Journal of Molecular Graphics and Modelling

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1 Joual of Molecula Gaphics ad Modellig 38 (2012) Cotets lists available at SciVese ScieceDiect Joual of Molecula Gaphics ad Modellig j oua l h o me p age: New statistical bouday coditios fo ago tugste iteactios M.S. Ozhgibesov a,, T.S. Leu a, C.H. Cheg a, A.V. Uti b a Depatmet of Aeoautics ad Astoautics, Natioal Cheg Kug Uivesity, Taia 701, Taiwa, ROC b Khistiaovich Istitute of Theoetical ad Applied Mechaics SBRAS, Novosibis, Russia a t i c l e i f o Aticle histoy: Accepted 26 Jue 2012 Available olie 7 July 2012 Keywods: Molecula dyamics Maxwell distibutio Ago tugste iteactios Mose potetial Statistical aalysis Bouday coditios a b s t a c t I this study, scatteig pocesses of ago beam impigig o tugste suface ae ivestigated umeically by applyig molecula dyamics (MD) simulatios. Eegy tasfe, mometum chage, ad scatteig pocesses of ago gas atoms fom W(1 1 0) suface ae discussed. A ew model of ago tugste (A W) iteactio is poposed. Based o the ew poposed model, oe ca simplify the bouday coditios of this poblem. The ew bouday coditios ae poved to be i lie with pevious expeimetal ad theoetical esults. This pape demostates how to poceed omalizatio ad futhe covesio of the MD simulatio esults ito bouday coditios. Applicatio of the ew poposed bouday coditios fo A W iteactios povides a sigificat speedup of computatios. Cow Copyight 2012 Published by Elsevie Ic. All ights eseved. 1. Itoductio Nowadays a lot of mico/ao systems become a itegal pat of ou daily life, fo example: gavity seso o mico gyoscope ae applied i may hadheld electoic gadgets. Recet techological advaces allowed oe to ceate mico/ao devices with chaacteistic size of seveal aometes. The desig pocesses of such mico/ao devices ae vey expesive ad complicated ad ivolve both expeimetal ad umeical studies of solid ad fluid mechaics iside the device. It should be oted that mico/ao flow is chaacteized by a high degee of aefactio. It meas that assumptio of classical cotiuum theoy caot be applied fo aalysis of mico/ao flow. Fotuately thee is a umeical method, called molecula dyamics (MD) simulatio method. It is based o the fudametal piciples of mechaics ad allows oe to simulate pocesses o molecula level, i.e., the simulated system is cosideed as esemble of molecules, athe tha a cotiuous medium. Oe of the most challegig pats of umeical simulatios of a mico/ao flow is bouday coditios o the solid liquid iteface. The choice of bouday coditios (BC) iflueces o a flow sigificatly [1]. The icoect BC might cause the wog estimatio of fluid behavios i a mico/ao system. Thee ae two ways to set up the BCs fo the MD simulatios: (1) to epeset the solid pats of the simulated system by molecula stuctues ad coside iteactios betwee fluid molecules ad solid bouday molecules; (2) to Coespodig autho. Tel.: x63638; fax: addess: omise@gmail.com (M.S. Ozhgibesov). use elatioships which state depedece betwee paametes of fluid molecule impiged o the solid bouday ad the scatteed oe. The fist appoach is the most compehesive oe, but oe should ealize it also iceases the umbe of simulated molecules sigificatly. Cosequetly the time of computatios ises as well. The secod way is widely used due to its elative simplicity. The most commoly used model [1 3] is specula-diffusive model poposed by Maxwell [4] i his studies of gas suface iteactios. The ey poit of Maxwell s BC is that a gas molecule impigig o a suface is scatteed ito two factios, oe that eflects speculaly ad exchages o eegy ad the othe that accommodates completely ad desobs with a equilibium distibutio coespodig to the suface tempeatue. Maxwell s BC was extesively used fo studies of gas flow though micochaels. Recetly we have show [5] that this specula-diffusive model is too ough to epoduce all the pocesses accompay the iteactios of ago molecules with tugste substate. The eegy tasfe ad othe pocesses accompayig the scatteig of aefied gases fom solid sufaces have bee the subject of a seies of studies. Weibeg ad Meill [6] detemied agula distibutios fo gas atoms scatteed by a sigle-cystal W(1 1 0) suface. The expeimetal esults of Jada et al. [7] allowed the eseaches to elate the aveage ietic eegy of scatteed ago atoms to the suface tempeatue, as well as to the icidet ietic eegy. The theoetical explaatios fo ago atoms scatteig fom a self-assembled moolaye o Ag(1 1 1) have bee poposed ecetly by Fa ad Maso [8]. Futhemoe, Gibso et al. [9] coducted a detailed study of A scatteig fom a odeed 1- decaethiol Au(1 1 1) moolaye. Recetly, Chase et al. [16] have coducted expeimetal ad molecula dyamics studies of ago /$ see fot matte. Cow Copyight 2012 Published by Elsevie Ic. All ights eseved.

