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Physica A 363 (2006) 307–314

www.elsevier.com/locate/physa

A quasi-molecular dynamics simulation study on the effect ofparticles collisions in pulsed-laser desorption

Xinyu-Tana,b, Duanming-Zhanga,�, Shengqin-Fengb, Zhi-hua Lia,Guan Lia, Li Lia, Liu Dana

aDepartment of Physics, Huazhong University of Science and Technology, Wuhan 430074, ChinabDepartment of Physics, Three Gorges University, Yichang 443002, China

Received 4 August 2005; received in revised form 28 October 2005

Available online 12 December 2005

Abstract

The dynamics characteristic and effect of atoms and particulates ejected from the surface generated by nanosecond

pulsed-laser ablation are very important. In this work, based on the consideration of the inelasticity and non-uniformity of

the plasma particles thermally desorbed from a plane surface into vacuum induced by nanosecond laser ablation, the one-

dimensional particles flow is studied on the basis of a quasi-molecular dynamics (QMD) simulation. It is assumed that

atoms and particulates ejected from the surface of a target have a Maxwell velocity distribution corresponding to the

surface temperature. Particles collisions in the ablation plume. The particles mass is continuous and satisfies fractal theory

distribution. Meanwhile, the particles are inelastic. Our results show that inelasticity and non-uniformity strongly affect

the dynamics behavior of the particles flow. Along with the decrease of restitution coefficient e and increase of fractional

dimension D, velocity distributions of plasma particles system all deviate from the initial Gaussian distribution. The

increasing of dissipation energy DE leads to density distribution clusterized and closed up to the center mass. Predictions of

the particles action based on the proposed fractal and inelasticity model are found to be in agreement with the

experimental observation. This verifies the validity of the present model for the dynamics behavior of pulsed-laser-induced

particles flow.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Laser ablation; Non-uniform particles system; Computer simulation; Fractal dimension

1. Introduction

Pulsed-laser deposition (PLD) or ablation (PLA) is a versatile technique to bring atoms or particles into thegas phase. A prime application is the deposition of functional thin film [1–3]. The fundamental processes passthrough a number of stages. During the first stage, the laser heats the surface region and induces ejection ofmaterial, which forms a plume in front of the surface as a dense cloud of particles [4]. The plume particles canbe deposited on a substrate as a growing thin film. An acceleration or deceleration processes occurring in this

e front matter r 2005 Elsevier B.V. All rights reserved.

ysa.2005.11.022

ing author.

esses: tanxin@ctgu.edu.cn ( Xinyu-Tan), zhangd@public.wuhan.cngb.com ( Duanming-Zhang).

ARTICLE IN PRESSXinyu-Tan et al. / Physica A 363 (2006) 307–314308

flow will alter the energy with which particles impinge on a substrate and hence influence the quality of thethin film forming. So it is important to know the mechanism of the dynamics behavior of the particles flow.The fundamental aspects of these have been studied by experimental and analytical methods [5].

Recently, the numerical methods such as Monte Carlo and Molecular-dynamics simulations were applied toinvestigate the dynamics behavior of particles flow of PLD by some researchers [6–8].They found that acombined numerical study can roughly describe a system which corresponds to real ablation in the thermalregime. Information on the particles distribution at the surface, and hence on the desorption mechanism, issought by measuring the properties of the desorbed particles at some distance from the surface and byextrapolating the measured values towards the surface [9]. Because the ablation plume may have a complexcomposition, with clusters and granular mixture of different sizes comprising a particles flow of the ejectedmaterials, both the particle mass m and the size of particles change in a large range. Therefore, the density ofthe desorbed particles in front of the surface may be so high that collisions among them will change theparticles distribution considerably. It is a pity that all the present theories are based on the assumptions thatthe ejected particles have same mass and collisions are elastic and they did not consider number of moleculesin the ejected clusters which can be as high as tens or even hundreds of thousands [10]. Until now the effects ofparticle size and elastic degree (defined as restitution coefficient in our paper) on particle kinetic behaviorgenerated by laser ablation were neglected by most of the previous works [4–8] and little attention was paid tostudy the non-uniformity of particle mass and inelasticity. These are limiting cases in contrast to high kineticenergy particles present in actual PLD processes. Especially during the radiation of high-power nanosecondpulsed-laser radiation and phase explosion, the target material makes an abrupt transformation fromsuperheated liquid into a mixture of cluster droplets and vapors, and then these particles are ejected from thetarget, both the particles size r so the particles mass m and the sizes of particles change considerably.

