multistage gas–surface interaction model for the direct simulation monte carlo method

Multistage gas–surface interaction model for the direct simulation Monte Carlo methodNobuhiro Yamanishi, Yoichiro Matsumoto, and Kosuke Shobatake

Citation: Physics of Fluids (1994-present) 11, 3540 (1999); doi: 10.1063/1.870211 View online: http://dx.doi.org/10.1063/1.870211 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/11/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A washboard with moment of inertia model of gas-surface scattering J. Chem. Phys. 120, 1031 (2004); 10.1063/1.1628674 GasSurface Interaction Model Evaluation for DSMC Applications AIP Conf. Proc. 663, 965 (2003); 10.1063/1.1581644 Open-system quantum dynamics for gas-surface scattering: Nonlinear dissipation and mapped Fourier gridmethods J. Chem. Phys. 113, 8753 (2000); 10.1063/1.1318902 Scattering and trapping dynamics of gas-surface interactions: Theory and experiments for the Xe-graphitesystem J. Chem. Phys. 109, 10339 (1998); 10.1063/1.477689 Collisional energy loss in cluster surface impact: Experimental, model, and simulation studies of some relevantfactors J. Chem. Phys. 108, 10262 (1998); 10.1063/1.476487

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

37.191.221.36 On: Sat, 10 May 2014 09:29:12

http://scitation.aip.org/content/aip/journal/pof2?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/754833240/x01/AIP-PT/PoF_ArticleDL_050714/Awareness_LibraryF.jpg/5532386d4f314a53757a6b4144615953?x

http://scitation.aip.org/search?value1=Nobuhiro+Yamanishi&option1=author

http://scitation.aip.org/search?value1=Yoichiro+Matsumoto&option1=author

http://scitation.aip.org/search?value1=Kosuke+Shobatake&option1=author

http://scitation.aip.org/content/aip/journal/pof2?ver=pdfcov

http://dx.doi.org/10.1063/1.870211

http://scitation.aip.org/content/aip/journal/pof2/11/11?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jcp/120/2/10.1063/1.1628674?ver=pdfcov

http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.1581644?ver=pdfcov







PHYSICS OF FLUIDS VOLUME 11, NUMBER 11 NOVEMBER 1999

This a

Multistage gas–surface interaction model for the direct simulationMonte Carlo method

Nobuhiro Yamanishi and Yoichiro MatsumotoDepartment of Mechanical Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Kosuke ShobatakeDepartment of Material Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

~Received 13 January 1999; accepted 30 June 1999!

A multistage ~MS! gas–surface interaction model for a monatomic/diatomic gas moleculeinteracting with a solid surface is presented, based on the analysis of molecular dynamics~MD!simulations and a model equation derived from the classical theory of an ellipsoid hitting a hardcube. This model is developed for use with the direct simulation Monte Carlo~DSMC! method andbelongs to the thermal scattering regime. The molecular dynamics method is used for themolecular-level understanding of the scattering of O2, N2, and Ar from a graphite surface. The basicidea of the model is to separate the collision into three stages. At stages 1 and 2, the energy andscattering direction are determined by the model equation. At stage 3, according to the translationalenergy, the molecule is determined to scatter, re-enter or be trapped by the surface. Re-enteringmolecules return to stage 1. The model parameters are determined from our MD database.Experiments are also performed by scattering a supersonic O2 molecular beam from a clean graphitesurface in an ultrahigh vacuum chamber. The in-plane scattering distribution, out-of-plane scatteringdistribution, and in-plane velocity distribution of the model show good agreement with those ofmolecular beam experiments. A model equation was included in the MS model to maintain thermalequilibrium between a gas and a surface at the same temperature when applied to DSMCsimulations and the results are also shown. The high accuracy of the model clearly shows that suchmultiple-scale analysis can lead to the development of realistic models of the gas–surfaceinteraction. © 1999 American Institute of Physics.@S1070-6631~99!03110-4#

te-g

munheecin

s aThnd

laauaucnsich

els

ee

rat

Datethesticithyd

cu-to

ol-ssrgyn.er

achy to

–. In-

togese-

a-llly

I. INTRODUCTION

The successful application of direct simulation MonCarlo ~DSMC! simulations1 requires the development of accurate gas–surface interaction models, in addition to gas–molecular collision models. Since most practical probleinvolve gas–surface interaction phenomena, a proper boary condition is required to obtain reliable results from tnumerical analysis of rarefied gas flows. The diffuse refltion model, where the scattering distribution obeys the coslaw and a Maxwellian velocity distribution, is often used apractical model for purposes of engineering surfaces.Maxwell model is a combination of specular reflection adiffuse reflection. The Cercignani–Lampis–Lord~CLL!model2–5 satisfies reciprocity and reproduces the lobuscattering distribution, but one must use an adjustable pareter for energy accommodation and it does not present qtitative agreement with molecular beam experiments. Slimitations of the current models for boundary conditioclearly show the need for a more realistic model whagrees with experimental results.

One effective approach for developing collision modis the use of the molecular dynamics~MD! method. MDstudies are able to give quantitative agreement with expmental data and the method is an important tool for undstanding the dynamics of a molecular phenomenon. Kou6,7

and Yamamoto and Yamashita8 have employed a direc

3541070-6631/99/11(11)/3540/13/$15.00

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

37.191.221.36 On: Sat, 1

assd-

-e

e

rm-n-h

ri-r-

method by simply coupling the DSMC method with Mcalculations. This coupled method may seem as the ultimanswer, but the enormous time required for calculatingcollision process of the molecules makes it an unrealitechnique. This is especially so for molecular systems wsolid crystals. A more efficient approach was taken by Boet al.9 and Tokumasu and Matsumoto10 in which the MDmethod was used to calibrate or construct their intermolelar collision models. Such multiple-scale analysis can leadthe construction of a physical model at the microscopic clision level, i.e., detailed description of the collision proceby including the effects of such factors as potential enesurface, crystal lattice, bond length, and multiple collisioAlthough the reduction of computation time and the numbof model parameters will be required to make this approuseful to the general user, the approach is an effective wadevelop realistic models.

In our previous papers,11–13such an analysis on the gassurface interaction was performed to construct a modelour first paper,11 MD simulations and molecular beam experiments of the O2/graphite system were carried outverify the numerical method and to obtain basic knowledof the dynamics of the gas–surface interaction. In the subquent paper,12 based on the data obtained from MD simultions, the multistage~MS! gas–surface interaction modewas introduced. The basic idea of the model was initia

0 © 1999 American Institute of Physics

ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

0 May 2014 09:29:12

ntorabdde

tiul

uerth

teerd

icagel

he

canas

aea

posugtauu

hareteeramuce.cuet

chon

onl

derdes

as–m-

amof

hethebe

arenotingtheedthe

theing

le

of

v-h

ec-

h isrdicf

3541Phys. Fluids, Vol. 11, No. 11, November 1999 Multistage gas–surface interaction model . . .

This a

presented here, in which the collision process is divided idifferent stages. At this point, the model was only fO2/graphite at a surface temperature of 298 K and was cbrated by a small database. Also, the model was limitedits inadequate rotational energy distributions, unclear moparameters, and lack of physical meaning for the moequation. Therefore the model was not comparable withperimental results. In our recent paper,13 the model madesome progress, an adequate rotational energy distribuwas given, a simple comparison with experimental reswas shown, and the model was applied to the DSMC simlation of a narrow channel flow for comparison with othmodels. But at this stage, the model was only forO2/graphite system, unclear parameters remained andmodel had not proven that it could maintain equilibrium btween a gas and a surface at the same temperature. Ovthese series of papers presented only a prototype of thesired MS model.

