Introduction Motivation Objective DSMC (The Direct Simulation Monte Carlo method) is often used to analyse the plasma kinetics, since the behaviour of electrons and ions can be calculated correctly by this model even in the non-equilibrium region like electrode sheath. In order to improve the accuracy of DSMC, it is necessary to describe the density and the velocity distributions in detail. A number of slabs and the mesh points in the velocity distributions are needed, and this causes considerable computing time. Objective To develop DSMC coupled with LPWS in order to improve the accuracy of DSMC without increasing its computing time. LPWS is adopted to describe the density distribution of the particles in discharge space. The behaviour of electron swam in a parallel plate gap is traced and the swarm parameters calculated by this method, and the obtained parameters are compared with those calculated by conventional particle models.

DSMC (The Direct Simulation Monte Carlo method) Discharge space is divided into slabs, and velocity distributions of the charged particles are defined in the slabs. Each of the particles in the distributions is picked up, and its drift, acceleration and collision in unit time Dt are calculated by Monte Carlo technique, then the particle is stored in a new velocity distribution. This procedure is repeated and the spatiotemporal evolution of the particles is given. BEFORE AFTER n(z) n(z) z z E slab E slab f (vz,vr) Cathode Cathode Anode Anode f (vz,vr) Change in Dt is calculated by MCS vr vz vz vr

EEDF sampling by conventional method and LPWS Legendre Polynomial 　　　　　　　　　　　　　 Weighted Sampling (simple counting) -１≦χ≦１ Number of electrons in a bin is sampled. To obtain EEDF in detail fine mesh(bin) is needed statistical fluctuation a number of electron must be sampled considerable computing time Density gradient Gi(x) in a bin is described 　using Legendre polynomial. To obtain EEDF in detail accurate G(x) is needed independent on bin size

Method of LPWS Density gradient in the i th bin ei ei+1 ei+2 Gi Gi+1 At boundary , Gi and Gi+1 are not necessarily continuous. Solutions To sample enough number of electron To superpose Gi using B-spline Superposition Secondary B-spline in general, m : the order of B-spline Si : B-spline 0≦x≦１ ( K.Satoh et al, T.IEE, Japan Vol.120-A, No.2, pp.147-53 (2000) )

Direct Simulation of Monte Carlo with LPWS LPWS gives detailed information and good statistics of density and velocity distributions with smaller number of particles and with wider size of bin than the conventional sampling. n n(z) 1D LPWS n(z) 1D LPWS can give accurate density distribution n(z) z E slab Cathode Anode 2D LPWS f (vz,vr) vr vz 2D LPWS can give accurate velocity distribution f(vz,vr) f (vz,vr) vr vz

Calculation conditions Gas : CF4 Electrodes : fully absorbing walls Gap length : 3cm Gas temperature : 0 ℃ Gas pressure p : 1Torr Reduced electric field : 400Td Initial electron number : 10,000 Sampling interval Dt : 0.1ns Legendre polynomial : 8 terms B-spline : primary Number of slabs : 30 Description of f(vz,vr) :conventional method Collision cross sections of CF4 H. Itoh et al, Proceedings of 21st ICPIG, Vol.1, p.245 (1993) Initial density distribution z n(z) 1cm 2cm 3cm 0cm Gauss distribution with the center of mass of 0.15cm and s = 0.0167 is used.

Swarm parameters by conventional DSMC #1 Mean energy Electron number Velocity Wv Ionization frequency

Swarm parameters by conventional DSMC #2 Center of mass (1st order moment) 2nd order moment Mz The temporal variations of the mean energy, the drift velocities and the centre of mass (1st order moment) are not sensitive to the mesh number of the density and the velocity distributions. The error in the ionisation frequency of the coarsest mesh (50x50x50) is not small, however, the error decreases when finer mesh is used. The 2nd order moment is strongly affected by the number of the meshes.

Swarm parameters by DSMC with LPWS 2nd order moment Mz Density distributions (T=40.0ns) Although only 30 slabs are used for the density distribution in the DSMC coupled with LPWS, the temporal variation of the 2nd order moment approaches to that of the MCS. Density distributions calculated by the conventional DSMC with coarse mesh are widened, so that they are strongly affected by numerical diffusion. However, density distribution calculated by DSMC coupled with LPWS is not affected by the numerical diffusion.

Electron swarm parameters and calculating time The longitudinal diffusion coefficient DL is strongly affected by the number of the meshes and the value of error in DL is about 17% in the result of the conventional DSMC with the finest mesh (200x200x200) studied here. However, the value of error in DL obtained by DSMC coupled with LPWS descends to about 10%. Computing time of DSMC coupled with LPWS is shorter than a quarter of that of the conventional DSMC with the finest mesh. Computing time of DSMC coupled with LPWS enables improving the accuracy and reducing the computing time of DSMC.

DSMC with 1D-LPWS for n(z) and 2D-LPWS for f(vz,vr) Ionisation frequency Ri Longitudinal moment Mz vz × vr × z CPU Time Ri (s-1) DL(cm2/s) MCS(Thomas) 3.3min(DEC) , 59.2( - ) , 0.94( - ) 30× 30× 30 (1D+2D) ,12.5min(DEC) , 59.2( 0% ) , 0.99( +5.3%) 200×200× 30 (1D) , 9.3min(DEC) , 59.7( +0.8% ) , 1.04( +10.3%) By using 2D-LPWS for describing velocity distribution f(vz,vr) and 1D-LPWS for n(z), accuracy is also improved, however computing time increases. (about 1.5 times long) 1D-LPWS for n(z) + 2D-LPWS for f(vz,vr) is not suited practically for plasma simulation.

Conclusion DSMC coupled with LPWS is developed in this work. This method reduces the error of swarm parameters and computing time. Poisson’s equation and scaling technique for simulation of processing plasma are easily coupled with this method. This method makes a contribution of improving the accuracy and accelerating calculation of particle model simulation for processing plasmas.

Thank you for your comments