2 376 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) scatteig fom liquid idium. They have show how the agula ad eegy distibutios of scatteed atoms deped o icidet eegy. Iapplicability of the simple had-sphee model fo the desciptio of gas suface iteactios is also peseted. While thee ae may publicatios elated to gas suface iteactios, thei esults ae still isufficiet to defie bouday coditios that ca descibe the gas flow i mico/ao systems. The best way to study gas suface iteactios is to coduct a expeimet, but oe should ealize that accuate measuemets of gas suface iteactios o a micoscopic level ae vey expesive ad time cosumig. Fotuately, the ecet achievemets i compute sciece ad umeical methods made it possible to ivestigate such pocesses usig MD simulatio method. I this pape, MD method was applied to study the ago gas scatteig pocesses o a W(1 1 0) suface. This appoach made it possible to pecisely descibe the iteactio betwee ago gas ad tugste. The aims of this wo wee to study effects of ago scatteig o the tugste suface ad to popose bouday coditios descibig coelatios betwee the paametes of icidet ad scatteed atoms. The method applied i the peset pape ca be simply expessed as the bombadmet of a tugste suface with ago atoms, whee futhe aalysis of the scatteed atoms tajectoies was coducted. Aalysis of both agula distibutios ad distibutios of velocities of scatteed atoms wee pefomed usig mea values ad oot mea squae deviatios (RMSDs). The combiatios of these paametes povide complete ifomatio about pocess of gas atoms scatteig pocess. It is show that esults of cuet study ae i lie with expeimetal ad theoetical esults obtaied by the othe eseaches. The ifomatio obtaied i simulatios was statistically aalyzed ad epeseted by polyomial fuctios of icidet eegy ad agle of icidece. All the fuctios that state elatioship betwee paametes of impigig gas atoms ad scatteed atoms have bee obtaied usig the Least Squaes Method (LSqM). As a coclusive step of the wo, we have poposed a algoithm descibig a implemetatio of the elatios metioed above to a eal study of gas flow with tugste bouday. These elatios ca be used to specify bouday coditios fo ago tugste iteactios. The mai esult of this wo is poposed ew bouday coditios which ae able to epoduce mechaism of ago tugste iteactios ad allow oe to sigificatly educe computatio time equied fo the studies of gas flow aoud metal suface. 2. Methodology ad computatio The physical system ivestigated i this study cosisted of tugste W(1 1 0) substate with a tempeatue of T suf ad ago atoms with a iitial velocity vecto magitude V i ad velocity vecto diectio detemied by the azimuthal (i hoizotal plae) ad pola (i vetical plae) agles i ad ˇi, espectively. The simulatio pocedue cosisted of a substate bombadmet of ago atoms, afte which the scatteed atoms paametes V s, s, ad ˇs wee detemied, as show i Fig. 1. Cuet eseach was pefomed umeically by usig MD simulatio method. All algoithms descibed i this study wee implemeted usig Fota code developed by the authos of this pape. I ode to evaluate the gas scatteig effects, diffeet collimated beams of ago atoms with itesity of Pa ae set up to impige o the (1 1 0) face of a tugste cystal. The iteactios amog tugste atoms wee tae as beig sums of paiwise Mose potetial: D W [e 2B W( R W ) 2e B W( R W ) ], 0 < < 2.3R W W W, (1) 0, 2.3R W Fig. 1. Coodiate system, whee i ad ˇi ae the icidet azimuthal ad pola agles, espectively; s ad ˇs ae the scatteed atom s azimuthal ad pola agles, espectively; S i ae the azimuthal agle chage caused by the scatteig of gas atom; V i ad V i ae the omal ad tagetial velocity compoets of the icidet atom; V S ad V S ae the omal ad tagetial velocity compoets of the scatteed atom. whee the potetial s paametes [10] wee: D W ev, B W m 1, R W m. To descibe the iteactio betwee both ago tugste ad ago ago atoms, Lead Joes potetial