In the present paper, based on the analysis of MD simlations and a model equation derived from the classtheory of an ellipsoid hitting a hard cube, the multistagas–surface interaction model is presented. The modemainly developed for the use of DSMC simulations. Tapplicable systems of the model are O2/graphite, N2/graphite,and Ar/graphite and the model belongs to the thermal stering regime.14 Graphite was selected for experimental coditions because a clean and perfect surface can be eobtained.

The significant advances from our previous works arefollows. First, a full description on the derivation of thmodel equation is given. Second, unclear parameters hdisappeared or are clarified. Third, the modeling of thetential energy surface has been revised. Fourth, new reof molecular beam experiments are obtained by usin3-axis manipulator. The manipulator has enabled us to obin-plane and out-of-plane scattering data of the gas molecThe present paper shows that the MS model can reprodthe out-of-plane scattering of the gas molecule, whichgiven the present model a new dimension of reliability, pvious papers having only shown results for in-plane scating. Fifth, a model equation was included to maintain thmal equilibrium between a gas and a surface at the stemperature when applied to DSMC calculations and sresults are also shown. Finally, the present model wastended for the use of the N2/graphite and Ar/graphite system

The present paper is organized as follows. First, molelar dynamics simulation of the gas–surface interaction is pformed. The main purpose of the numerical simulation isanalyze the translational (Etr) and rotational energy (Erot)distribution of the gas molecule after its first collision. Eadistribution is analyzed by fitting the Gaussian distributifunction for Etr and Danckert’s distribution function15 forErot . This procedure leads to the database of the initial cditions of the gas molecules~translational energy, rotationaenergy, incident angle, and surface temperature! and the dis-tribution function parameters. Next, a model equation isrived from the classical theory of an ellipsoid hitting a hacube. The equation is used to determine the relation betwthe initial condition and the function parameters. Finally, u


37.191.221.36 On: Sat, 1

o

li-yelelx-

onts-

ehe-all,e-

u-l

is

t--ily

s

ve-ltsainle.ces-r--e

hx-

-r-o

-

-

en-

ing the database and the model equation, the multistage gsurface interaction model is constructed. The model is copared with the results obtained from our molecular beexperiments. It was also applied to the DSMC simulationa gas surrounded by wall boundaries.

II. MOLECULAR DYNAMICS SIMULATION OF THEGAS–SURACE INTERACTION

A. Governing equations

The molecular dynamics simulation is carried out in tCartesian coordinate system for the numerical analysis ofgas–surface interaction. All molecules are assumed toclassical particles. The oxygen and nitrogen moleculesassumed to be rigid rotors with a fixed bond length andvibrational degree of freedom. The forces and torques acon a diatomic molecule are the sum of those acting ontwo atoms of the molecule. The rotational energy is assumto take a continuous value. The surface graphite layer isxy plane andz is positive heading outward from it~Fig. 1!.

The trajectories of the monatomic gas molecule andsurface molecules are calculated according to the followequation:

mi

d2xi

dt25(

j Þ iFi j 52(

j Þ i

df~r i j !

dri j

r i j

r i j, ~1!

where t is time, f is intermolecular potential,mi and xi

5(xi ,yi ,zi) are mass and position of moleculei , Fi j andr i j

are intermolecular force and distance between molecuiand j , respectively.

The governing equation for the translational motionthe diatomic gas molecule is

mg

d2xg

dt25(

i 51

2

(j 51

Ns

Fi j , ~2!

whereNs is number of solid molecules,mg andxg are massand position of the weight center of moleculeg, respec-tively.

The rotational motion of the diatomic molecule is goerned by torqueTp about the center of mass. Althougtorque (T) can be evaluated with angular momentum (L ) inspace-fixed axes, it is most convenient to make the conn

FIG. 1. Molecular dynamics simulation system. The gas molecule whicplaced above the surface is O2 and the solid crystal is graphite with a Milleindex of ~0001!. 288 carbon atoms are placed in 3 layers with perioboundary conditions in thex andy directions to create a wide thin film ographite.


0 May 2014 09:29:12

ren

fm

ton

e.

s

an

mthga

thet

the

,,

msthe

e-s

ingt

rem

hin

of-

f

ical

-ve-m-

3542 Phys. Fluids, Vol. 11, No. 11, November 1999 Yamanishi, Matsumoto, and Shobatake

This a

tion with angular velocity vectorvp , through angular mo-mentumL p , in the body-fixed principal axis system whethe inertia tensorI p is diagonal. Thus the governing equatiois as follows:

dL

dt5T, ~3!

dL p

dt1vp3L p5Tp . ~4!

The resulting equations for the components ofvp in thebody-fixed frame are

vpx5$Tpx1~ I px2I pz!vpyvpz%/I px , ~5!

vpy5$Tpy1~ I pz2I px!vpzvpx%/I py , ~6!

vpz5$Tpz1~ I px2I py!vpxvpy%/I pz , ~7!

where I px , I py , and I pz are the three principal moments oinertia. For the diatomic molecule of our molecular systeI px52mar a

2 , I py52mar a2, and I pz50, where ma is the

atomic mass andr a is the distance from the center of massthe atomic site which was taken to be 0.60 Å for oxyge16

and 0.55 Å for nitrogen.17 Since thez principal axis is on themolecular bond,vpz50.

The orientational equations of motion can be describin terms of Euler angles~u,f,c!, but contain singularitiesQuaternion parameters~j,h,z,x! ~Ref. 18! are used to avoidsuch a problem. The parameters are defined by

j5sinS u

2D sinS c2f

2 D , ~8!

h5sinS u

2D cosS c2f

2 D , ~9!

z5cosS u

2D sinS c1f

2 D , ~10!

x5cosS u

2D cosS c1f

2 D , ~11!

with the constraint

j21h21z21x251. ~12!

The quaternions of the gas molecule satisfy the equationmotion,

S jh

zx

D 51

2 S 2z 2x h j

x 2z 2j h

j h x z

2h j 2z x

D S vpx

vpy

vpz

0D . ~13!

The intermolecular potential between the gas atomthe solid atom is the Lennard-Jones potential in the form

f i j ~r i j !54«LJH S r LJ

r i jD 12

2S r LJ

r i jD 6J , ~14!

where«LJ andr LJ are the potential parameters. Both paraeters were determined from the depth and position ofphysisorption well of the gas–surface system. First, the


37.191.221.36 On: Sat, 1

,

d

of

d

-es

molecule is placed above the center of the unit cell atheight of the physisorption well depth.r LJ is adjusted so thathe total potential energy takes a minimum. Next,«LJ is ad-justed so that the total potential energy takes the value ofphysisoprtion well depth.

For oxygen–carbon,«LJ and r LJ will take the value of6.55 meV and 2.47 Å,16 respectively. For nitrogen–carbon«LJ53.59 meV andr LJ53.17 Å,19 and for argon–carbon«LJ55.58 meV andr LJ52.13 Å.20

The intermolecular potentials between the carbon atoare all assumed to be those of the harmonic oscillator inform

f i j ~r i j !5 12 K~r i j 2r 0!2, ~15!

whereK is the oscillator constant andr 0 is the equilibriumspacing. The oscillator constant is determined from the Dbye temperature of graphite.21 For the nearest carbon atomwhich are in the same plane,K544.44 eV/Å2 and r 0

51.42 Å. For the nearest carbon atoms in the neighborplane,K56.39 eV/Å2 and r 053.40 Å. For the next nearescarbon atoms in the neighboring plane,K52.13 eV/Å2 andr 053.68 Å. The verification of the potential parameters wedone previously by comparison with molecular beaexperiments.11

B. Numerical procedure

Prior to the calculation of gas–surface interaction, a tfilm of graphite at thermal equilibrium ofTS is obtained. 288carbon atoms are placed in 3 layers, with the Miller indexthe surface at~0001!. Periodic boundary conditions are applied in thex andy directions to construct a wide thin film ographite. The initial velocity (us ,vs ,ws) for the carbon at-oms are

us5A3kBTS

mscosQ cosF, ~16!

vs5A3kBTS

mssinQ cosF, ~17!

ws5A3kBTS

mssinF, ~18!

wherems is the mass of the carbon atom,TS is the targetsurface temperature,kB is the Boltzmann constant, andQand F are the polar and azimuthal angles in the sphercoordinate system, respectively.Q andF are chosen to be

Q5cos21~122U !, ~19!

