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Transport Properties of Lennard-Jones Mixtures: A Molecular Dynamics Simulation Study

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EMD Smulatons of LJ Mxtures Bull. Korean Chem. Soc. 2008, Vol. 29, No Transport Propertes of Lennard-Jones Mxtures: A Molecular Dynamcs Smulaton Study Song H Lee Department of Chemstry, Kyungsung
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EMD Smulatons of LJ Mxtures Bull. Korean Chem. Soc. 2008, Vol. 29, No Transport Propertes of Lennard-Jones Mxtures: A Molecular Dynamcs Smulaton Study Song H Lee Department of Chemstry, Kyungsung Unversty, Busan , Korea. E-mal: Receved February 14, 2008 Equlbrum molecular dynamcs smulatons n a canoncal ensemble are performed to evaluate the transport coeffcents of several Lennard-Jones (LJ) mxtures at a lqud argon states of 94.4 K and 1 atm va modfed Green-Kubo formulas. Two component mxture of A and B s bult by consderng the nteracton between A and A as the attractve (A) potental, that between A and B as the attractve potental (A), and that between B and B as the repulsve potental (R), labelled as AAR mxture. Three more mxtures - ARA, ARR, and RAR are created n the same way. The behavor of the LJ energy and the transport propertes for all the mxtures s easly understood n terms of the porton of attractve potental (A %). The behavor of the thermal conductvtes by the translatonal energy transport due to molecular moton exactly concdes wth that of dffuson constant whle that of the thermal conductvtes by the potental energy transport due to molecular moton s easly understood from the fact that the LJ energy of AAR, ARR, and RAR mxtures ncreases negatvely wth the ncrease of A % from that of the pure repulsve system whle that of ARA changes rarely. Key Words : Dffuson, Shear vscosty, Thermal conductvty, Lennard-Jones mxtures, Molecular dynamcs smulaton Introducton Two-component fluds can undergo segregaton or phase separaton 1 whch s akn to that n bnary clusters. 2-5 Whle some gross features of the late-stage of the demxng process have ther bulk phase analogs, the dynamcs s strongly nfluenced by fluctuatons and other fnte sze effects. The morphology of the segregated fluds arses from an often delcate balance of nternal and surface force as well as entropc contrbutons. The nterfaces separatng stable phases may have thckness that are comparable to the flud dmensons, and strong partcle correlatons may exst wthn the flud as a result of surface forces. 2,3 The frst computer smulaton study for the equaton of state of an equmolar bnary mxture of nearly equal hard spheres was carred out by Rotenberg n the 1960s usng Monte Carlo method. 6 An extensve seres of computatons for Lennard-Jones mxtures followed n the 1970s to determne the excess thermodynamc functons of mxng. 7,8 After that, a number of molecular dynamcs studes on the transport coeffcents n bnary flud mxtures have been reported There was good agreement among these studes, and therefore t can be sad that the basc method to calculate the transport coeffcents by MD smulatons has been establshed. Transport coeffcents - self-dffuson coeffcent, D, shear vscosty, η, and thermal conductvty, λ - of pure fluds can be calculated from equlbrum molecular dynamcs smulaton by the nfnte tme ntegral of an equlbrum correlaton functon of the form known as the Green-Kubo formulas Assocated wth any expresson of the Green- Kubo formulas there s also the Ensten formula to calculate the transport propertes. In recent years, non-equlbrum molecular dynamcs (NEMD) smulatons have emerged as a powerful tool for the study of transport coeffcents of both smple and molecular fluds In the present paper, we report new results of equlbrum molecular dynamcs (EMD) smulatons of mxtures of two LJ partcles n a canoncal ensemble (NVT fxed). The goal of ths study s to elucdate the dependence of transport propertes of LJ mxtures on the mole fracton of component A, xa. Ths paper s organzed as follows: We present the molecular models and detals of MD smulaton methods n next secton, theores for transport propertes n Secton III, our smulaton results n Secton IV, and concludng remarks n Secton V. Molecular Models and NVT