In this paper, we present a non-uniform granular system in one-dimensional case and by use of quasi-molecular dynamics (QMD) [11,12] simulation study the movement of particles.We establish a new model tostudy the flow of particles evolvement away from the desorption spot, namely considering the fact that thenumbers of molecules of ejected particles are not the same and supposing the particles mass distribution iscontinuous and satisfies the fractal theory [13–15] (Section 2). The influence of the inelasticity is first discussed(Section 3.1) with QMD method. The effect of the fractal dimension D ð2pDp3Þ influenced the carryingcharacteristics of particles because different granularity distribution is then studied (Section 3.2). Based on therelevant statistical results, theoretical analysis of the influence of non-uniformity and the inelasticity on thedynamic actions of plasma particles is also given.

2. Theoretical model

2.1. Characteristic of matter rapidly heated by nanosecond pulsed laser

When a high-powered short-pulsed-laser beam irradiates on a target surface, the target will absorb the laserenergy, part of the solid target will be liquated immediately. Heating the sample above the boiling point ispossible because the laser pulse is too short for bubbles to nucleate, a significant superheating above theboiling point will occur, namely the Knudsen layer. Then the target will make a rapid transition fromsuperheated liquid to high-temperature and high-pressure plasma. Mixture of vapor and liquid droplets willsputter from the target surface [2,16,17]. After a series of collisions and flights, the high-temperature and high-pressure particles flow gets to the substrate, and then the thin film begins to grow. This is the whole process oflaser deposition.Because of having different initial velocity, collision is inevitably acute between the particlesflow and the particle density is not homogeneous in the vertical direction to the target.

2.2. Kinetic QMD model—inelastic hard-sphericity one-dimensional model of non-uniform particle system with

fractal characteristic

We consider the particles flow as a non-uniform granular mixture and the granularity of the system isfractal. The mass distribution of N particles is continuous (mminpmpmmax), mmin is the minimal mass of theparticles, and mmax is the maximal mass of the particles). Ni is the number of particles of component, and

ARTICLE IN PRESSXinyu-Tan et al. / Physica A 363 (2006) 307–314 309

N1 þN2 þ � � � þNi þ � � � ¼ N . The mass of the particle i is mi . For simplicity, the surface of the hard andglobal particles is smooth and the material per particle is identical, but the size of particles is different and isdetermined by the numbers of molecules in the ejected particles. The mass of a particle

m ¼4

3pr3rp, (1)

where rp and r are mass density and radius of particles, respectively. Experiments show that granular materialsexhibit some fractal characteristics. The size distribution of particles in granular system satisfies the size-frequency character by fractal theory [12,13] when n0=N51:

Y NrðrÞ ¼ 1�N�1n0

r

rmax

� ��D

, (2)

where Y Nris the ratio of Nr to N, Nr is the number of particles whose size is smaller than r, n0 is the number of

particles with the maximum size rmax, and D is the fractal dimension and 2oDo3, collected with Eqs. (1) and(2), the mass of any particle in the mixture is given by

m ¼ mmaxN

n0ð1� Y Nr

Þ

� �3=D

. (3)

If the values of N, mmax, n0 and D are given, according to Eq. (3), we can randomly evaluate the mass of perparticle in the non-uniform granular system with fractal distribution generated by PLA.