F52pU, ~20!

whereU is an uniform random number in the range of~0,1!.Time integration is done by the leapfrog method18 at

time intervaldt50.1 fs. The graphite crystal is under temperature control for the first 20 000 steps by scaling thelocity of the carbon atoms towards the target surface teperature. This is done by using the scaling coefficientgdescribed as


0 May 2014 09:29:12

00acoow

cartia

na

ea

s–

f-itiaitianio

nsda

h

omthcehe

–

yheandn, aing

vi-pe

herthe

ureter

at

rgythe

ur.l

ine ofis


This a

g5ATS

TS8, ~21!

where TS8 is the actual surface temperature. After 20 0steps without temperature control, the graphite surfachieves thermal equilibrium. The position and velocitythe carbon atoms are then recorded for the use of the folling procedure.

Next, the gas molecules collide with the graphite surfaone at a time. The initial conditions of the carbon atomsgiven from the data of the previous procedure. The initranslational temperatureTtr , rotational temperatureTrot andincident angle towards the surface normalu in are the initialconditions of the gas molecule. Thus, the initial translatiovelocity (ug ,vg ,wg) and the initial angular velocity(vpx ,vpy) are given by

ug5A3kBTtr

mgcos 2pU sinu in , ~22!

vg5A3kBTtr

mgsin 2pU sinu in , ~23!

wg5A3kBTtr

mgcosu in , ~24!

vpx5A2Trot

I pcos 2pU, ~25!

vpy5A2Trot

I psin 2pU, ~26!

whereI p5I px5I py . The initial position is varied so that thgas molecule approaches the surface at different timingsthe initial orientation is taken to be isotropic. The initial gasurface separation is larger thandc55.7 Å a distance atwhich the intermolecular force is negligible.

The simulation is terminated after the first collision. Ater each collision, the graphite surface is set back to its incondition, i.e., the carbon atoms are restored to the inposition and velocity. For each set of incident energy aangle, 4000 trajectories are calculated. The initial conditranges from 500 to 2000 K forEtr , 0 to 2000 K forErot and30° and 60° foru in . A total of 64 cases for O2, 42 cases forN2, and 24 cases for Ar were calculated. All calculatiowere done by nondimensional quantities where the stanvalue for energyER51.435 70•10217J, massmR51.994 50•10226kg, and lengthxR51.42 Å. The procedures whicinvolve rotation are not used for Ar gas molecules.

C. Potential energy surface

The potential energy contour for an oxygen gas atinteracting with graphite in the plane perpendicular tosurface and passing above the diagonal line of the unit~Fig. 2! is shown in Fig. 3. The contours are drawn by tintermolecular potential of Eq.~14!. The potential energy fora given sitei is to be


37.191.221.36 On: Sat, 1

ef-

eel

l

nd

ll

dn

rd

ell

F i5(j 51

Ns

f i j , ~27!

where F i is the sum of the potential energy of oxygencarbon.

The potential energy is zero atz5` and gradually de-creases withz. It takes a minimum at the potential energwell which is located above the center of the unit cell. Tpotential energy suddenly increases beyond the welltakes a large positive value near the surface. At this regiostrong repulsive force will act on the gas molecule, creata hard wall@potential energy surface~PES!# which reflectsimpinging gas molecules.

The lattice pattern, potential parameter, and thermalbration of the solid crystal determine the geometric shaand vibration of the PES. A wider lattice, largerr i j , smaller«LJ, and lower surface temperature will lead to a smootand stable surface. For the quantitative understanding ofcorrugation, MD calculations were performed to measheightz of the PES. Measuring points were set at the cen~1!, on-top~2!, and bridge~3! of the unit cell.

The gas atom is moved above the graphite surfacethermal equilibrium ofTS5199 K, 298 K, and 500 K. If thepotential energyF i is equal toE0 ~1200 K! at site (xi ,yi ,zi)

FIG. 2. Unit cell of graphite and the sites for measuring the potential enesurface. Carbon molecules are shown by circles. The dashed line isdiagonal line of the unit cell, used for illustrating the equal energy contoThe shaded region and thex,y axes are used for illustrating the modepotential energy surface.

FIG. 3. Equal energy contours~eV! for oxygen atom interacting with thegraphite surface according to the intermolecular potential of Eq.~14!. Theplane is perpendicular to the surface and passes above the diagonal lthe unit cell shown in Fig. 2. Perpendicular distance from the surfacez~Å!.


0 May 2014 09:29:12

uwe

-

h

pe

ome

oc-t

om.

ans-ur-

feraperce-

ghore

pe-

ulef

ace.ationeat

ter

-t

ter

ce

ehgy, af t

the


This a

then z5zi . The trajectory of the carbon atoms were calclated without reference to the gas atoms. Measurementsdone every 100 steps for a total of 10 000 points.

Figure 4 is the relation between surface temperatureTS

andz where the open symbols are the mean^z& and the errorbars are the standard deviations(z) of the measured distribution. It was found thatTS does not effectz& as the differ-ence is less than 1%.TS does have a positive correlation wits(z), but s(z) is 3.4% of^z& at maximum. Thus, it can besaid thatz is constant ands(z) is small enough that thevibration of the PES is negligible.

Similar measurements were done for the N2/graphite andAr/graphite systems and lead to the same conclusion. Scific values and modeling of the PES are presented in SIV.

D. Time history of a scattering gas molecule

An example of the scattering process of O2 with initialconditions ofTtr51600 K, Trot51600 K, u in535° andTS

5298 K is shown in Fig. 5. The typical scattering processthis system can be observed. First, the approaching gasecule is accelerated by the attractive force of the surfac

FIG. 4. Relation between the graphite surface temperatureTS and the meanheight z of the potential energy surface at each site. Error bars denotestandard deviation of measuredz. The values for on-top, bridge, and centsites are shown byh, n, ands, respectively.

FIG. 5. Time history of an O2 molecule scattering from a graphite surfawith initial conditions of Ttr51600 K, Trot51600 K, u in535° and TS

5298 K. The abscissa is nondimensional time. The right hand ordinatthe upper part of the figure is the nondimensional distance between eacatom and the surface plane, while the left hand ordinate is the total enerthe gas molecule. The total energy is the sum of translational, rotationalpotential energies. The lower ordinate is the nondimensional energy ogas molecule.


37.191.221.36 On: Sat, 1

-re

e-c.

fol-as

both Etr and Erot are excited. The excitation ofEtr occurswhen it passes through the potential well, andErot is excitedwhen it is decelerated. The gas–surface energy transfercurs during this interaction.Etr and potential energy exhibithis typical time history for most cases, butErot variesgreatly, depending largely on the initial condition.

After such interaction, the gas molecule escapes frthe surface as bothEtr andErot converge to constant valuesThe decrease of total energy shows that energy was trferred from O2 to the graphite surface. The increase of sface temperature for our system is 2 K at maximum. Theincrease ofErot also shows that there is an energy transfrom translation to rotation. The gas molecule can escfrom the surface to a distance where the intermolecular fois negligible if it has enoughEtr . In this case the gas molecule re-enters. At the second collision,Etr increases fromthe energy transfer of rotation to translation which is enouto escape from the surface. Most gas molecules scatter mor less in the specular direction, but molecules which exrience more collisions scatter more diffusely.