fuctios ae applied ad depicted as followig: [ ( ) 12 ( ) 6 ] RWA RWA 4ε WA, 0 < < 2.5R WA W A, (2a) 0, 2.5R WA [ ( RA ) 12 ( RA ) 6 ] 4ε A, 0 < < 2.5R A A, (2b) 0, 2.5R A whee the paametes values used fo the case of ago tugste ad ago ago, espectively, wee: ε WA / B K, R WA 2.93 Å ad ε A / B K, R A 3.4 Å. The iitial lateal positio fo the impigig atom was selected adomly o a plae 17 Å above the aveage positio of atoms of the uppemost solid laye. I ode to model the scatteig of a velocity-selected, collimated beam, the iitial mometum of the icidet atom was tae to be the same fo each tajectoy of a give set. The secod ode velocity Velet scheme [10,11] with time step of t s (smalle tha the chaacteistic time of atom iteactios) was used fo a itegatio of equatios of motio. The computatioal pocess was cotiued util all the gas atoms wee withi the foce field of the tugste atoms. Scatteed gas atoms that wet beyod this distace wee excluded fom the system (ago atoms did ot pass though the tugste substate), ad ifomatio o thei velocities ad coodiates wee stoed i a file. The exclusio of these scatteed atoms educes the time equied fo the calculatio ad pevets the adomizatio of velocities of ago atoms as a cosequece of thei collisio with each othe. 3. Results ad discussio This study was pefomed fo a wide age of iitial paametes of the impigig beam ad metal suface as well. Total of 1920 cases wee computed. The cuet ivestigatios wee coducted fo vaious agles of icidets (fom ˇi 0 70 with 5 steps), a seies of suface tempeatues (T suf 350 K, 400 K, 450 K, 500 K), ad vaied velocities of impigig A atoms (fom V i m/s with 50 m/s steps). Ay aalytical o umeical study based o mathematical desciptios has assumptios ad simplificatios. Cosequetly,

3 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) Fig. 2. Mea eegy of scatteed atoms vesus eegy of impigig atoms at ˇi 45. esults of such wos must be compaed with elated expeimetal esults i ode to pove the easoability of assumptios. Fig. 2 shows the coelatio of the mea ietic eegies of the impigig atoms (iitial eegy of the beam) with the mea ietic eegy of the beam scatteed (aveage eegy of scatteed atoms) by the tugste suface. Solid lie i Fig. 2 is the esult fom Jada expeimets [7] ad blac coss symbols epeset MD esults obtaied i the cuet study. Both esults show the liea elatioship betwee the mea ietic eegies of the impigig atoms ad mea ietic eegy of the scatteed atoms whe the agle of the icidets was ˇi 45. The liea elatioship fom MD esults ca be fitted by the followig fuctio: E s b B E i + b S (2 B T suf ), (3) whee B is the Boltzma s costat, ad b S 0.18 ad b B 0.77 ae the popotioality factos. The diffeece betwee the Jada expeimetal data ad the MD simulatio esults does ot exceed 9%. This diffeece ca be explaied by the fact that eve the best MD models ae a ough appoximatio of eal pocesses. Fo example, the simulated substate was ideal, i.e., it had o iegulaities o scatches, while eal substate may have some faults. Based o the above, we coclude that achievig 9% diffeece is a easoable esult. Aothe impotat paamete of the gas suface iteactio is the mometum chage of the atoms, which ca be chaacteized by a agula distibutio of scatteed atoms. Fig. 3 illustates the omalized distibutios of pobability desity of the pola agle of the scatteed atoms (omalizatio was caied out by the pea value i ode to impove visibility). Cicles ad cosses coespod to two cases of suface tempeatue, T suf 350 K ad T suf 500 K, espectively. Futhemoe, the tempeatue ad icidet agle of the ago beam wee equal to T G 295 K ad ˇi 45, espectively. As ca be see, the ago scatteig distibutio peas iceased i itesity with the icease of suface tempeatue. This fidig is i lie with the esults of Weibeg [6]. Fig. 4 shows the aveage eegy of the scatteed beam as a fuctio of icidet beam eegy E i fo diffeet values of the icidet agle ˇi of the