Effect of gas flow on the nanoparticles transport in dusty acetylene plasmas

Xiangmei LIU (劉相梅), Wenjing LIU (劉文靜), Xi ZHANG (張茜),Xiaotian DONG (董曉天) and Shuxia ZHAO (趙書霞)

1 School of Science, Qiqihar University, Qiqihar 161006, People’s Republic of China

2 School of Physics, Dalian University of Technology, Dalian 116024, People’s Republic of China

Abstract

Keywords: gas flow, dusty acetylene plasmas, nanoparticles transport

1.Introduction

In recent years,researchers have shown that gas flow has an essential influence on plasma properties, nanoparticle formation,and nanoparticle transport[17-23].Cole et al[17]studied nanoparticle formation using a dielectric barrier discharge plasma system.Their experimental results showed that the flow rate significantly impacts the nanoparticle density and production rate.De Bleecker et al[18]studied the role of gas temperature differences on nanoparticle transport and found that thermophoresis can significantly influence nanoparticle spatial distribution.Hasan et al [19] investigated the impact of gas flow rate effect on the transport of chemical species in an atmospheric-pressure plasma discharge.They observed that a reasonable flow rate could enhance mass transport and manipulate the plasma density.However, the complexity of the neutral gas system has caused most researchers to consider the density, velocity, and gas temperature distribution of the neutral gas as input data [18, 24],and did not fully self-consistently study the flow and heat transfer process of neutral gases.Therefore, it is necessary to carefully study the neutral gas flow effect on nanoparticle growth and transport in acetylene microdischarge.

In this work, a two-dimensional multi-fluid model is developed to investigate the gas flow effect on nanoparticle transport in dusty acetylene plasmas.The neutral gas density,momentum,and energy balance equations are introduced and studied using the fluid model in section 2, and the transport and nanoparticle growth are described.The simulation results are presented in section 3, and the gas flow effects on the C2H2microplasmas properties and nanoparticle behaviors are carefully discussed.Finally, the conclusions are summarized in section 4.

2.Model description

2.1.Fluid module

In the 2D fluid model, 48 different particles are introduced,which can be seen in table 1,and the corresponding chemical reaction coefficients are obtained from [7, 25].The background gas densitynnand velocityunare calculated using the continuity and momentum balance equations

where Rnand Mnrepresent the transfer of mass and momentum to background gas from collisions with other particles, mnis the neutral gas mass andpnis the pressure.The neutral viscous stress tensoris assumed to be a Newtonian formwhereη=1.0× 105Pa sis the viscosity coefficient of acetylene gas.

Assuming that the plasma is in local thermal equilibrium,thus the neutral gas temperature Tnis characterized by a single energy balance equation

where n is the total neutral gas density andCvis the heat capacity when the volume is constant.The term ? ·qnis the energy transfer due to thermal conduction, whererepresents the heat flux,andk= 0.0233 Wis the thermal conductivity.The termsand Enare the energy transfer due to pressure volume work and collisions,respectively.

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Table 1.The particles calculated in the model.

The densitynjfor each species (electrons, ions,nanoparticles, radicals and molecules) and flux Gjof small species (electrons, ions, radicals and molecules) are described by the continuity and momentum equations, and the momentum equation is estimated by the drift-diffusion approximation

whereμjandDjare the mobility and diffusion coefficients,andRjrepresents the particle’s formation and loss terms.For ions, the electric fieldE is replaced by an effective electric fieldwhich accounts for the inertia effects.

The electron temperatureTeis solved by the electron energy balance equation

where Gwis the electron energy density flux andRwis the loss of electron energy due to electron impact collisions.

Poisson’s equation makes the model fully self-consistent

Here,φis the potential,ε0is the vacuum permittivity, andne,ndare the electron,ion and nanoparticle densities.Qdis the nanoparticle charge.

The dust particle formation (nucleation) can occur through successive reactions between acetylene molecules and anions,with the primary anions being H2CC?and C2H?

The largest anion stops at C12H?, which is taken as the production term for the smallest nanoparticle volume section in the coagulation stage.