To understand the energy variation of the gas molecafter the first collision,DEtr is defined as the difference otranslational energy at timeta and tb , before and after thecollision. Thus the translational energy loss is

DEtr5Etr~ ta!2Etr~ tb!, ~28!

whereta takes the initial value andtb takes the value whenthe gas molecule is at the farthest distance from the surfFor scattering gas molecules we use the value at separdistancedc which is far enough that the intermolecular forcis negligible; for re-entering molecules we use the valuethe moment the normal velocity reverses.

E. Energy distribution after the first collision

An example of the translational energy distribution afthe first collision is presented in Fig. 6, whereTtr51000 K,Trot50 K, u in560°, andTS5298 K. The open symbols denote MD results of O2, where as the solid line is the best fiusing the Gaussian distribution function defined as

f ~Etr, f !51

A2ps~Etr, f !2

expS 2~Etr, f2^Etr, f&!2

2s~Etr, f !2 D , ~29!

he

ofgasofndhe

FIG. 6. Translational energy distribution of O2 after the first collision withinitial conditions ofTtr51000 K, Trot50 K, u in560° andTS5298 K. Theopen circles denote MD results and the solid curve is the best fit usingGaussian distribution function.


0 May 2014 09:29:12

n,f-io

no

n

o

l-

st

d

a-fo-

ylidth

ci-

cityntz-sur-ns-

-

rgyledI.

ar-s intion

oid

a

ed

an-

g

hecity

r


This a

whereEtr, f is the translational energy after the first collisio^Etr, f& is the mean, ands(Etr, f) is the standard deviation othe distribution. The distribution for highEtr cases are completely symmetric and can be fitted by a Gaussian functAlthough the distribution for lowEtr will not be symmetricbecause of the fall off of theEtr, f,0 region, its distributioncan also be fitted by the Gaussian. The final translatioenergy distribution can be transferred to the distributionthe translational energy loss by Eq.~28!.

Such analysis for the entire database will give^DEtr&ands(Etr, f) for each initial condition. If the relation betweethe initial condition~Etr , Erot , u in , andTS! and both quan-tities can be found, the energy distribution after the first clision can be easily estimated.

Next, the rotational energy distribution after the first colision is presented in Fig. 7, whereTrot50 K and 1000 K(Ttr5500 K, u in530°, TS5298 K!. The MD results of O2are shown by open symbols and the solid lines are the beusing Danckert’s distribution function defined as

f ~Erot,f !512k

Ew1expS 2

Erot,f

Ew1D1

k

Ew2x~AEp /Ew2!

3expS 2~AErot,f2AEp!2

Ew2D , ~30!

whereErot,f is the rotational energy after the first collision,k,Ew1 , Ep , and Ew2 are the fitting parameters andx is thenormalization function defined as

x~x!5exp~2x2!1Apx~11erf~x!!. ~31!

In Eq. ~30!, the first term is the Boltzmann distribution anthe second one is the non-Boltzmann distribution.

At low Erot , the distribution can be fitted to a combintion of the Boltzmann and the non-Boltzmann terms, buthigh Erot , the distribution is only fitted by the nonBoltzmann term. Thus, Eq.~30! will be taken as

f ~Erot,f !51

Ew2x~AEp /Ew2!expS 2

~AErot,f2AEp!2

Ew2D .

~32!

When Erot is low, Erot,f is dominated by the energtransfer from translation or from the vibration of the somolecules. From the data base obtained, it was revealed

FIG. 7. Rotational energy distribution of O2 after the first collision withinitial conditions ofTtr5500 K, u in530°, andTS5298 K. s denotesTrot

50 K andn denotesTrot51000 K; the solid curves are the best fits usinDanckert’s distribution function.


37.191.221.36 On: Sat, 1

n.

alf

l-

fit

r

at

Ew1 has a positive correlation withTS and a small positivecorrelation withEtr,n , whereEtr,n is the translational energyassociated with the normal velocity component of the indent gas molecule. On the other hand,Ep had a positivecorrelation with Etr,n and almost no correlation withTS .Also, both parameters had a small correlation withEtr,t , thetranslational energy associated with the tangential velocomponent. Since parameterEw1 represents the Boltzmanterm andEp represents the non-Boltzmann term, the Bolmann distribution stands as the energy transfer from theface and the non-Boltzmann distribution stands as the trafer from the translational degree of freedom.

WhenErot is high, parameterEp has a positive correlation with Erot and a small correlation withEtr,n and Erot .Thus, the non-Boltzmann distribution stands as the enetransfer from rotation to translation or the surface. A detaidescription on the correlation will be discussed in Sec. II

III. MODEL EQUATION

A simple analysis of the gas–surface interaction is cried out to derive a model equation. The system is set aRef. 22, but the analysis is extended to obtain an equawhich gives the relation betweenEtr , Erot , u in , TS , and^DEtr&.

The diatomic gas molecule is assumed to be an ellipswith incident velocityV5(Vn ,Vt), incident angular velocityv, and massM as in Fig. 8. The surface is assumed to behard cube with velocityv5(vn ,v t) and massm. The inci-dent and scattering angles areu i and uo , respectively. Thevelocity and angular velocity after the collision are indicatby 8. The system is two dimensional for thex(t) and y(n)direction. The conservation of energy, momentum, andgular momentum is taken as

12 Iv21 1

2 MV21 12 mv25 1

2 Iv821 12 MV821 1

2 mv82,~33!

MVn1mvn5MVn81mvn8 , ~34!

FIG. 8. Schematic of an ellipsoid hitting a hard cube as in Ref. 22. Tdiatomic gas molecule is assumed to be an ellipsoid with incident veloV5(Vn ,Vt), incident angular velocityv and massM . The surface is as-sumed to be a hard cube with velocityv5(vn ,v t) and massm. The incidentand scattering angles areu i anduo , respectively. The velocities and angulavelocity after the collision are indicated byV8, v8, and v8, respectively.The system is two-dimensional for thex(t) andy(n) direction.


0 May 2014 09:29:12

-l-

gc

ath

he

-

anD

nd

of

hethe

f

al-di-al

and

ofrd

nsrac-ter-cted

s tol isthe

theout-tialthe

eotethe


This a

MVt1mv t5MVt81mv t8 , ~35!

Iv1MVL5Iv81MV8L8, ~36!

where Eqs.~34! and ~35! are the normal and tangential momentum, respectively,I is the moment of inertia of gas moecule about its center of mass, andL and L8 represent, re-spectively, the moment arms of the center of mass of themolecule about the point of collision before and after impaEquation~33! can also take the form

Iv21M ~Vn21Vt

2!1m~vn21v t

2!

5Iv821M ~Vn821Vt8

2!1m~vn821v t8

2!, ~37!

as a sum of the tangential and normal component.If it is assumed that the energy change of the norm

translational energy and rotational energy is greater thanof the tangential translational energy, that is,

I ~v22v82!1M ~Vn22Vn8

2!1m~vn822vn8

2!

@M ~Vt822Vt

2!1m~v t822v t

2!, ~38!

then Eq.~37! can be simplified to

I ~v22v82!1M ~Vn22Vn8

2!1m~vn22vn8

2!50. ~39!

Equation~36! is

Iv1MlVn5Iv81MlVn8 , ~40!

as in Ref. 22, wherel is the tangential separation between tcenter of mass and the collision point.