At the beginning, we assume that particles leave the surface in thermal equilibrium, i.e. every velocitycomponent obeys a Gaussian distribution. On the vertical direction with temperature T0, with the obvious butnotable exception, particles must leave the surface in outward direction, i.e. [5,7,18,19]:

f ðvÞ ¼ Nm

2pkT0

� �3=2

exp �mðv� v0Þ

2

2kT0

� �, (4)

where k is the Boltzman’s constant, m is the particle mass, T0 is the initial surface average temperature, v0 isthe initial average velocity for particles. Because the displacement and velocity on the vertical direction aremuch larger than that of the parallel direction, it is reasonable to suppose that the pulsed particlesperpendicularly desorb at the Knudsen layer with the temperature T0 and the whole movement can be takenas a one-dimensional model of non-uniform particle system. For a thermal desorption mechanism, the initialparticles average velocity v0 is the thermal average velocity of the desorbed particles at the surface averagetemperature T0:

v0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8kT0=pm

p. (5)

To study the actual granular gas with fractal characteristic, we present the following one-dimension flowmodel: the initial flow is divided into cells with dimensions of about one mean free path. Each cell is filled withsimulated particles that are characterized by spatial coordinates, velocities, and mass. The mass distribution ofN different particles which are evenly located on the grid satisfies Eq. (3). The value L of the whole size isseveral orders of the depth of Knudsen layer. In our simulation, it is in the order of 10�5 m with a wall at i ¼ 0and an open boundary at i ¼ N þ 1 [20]. Between each grid, particle may collide and leave its initial location.After a time step Dt, the collision procedure restarts, etc.

It should be emphasized that the length L=N of spacing between particles is more important than the size ofparticles and the particles never deform (this effect can be taken into account in the restitution coefficient)during expansion, the particles dissipate the kinetic energy through inelastic collisions.

The characteristic time tc should be considered when collisions are taken into account. It is the averagecollision time between two successive encounters. An estimate of tc as a function of average density andtypical velocity is [21]

tc�L

2Nffiffiffiffiffiffiffiffihv2i

p , (6)

ARTICLE IN PRESSXinyu-Tan et al. / Physica A 363 (2006) 307–314310

where hv2i is the statistical average of velocity square. When collisions occur, the mutual collisions betweenparticles follow the following rules: (a) only binary collisions are considered, (b) each collision isinstantaneous, and (c) the collision is inelastic and the post-collisional velocities are related to the pre-collisional ones by

v0i ¼ðmi � emjÞvi þ ð1þ eÞmjvj

mi þmj

, (7)

v0j ¼ð1þ eÞmivi þ ðmj � emiÞvj

mi þmj

, (8)

where i; j mean the ith and jth particles, e is the restitution coefficient. We suppose that any particle that isscattered back to the surface is treated as an absorbed particle, and the heat of recondensation has beenneglected so that the temperature calculations in the solid can be decoupled from the QMD simulation. Theoverall simulation parameters are summarized in Table 1. The particles undergo scattering primarily in aKnudsen layer up to a distance x in front of the surface.

Then we simulate the stochastic movement of the particles. First, we arrange the particles with initialvelocity stochastically according to Eqs. (4) and (5). Supposing Dt ¼ 0:01tc, The collisions dominate thedynamics and the simulations have been performed using a fixed step Dt. After a long time t ¼ 1:0� 103Dt, wemake a statistic for the velocity and position of particle.