MD Smulatons One of smpler, more dealzed, par potentals commonly used n computer smulatons s a smple Lennard-Jones (LJ) 12-6 potental : v LJ σ = 4ε -- r 12 σ -- r 6. (1) Values of the LJ potental parameters of ε/k = 120 K and σ = 0.34 nm provde reasonable agreement wth the expermental propertes of lqud argon. Ths s the typcal LJ potental for the attractve (A) potental used n ths EMD smulaton study wth the LJ potental parameters. It s often useful to dvde more realstc potentals nto separate attractve and repulsve components, and the separaton proposed by Weeks et al. 18 nvolves splttng the potental at the mnmum (rm). For the LJ potental, the repulsve part s called the WCA potental: 642 Bull. Korean Chem. Soc. 2008, Vol. 29, No. 3 Song H Lee v WCA. (2) The WCA potental wth the same LJ potental parameters for the attractve potental (A) s used for the repulsve (R) potental n ths study. We begn by consderng mxtures of two LJ partcles nteractng through the above attractve (A) or repulsve (R) potental at a lqud argon state of constant temperature and volume K and g/cc. For two component mxture of A and B, the nteracton between A and A s chosen as the attractve (A) potental, that between A and B as the attractve potental (A), and that between B and B as the repulsve potental (R). Ths mxture s labelled as AAR. Three more mxtures - ARA, ARR, and RAR are created n the same way. The mole fracton of partcle A, x A, s chosen as 0.125, 0.25, 0.375, and 0.5. Accordngly the mxtures are further labelled as AAR1, AAR2, AAR3, etc. The prelmnary canoncal ensemble (NVT fxed) EMD smulatons for 8000 LJ partcles of several mxtures were started n the cubc box of length L = nm, of whch the densty s equal to g/cm 3 at 94.4 K and 1 atm. The nter-partcle potental was truncated at 2.5 σ, whch s the cutoff dstance used n many other smulatons. Long range correctons to the energy, pressure, etc. due to the potental truncaton were ncluded n these propertes by assumng that the par dstrbuton functon was unform beyond the cutoff dstance. 19 The equatons of moton were solved usng the velocty Verlet algorthm 20 wth a tme step of second. The systems were fully equlbrated and the equlbrum propertes were averaged over fve blocks of 10,000 tme steps. The confguratons of LJ partcles were stored every tme step for further analyss. Green-Kubo Formula As dynamc propertes, we consder dffuson constant (D), shear vscosty (η), thermal conductvty (λ), and frcton constant (ζ) of LJ mxture systems. Dffuson constant can be obtaned through two routes: the Green-Kubo formula from velocty auto-correlaton functons (VAC): D s = 1. (3a) 3 -- v ( t) v ( 0) dt and the Ensten formula from mean square dsplacements (MSD): D s = 1. (3b) 6 -- lm d r( t) r( 0) t dt Shear vscosty s calculated by the modfed Green-Kubo formula for better statstcal accuracy 21,22 : η = V -----, (4) kt dt P 0 αβ ( 0) P αβ ( t) where ( r) = P αβ 4ε σ -- σ -- 6 r r + ε, r r m = 2 1/6 σ 0, r r m 0 ( t) = 1 V -- mv α ( t) v β ( t) + r α ( t) f β ( t) j wth αβ = xy, xz, yx, yz, zx, and zy. Thermal conductvty s also calculated by the modfed Green-Kubo formula for better statstcal accuracy 21,22 : V λ = dt q, (5) kt 2 0 α ( 0) q α ( t) where α = x, y, and z. The heat flux by each molecule s Here, the energy of molecule s gven by. (6) ( t) = 1. (7) 2 --m v ( t) Φ[ r j ( t) ] The heat flux by each molecule, Eq. (6), wth the energy of molecule, Eq. (7), conssts of three contrbutons : where and tm q α 1 1 q α ( 0) = -- ε ( t) v α ( t) + -- r jα ( t) [ v ( t) f j ( t) ] V 2 j ε pm q α j q α = q tm α + q pm t α + q α, (8) q tm α = 1 V m v 2 q pm α = 1 V t q α, (9) Φ( r j ) v α, (10) j, (11) and are the translatonal and the potental energy t transport, respectvely, due to molecular moton and q α s the translatonal energy transfer due to molecular nteracton. Hence, the thermal conductvty, Eq. (5), conssts of three contrbutons: λ t = λ tm + λ pm + λ t. (12) Fnally the frcton constant s defned as ζ = Results and Dscusson Mxtures wth the repulsve potental between components A and B, ARA and ARR, exhbt segregaton as shown n Fgure 1 whle those wth the attractve potental between A and B, AAR and RAR, are a mxed state of two LJ partcles. Apparently the segregaton