A nonequilibrium molecular dynamics simulation of evaporation

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1 Intrnational Confrnc Passiv and Low Enrgy Cooling 543 A nonquilibrium molcular dynamics simulation of vaporation Z.-J. Wang, M. Chn and Z.-Y. Guo Dpartmnt of Enginring Mchanics, Tsinghua Univrsity, Bijing , P. R. China ABSTRACT Th vaporation procss is invstigatd by prforming non-quilibrium molcular dynamics simulations. Th transint dnsity, tmpratur and prssur profils ar statistical calculatd, and th vaporation flux is obtaind from molcular lvl. With Hrtz-Knudsn- Labuntsov quation, th vaporation cofficints basd on nonquilibrium molcular dynamics simulations ar computd and compard with th data calculatd by th quilibrium molcular dynamics simulations and th transition stat thory. Th vaporation cofficints of th nonquilibrium molcular dynamics agr with thos from th transition stat thory and som data of quilibrium molcular dynamics calculations in litratur within 15%. Howvr, if using Hrtz-Knudsn-Labuntsov quation to prdict th vaporation/condnsation mass flux, th statistical modl prsntd by Matsumoto t al., (1995) dos not show rasonabl agrmnt with prsnt simulation and som othr data in litratur, spcially in th high tmpratur rgion. 1. INTRODUCTION Evaporation and condnsation at th liquidvapor intrfac hav drawn considrabl attntion from th physical and chmical community for many yars. Extnsiv rsarchs hav bn carrid out, which involv th liquid-vapor intrfac, th intrphas transport dynamics, vaporation/condnsation cofficint, tc. Howvr, du to th comxity of its physical mchanism, thr ar still many challngs rmaind unsolvd. For instanc, it is still difficult to accuratly prdict th phas chang flux in many situations. From th kiic thory viw (Collir and Thom, 1994; Cammnga, 1980), assuming an idal gas and nglcting intractions btwn individual molculs, th mass flux of th vaporation and condnsation can b stimatd as: M m& = (1) whr m& is th mass flux of phas chang, M is th molar mass, R is th gas constant, p v and p l ar th vapor and th liquid prssurs sparatly, T v and T l ar th vapor and th liquid phass tmpraturs sparatly. Considring th ral gas ffcts, th classical thoris of vaporation and condnsation has to introduc th vaporation/condnsation cofficints, which ar dfind as th ratio of th vaporatd/condnsd molculs to th incidnt ons into th intrfac. Gnrally, th vaporation/condnsation cofficints ar put into Equation (1) to adapt th xprimntal data to th simifid thortical prdiction, in th form of: M m& = c (2) Equation (2) is th so-calld Hrtz-Knudsn- Langmuir quation (Cammnga, 1980), whr c is th condnsation cofficint and is th vaporation cofficint. Schrag (Schrag,

2 544 Intrnational Confrnc Passiv and Low Enrgy Cooling 1953) and Labuntsov (Labuntsov, 1967) rspctivly modifid Equation (2) for narquilibrium and non-quilibrium conditions as following, 2 M m& = c (3) 2 c 2 M m& = c (4) c Equation (3) is known as Hrtz-Knudsn- Schrag Equation and Equation (4) is so calld as Hrtz-Knudsn-Labuntsov Equation. Furthrmor, a mor comx kiic vaporation and condnsation quation taking into account th vlocity distribution of th rflctd molculs is givn by Ytrhus (Ytrhus, 1997), and Ros gav an ovrviw on th liquid-vapor intrphas mass transfr (Ros, 1998). For th vaporation/condnsation cofficints, a numbr of xprimntal invstigations hav bn carrid out, but thr ar oftn discrpancis among th masurd vaporation/condnsation cofficints from th diffrnt xprimnts (Cammnga, 1980; Eams t al., 1997; Bdaux and Kjlstrup, 1999; Mark and Straub, 2001). As an xam, chcking th vaporation/condnsation cofficints of watr, at th sam tmpratur, th masurd data vary ovr mor than thr ordrs (Eams t al., 1997; Bdaux and Kjlstrup, 1999; Mark and Straub, 2001). In rcnt yars, by using molcular dynamics (MD) simulations, th vaporation/condnsation cofficints undr quilibrium wr statistically computd by Matsumoto t al. (1995), Tsuruta and Nagayama (2000), and us (Wang t al., 2003) rspctivly, with diffrnt statistical mthods and physical modls, and diffrnt valus wr obtaind. Manwhil, th condnsation cofficints can b thortically prdictd. Th vaporation/condnsation cofficints calculatd by a modifid transition stat thory by us (Wang t al., 2002) agr wll with th simulatd data in th molcular dynamics calculations by both Tsuruta and Nagayama, (2000) and us. Howvr, it is difficult to giv mrit for ths simulatd and thortical rsults sinc ths currntly availabl data ar undr quilibrium conditions, whil in practic th vaporation and condnsation procsss tak ac mostly undr quasi-quilibrium or nonquilibrium conditions. In this contribution, nonquilibrium molcular dynamics simulations (NEMD) ar prformd to study th vaporation and condnsation procsss. Evaporation cofficints ar obtaind undr nonquilibrium and compard with th data from quilibrium molcular dynamics simulations. 2. SIMULATION DETAILS 12,000 molculs with th non-polar intracting forc xprssd by th 12-6 Lnnard-Jons potntial ar moyd to simulat th vaporations of th liquid argon. Th Lnnard-Jons potntial has th form (Alln and Tildsly, 1987), 12 6 σ σ U ( r) = 4ε (5) r r whr r is th molcular pair sparation, and 10 paramtrs of th potntial ar σ = m, ε/k B =119.8K, in which k B is Botlzman s constant. Th tim stp is 1.19fs, and th cut-off radius is 4.0σ. Priodic boundary conditions ar apid to all thr dirctions. Firstly, th molculs ar nclosd in an mpty cubic box, and canonical nsmbl (th numbr of th molculs N, th volum of th systm V and th tmpratur T ar fixd) MD simulations ar carrid out to modl th saturatd liquid systm at 100K and 120K rspctivly. Th initial distanc btwn molculs is th avrag distanc of th saturatd liquid molculs. Th constant bulk tmpratur is prformd with th constraint mthod (Alln and Tildsly, 1987). Th static paramtrs, th prssur P and th dnsity ρ, of th simulatd liquid at saturatd tmpratur T ar compard with othr rportd data (Dunikov t al., 2001) to judg whthr th simulatd liquid is th saturatd liquid undr quilibrium. Scondly, by longating th simulatd systm in z dirction, xpand th saturatd liquid systm undr quilibrium from th vol-