3. Results

3.1. Influence on the dynamic action of inelasticity

Fig. 1 visualizes the velocity distribution of the system in the one-dimensional case when the fractaldimension D ¼ 2:8 and the restitution coefficient e equals (a) 0:99, (b) 0:7 and (c) 0:4, respectively, after theprocedure runs 1000 times. Symbols are the simulation results. The dash-dot line is the initial velocitydistribution and the line is the fitting curve for e ¼ 0:99. We can see, vx40 everywhere for initial velocity.However, after the procedure runs 1000 steps, the minus velocity all appear in three different conditions. Thismeans there are sufficient collisions during the flow flight. When e ¼ 0:99, the velocity distribution is nearlyconsistent with a renewed Gauss distribution. However, on the other two conditions (e ¼ 0:7 and 0.4), thevelocity distributions all deviate from the initial Gaussian distribution. Furthermore, the less the restitutioncoefficient e, the more likely the distribution deviates to Gauss. Because of the influence of the inelasticcollision force, the non-Gaussian behavior of the velocity distribution gets stronger as the inelasticityincreases.

We display the instantaneous density of the system versus the restitution coefficient e in one-dimensionalcase in Fig. 2. Likewise, fractal dimension D ¼ 2:8 and restitution coefficient e are (a) 0:99, (b) 0:7 and (c) 0:4.The symbols represent the non-equilibrium regime with particles and the solid line is the fitting curve fore ¼ 0:4. We can see, close to the surface, a considerable backflow of desorbed particles towards the surface isestablished, as evidenced by the nonvanishing density at the surface and the negative flow velocity there (seeFig. 1).

Table 1

Thermal and optical properties of silicon

Number of articles Minimum mass of particles (g) Maximum mass of particles (g)

N mmin mmax

2� 105 o10�15 10�12

Average collision time (s) Time step (s) Number of particles with the maximum

mass

tc Dt n0

1:0� 10�10 1:0� 10�12 1

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-400 -200 0 200 400 600 800 1000 1200 1400

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

e=0.99 e=0.7 e=0.4

Inte

nsi

ty,a

rb.u

nit

s

Velocity,m/s

Fig. 1. The instantaneous velocity probability distribution after running 1000 times when D ¼ 2:8; e ¼ 0:99; 0:7 and 0:4. Symbols are

simulation results. Dash–dot line is initial velocity probability distribution and solid line represents the fitting curve when e ¼ 0:99.

0 5 10 15 200.00

0.01

0.02

0.03

0.04

e=0.99e=0.7e=0.4

Inte

nsi

ty, a

rb.u

nit

s

Distance from the surface, um

Fig. 2. The instantaneous density distribution profile when the calculation parameters are the same as in Fig. 2. Symbols are simulation

results and solid line represents the fitting curve when e ¼ 0:99.

Xinyu-Tan et al. / Physica A 363 (2006) 307–314 311

We can also see, considering the effect of the inelasticity, the instantaneous density is no more spatialhomogeneous, and clusterizations appear. Whole particles density showed an interior dense outer attenuatefusiform profile. This result is entirely in agreement with the experimental observation reported in Refs. [5–7].From e ¼ 0:99–0.4, the spatial distribution asymmetric characteristic is more evident with the decreasing valueof the restitution coefficient.

The velocity and density distribution fitting curve in the flow direction for e ¼ 0:99 show that they may befitted well by a Maxwellian distribution, indicates that if ones do not consider the energy dissipation of theparticles flow, the distribution can get a Maxwellian distribution equilibrate thermally in flow direction viacollisions. This result is in agreement with the study in Refs. [9,10].

ARTICLE IN PRESSXinyu-Tan et al. / Physica A 363 (2006) 307–314312

It is believed that the energy dissipation due to the inelastic collisions causes the instantaneous energybalance invalid, which leads to the non-Gaussian velocity distribution and the spatial distribution that isasymmetric. The dissipation of energy during one collision [21]

DE ¼ð1� e2Þðvi þ vjÞ

2

2

mimj

mi þmj

. (9)

Apparently, in the system the lower the value of the restitution coefficient e, the more the dissipation of energywhich is caused by collisions among particles. Therefore, with the decrease value of the restitution coefficient e

the movement of the particles deviates from the collision-free movement more distinctly. Then the velocitydistribution deviates more obviously from the Gaussian and the spatial asymmetry is more pronounced.