If Eqs. ~34!, ~35!, and~40! are substituted into Eq.~39!,the normal velocity is taken to be

Vn85Vn~12m1m l 2/K !12m~vn1 lv!

11m1~m l 2/K !, ~41!

wherem5m/M and K5I /M . The equation can be simplified as

Vn85c1Vn1c2vn1c3v. ~42!

If it is assumed thatv t85aVt1bv t , the tangential velocity istaken to be

Vt85c4Vt1c5v t . ~43!

Thus, the translational energy after the collisionEtr, f shall be

Etr, f512 M ~Vn8

21Vt82!, ~44!

5 12 M $~c1Vn1c2vn1c3v!21~c4Vt1c5v t!

2%, ~45!

5ca12 MVn

21cb12 MVt

21cc12 Iv21cd

12 Mv21d,

~46!

5caEtr,n1cbEtr,t1ccErot1cdEs1d. ~47!

If it is assumed thatd is negligible when the mean ofEtr, f istaken,^Etr, f& is to be

^Etr, f&5caEtr,n1cbEtr,t1ccErot1cdEs . ~48!

Thus, the translational energy loss^DEtr& is to be

^DEtr&5~12ca!Etr,n1~12cb!Etr,t2ccErot2cdEs ,~49!


37.191.221.36 On: Sat, 1

ast.

lat

^DEtr&5cnEtr,n1ctEtr,t1crErot1csEs , ~50!

where cn512ca , ct512cb , cr52cc , cs52cd . Equa-tion ~50! shows that the linear sum of the initial energies cdescribe DEtr&, and if the assumptions are appropriate, Mresults should obey the equation.

Figure 9 is the relation between the initial condition a^DEtr& according to model equation~50!. Parameterscn

;cs were adjusted by singular value decomposition23 for thebest fit of MD results. Open symbols denote MD resultsO2, the error bars are the standard deviation of theDEtr

distribution, and the solid line is the model equation. TMD results obey the model equation and indicates thatassumptions above were appropriate.

The values of parameterscn;cs indicate the strength othe correlation of each energy, showing thatEtr,n has thestrongest correlation withDEtr& and it is relatively small forEtr,t . The strong bond inside the graphite layer createsmost no freedom for the carbon atoms in the tangentialrection. Thus most of the interaction occurs at the normdirection. The negative correlation withErot andEs indicatesenergy transfer between the rotational degree of freedomthe graphite surface.

The model equation was extended for the correlationparameterss(DEtr), Ep andEw2 and have obtained similaresults. The values ofcn;cs for each parameters were usefor the modeling, as shown in Sec. IV.

IV. MULTISTAGE GAS–SURFACE INTERACTIONMODEL

In this section, based on the analysis of MD simulatioand the model equation, our multistage gas–surface intetion model is presented. The present model is for the inaction of O2, N2, and Ar gas molecules and a clean perfegraphite surface. Since the MD calculations were performat the energy range mentioned earlier, the model belongthe thermal scattering regime. The basic idea of the modeto separate each collision into three stages. At stage 1,translational and rotational energy is determined bymodel equation and model parameters. At stage 2, theof-plane scattering direction is determined from the potenenergy surface. At stage 3, from the translational energy,

FIG. 9. Relation between the initial condition and^DEtr& according tomodel Eq. ~50!. Parameterscn;cs were adjusted by the singular valudecomposition method for the best fit of MD results. Open symbols denMD results of O2/graphite, the error bars are the standard deviation ofDEtr distribution, and the solid line is the model equation.


0 May 2014 09:29:12

Rfleanlepeetr

vex-

n

thdels

e

th

c-

annenite

-de-

’s


This a

molecule scatters, re-enters or is trapped to the surface.entering molecules go back to stage 1 and the diffuse retion is applied for trapped molecules. The proceduresparameters for rotation can be ignored for Ar gas molecu

Experiments were also carried out by scattering a susonic O2 molecular beam against a clean graphite surfacan ultrahigh vacuum chamber. The in-plane scattering disbution, out-of-plane scattering distribution, and in-planelocity distribution of the model were compared with the eperimental results.

A. Stage 1

First, the energy transfer stage determines the final tralational energyEtr* and the final rotational energyErot* . Thefinal translational energy is determined by estimatingtranslational energy lossDEtr . The energy loss is separateinto DEtr,n and DEtr,t , the components normal and parallto the surface. Thus, the mean translational energy lostaken to be

^DEtr,n&5cnEtr,n1crErot1csEs , ~51!

^DEtr,t&5ctEtr,t , ~52!

where Es51.5kBTS , Etr,n5Etr cos2 uin and Etr,t

5Etr sin2 uin . Such separation can set the normal componto receive the energy transfer from the rotational degreefreedom. The standard deviations of DEtr also can be ex-pressed by a similar equation;s(DEtr) is taken to be

s~DEtr!5cnEtr,n1ctEtr,t1crErot1csEs . ~53!

s(DEtr) is not separated, with both components usingsame values. The translational energy loss of the gas mecule is taken to be

DEtr,n5 f G~^DEtr,n&,s~DEtr!2!, ~54!

DEtr,t5 f G~^DEtr,t&,s~DEtr!2!, ~55!

where f G(m,s2) gives a random Gaussian distribution acording to meanm and variances2. Thus,Etr* is taken to be

Etr,n* 5Etr,n2DEtr,n , ~56!

Etr,t* 5Etr,t2DEtr,t . ~57!

The velocity vector of the gas moleculeva5(ua ,va ,wa) istaken to be

ua5&Etr,t*

A4 mg2~Etr,n* 21Etr,t* 2!

, ~58!

va50, ~59!

wa5&Etr,n*


. ~60!

The present stage will determine the ratio of the normaltangential velocity component, which is only the in-plavelocity vector at this point. The effects of the potential eergy surface at stage 2 will distribute the tangential velocto the x and y directions. The model parameters for thpresent procedure are listed in Tables I and II.


37.191.221.36 On: Sat, 1

e-c-ds.r-ini--

s-

e

is

ntof

eol-

d

-y

Next, the final rotational energy follows the similar procedure except that two different distributions are givenpending on parameterEp , which is taken to be

Ep5cnEtr,n1ctEtr,t1crErot1csEs . ~61!

If Ep<0K, Erot* is determined according to Danckertdistribution function which is taken to be

f ~Erot* !512k

Ew1expS 2

Erot*

Ew1D 1

k

Ew2x~AEp /Ew2!

3expS 2~AErot* 2AEp!2

Ew2D , ~62!

where

x~x!5exp~2x2!1Apx~11erf~x!!. ~63!

Thus,Erot* is taken to be

TABLE I. MS model parameters for stage 1:^DEtr&.

Parameter O2 /graphite N2 /graphite Ar/graphite

cn 0.611 0.287 0.300ct 0.008 0.036 20.004cr 20.051 20.072 –cs 20.035 0.127 20.100

TABLE II. MS model parameters for stage 1:s(DEtr).


cn 0.243 0.278 0.194ct 0.111 0.055 0.069cr 0.088 0.073 -cs 0.315 0.196 20.177

TABLE IV. MS model parameters for stage 1:Ep .


cn 20.086 20.013 –ct 20.010 0.015 –cr 0.772 0.871 –cs 20.030 0.029 –

TABLE V. MS model parameters for stage 1:Ew2 .


cn 0.097 0.025 –ct 0.044 0.005 –cr 20.022 0.023 –cs 0.181 0.043 –

TABLE III. MS model parameters for stage 1:Ep<0.


k 0.241 0.416 –Ew1 (ER) 0.194•1023 0.404•1024 –Ep (ER) 0.925•1023 0.429•1023 –Ew2 (ER) 0.596•1024 0.195•1023 –


0 May 2014 09:29:12

ytio

-is

idoc

pop

l

tor

rec-

uleeter-

thees.ve

ota-s the

nsmol-

gi


This a

Erot* 5 f D1~k,Ew1 ,Ep ,Ew2!, ~64!

wheref D1() gives a random value according to Eq.~62!. Thefour parametersk, Ew1 , Ep , andEw2 at the present energrange were almost constant and had no obvious correlaThus the values are kept constant as in Table III.