2.2.Nanoparticle module

Nanoparticle charge is directly proportional to the floating potentialwhere the floating potentialVflis described by equalizingIi=Ie.The ion currentIiand the electron currentIeare calculated from the orbital motion limited theory [26]

wherekBis the Boltzmann constant,m i,meare the ion and electron masses andare their temperatures, respectively.It should be noted that ion-neutral collisions significantly affect particle charging [27, 28], thuskBTiin equation (8) is replaced by the mean energywhich accounts for the drift velocity of ionsυ.iThe effect of ion-neutral collisions on the ion drift velocity is included through the momentum equation of ions(a re-written equivalent form of equation(4)),whereνiis the momentum transfer frequency of the ion i.It is noted that the drift and diffusion approximation is applied onto the momentum equation.When the diffusion term is further omitted or excluded, we havewhich will then be very similar to the analytical approximation model used in[28].To better understand the particle charging,a molecular dynamics simulation is better to be used as in the[28].Nevertheless, we believe that it will not significantly influence the change trend of particle charge, as estimated.

Nanoparticles are subject to ion drag, thermophoresis,and neutral drag forces besides the electrostatic force[25,26].Thus, the nanoparticle flux Gdis given by

whereμd,are the mobility and diffusion coefficients of nanoparticles,Eeffis the effective electric field.mdare the ion and nanoparticle masses, rdis the radius of nanoparticles andνmdis the momentum loss frequency.Giis the ion flux,υsis the mean speed andυthis the background gas’s thermal velocity.bcis the collection parameter, Γ is the Coulomb logarithm, andπb/2is the impact parameter for the deflection angle of π/2.Note that,different from our previous studies, the thermophoretic force acting on the nanoparticles could significantly affect the profile of nanoparticle density.

In the coagulation phase, the nanoparticles will quickly grow from several to tens of nanometers.To more effectively study the nanoparticle growth mechanism, an aerosol dynamics equation [29] is introduced and the nanoparticle densityn(v) is described in the volume range ofv～v+dvas

Figure 1.Schematic diagram of the reactor.

where the first item on the right of equation(10)describes the formation of nanoparticles in the volume range ofdv, and the second item represents the loss of particles.is the coagulation frequency between the interacting nanoparticles with the volumeuandv-u,J0(v) is the new particle formation rate by nucleation, andis noted that the collisions are calculated twice in the integral, which is why 1/2 is introduced.

2.3.Numeical approach

Plasma and nanoparticle modules with two separate computation cycles are used to describe the dusty C2H2discharges.In the first computation cycle,a time step of 3.7×10?12s is used to describe the plasma module, which includes the computation of particle balance equations, electron energy equations, and Poisson’s equation.It can be noticed that the nanoparticle generation (nucleation) is implemented in the plasma module.In the second computation cycle, a larger time step of 3.7×10?8s is used to describe the nanoparticle module, which includes the computation of nanoparticle charging, transport and growth.

Two computing cycles are coupled together by an iterative process.First the fluid module is calculated for several radio-frequency (RF) cycles while the nanoparticles are assumed to be immobile.In the second computation cycle,the nanoparticle module begins with the time-averaged electron flux,positive ion flux,and electric field,which are calculated from the fluid module.The resulting nanoparticle density and charge are coupled to Poisson’s equation.

3.Results and discussion

The reactor configuration is illustrated in figure 1, where the RF (13.56 MHz) source is applied to the top electrode with a voltage ofand the bottom electrode is grounded.In figure 1, the acetylene gas flows in from the top showerhead electrode and flows out from the sidewall.For neutral gas,the inlet condition is set as the velocity-inlet while the outlet condition is the pressure-outlet.The input voltagethe ion temperature of 300 K and the pressure of 500 Torr are fixed.

Figure 2.Calculated pressure force (a) and viscous force (b) acting on background gas.

Figure 3.The spatial distributions of neutral gas velocity.

In the capacitively-coupled RF atmospheric-pressure electronegative gas(C2H2)discharges,the drift and ambipolar fields play a dominant role in sustaining the discharges, and the electronegativity increases with pressure.Thus,the results are restricted to the drift-ambipolar regime [30-32].The gas flow effect on the plasma properties and nanoparticle behavior is a critical problem, thus the neutral gas inlet velocity varies from 0 to 4.0 m s?1.

The spatial distribution of pressure and viscous forces is shown in figure 2, with an inlet velocity of 4.0 m s?1.The arrows in the figure indicate the direction of the force,and the colors represent the magnitude of the force, since the force varies by several orders of magnitude.It can be seen from figure 2(a) that the pressure and viscous forces are in the vertical direction,which will prevent the background gas flow out from the wall.Furthermore, the pressure force is much stronger than the viscous force, about two orders of magnitude larger.This means that pressure plays a dominant role in the transport of neutral gas.It can be noticed that the viscous force at the inlet is much larger(about 1.0×106Pa cm?1)but decreases sharply in the bulk plasma (about 2.0 ×102Pa cm?1).The collision drag term is of no influence as indicated in the simulation(due to the tiny sheath)and hence is not drawn herein.