3.2. Influence on the dynamic action of ununiformity

We further investigate how the non-uniformity influences dynamic properties of the granular system. Figs. 3and 4 describe the numerical results of our simulation when it is in a quasi-elastic case of e ¼ 0:99, fractaldimension D ¼ 2:8 and 2.1.

Fig. 3 shows the distribution of velocities after all particles run 1000 times. In the two different fractiondimension, velocity distributions which are described by rotundities and triangle all cease to be the initialGaussian profile. Moreover, the deviation becomes more pronounced as the fractal dimension D increases.Similarly, we simulate the density distribution in three different fraction dimensions in Fig. 4 while the otherparameters are same. We can see the fraction dimensions may affect the particles flight. However, it does notchange the whole distribution trend.

Now, let us analyze the simulation results. From Eq. (9) we get

DE ¼ð1� e2Þðvi þ vjÞ

2

2

mimj

mi þmj

¼ð1� e2Þðvi þ vjÞ

2

2

miðmi þ DmijÞ

2mi þ Dmij

, (10)

where Dmij ¼ mi �mj. Obviously, if the restitution coefficient e does not change, when the difference of themass between the two colliding particles is greater, the dissipation of energy DE is higher. As D is a reflectionof the non-uniformity of the mass in the system, we see with the increasing value of D, the average dissipatedenergy per particle increases and the velocity probability distribution deviates more obviously from theGaussian one, which is just consistent with the experimental phenomenon [22]. It can be seen that usingfraction theory can describe the plasma-particles flow dynamics action generated by PLA very well.

-200 0 200 400 600 800 10000.000

0.005

0.010

0.015

0.020

Intial velocity distribution

D=2.1 D=2.8

Inte

nsi

ty, a

rb.u

nit

s

Velocity, m/s

Fig. 3. The instantaneous velocity distribution profile in different fraction dimension when e ¼ 0:99.

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0 2 4 6 8 10 12 14 16 18 200.00

0.02

0.04

0.06

0.08

0.10

Inte

nsi

ty,a

rb.u

ints

distance from the initial surface,um

d=2.1 d=2.5 d=2.8

Fig. 4. The instantaneous density distribution profile in different fraction dimension when e ¼ 0:99.

Xinyu-Tan et al. / Physica A 363 (2006) 307–314 313

4. Conclusion

In this paper, based on the consideration of the inelasticity and non-uniformity of the plasma-particlesinduced by laser ablation, we introduce a perfect one-dimensional dynamic model to simulate the plasma-particles expansion and explain the experimental observation by using a QMD method. Fractal theory givenby Eqs. (2) and (3) is first used in PLA to describe the non-uniform particles system. The influences ofinelasticity and non-uniformity on the dynamic action of the system are investigated in detail. The results fromthe present model indicate that the dynamic characteristics of the non-uniform particles system can beevidently influenced by the inelasticity and the non-uniformity. Along with the decrease of restitutioncoefficient e and increase in fraction dimension D, the velocity distributions of the plasma-particles system alldeviate from the initial Gaussian distribution. Because of the influence of the inelastic collision and thedifference of the particles mass, when e decreases and D increases, the dissipation of energy DE increases. Thisleads to density distribution being not uniform and becomes quasi-Gauss distribution in appropriate e and D.The predictions of the particles action based on the proposed fractal and inelasticity model are found to be inagreement with the experimental observation. This verifies the validity of the present model for pulsed laser-induced particles.

Acknowledgements

This work was supported by the National Nature Science Foundation of China through Grant no.50272022, Nature Science Foundation of HUBEI Province through Grant no. 2001ABB099, SunshineFoundation of Wuhan city through Grant no. 20045006071-40, and Major Science Foundation of EducationDepartment of Hubei Province of China, Grant no. 2003Z002.

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