the probability of the number of vapor argon atoms increased and the evaporation ratios were evalu-ated to have been approximately 3.06–27.33% depending on the temperature, but the contact angle decreased during evaporation. Summarizing, it was further conﬁrmed that the equilibrium state of the argon clusters at various temperatures can provide qualitative results and the entire evapora-tion process can be successfully simulated by molecular dynamics.

Acknowledgements

This work was partially supported by the National Science Council of Taiwan, under Grant Nos. NSC91-2218-E218-001 and NSC91-2212-E-218-007.

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