beam. It is clea that all plots peseted i Fig. 4 have two egios: (1) a liea egio, whee the mea eegy of the scatteed atoms liealy depeded o the icidet eegy; ad (2) a oliea egio, coespodig to elatively low eegies of impigig A atoms. The lieaity of depedece betwee E s ad E i ca be claified by usig coelatio coefficiet peseted i Table 1. Fig. 3. Agula distibutio of omalized pobability desity fuctio i case of A scatteig fom W(1 1 0) at T G 295 K ad icidet agle ˇi 45. +, T suf 500 K;, T suf 350 K. If the distibutio of adom umbes coespoded to a Maxwell Boltzma distibutio, the elatioship betwee oot-mea-squae deviatio (RMSD) ad the mea value would be descibed by the followig elatio (see e.g., [12]): (4) 2 The atio ( V / V ) of RMSD to the mea velocity of the atoms scatteed by the suface as a fuctio of both icidet beam eegy ad icidet pola agle is peseted i Fig. 5. Oe ca see that the value of V / V appoaches 0.42 (show i Eq. (4)) with deceasig icidet eegy egadless of the icidet pola agle ˇi, i.e., the distibutio of velocities of scatteed ago atoms teds towads a Maxwell Boltzma distibutio. A simila effect was metioed by Fa ad Maso [8] ad obseved expeimetally i Sectio 2 of Ref. [15] ad i Sectio 3.D of Ref. [16]. The moe detailed desciptio of the methodology ad method of aalysis ca be foud i the pevious wo [5]. Coelatio betwee aveage eegy of scatteed atoms ad paametes of icidece was appoximated usig the least squae method (this method was used to obtai all appoximatio fuc- Fig. 4. Mea eegy of scatteed atoms vesus eegy of impigig A atoms. +, ˇi 0 ;, ˇi 20 ;, ˇi 40 ;, ˇi 60.

4 378 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) Table 1 Coelatio coefficiets betwee E i ad E s fo diffeet agle of icidece ˇi. ˇi Coelatio coefficiet ˇi Coelatio coefficiet tios discussed i cuet wo). Eq. (5) epesets the depedece betwee E s, E i, ad ˇi. It should be oted, that the uits of E i ad ˇi ae Joule ad degee, espectively. E s f (E i, ˇi) E i 3 A0 (ˇi) + (2 B T suf ) b B 5 B0 (ˇi), b S (5) whee b S ad b B have the same physical meaig as i Eq. (3). Coefficiets A0 ad B0 i Eq. (5) deped o pola agle ˇi ad thei values ae show i Table 2. Oe ca easily chec that if ˇi 45 the coefficiets b S ad b B i Eq. (5) have the values of ad 0.837, espectively, which ae less tha 9% diffeece fom the values show i Ref. [7]. It should be oted that the facto b S of Eq. (5) iceases with iceasig icidet agles of A atom beams. The same effect was also oted by Agawal ad Raff [13]. Coelatio of the omalized oot mea squae deviatio of scatteed atom s velocities with paametes of icidece show i Fig. 5 ca be appoximated by Eq. (6) ad the coefficiets A1, ae listed i Table V f (E i, ˇi) A1, (ˇi) (E i ) (6) 0 Both pola ad azimuthal agles of the scatteed atoms wee vaied withi ages of [ /2; /2], thus the omalized RMSD coespodig to a uifom distibutio fo adom umbes withi this age was [14]: ui b a (7a) Fig. 6. Nomalized oot mea squae deviatio of the azimuthal agle s of scatteed atoms vesus ietic eegy of impigig atoms ad icidet pola agle. +, ˇi 0 ;, ˇi 15 ;, ˇi 30 ;, ˇi 45 ;, ˇi 60 ;, ˇi 70. Thus, the omalized RMSDs of scatteig of the scatteig agles s ad ˇs, espectively, ae defied as follows: ui 30 3 (7b) ˇ ˇ ˇ ui 30 (7c) 3 RMSDs of the scatteig agles s ad ˇs as peseted i Figs. 6 ad 7, espectively, wee omalized by the ui fom Eq. (7a). Based o the esults show i Fig. 6, oe could coclude that the mea azimuthal agle of the scatteed beam s was idepedet of icidet ietic eegy ad icidet agle ˇi whe a Fig. 5. Root mea squae deviatio of scatteed atom s velocities V ove mea velocity of scatteed atoms V vesus ietic eegy impigig atoms. +, ˇi 0 ;, ˇi 15 ;, ˇi 30 ;, ˇi 45 ;, ˇi 60 ;, ˇi 70. Fig. 7. Nomalized oot mea squae deviatio ˇ of the pola agle ˇs of scatteed atoms vesus ietic eegy of impigig atoms ad icidet pola agle. +, ˇi 0 ;, ˇi 15 ;, ˇi 30 ;, ˇi 45 ;, ˇi 60 ;, ˇi 70.

