Non-equilibrium structure of the vapor-liquid interface of a binary fluid

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1 Non-equlbrum structure of the vapor-lqud nterface of a bnary flud Aldo Frezzott Dpartmento d Matematca del Poltecnco d Mlano - Pazza Leonardo da Vnc 32 - I233 Mlano - Italy Abstract. The evaporaton of a bnary lqud mxture of monatomc speces nto near vacuum has been nvestgated by molecular dynamcs smulatons. It has been assumed that atomc nteracton forces can be derved by Lennard-Jones potentals. Results are presented about surface composton changes nduced by evaporaton, the shape of the dstrbuton functons of evaporatng atoms. Estmates of evaporaton coeffcents are gven. Keywords: mxtures, evaporaton, molecular dynamcs PACS: 5..+y,64.7.fm INTRODUCTION A relatvely small number of studes have nvestgated the structure of the one-dmensonal Knudsen layer whch forms n the vapor phase above the lqud surface durng the evaporaton of a bnary substance[, 2, 3]. The common bass of these nvestgatons s the followng system of coupled, steady and spatally one-dmensonal Boltzmann equatons[4]: 2 f v x x = Q j ( f, f j ) =,2 () j= f (x,v) beng the dstrbuton functon of molecular veloctes v of speces at the spatal locaton x, whereas Q j ( f, f j ) s the collson ntegral representng the nteracton between speces and j. Eqs.() have been solved by a varety of technques, rangng from moment methods[] to DSMC smulatons[2] and determnstc schemes for the lnearzed form of the equatons[3]. Snce the man am of the studes mentoned above was understandng the gas dynamcs n the vapor knetc regon, the smplest form of boundary condtons at the vapor-lqud nterface was adopted. Accordngly, Eqs. () were solved assumng that followng boundary condtons hold at the evaporatng surface located at x = : f (,v) = n w exp (2πR T w ) 3/2 ( v2 2R T w ),v x > (2) In Eq. (2) T w s the surface temperature whereas n w s the saturated vapor densty of the th speces at temperature T w. As s clear from Eq. (2), a unt evaporaton coeffcent was assumed for both speces. Moreover, both the surface temperature T w and the partal vapor denstes n w were kept constant. Both assumptons present problematc aspects whch deserve a deeper analyss. In the case of a mono-component flud, the functonal form of the dstrbuton functon of evaporatng molecules and the determnaton of evaporaton coeffcents has been the subject of several nvestgatons based on molecular dynamcs (MD) smulatons [5, 6, 7] and mean feld approxmatons of smple lquds[8]. The results show that the half-range Maxwellan specfed by Eq. (2) provdes a good approxmaton of smulaton results, n the case of near vacuum evaporaton. However, all smulaton methods predct evaporaton coeffcents slghtly below unt. Devatons from the functonal form of Eq. (2) have been reported n presence of a sgnfcant molecular flux mpngng on the lqud surface [9]. Consderng constant surface propertes and a steady analyss of the vapor flow n the Knudsen layer s not unreasonable. As a matter of fact, changes of the lqud surface physcal condtons mght be slow when compared wth the characterstc knetc tme scales n the vapor. However, the steady analyss predcts that, as expected, the net number fluxes φ of the two vapor components are not the same, the dfference beng determned by the speces mass rato. Hence, evaporaton should cause a contnuous change n the surface composton of the condensed phase. In the lght of the consderatons expressed above, the research work descrbed n the present paper ams at mprovng