If Ep.0 K, Erot* is determined according to the nonBoltzmann term of Danckert’s distribution function whichtaken to be

f ~Erot* !51

Ew2x~AEp /Ew2!expS 2

~AErot* 2AEp!2

Ew2D ,

~65!

where parameterEp is determined from Eq.~61! andEw2 istaken to be

Ew25cnEtr,n1ctEtr,t1crErot1csEs . ~66!

Thus,Erot* is taken to be

Erot* 5 f D2~Ep ,Ew2!, ~67!

wheref D2() gives a random value according to Eq.~65!. Themodel parameters forEp andEw2 are listed in Tables IV andV.

B. Stage 2

Next, the effect of the potential energy surface is consered to determine the out-of-plane scattering. Stage 2 dnot interfere with stage 1, since the gas molecule reflespecularly, where the velocity component parallel to thetential energy surface is conserved and the normal comnent reversed. PES is simplified byz5L(x,y) which istaken to be

L~x,y!5a1

2 H cosS 2p

)LyD 21J

1a2

2 H 12cosS 2p

LxD J 1a3 , ~68!

where parametersa1 , a2 , and a3 are set to fit the actuapotential energy surface,L51.42 Å and (x,y) is the colli-sion point in a unit cell~Fig. 10!.

FIG. 10. Schematic of model potential energy surface of the shaded rein Fig. 2. The axes also correspond to those of Fig. 2.


37.191.221.36 On: Sat, 1

n.

-ests-o-

First, the impact point (xb ,yb) is chosen randomly fromthe unit graphite cell as in Fig. 2. The surface normal vecn can be taken as

n521

A11S ]L

]xD 2

1S ]L

]yD 2F ]L

]x]L

]y21

G . ~69!

Since the lattice pattern is random against the incident dition, thexy direction ofn is rotated by 2pU. The specularreflection gives the relations taken to be

uvinu5uvbu, ~70!

vin•n1vb•n50, ~71!

wherevin is the incident velocity andvb is the reflected ve-locity. Thus,vb is to be

vb5vin22~n•vin!n. ~72!

Finally, the tangential velocity is distributed to thex and ydirections and the final velocity vector vout

5(uout,vout,wout) is to be

uout5&Etr,t*


ub

Aub21vb

2, ~73!

vout5&Etr,t*


vb

Aub21vb

2, ~74!

wout5&Etr,n*


. ~75!

The model parameters forz1 , z2 , andz3 are listed in TableVI.

C. Stage 3

At stage 3, from the translational energy, the molecscatters, re-enters or is trapped to the surface. This is dmined from the value ofEtr,n* and Etr,t* . If W1<Etr,n* andW1<Etr,t* , the gas molecule scatters, and in this casevelocity and rotational energy at stage 2 are the final valu

If the gas molecule does not satisfy the condition aboand W2<Etr,n* 1Etr,t* 1Erot and collision numberNcoll<10,the gas molecule re-enters to the surface. The velocity, rtional energy, and scattering angle at stage 2 are used ainitial values with the process returning to stage 1.

If the gas molecule does not satisfy any conditioabove, the gas molecule is determined as a physisorbed

on

TABLE VI. MS model parameters for stage 2.


z1 (Å) 1.815 2.466 2.376z2 (Å) 2.046 2.570 2.497z3 (Å) 2.021 2.561 2.485


0 May 2014 09:29:12

in

i-rehet-

-

.tuty

ithpe-ttndbeoff

oleent

2.3

tor.ci-

e

ringthendd

thatcu-wide

e-s

aialondplenterr isby

s a

eadTo-cehere

a

glerface


This a

ecule and the diffuse reflection for temperatureTS is applied.Model parametersW1 and W2 are determined from theLennard-Jones parameter«LJ and physisorption well depthWP , respectively. MD analysis has shown that scattermolecules have a potential energy of 2«LJ on average. ThusW152«LJ andW25WP ~see Table VII!.

Finally, to maintain equilibrium when a stationary equlibrium gas interacts with the solid surface, the diffuseflection is applied to all reflecting molecules inside tDSMC calculation cell when the following condition is saisfied:

Etr1Erot,Etr* 1Erot* . ~76!

Here,Etr is the mean incident translational energy,Erot is the

mean incident rotational energy,Etr* is the mean final trans

lational energy, andErot* is the mean final rotational energySuch condition is applied to the model because the strucof the model will not permit satisfaction to the reciprocicondition.

D. Comparison with molecular beam experiment

The verfication of the model is done by comparison wmolecular beam experiments. Such experiments areformed in the ultrahigh vacuum~UHV! molecular beam scattering system, constructed to study molecule–surface scaing processes and photoexcited surface reactions uultrahigh vacuum conditions, whose details are descrielsewhere.24 Figure 11 illustrates the schematic top viewthe molecular beam apparatus. This apparatus consists o~a!

FIG. 11. Schematic top view of the molecular beam–surface scatteringparatus. QMF denote a quadrupole mass filter.

TABLE VII. MS model parameters for stage 3.


W1 (eV) 0.0131 0.00718 0.0112W2(eV) 0.0897 0.10400 0.0787


37.191.221.36 On: Sat, 1

g

-

re

r-

er-erd

a main chamber,~b! a 3-axis manipulator,~c! a triply differ-entially pumped molecular beam source,~d! a triply differ-entially pumped rotatory detector chamber with a quadrupmass spectrometer equipped with an electron bombardmionizer. The base pressure of the main chamber is31029 Torr.

The sample is mounted on a 3-axis manipula~Musashino Engineering! in the center of the UHV chamberThe manipulator was designed to have control over the indent angle, the scattering angle~measured with respect to thsurface normal!, and the azimuthal angle~Fig. 12!. The azi-muthal angle is defined as the angle between the scattedirection over the surface and the plane of incidence ofmolecule~in-plane!~the plane through the surface normal athe incident direction of the molecule!. The beam source anthe detector chambers were made as slim as possible sothe angle subtended by the directions of the incident molelar beam and scattered beam to be detected spans aangle range from 0° to 145°.

To determine the velocity distributions of scattered spcies the time-of-flight~TOF! measurement technique waemployed. The continuous supersonic O2 molecular beam ismodulated in the first differential pumping chamber usingrotating chopper disk with four symmetrically spaced radslits. The pulsed molecular beam goes through the secdifferential pumping chamber and then enters the samchamber. The distance from the chopper to the sample ceis 123 mm and that from the sample center to the ionize353 mm. The scattering angle distributions were obtainedcounting the ions at each angle.

The crystalline graphite sample which was used asubstrate is high-quality, highly oriented graphite~Commer-cial name: Super Graphite, MB grade with a mosaic sprof 0.6°, purchased from Matsushita Research Institute,kyo, Inc.!. The topmost layer of the graphite crystal surfawas peeled off using Scotch tape in the ambient atmosp

p-

FIG. 12. Definition of in-plane and out-of-plane angles. The azimuthal anf f is defined as the angle between the scattering direction over the suand the plane of incidence of the molecule.

FIG. 13. Time of flight distribution of incident molecular beam of O2. Dotsare the time of flight data and the solid curve is the best fit.