Figure 3 shows the spatial distributions of background gas velocity at different inlet velocities.Herein, the arrows show both the velocity direction and magnitude(indicated by the arrow length).The velocity direction of background gas is vertically downward except for certain local regions,which is opposite to the direction of forces.Upon increasing the gas inlet velocity, from 0.5 to 4.0 m s?1, the gas velocity in the discharge chamber is increased.Meanwhile, the velocity profile is more squeezed up to the shower head at increasing the inlet velocity and so the gas velocity decreases faster when traveling from the shower head to the chamber bottom.As seen from equation (2)and figure 2, the pressure gradient force is several orders higher than the viscous term, but the background gas velocity in the discharge region is decreased herein in figure 3.It implies the pressure gradient force hinders the transport of neutrals, rather than accelerating them.The reason is given below when explaining the data in figure 5.The peak of velocity at the shower head forms at the trigger of strong inlet velocity in the neutral transport, i.e.determined by the boundary condition of the momentum equation and the significant pressure gradient at the inlet shown in figure 2(a).This process we analyzed can be recognized when carefully observing the green part of the color legend, since it represents both the inlet boundary velocity and the bulk velocity near the shower head.

Figure 4.The spatial distributions of neutral gas temperature Tn (a) and density nn (b) for different inlet velocities.

Figure 5.The spatial distributions of heat conduction (a) and pressure volume work (b).

Figure 4 presents the spatial distributions of neutral gas temperature and background density at different inlet velocities.In figure 4(a), the gas temperature in the bulk plasma increases with the inlet velocity at the influence of gas advection.Furthermore, the gas temperature drops sharply very close to the shower head, e.g.from 380 to 120 K at the inlet velocity of 4.0 m s?1.In the other region, the gas temperature is quite uniform.Correspondingly, the neutral density in figure 4(b) is peaked at the shower head once the gas transport is added,and the peak density is higher at larger inlet velocity.This is because the gas transport is in a whole incompressible at such low subsonic velocities and so the total pressure is conserved, withThe spatial variation of gas temperature caused by the gas advection is explained next in figure 5.As seen further, this very local change of gas temperature profile at the inlet will lead to a strong global change in both the profiles of nanoparticles and plasma parameters.

To better understand gas temperature, the heat conduction and pressure work terms are shown in figure 5, with the inlet velocity of 4.0 m s?1.The thermal conduction and pressure volume work terms play an important role in the neutral gas energy balance equation, since they are much larger than the other terms.By referring to equation (3), the heat conduction term in figure 5(a) is determined by the gas temperature gradient shown in figure 4(a).The value of this term is negative and only spatially varied at the inlet due to the strong gas temperature gradient therein, which is an equivalent energy loss term to the temperature variable.The thermal conduction almost does not influence on the rest of the discharge region since the gas temperature is smooth when it is plotted at the present legend scale and resolution.The pressure work term in figure 5(b), represented by the second term at the right side of equation (3), is also negative at the inlet because,as mentioned before,the gas advection is triggered by the inlet velocity boundary condition in figure 3 and so the thermal energy of gas medium is transferred into kinetic energy.This energy transfer process is the major loss term of gas internal energy at the inlet and that is why gas temperature sinks at that location.It is interesting to note that except for the inlet position,the pressure work is changed into a positive value in the rest region.This is again related to figure 3, where the gas velocity is reduced with the distance from the inlet to the chamber bottom.Obviously, here in the major discharge area, the reverse energy transfer process occurs, i.e.the kinetic energy is transferred into thermal energy.This explains well the gas temperature increase in the bulk discharge area with inlet velocity value in figure 4(a).In our opinion,this is one equivalent isovolumetric compression process as we learned in the curriculum of thermodynamics.

Figure 6.The spatial distributions of electron temperature and density for different inlet velocities.

Figure 7.The spatial distributions of nanoparticle density for different inlet velocities,with the nanoparticle diameters of 1 nm(a)and 10 nm(b).