5 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) Table 2 Coefficiets A0 ad B0 of the appoximatio polyomial (5) A * *10 5 4*10 7 B * * * * *10 10 Table 3 Coefficiets A1, of the appoximatio polyomial (6) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 13 atom s beam impiged omally o the suface (data poits epeseted by coss symbols). Fig. 7 pesets coelatio simila to those show i Fig. 6, but fo the pola agle ˇs of the scatteed beam. Nomalized RMSDs of the azimuthal agle s ad pola agle ˇs of scatteed atoms vesus ietic eegy of impigig atoms ad icidet pola agle show i Figs. 6 ad 7 ae appoximated by the polyomials (8) ad (9), espectively: 7 7 f (E i, ˇi) A2, (ˇi) (E i ) (8) ˇ f (E i, ˇi) A3, (ˇi) (E i ) 0 It is obviously fom Fig. 7 that the RMSD of pola agle of scatteed atoms is ivesely popotioal to the icidet eegy, i.e., the agula distibutio becomes aowe while the icidet eegy iceases. Fig. 8 shows that factio of atoms scatteed i bacwad diectio is sufficiet i case of low icidet eegy A atoms ad it deceases whe the eegy of fallig beam becomes highe. These coclusios ae i lie with expeimetal esults show i Refs. [15,16]. Coefficiets A2, ad A3, fo Eqs. (8) ad (9) ae listed i Tables 4 ad 5, espectively. Fig. 9 illustates the coelatio of the aveage pola agle ˇs of the scatteed atoms with icidet ietic eegy ad pola agle ˇi. This is appoximated by the polyomial Eq. (10) havig coefficiets A4, which ae listed i Table ˇ f (E i, ˇi) A4, (E i ) (ˇi) (10) 0 The esults peseted i Fig. 8 shows that a icease i icidet eegy caused lieaizatio of depedece betwee the icidet (9) Fig. 8. Factio of atoms scatteed i bacwad diectio vesus eegy of impigig A atoms. +, ˇi 10 ;, ˇi 20 ;, ˇi 40 ;, ˇi 60. ad the scatteed pola agle, while the aveage pola agle of the scatteed ago atoms that had low icidet eegy teded towad zeo. Coelatio coefficiet betwee aveage value of pola agle ˇs of scatteed atoms ad icidet agle ˇi of the beam is show i Fig. 10. Fig. 10 poves that coelatio betwee agles metioed above becomes liea whe the icidet eegy is high eough. It is clea that the distibutio of the agle ˇs did ot become fully uifom (the agle did ot become idepedet of othe paametes) i the age of studied icidet paametes of A beam. O the othe had, the distibutio of the agle s became uifom (as show by the coss symbols i Fig. 11) whe the icidet beam had low eegy ( E i /(2 B T suf ) 0) o was a omal fallig beam (ˇi 0), ad became Gaussia whe the icidets had high Table 4 Coefficiets A2, of the appoximatio polyomial (8) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 15

6 380 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) Table 5 Coefficiets A3, of the appoximatio polyomial (9) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 14 Fig. 9. Aveage pola agle ˇs of scatteed atoms vesus ietic eegy of impigig atoms ad icidet pola agle. +, ˇi 0 ;, ˇi 15 ;, ˇi 30 ;, ˇi 45 ;, ˇi 60 ;, ˇi 70. eegy (show as cosses ad a dashed lie i Fig. 12). Compaig agle ˇs i Figs. 11 ad 12, oe ca see that the scatteig of A atoms was moe specula i the case of the high eegy fallig beam (Fig. 12) tha i the case of the low eegy oe (Fig. 11). It should be oted that scatteig does ot become fully specula i the age of studied paametes, because A atoms chage thei eegy ad mometum upo collisio. This issue ca also be explaied i tems of elatio betwee E i mi ad ε WA. Pobability distibutio fuctios (PDF) peseted i Figs. 11 ad 12 have 30 bis ad the citical value of Chisquae coespodig to 0.05 level of sigificace is Values of 2 fo the cuves fittig the data poits peseted i Fig. 11 ae 9.18 ad 13.01, coespodig to uifom PDF of s ad omal