0 May 2014 09:29:12

ring

othnhet-

igcoh

ethed

din

t

otic

buc

i-n

oat

n-e

echying

n-enre

ules

atngles

at-ityri-g

elvesastSs at

o


This a

before inserting it into the sample chamber. Before scatteexperiments the sample was cleaned by resistive heatin773 K.

The incident translational energyEti , angle of the inci-dent beamu i and the substrate temperatureTS is 0.09 eV,35° or 60° and 303 K, respectively. The TOF distributionincident molecular beam, as in Fig. 13, was transferred tovelocity distribution and was directly used as the incidevelocity of the gas molecule for model calculations. Also, tincident rotational energyEri was assumed to obey the Bolzmann distribution taken to be

f ~Eri !51

kBTriexpS 2

Eri

kBTriD , ~77!

whereTri was estimated to be 50 K.The TOF distribution of scattered molecules, as in F

14, has been analyzed assuming that each spectrum isposed of a linear sum of two velocity components, tshifted Maxwell–Boltzmann~MB! distribution@eq.~78!# anda MB distribution@eq. ~79!#. The shifted-MB distribution isassumed to have the form

f i~v !5Cv3 expS 2mg

2RTi~v2ui !

2D , ~78!

whereui , Ti , andC are the drift velocity, the temperaturrepresenting the velocity spread of the distribution, andnormalization constant. The second component is assumobey a MB form

f i~v !5Cv3 expS 2mgv2

2RTiD , ~79!

whereTi takes the value of the surface temperatureTS . Thisis so because the first component exhibits a rather sharptribution and must be due to direct inelastic scatterwhereas the slowest component may be interpreted asmolecules scattered via trapping-desorption.25 The compari-son with the model was done using only the distributionthe first component for the varification of the direct inelasscattering of the model.

Figures 15 and 16 are the in-plane scattering distrition, out-of-plane scattering distribution, and in-plane veloity distribution for u in535° and 60°, respectively. The incdent energies were set as mentioned above. As showeach figure, results from the CLL model3 were also included.CLL model calculations require the values of the accommdation coefficient for kinetic energy associated with normvelocity component (an) and tangential velocity componen

FIG. 14. Time of flight distribution of scattered O2. Dots are the time offlight data and the solid curve is the best fit by the linear combinationEqs.~78! ~chain curve! and ~79! ~dashed curve!.


37.191.221.36 On: Sat, 1

gto

fet

.m-

e

eto

is-ghe

f

--

in

-l

(a t), and the accommodation coefficient for rotational eergy (a r). a was varied from 0.0 to 1.0 by 0.1 under threstriction ofan5a t or an1a t51. Also,a r was kept equalto an . The value ofa r does not have any effect on thscattering distribution or the velocity distribution, since eadegree of freedom is treated separately. The most satisfresults were obtained froman50.9, a t50.1, anda r50.9.

The top figure illustrates the polar plot of the flux intesity where the filled symbol is from the model and the opsymbol is from the experiment. Experimental results weobtained by plotting the flux intensity for angles atf f50°,whereas model results were obtained by sampling molecscattering towards the in-plane with a sampling area of65°.The middle figure is the flux intensity of the out-of-planeu f535° or 65°. The model results were obtained by dividithe whole sampling area into cells with equal solid angand using the ones atu f535° or 65°. The bottom figure isthe velocity distribution of the molecules which were sctered towards the in-plane. Experimental in-plane velocdistributions were obtained by summing the velocity distbution of angles withf f50° according to the scatterinintensity.

The in-plane scattering distribution of the MS modagrees well with experiments whereas the CLL model gia distribution shifted towards the surface normal by at le10°. The out-of-plane scattering distribution of the Mmodel agrees with experiments although it overestimate

f

FIG. 15. In-plane scattering distribution~top!, out-of-plane scattering~middle! distribution, and in-plane velocity distribution~bottom! for initialconditions ofEti50.09 eV, u i535° andTS5303 K. Eri was assumed toobey the Boltzmann distribution withTri 550 K. s: experiment; solid line:MS model, dashed line: CLL model~an50.9, a t50.1, a r50.9!.


0 May 2014 09:29:12

edinmate

famens

saa

-ioauee

acee ata-Fig.ith

naltheis

ol-

is-heell

eldas

ceandannly

aceon

ans-ero-ofu-as

ob-

.

he

les

-


This a

f f.30° of case 1. The in-plane velocity distribution of thMS model agrees with experiments whereas the CLL mooverestimates and gives a shifted broader distribution. Sthe CLL model does not give sufficient normal momentuloss, the velocity distribution is overestimated and the sctering distribution shifts towards the surface normal. A betvelocity distribution can be given by adjustingan and a t ,but the shift of the scattering distribution will remain.

The experiment includes uncertainties such as surirregularity26 and it is believed that the slight difference frothe MS model occurs for this reason. A better agreembetween the two can be obtained by performing experimewhich do not include such uncertainties, but our purpohere was to demonstrate that the multiple-scale analysigas-surface interaction will lead to the construction of reistic gas–surface interaction models. A more accurate qutitative comparison of the O2, N2, and Ar–graphite interaction must await experiments which are closer to simulatconditions; i.e., no surface irregularity and a molecular bewith well defined translational and rotational energy. Boverall it can be said that the MS model gives good agrment with experimental results.

FIG. 16. In-plane scattering distribution~top!, out-of-plane scattering~middle! distribution, and in-plane velocity distribution~bottom! with simi-lar conditions as in Fig. 15, except thatu i560°. Symbols are as in Fig. 15

FIG. 17. Schematic of the calculation domain for DSMC simulation. Tgas is surrounded by test boundary walls with 1mm separation.


37.191.221.36 On: Sat, 1

elce

t-r

ce

nttseofl-n-

nmt-

E. Application to the DSMC simulation „gas–surfacethermal equilibrium …

One of the most important properties that a gas–surfinteraction model must have is to satisfy detailed balancequilibrium. Our MS model is applied to the DSMC simultion of a gas surrounded by wall boundaries, as shown in17. The DSMC calculation domain is one-dimensional wthe separation of wall boundaries set at 1mm. The wall sur-face is graphite at 300 K, the gas molecule is O2 and theKnudsen number is 0.07. The translational and rotatiotemperature of the gas molecule is given according toMB distribution at 300 K. The molecular collision processdetermined by the null-collision technique.27 The frequentlyused combination of the Larsen–Borgnakke~LB! model28

and the variable hard sphere model1 are employed for thecollision between diatomic gas molecules. The inelastic clision probability was set at 0.4 for the LB model.

Figure 18 is the translational and rotational energy dtribution of gas molecules inside the calculation cell at tborder of both wall boundaries. Both distributions agree wwith the MB distribution at 300 K, showing that our modwill maintain equilibrium. Similar results were obtainewhen the system was set at 500 K or when the initial gtemperature was set higher than the wall temperature.

V. CONCLUSION

A multistage gas–surface interaction model for an O2,N2, and Ar gas molecule interacting with a graphite surfawas presented, based on the analysis of MD simulationsa model equation derived from the classical theory ofellipsoid hitting a hard cube. The present model was maideveloped for the use in DSMC simulations.

MD analysis has shown that the potential energy surfis relatively stable despite the thermal vibration of the carbatoms and can be modeled by the cosine function. The trlational energy distribution after the first collision can bfitted by the Gaussian distribution function, whereas thetational energy distribution can be fitted by a combinationthe Boltzmann distribution and the non-Boltzmann distribtion. A simple analysis of the gas–surface interaction wcarried out to derive a model equation. The equation

FIG. 18. Translational and rotational energy distribution of gas molecuinside the calculation cell at the border of both wall boundaries.s: transla-tional energy;h: rotational energy; solid curve: Maxwell–Boltzmann distribution.