2 the boundary condtons to be used at the vapor lqud nterface of a mult-component flud. Followng the research lnes of the sngle component case, the near vacuum evaporaton of a lqud mxture of two monatomc speces has been studed by non-equlbrum MD smulaton, n order to obtan the form of the dstrbuton functon of evaporatng molecules as well as estmates of evaporaton coeffcents followng the surface composton tme evoluton. EQUILIBRIUM MD SIMULATIONS AND PHASE COEXISTENCE DATA Equlbrum and non-equlbrum classcal MD smulatons[] of a bnary lqud n contact wth ts vapor phase have been performed on system of atoms belongng to two dfferent speces characterzed by atomc masses m and m 2, respectvely. It s assumed that atoms of speces a (a =,2) nteract parwse wth atoms of speces b (b =,2) through forces derved from the followng Lennard-Jones potental [ (σab ) 2 ( ) ] 6 σab φ ab (ρ) = 4ε ab (3) ρ ρ beng ρ the dstance between two nteractng atoms. As s well known, ε ab s the depth of the potental well, whereas ρ = 2 /6 σ ab s the poston of the potental mnmum. The quanttes m, σ and σ m ε have been adopted as reference mass, length and tme unts, respectvely. In all smulatons descrbed below, the followng set of nondmensonal parameters has been adopted m 2 = σ 2 =.33396, σ 22 = ε 2 =.66858, ε 22 = The parameters choce has been dctated by the avalablty of a number of prevous studes of the phase coexstence dagram of Argon-Krypton mxtures, based on the same (or smlar) parameters set[]. The dmensonless values gven above have been obtaned by selectng Argon as reference speces and applyng Lorentz-Berthelot mxng rule[] σ 2 = σ + σ 22 (4) 2 ε 2 = ε ε 22 (5) to compute cross nteracton parameters. The system dynamcs has been obtaned by computng the moton of a few thousand atoms n paralleleppedal box of sdes L x, L y and L z. To smulate an nfnte system, perodc boundary condtons have been appled along x and y drectons whch are parallel to the lqud surface. When smulatng systems n equlbrum, perodc boundary condtons have also been appled along z drecton, normal to the vapor-lqud nterface. Newton s equatons have been ntegrated numercally by a smple velocty Verlet scheme[]. Forces have been computed followng the mnmum mage conventon[] and truncatng atomc nteractons when the relatve dstance ρ exceeded a specfed truncaton radus ρ c. Atoms have also been sorted nto a cell system, n order to make the search of nearest neghbors more effcent. The knowledge of the phase coexstence dagram of the bnary flud s of fundamental mportance for the nterpretaton of non-equlbrum smulatons. The equlbrum propertes of the lqud-vapor system n bnary mxtures has been extensvely nvestgated by Monte Carlo, Gbbs ensemble Monte Carlo and MD methods[]. However, the results for a specfc system slghtly depend on the adopted numercal technque and computatonal parameters. In partcular, the choce of the truncaton radus ρ c strongly affects phase coexstence data[]. Therefore a few reference ponts of the phase coexstence dagram have been computed afresh by runnng a number of equlbrum MD smulatons at constant atom numbers N, total volume V = L x L y L z and temperature T. The desred temperature level has been obtaned by a smple Gaussan thermostat[]. Reference equlbrum results have been obtaned for three values of the reduced temperature T = k BT ε, beng k B the Boltzmann constant. For each temperature value, three dfferent system compostons have been consdered by assgnng the overall Argon (speces ) concentraton x = N /(N + N 2 ). As shown below, a more refned composton resoluton has been obtaned for the specal case T =.8. In order to keep