3 Intrnational Confrnc Passiv and Low Enrgy Cooling 545 Tabl 1: Th siz paramtrs of th simulatd systm. T (K) S (10-20 m 2 ) L (10-10 m) L (10-10 m) um V=SL to th volum S(L+ L), in which S is th ara of th surfac prpndicular to z dirction, L is th lngth in z dirction of th original systm, and L+ L is th lngth in z dirction of th xpandd systm. S, L, L at two diffrnt simulation tmpraturs ar givn in Tabl 1. Th saturatd liquid film is sttld in th middl of th xpandd systm. Thn in th simulatd systm, th vaporation bgins at th two surfacs of th saturatd liquid film. During th simulation of th vaporation, th numbr of molculs N and th volum of th systm V ar sttld constant. In ordr to prvnt th wandring of th liquid film, adjusting th mass cntr of th systm to b in th middl of th systm is prformd at vry MD stp. Th intial 10,000 MD stps (11.9ps) ar usd to produc th liquid-vapor intrfac. Aftr xpanding th systm for MD stps (11.9ps), w bgin to rcord th numbr of th vapor molculs N to obtain th vaporation flux m& from th molcular lvl. Morovr, th prssur profil P(z), th tmpratur profil T(z) and th dnsity profil ρ(z) ar statistically calculatd vry 50,000 MD stps (55.9ps), and th liquid and vapor avrag thrmodynamic paramtrs P l, T l, and P v, T v, ar obtaind vry 50,000 MD stps (55.9ps). Th local prssur is computd with th mthod prsntd by Hoovr t al. (Hoovr t al., 1991). Assuming c =, with Equation (4) th vaporation cofficint can b calculatd. 3. RESULTS AND DISCUSSIONS Th dnsity profil, tmpratur profil and th prssur profil in th dirction normal to th intrfac ar statistically calculatd vry 59.5ps from 11.9ps. For an initial tmpratur of 100K, th chang of th dnsity, th tmpratur, and th prssur profils with rspct to tim ar shown in Figurs 1, 2 and 3. Th dnsity of vapor incrass with tim as ρ (kg/m 3 ) Dnsity t: [11.9, 71.4] ps Figur 1: Chang of th dnsity profil with rspct to tim. T (K) 300 Tmpratur t: [11.9, 71.4] ps Figur 2: Chang of th tmpratur profil with rspct to tim. P (bar) Prssur t: [11.9, 71.4] ps Figur 3: Chang of th prsur profil with rspct to tim. xpctd. Th tmpratur gradually dcrass with tim bcaus of th latnt hat absorption, and bcaus of th rapid hat conduction of liquid argon, no clar tmpratur gradint was sn in th tmpratur profil of th liquid