0 May 2014 09:29:12

ow

ercor

andA

cura

nHVnof

ulno

plede

oano

feu-dan

n

as

ac

to

fac

s

i-oc

i-jec-

on

ofs,’’

ys.

ics

es

or

er-

n

of

l

the

en

m

g,

T.

s

. J.of

enem.

o

nte


This a

tained gave the correlation between the initial conditionthe system and the mean translational energy loss andextended for the estimation of other distribution paramet

The basic idea of the MS model is to separate eachlision into three stages. At stage 1, the translational andtational energy is determined by the model equationmodel parameters. At stage 2, the out-of-plane scatteringrection is determined from the potential energy surface.stage 3, depending on the translational energy, the molescatters, re-enters or is trapped to the surface. Re-entemolecules return to stage 1, and the diffuse reflection isplied for trapped molecules.

Experiments were carried out by scattering a supersoO2 molecular beam from a clean graphite surface in an Uchamber. The in-plane scattering distribution, out-of-plascattering distribution, and in-plane velocity distributionthe model were compared with the experiment. The Mmodel obtained good agreement with experimental resshowing that the model can reproduce such phenomewith high accuracy. The quantitative agreement with mlecular beam experiments clearly shows that such multiscale analysis can lead to the development of realistic moof the gas–surface interaction.

The model was also applied to the DSMC simulationa gas surrounded by wall boundaries. The translationalrotational distribution agreed well with the MB distributioat wall temperature, showing that the model is capablemaintaining thermal equilibrium.

ACKNOWLEDGMENTS

The authors wish to express their appreciation to Prosor J. Matsui of Yokohama National University for his valable suggestions and comments. This study was supporteGrants of the Japanese Ministry of Education, Science,Culture ~07555060 and 08004026!. One of the authors~N.Y.! is supported by a JSPS Fellowship for Japanese JuScientists.

1G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of GFlows ~Clarendon, Oxford, 1994!.

2C. Cercignani and M. Lampis, ‘‘Kinetic models for gas–surface intertions,’’ Transp. Theory Stat. Phys.1, 2, 101~1971!.

3R. G. Lord, ‘‘Application of the Cercignani–Lampis scattering kerneldirect simulation Monte Carlo simulations,’’ inRarefied Gas Dynamics,edited by A. E. Beylich~VCH, Weinheim, 1991!, p. 1427.

4R. G. Lord, ‘‘Some extensions to the Cercignani–Lampis gas–surscattering kernel,’’ Phys. Fluids A3, 706 ~1991!.

5R. G. Lord, ‘‘Some further extensions of the Cercignani–Lampis gasurface interaction model,’’ Phys. Fluids7, 1159~1995!.

6K. Koura, ‘‘Monte Carlo direct simulation of rotational relaxation of datomic molecules using classical trajectory calculations: Nitrogen shwave,’’ Phys. Fluids9, 3543~1997!.


37.191.221.36 On: Sat, 1

fass.l-

o-di-tle

ingp-

ic

e

Stson--ls

fnd

f

s-

byd

ior

-

e

–

k

7K. Koura, ‘‘Monte Carlo direct simulation of rotational relaxation of ntrogen through high total temperature shock waves using classical tratory calculations,’’ Phys. Fluids10, 2689~1998!.

8K. Yamamoto and K. Yamashita, ‘‘Analysis of the Couette flow basedthe molecular dynamics study for gas-wall interaction,’’ inRarefied GasDynamics 20, edited by C. Shen~Peking University Press, Beijing, 1997!,p. 375.

9I. D. Boyd, D. B. Bose, and G. V. Candler, ‘‘Monte Carlo modelingnitric oxide formation based on quasiclassical trajectory calculationPhys. Fluids9, 1162~1997!.

10T. Tokumasu and Y. Matsumoto, ‘‘Dynamic molecular collision~DMC!model for rarefied gas flow simulations by the DSMC method,’’ PhFluids 11, 1907~1999!.

11Y. Matsumoto, N. Yamanishi, and K. Shobatake, ‘‘Molecular dynamsimulation of O2 scattering from a graphite surface,’’ inRarefied GasDynamics 19, edited by J. Harvey and R. G. Lord~Oxford UniversityPress, Oxford, 1995!, Vol. 1, p. 995.

12N. Yamanishi and Y. Matsumoto, ‘‘A new model for diatomic moleculscattering from solid surfaces,’’ inRarefied Gas Dynamics 20, edited byC. Shen~Peking University Press, Beijing, 1997!, p. 381.

13N. Yamanishi and Y. Matsumoto, ‘‘The multistage reflection model fDSMC calculations,’’ inRarefied Gas Dynamics 21~in press!.

14F. O. Goodman and H. Y. Wachman,Dynamics of Gas–Surface Scatter-ing ~Academic, New York, 1976!.

15A. Danckert, ‘‘A reciprocal scattering kernel for molecule–surface intaction including rotational energy,’’ inRarefied Gas Dynamics 20, editedby C. Shen~Peking University Press, Beijing, 1997!, p. 404.

16S. Beran, J. Dubsky´, and Z. Slanina, ‘‘Quantum chemical study of oxygeadsorption on graphite,’’ Surf. Sci.79, 39 ~1979!.

17G. Herzberg,Molecular Spectra and Molecular Structure, I. SpectraDiatomic Molecules~Krieger, Florida, 1950!.

18M. P. Allen and D. J. Tildesley,Computer Simulation of Liquids~Claren-don, Oxford, 1987!.

19G. Vidali, G. Ihm, H.-Y. Kim, and M. W. Cole, ‘‘Potentials of physicaadsorption,’’ Surf. Sci. Rep.12, 133 ~1991!.

20M. W. Cole and J. R. Klein, ‘‘The interaction between noble gases andbasal plane surface of graphite,’’ Surf. Sci.124, 547 ~1983!.

21L. K. Cohen, ‘‘A lower bound on the loss of graphite by atomic oxygattack at asymptotic energy,’’ J. Chem. Phys.99, 9652~1993!.

22W. L. Nichols and J. H. Weare, ‘‘Homonuclear diatomic scattering frosolid surfaces: A hard-cube model,’’ J. Chem. Phys.62, 3754~1975!.

23W. H. Press, B. P. Flannery, S. A. Teukoslsky, and W. T. VetterlinNumerical Recipes in C~Cambridge University Press, Cambridge, 1988!.

24K. Shobatake, K. Ito, H. Yoshikawa, T. Ogi, H. Ariga, H. Ohashi, andFujimoto, ‘‘Dynamics and energy transfer in the scattering of Xe, O2, andCl2 from the graphite surface,’’ inElementary Processes in Excitationand Reactions on Solid Surfaces, edited by A. Okiji, H. Kasai, and K.Makoshi ~Springer, Tokyo, 1996!, p. 112.

25J. E. Hurst, C. A. Becker, J. P. Cowin, K. C. Janda, L. Wharton, and DAuerbach, ‘‘Observation of direct inelastic scattering in the presencetrapping-desorption scattering: Xe on Pt~111!,’’ Phys. Rev. Lett.43, 1175~1979!.

26P. Pfeifer and D. Avnir, ‘‘Chemistry in noninteger dimensions betwetwo and three. I. Fractal theory of heterogeneous surfaces,’’ J. ChPhys.79, 3558~1983!.

27K. Koura, ‘‘Null-collision technique in the direct-simulation Monte Carlmethod,’’ Phys. Fluids29, 3509~1986!.

28C. Borgnakke and P. S. Larsen, ‘‘Statistical collision model for the MoCarlo simulation of the polyatomic gas mixture,’’ J. Comput. Phys.18,405 ~1975!.


0 May 2014 09:29:12

multistage gas–surface interaction model for the direct simulation monte carlo method

Documents