3 ,8,8,6,6 nσ 3,4 n a σ 3,4,2, z/σ z/σ FIGURE. Left- Equlbrum densty and concentraton profles. T =.8, x =.5. Sold lnes represent averaged data over sx ndependent runs: (black) total reduced densty nσ ; (blue) n σ ; (red) n 2 σ. Blue dot-dashed lne represents averaged Argon concentraton, x = n /n. Dashed lnes represent sngle run densty profles. Rght- Evaporaton nto near vacuum. Tme evoluton of total and partal denstes profles. Color ndcates tme level: t = 2 2 (black); t = 2 3 (red); t = 4 3 (blue). Sold lnes: total number densty, nσ 3 ; dashed lnes: number densty of speces ; dotted lnes: number densty of speces 2. the computatonal effort wthn acceptable lmts, n vew of non-equlbrum smulatons, the truncaton radus ρ c has been set equal to 3σ 22. The smulaton box dmensons L x, L y and L z have been set equal to 8ρ c, 8ρ c and 4ρ c, respectvely, whereas the dmensonless tme step has been set equal to.5. The total number of atoms was vared between 8 and 2. Each smulaton has been started from an ntal system confguraton n whch atoms are located at the corners of a smple cubc lattce wth random veloctes sampled from a Maxwellan wth zero mean velocty and temperature equal to T. Then, the system dynamcs s followed for 4 tme steps. Tme averaged macroscopc quanttes are obtaned from the last tme steps. Test computatons have been successfully compared wth the results n Ref.[] by settng the truncaton radus ρ c equal to 4.4σ 22. Densty and concentraton profles are shown n Fgure for the case T =.8, x =.5. The total number densty profle n(z) quckly reaches ts fnal shape and exhbts very lttle run to run varatons, as shown by the standard devaton assocated wth the total lqud and vapor denstes (n (l) and n (v) )) reported n Table. Partal denstes profles are less unform n the lqud regon and dsunformtes exhbt both a longer lfe tme and a larger run to run varaton[]. Smoothed densty profles have been obtaned by averagng the results of sx ndependent runs. The equlbrum partal and total lqud and vapor denstes reported n Table have been obtaned by spatal averages of densty profles n the vapor and lqud bulk regons. As s clear from the nteracton parameters settng, Argon s the more volatle speces. The system composton s not constant, the vapor beng Argon rch. It s also worth observng that Argon s adsorbed on the lqud surface[], as t can be seen from the relatve dsplacement of Argon and Krypton densty profles and the Argon densty local maxma n proxmty of the vapor lqud nterface. NON-EQUILIBRIUM MD SIMULATIONS After obtanng the necessary reference ponts on the phase coexstence dagram, a seres of non-equlbrum MD smulatons has been performed to study the evaporaton of the lqud n near vacuum condtons. As noted n prevous nvestgatons[5], n these numercal experments the back scattered vapor component s vrtually absent. Hence t s possble to obtan the shape of the dstrbuton functon of evaporatng molecules as well as evaporaton coeffcents estmates. MD smulatons have been performed by the same general numercal method used for equlbrum smulatons. However, followng prevous studes[6, 8], perodc boundary condtons along z drecton have been replaced wth a smple open end condton whch removes from the smulaton box those atoms whose dstance from the recedng vapor-lqud nterface exceeds a specfed threshold. The dstance should be large enough not to nfluence the nterface structure, but small n comparson wth the vapor mean free path λ v. In non-equlbrum smulatons, the thermostat has been appled only n a central strp of the lqud slab. The strp thckness has been gradually reduced n tme followng the nterface moton. Each smulaton has been started from a specfed equlbrum condton n

4 TABLE. overall speces composton x. n (l) σ 3 reduced densty of speces n the vapor phase; x (l) Equlbrum lqud and vapor denstes as a functon of reduced temperature T = k b T /ε and s the reduced densty of speces n the lqud phase; n(v) σ 3 s the and x(v) are the lqud and vapor concentratons of speces. Standard errors assocated wth reduced denstes are wrtten below each value. T x n (l) σ 3 n (l) 2 σ 3 n (l) σ 3 x (l) n (v) σ 3 n (v) 2 σ 3 n (v) σ 3 x (v) e ±2.5e-3 ±2.2e-3 ±4.e-4 ±.5e-4 ±.e-4 ±2.e e ±4.4e-3 ±4.2e-3 ±3.e-4 ±3.e-4 ±.3e-4 ±4.e e ±5.e-3 ±5.e-3 ±4.e-4 ±5.e-4 ±.9e-5 ±5.e e ±4.e-3 ±3.e-3 ±4.6e-4 ±3.2e-4 ±2.e-4 ±4.e e ±2.9e-3 ±2.8e-3 ±4.2e-4 ±4.6e-4 ±.4e-4 ±5.e e ±4.5e-3 ±4.5e-3 ±7.e-4 ±8.e-4 ±.5e-4 ±8.e e ±2.e-3 ±.9e-3 ±2.4e-4 ±4.2e-4 ±2.7e-4 ±6.3e e ±4.3e-3 ±4.3e-3 ±4.8e-4 ±7.3e-4 ±2.4e-4 ±9.3e e ±.8e-3 ±.2e-3 ±6.9e-4 ±.4e-3 ±.7e-4 ±.3e-3 whch the lqud slab thckness has been set approxmately equal to 5σ. The smulaton box secton area S = L x L y has been set equal to 36ρc 2. Large ntal slab thckness and box secton area are necessary to run suffcently long smulatons and collect enough atoms n the vapor phase to compute the quanttes of nterest. The results presented here have been obtaned by followng the evoluton of the ntally equmolar system (x =.5) at T =.8, the ntal total number of atoms beng equal to 366. The temperature value has been selected as a compromse between the two contrastng needs of havng a rarefed vapor and a suffcent number of atoms of both speces n the vapor phase. In the condton specfed above the rato λ v σ s of the order of n (v) σ whereas the average atomc dstance n the vapor phase s 3 n (v) σ 3 whch, accordng to Table, takes the value 5.6, about twce the value of the truncaton radus ρ c. Total and partal denstes snapshots at dmensonless tmes t = 2, 2, 4 are shown n Fgure. Durng evaporaton, the total densty profle recedes whle keepng almost constant shape. In the prescrbed condtons, the evaporaton rate s small, therefore the temperature dfference between the thermostatted central slab regon and the lqud surface s of the order of.2 n reduced unts. The preferental evaporaton of Argon causes a contnuous change of the system composton both n the lqud and n the vapor phase, as expected. In partcular the surface composton becomes Krypton rch, whereas the Argon densty profle n the nterface regon keeps the local maxma already observed n equlbrum smulatons. The dstrbuton functons of evaporatng atoms has been obtaned by recordng events of atoms crossng two control surfaces located n the vapor regon at a dstance of 3σ from the nterfaces, whose postons have been obtaned from the maxmum slope of the total densty profle. As shown n Fgure 2, the normalzed reduced dstrbuton functons of velocty components normal and parallel to the lqud surface are well approxmated by Maxwellans havng the lqud bulk temperature, as found for a mono-component flud n smlar condtons[5, 6, 8]. The evaporaton coeffcents α of the two evaporatng speces have been obtaned from the expresson: φ = α n (eq) kb T s (T s,x s ) (6) 2πm In Eq. 6, φ s the evaporatng flux of speces n free molecular regme, T s and x s are the surface temperature and Argon concentraton, respectvely. The quantty n (eq) s the equlbrum vapor densty of speces at temperature T s and lqud concentraton x s. The fluxes φ have been obtaned from the MD smulaton by recordng the numbers N (out) (t) of atoms of speces removed from the doman after crossng a control surface placed at a dstance d r of 5σ from the recedng nterfaces. Smulatons wth larger values of d r dd not show apprecable changes of N (out) (t) hstores, thus confrmng

5 ,5,4,3,2, ,7,6,5,4,3,2, (a) (c) v x,y,7,6,5,4,3,2, 2 3 4,8,6,4,2 (b) (d),5,5 2 2,5 3 v z N (out) (t) t FIGURE 2. Left - Reduced normalzed dstrbuton functons of evaporatng atoms: (a) Argon f x,y (v x,y ); (b) Argon v z f z (v z ); (c) Krypton f x,y (v x,y ); (d) Krypton v z f z (v z ). Sold lnes: Maxwellan dstrbutons at T =.8. Rght - Tme hstores of number of evaporated atoms N (out) (t) (superposton of sx ndependent MD smulatons) the assumpton of nearly free molecular flow n the vapor. As shown n Fgure 2, fluxes are not constant because of the surface composton change. The Argon flux decreases, whereas the Krypton flux ncreases. The equlbrum vapor denstes n (eq) have been obtaned by settng T s =.8 and by assgnng to x s the nstantaneous value taken by Argon concentraton at the spatal locaton just behnd the local densty maxma n proxmty of the vapor-lqud nterface. Ths choce s based on the assumpton that