Figure 6 presents the spatial distributions of electron temperature and density with various inlet velocities.As shown in figure 6(a), the electron temperature exhibits a much lower value in the bulk plasma but begins to rise quickly in the sheath regions.Note that the electron density and temperature are very sensitive to the inlet velocity.As the inlet velocity increases from 0 to 4.0 m s?1,the electron temperature at the inlet moves to the lower electrodes, thus the electron temperature at the bottom electrode increases quickly from 1.69 to 1.94 eV,while the electron temperature at the inlet decreases from 1.69 to 1.27 eV.This is because, as inlet velocity increases, more collisions between the neutral gases and electrons occur,resulting in more electron energy loss at the inlet and less energy loss at the lower electrode (refer to figure 4(b)).The electron density in figure 6(b) presents two dominant peaks near the presheaths due to the strong ambipolar electric field and high electron energy,while a much lower value is found in the bulk plasma due to the strong drift electric field.Like the relation of gas temperature and density shown in figure 4, the electron temperature and density are found to vary in inverse proportion as well.It is found when the inlet velocity increases from 0 to 4.0 m s?1, the electron density at the inlet increases from4.7× 1011to5.4× 1011cm?3, because more electrons are produced at more frequent inelastic collisions between background gases and electrons when the inlet velocity increases.As seen, the plasma parameters and the gas advection in the present chamber configuration are tightly coupled.In another work of ours, it is observed that the two processes will be loosely coupled when the gas is set to flow through the sidewall tunnel of the chamber.

Figure 7 illustrates the spatial distributions of nanoparticle density for different inlet velocities, with the nanoparticle diameters of 1 nm (a) and 10 nm (b).It can be observed from figure 7 that, nanoparticles are mainly accumulated at presheaths due to the action of ion drag force when the gas flow is not considered.Once the gas flow is considered, a gas temperature gradient is presented at the inlet (see figure 4(a)),and thus nanoparticle begins to experience the action of thermophoresis force,represented by the last term on the right side of equation(9).It can be observed from figure 7(a)that,as the inlet velocity increases, the density of nanoparticles that are accumulated near the upper electrode is decreased significantly.This is because the thermophoretic force pushes nanoparticles to the upper electrode and then particles attach to the electrode surface and disappear.This process simulated by our simulation shows the same tendencies with the experimental observations[33].The experiment[33]observed that a large fraction of the particles was forced out of the interelectrode gap and trapped near the base of the cooled electrode(where the temperature difference is small) when applying a temperature difference of 38 K to one of the two electrodes.We predict that the thermophoretic force pushes nanoparticles to the electrodes and particles are trapped near the electrode where the temperature difference is small.Furthermore, the thermophoretic force is proportional to the gradient of the gas temperature,which increases with the inlet velocity(as shown in figure 4(a)) and so the density peak shift is more obvious with inlet velocity.Besides, this phenomenon is also more apparent when the nanoparticle size is increased,i.e.to 10 nm,as shown in figure 7(b).It is observed that the thermophoretic force drives nearly all particles to the upper electrode when the gas velocity is 4.0 m s?1, and a single-peak distribution is produced near the lower electrode.This is because the thermophoresis force is proportional to the square of the particle size, as shown in equation (9).

4.Conclusions

In summary, the effect of gas flow on plasma properties and nanoparticle behaviors is investigated by combining the selfconsistent 2D hydrodynamic model with the aerosol dynamics model.The neutral gas transport is mainly determined by the combined action of pressure and viscous forces, and the pressure force is much larger than the viscous force.The direction of pressure is opposite to the direction of velocity except at the inlet, hindering the propagation of gas.The pressure volume work and heat conduction significantly influence the neutral gas temperature.It is found that, the neutral gas temperature decreases and background gas density increases at the inlet, since the gas flow is approximately incompressible and the transfer process of thermal energy to kinetic energy occurs therein.On the other hand, the gas temperature in the main bulk discharge area is increased because an inverse energy transfer process has happened,which can be represented by the isovolumetric compression.

Under the influence of neutral gas density, the electron temperature at the inlet decreases and the electron density increases with increasing inlet velocity, since more collisions between electron and background gas take place.This leads to the peak of electron temperature near the upper electrode moving to the lower electrode and the peak of electron density near the lower electrode moving to the upper electrode(reversely).On the other hand, when a gas temperature gradient occurs, nanoparticle transport is dominated by thermophoretic force, which pushes nanoparticles toward the electrodes and particles are trapped near the electrode where the temperature difference is small.As nanoparticles grow to 10 nm, the action of thermophoretic force becomes more obvious than the other forces,which drives nearly all particles to the upper electrode and causes the nanoparticle density to change from a double-peak structure to a single-peak one.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Nos.11805107 and 12275039), the Fundamental Research Funds in Heilongjiang Provincial Universities of China (No.135509124), and the Graduate Innovation Foundation of Qiqihar University (No.YJSCX2022014).