distibutio of ˇs, espectively. The values of 2 fo fuctios Fig. 10. Coelatio coefficiets betwee ˇi ad ˇs vesus eegy of icidet beam. fittig the distibutios of agles s ad ˇs of scatteed atoms ae ad 12.94, espectively. All these values do ot exceed the citical value of the Chi-squae. It meas that these distibutio fuctios ca fit data poits well. By summaizig the esults ad discussio above, a ew algoithm fo bette descibig the scatteig behavios of ago atoms o tugste suface is poposed: 1. Paametes ˇi, i, V i ae to be detemied fom gas atom velocity compoets that eached stated bouday; 2. Paametes V, V,, ˇ, ad ˇ wee detemied by usig Eqs. (5), (6) ad (8) (10), espectively; Table 6 Coefficiets A4, of the appoximatio polyomial (10) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 09 Fig. 11. Desity of pobability of ˇs ad s. T suf 350 K, ˇi 20, V i 100 m/s. +, distibutio of s;, distibutio of ˇs; solid lie omal distibutio of ˇs.

7 M.S. Ozhgibesov et al. / Joual of Molecula Gaphics ad Modellig 38 (2012) flexible ad ealistic compaed with the simple specula-diffusive oe. I the pevious sectio we have poposed a algoithm to descibe how the polyomial ad pobability distibutio fuctios metioed above could be used as bouday coditios. Geeally, this wos explais how to implemet omalizatio ad futhe covesio of the expeimetal o simulatio esults ito bouday coditios. Applicatio of the ew poposed bouday coditios fo A W iteactios is pove to be able to sigificatly speed up computatios. The computatio time by use of this algoithm is oly about oe-twelfth of computatio time of MD simulatio ivolvig the eal tugste substate. This is implemeted by eplacemet of atoms o solid sufaces with sufaces followig the gas scatteig law descibed by the ew bouday coditios. Acowledgmets Fig. 12. Desity of pobability of ˇs ad s. T suf 350 K, ˇi 20, V i 1500 m/s. +, distibutio of s;, distibutio of ˇs; solid lie omal distibutio of ˇs; dashed lie omal distibutio of s. 3. If 0.9 the agle is to be detemied by usig a adom fuctio with uifom distibutio, othewise should be detemied by usig a samplig method based o the Gaussia pobability fuctio with the RMSD value obtaied i step 2 ad mea value equal to 0. s of the scatteed atoms is the defied as s + i. 4. Paamete ˇs was defied by usig a samplig method based o the omal pobability fuctio with the RMSD ˇ value obtaied i step 2 ad mea values equal to the ˇ of ˇs. 5. The velocity of the scatteed atom V s is to be defied by usig a samplig method based o Maxwell distibutio ad mea value V whe the icidet atom s eegy was E i /(2 B T suf ) 1, ad by usig the omal pobability fuctio with the RMSD V value obtaied i step 2 ad mea value V othewise. The peseted algoithm was tested by eplacig the tugste substate with a flat suface havig a tempeatue of T suf 350 K ad T suf 500 K ad applyig ew BCs descibed by the abovemetioed algoithm. We foud that the poposed model was capable of epoducig the pocesses of the iteactio betwee ago atoms ad the tugste substate descibed above. It should be oted that computatio time by use of this algoithm is oly oe-twelfth of computatio time of MD simulatio ivolvig the eal tugste substate. It is obvious that the amout of time savig will ise popotioally to the scale of solid sufaces epoduced by usig the algoithm metioed above. 4. Coclusio I this wo we developed ew bouday coditios which ae able to epoduce the behavios of ago atoms iteactios with tugste suface. The ew poposed bouday coditios ae based o Gaussia, uifom ad Maxwellia pobability distibutio fuctios coducted togethe by usig a set of polyomial fuctios. 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