such poston marks out the begnnng of a quas-equlbrum regon whose propertes could be descrbed by hydrodynamc equatons for the lqud phase. More precsely, the tme hstory of x s (t) has been obtaned from non-equlbrum smulatons. For each value of the lqud phase composton x s (t), the correspondng equlbrum partal vapor denstes n (eq) have been obtaned by applyng a smple cubc nterpolaton to the data n Table 2 whch contans ponts on the system coexstence dagram at T =.8 wth a greater resoluton on composton. TABLE 2. Equlbrum system at T =.8. Vapor densty and composton as a functon of lqud composton x x (v) x (l) n (v) σ n (v) σ The evaporaton coeffcents α are then obtaned by computng the fluxes φ by dfferentatng polynomal fts of N (out) data shown n Fgure 2. Tme hstores of surface composton x s (t) and evaporaton coeffcents are presented n Table 3. The numercal results seem to ndcate that the evaporaton coeffcents are not the same for the two speces. The evaporaton coeffcent of Argon appears to be slghtly below the value obtaned by MD smulatons of evaporaton of a mono-component LJ flud, whereas the evaporaton coeffcent of the less volatle speces s slghtly above the mono-component value. The results also show a surface concentraton dependence of evaporaton coeffcents. Ths s not unreasonable, snce surface composton determnes local bondng atomc forces. However the results presented n Table 3 are potentally affected by a few error sources, lke, for nstance the numercal dfferentaton of nosy N (out) data used to estmate nstantaneous fluxes.

6 TABLE 3. Evaporaton nto near vacuum. System ntal reduced temperature T =.8, ntal overall composton x =.5. Tme hstores of surface composton x s (t) and evaporaton coeffcents t σ ε m x s (t) α α CONCLUSIONS The evaporaton of a bnary lqud nto near vacuum has been nvestgated by molecular dynamcs smulatons. Numercal experments have been lmted to flud temperature and composton values whch gve rse to a dlute vapor phase. The tme evoluton of the lqud bulk and nterface regon has been obtaned. The dstrbuton functons of evaporatng atoms and evaporaton coeffcents have been obtaned. The results confrm prevous fndngs about the sngle component system, suggestng that a half-range Maxwellan wth zero drft velocty and temperature equal to the lqud surface temperature s a good approxmaton of emtted atoms dstrbuton functons shape, n the smulaton condtons. The computatons of evaporaton coeffcents s more delcate than n the sngle component case. The present results are stll based on a small number of smulatons, hence they have to be consdered as prelmnar tll the nfluence of all the possble error sources has been fully assessed. ACKNOWLEDGMENTS The author gratefully acknowledges the support receved from Fondazone Carplo wthn the framework of the project Surface Interactons n mcro/nano devces REFERENCES. A. Frezzott, Knetc Theory Study of the Strong Evaporaton of a Bnary Mxture, n Proceedngs of the 5th Internatonal Symposum on Rarefed Gas Dynamcs, edted by V. Boff, and C. Cercgnan, B.G. Teubner Stuttgart, 986, vol. 2, pp A. Frezzott, Knetc theory descrpton of the evaporaton of mult-component substances, n Rarefed Gas Dynamcs 2, edted by C. Shen, Pekng Unversty Press, 997, pp S. Yasuda, S. Takata, and K. Aok, Physcs of Fluds 7, 475 (25). 4. C. Cercgnan, The Boltzmann Equaton and Its Applcatons, Sprnger-Verlag, Berln, V. V. Zhakhovsk, and S. I. Ansmov, JETP 84 (997). 6. T. Ishyama, T. Yano, and S. Fujkawa, Phys. Fluds 6, 2899 (24). 7. R. Meland, A. Frezzott, T. Ytrehus, and B. Hafskjold, Phys. Fluds 6, 223 (24). 8. A. Frezzott, L. Gbell, and S. Lorenzan, Phys. Fluds 7, 22 (25). 9. T. Ishyama, T. Yano, and S. Fujkawa, Phys. Rev. Lett. 95, 8454 (25).. M. Allen, and D. Tldesley, Computer Smulaton of Lquds, Clarendon Press, C. D. Holcomb, P. Clancy, and J. Zollweg, Molecular Physcs 78 (993).