4 546 Intrnational Confrnc Passiv and Low Enrgy Cooling film. Th prssur of th liquid dcrass with tim, whil th prssur of th vapor incrass with tim, i.. th driving prssur of th vaporation dcrass. By calculat th chang of numbr of th vapor molculs vry 50,000 MD stps (dt=55.9ps), th vaporatd fluxs vry 50,000 MD stps (dt=55.9ps) ar obtaind as following, ( N( t + dt) N( t) ) m & = (6) 2 S dt m whr, m is th molcular mass, N(t+dt) and N(t) ar rspctivly th numbr of th vapor molculs at tim t and t+dt, S is th ara of th intrfac. Assuming c =, th vaporation cofficints ar prdictd with Equation (4) as shown in Tabl 2. Th vaporation cofficints obtaind from th NEMD simulations ar compard with th data from th MD simulations undr quilibrium and th transition stat thory as shown in Figur 4. Rasonabl agrmnts ar rachd among th vaporation cofficints from our NEMD simulations and thos from th modifid transition stat thory and th MD simulations undr quilibrium by Tsuruta and Nagayama, (2000) and by us. Btwn th NEMD simulations and th modifid transition stat thory or th quilibrium MD (EMD) calculations by us, th diffrnc of vaporation cofficints is within 10%, and btwn th NEMD simulations and th EMD calculations by Tsuruta and Nagayama it is about 15%. Howvr, btwn currnt NEMD calculations and th EMD simulations by Matsumoto t al., thr xist larg diffrncs of th vaporation cofficints at highr tmpraturs. If using Hrtz-Knudsn- Labuntsov quation to prdict th vaporation/condnsation mass flux, th statistical modl prsntd by Matsumoto t al. (Matsumoto t al., 1995) dos not show rasonabl agrmnt with prsnt simulation and som othr data in litratur, spcially in th high tmpratur rgion. Tabl 2: Nt vaporation flux and vaporation cofficint. T (K) Tim (ps) Evaporation Evaporation flux (kg/m 2 s) cofficint Evaporation Cofficint quilibrium MD by us quilibrium MD by Tsuruta t al. quilibrium MD by Matsumoto t al. Transition stat thory by us Non-quilibrium MD by us Tmpratur (K) Figur 4: Comparison of th vaporation cofficints. 4. CONCLUSIONS Molcular dynamics simulations ar carrid out to study th vaporation and condnsation undr non-quilibrium conditions. By using th Hrtz- Knudsn-Labuntsov quation for th nonquilibrium conditions, w calculat th vaporation cofficints of argon at four avrag liquid tmpraturs and compar thm with othr data availabl in litratur. Th comparisons show that: 1. Th vaporation cofficints from th nonquilibrium molcular dynamics can agr with th data from our modifid transition stat thory, th data from quilibrium molcular dynamics simulations by Tsuruta and Nagayama and by us. Th diffrncs ar within 15%. 2. For vaporation cofficints at highr tmpraturs, thr is a larg discrpancy btwn currnt non-quilibrium molcular dynamics and th quilibrium molcular dynamics calculation by Matsumoto t al. If using Hrtz-Knudsn-Labuntsov quation to calculat th vaporation/condnsation Ar

5 Intrnational Confrnc Passiv and Low Enrgy Cooling 547 mass flux, th statistical mthod proposd by Masumoto t al. dos not show rasonabl agrmnt with prsnt simulation and som othr data in litratur, spcially in th high tmpratur rgion. Alln, M.P. and D.J. Tildsly, Computr Simulation of Liquids, Arrowmith, Bristol, Bdaux, D. and S. Kjlstrup, Transfr cofficint for vaporation, Phys. A 270: Cammnga, H.K., Evaporation mchanisms of liquids, in Currnt Topics in Matrials Scinc 5, Kaldis d., Amstrdam, North-Holland. Collir, J.G. and J.R. Thom, Convctiv Boiling and Condnsation, Oxford: Clarndon, 3rd d.. Dunikov, D.O., S.P. Malyshnko and V.V. Zhakhovskii, Corrsponding stats law and molcular dynamics simulations of th Lnnard-Jons fluid, J. Chm. Phys. 115: Eams, I.W., N.J. Marr and H. Sabir, Th vaporation cofficint of watr: a rviw, Int. J. Hat Transfr, 4(12): Hoovr, W.G., Computational Statistical Mchanics, Elsvir, Amstrdam. Labuntsov, D.A., An analysis of intnsiv vaporation and condnsation, High Tmpratur (Engl. Transl.), 5: Mark, R. and J. Straub, Analysis of th vaporation cofficint and th condnsatio cofficint of watr, Int. J. Hat Transfr, 44: Matsumoto, M., K. Yasuoka and Y. Kataoka, Molcular mchanism of vaporation and condnsation, Thrmal Scinc & Enginring, 3(3): Ros, J.W., Intrphas mattr transfr, th condnsation cofficint and dropwis condnsation, Procdings of th 11 th Intrnational Hat Transfr Confrnc, Kyongiu, Kora, vol.1: Schrag, R.W., A Thortical Study of Intrphas Mass Transfr, Columbia Univrsity Prss, Nw York. Tsuruta, T. and G. Nagayama, A molcular dynammics approach to intrphas mass transfr btwn liquid and vapor, Procdings of th intrnational confrnc on hat transfr and transport phnomna in microscal, Banff, Canada, Wang, Z.J., M. Chn and Z.Y. Guo, Modifid Transition Stat Thory for Evaporation and Condnsation. Chins Phys. Ltt. 19(4): Wang, Z.J., M. Chn and Z.Y. Guo, A molcular study on th Liquid-vapor intrphas transport, Microscal Thrmophysical Eng., 7(4): Ytrhus, T., Molcular-flow ffcts in vaporation and condnsation at intrfacs, Multiphas Scinc and Tchnology 9(3): ACKNOWLEDGEMENTS Financial support from th National Natural Scinc Foundation of China (No ) is gratfully acknowldgd. REFERENCES

Principles of Humidity Dalton s law

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