affects the transport propertes of mxtures. Table 1 contans all the results of pure systems and several mxtures n ths equlbrum molecular dynamcs (EMD) smulaton study of canoncal (NVT) ensemble. Porton of attractve potental (A %) s defned as the number of attractve nteractng pars dvded by that of the total nteractng pars. Ths quantty of AAR, ARR, and RAR mxtures ncreases from the pure repulsve system as the mole fracton of component A, x A, ncreases but that of ARA v α = 1 V r jα ( v f j ) j (13) 3kT f 0 ( t) f ( 0) dt EMD Smulatons of LJ Mxtures Bull. Korean Chem. Soc. 2008, Vol. 29, No Fgure 1. Snap shot of ARA4 system at a state of 94.4 K and 1 atm. mxture decreases from the pure attractve system. The dependence of the transport propertes of several mxtures on the mole fracton of component A, x A, s easly understood by ths quantty n the below. We plot the LJ energy and the frcton constant n Fgure 2. The LJ energy of all the mxtures les between those of the pure attractve (A) and the pure repulsve (R) systems. As the mole fracton of component A, x A, ncreases, the LJ energy of AAR, ARR, and RAR ncreases negatvely wth the ncrease of porton of attractve potental (A %) from that of the pure repulsve system whle that of ARA changes rarely snce ths mxture conssts of two separate attractve systems. For deal lqud mxtures, one would expect that each of the propertes would depend lnearly on composton,.e., Fgure 2. LJ energy (E LJ n kj/mol) and frcton constant (ζ n g/ mol ps). and : LJ energy of the pure attractve and the pure repulsve systems, : AAR mxture, : ARA mxture, : ARR mxture, and : RAR mxture. The correspondng whte symbols are for the frcton constant. The error bars represent the uncertantes n the last reported dgt(s) gven n parenthess of Table 1. Y = x A Y A + 1 ( x A )Y B, (14) where Y s a property. The behavor of the LJ energy of ARR shows a perfect lnear model snce ths mxture conssts of one attractve system surrounded by repulsve partcles and the A % ncreases lnearly wth x A. On the other hand, the behavor of the LJ energy of AAR and RAR does not show a perfect lnear model but t seems close to an exponental model, gven by Y = exp [ x A lny A + ( 1 x A. (15) )lny B ] Ths model s an engneerng correlaton recommended for Table 1. Porton of attractve potental (A %), LJ energy (E LJ n kj/mol), dffuson constant (D(VAC) n 10 5 cm 2 /sec, Eq. (3a)), shear vscosty (η n mp, Eq. (4)), thermal conductvtes (λ n 10 4 cal/k cm sec), and frcton constant (ζ n g/(ps mol), Eq. (13)) of mxtures of LJ partcles at T = 94.4 K and p = 1 atm. Uncertantes n the last reported dgt(s) are gven n parenthess System A % E LJ D(VAC) η λ tm λ pm λ t λ t (Eq. 12) ζ Pure A (6) 2.45(5) 3.25(4) 0.143(2) 0.892(25) 0.852(29) (0) Pure R (1) 3.09(6) 2.69(2) 0.167(2) 0.021(1) 1.549(32) (1) AAR (1) 2.89(2) 3.42(3) 0.163(1) 0.128(2) 1.134(12) (0) AAR (1) 2.73(4) 3.59(3) 0.154(1) 0.264(6) 0.805(13) (0) AAR (1) 2.62(5) 3.40(5) 0.152(2) 0.420(11) 0.577(8) (0) AAR (1) 2.47(3) 2.90(3) 0.143(1) 0.551(9) 0.413(5) (0) ARA (4) 2.35(2) 3.18(7) 0.140(2) 0.794(7) 0.733(11) (0) ARA (5) 2.27(4) 3.13(3) 0.134(3) 0.718(14) 0.607(11) (0) ARA (3) 2.21(2) 2.57(4) 0.132(3) 0.700(8) 0.505(11) (1) ARA (4) 2.25(5) 1.92(4) 0.133(4) 0.712(13) 0.401(2) (0) ARR (1) 2.98(5) 2.31(2) 0.164(3) 0.051(2) 1.250(34) (0) ARR (1) 2.94(4) 1.90(4) 0.164(2) 0.095(2) 1.025(19) (0) ARR (1) 2.92(4) 1.40(1) 0.163(2) 0.144(5) 0.779(11) (1) ARR (2) 2.78(4) 1.01(1) 0.156(2) 0.187(4) 0.558(7) (0) RAR (1) 2.91(2) 3.44(5) 0.161(1) 0.110(1) 1.131(17) (0) RAR (1) 2.63(5) 3.51(2) 0.151(3) 0.176(2) 0.797(13) (0) RAR (2) 2.51(5) 3.26(2) 0.145(3) 0.228(4) 0.560(11) (0) RAR (1) 2.48(5) 2.80(5) 0.144(2) 0.241(5) 0.410(6) (1) 644 Bull. Korean Chem. Soc. 2008, Vol. 29, No. 3 Song H Lee Fgure 3. Dffuson constant (D n 10 5 cm 2 /sec: the black symbols) and shear vscosty (η n mp: the whte symbols) as a functon of the mole fracton of component A, x A, at a state of 94.4 K and 1 atm obtaned n ths NVT-EMD smulaton study. The legends are the same as n Fg. 2. predctng lqud mxtures n the absence of mxture property data. The frcton constant ncreases generally wth the ncrease of the mole fracton of A, x A. The trend of the frcton constant of AAR, ARR, and RAR, whch ncreases wth the ncrease of A % from that of the pure repulsve system, s smlar to that of the LJ energy. Note that the behavor of the frcton constant of AAR and RAR s almost equal each other snce these mxtures are a mxed state of two LJ partcles and the A % ncreases by almost equal amount wth the ncrease of the mole fracton of A, x A. The behavor of the frcton constant of AAR, ARR, and RAR shows a lnear model generally. In Fgure 3, we plot dffuson constant and vscosty for pure systems and several mxtures as a functon of x A, whch were obtaned through the Green-Kubo formulas (Eqs. 3(a) and (4)) n ths EMD smulaton study n NVT ensemble. The calculated dffuson coeffcent and shear vscosty for the pure attractve system at a state of 94.4 K and 1 atm are close to the expermental measures (D = cm 2 /sec at 90 K and g/cm 3, and η = 1.97 mp at 94.4 K and 1 atm for pure Ar). The dffuson constant for the pure repulsve system s larger than that for the pure attractve system and the vscosty s opposte as expected. As the mole fracton of component A, x A, ncreases, the dffuson constant of AAR, ARR, and RAR decreases wth the ncrease of A % from that of the pure repulsve system, but that of ARA decreases wth the decrease of A % from that of the pure attractve system snce ths mxture conssts of two separate attractve systems whch repel each other. The decreasng trend of the dffuson constant of AAR, ARR, and RAR wth the ncrease of A % from that of the pure repulsve system s smlar to the negatve ncrease of the LJ energy and the ncrease of the frcton constant. Here the smlarty of the behavor of the dffuson constant of AAR and RAR s notable agan. The vscosty of AAR and RAR shows a smlar behavor each other lke the dffuson constant of those mxtures, but Fgure 4. Thermal conductvtes (n 10 4 cal/k cm sec, λ tm: the black symbols and λ pm: the whte symbols) as a functon of the mole fracton of component A, x A, at a state of 94.4 K and 1 atm obtaned n ths NVT-EMD smulaton study. The legends are the same as n Fg. 2. t ncreases and then decreases wth the ncrease of A % from that of the pure attractve system. It s expected that mxng of attractve and repulsve partcles enhances the stress of the partcles to the wall larger than that of the pure attractve system and then dmnshes t over about 50% of A %. For the case of ARA mxture, the vscosty decreases wth the decrease of A % from that of the pure attractve system, but for the ARR mxture, the opposte s observed and the vscosty decreases almost lnearly wth the mole fracton of component A, x A, from that of the pure repulsve system. In Fgures 4 and 5, we plot three thermal conductvtes (λ tm, λ pm, and λ t ) and ther sum (λ t ) for pure systems and several mxtures. The expermental thermal conductvty s λ t = cal/kcmsec at a state of 94.4 K and 1 atm for pure Ar, whch s close to that of the pure attractve system n ths NVT-EMD smulaton study. Swtchng from attractve potental to repulsve potental causes to ncrease λ tm and λ t but to decrease λ pm and the total thermal conductvty, λ t, snce the decrement of λ pm overcomes the Fgure 5. Thermal conductvtes (n 10 4 cal/k cm sec, λ t: the black symbols and λ t: the whte symbols) as a functon of the mole fracton of component A, x A, at a state of 94.4 K and 1 atm obtaned n ths NVT-EMD smulaton study. The legends are the same as n Fg. 2. EMD Smulatons of LJ Mxtures Bull. Korean Chem. Soc. 2008, Vol. 29, No Fgure 6. Comparson of λ tm (n 10 4 cal/k cm sec: the black symbols) and D (n 10 5 cm 2 /sec: the whte symbols) as a functon of the mole fracton of component A, x A, at a state of 94.4 K and 1 atm obtaned n ths NVT-EMD smulaton study. The legends are the same as n Fg. 2. ncrement of λ tm and λ t. Energy transported va molecular moton governs heat conducton n gases, whle energy transfer between molecules due to molecular nteracton s a domnant factor n heat conducton n lquds. Lqud molecules transport energy by molecular moton and transfer ther energy to other molecules by molecular nteracton. Accordngly, λ tm and λ pm are the thermal conductvtes by the translatonal and the potental energy transport, respectvely, due to molecular moton, and λ t s that by the translatonal energy transfer due to molecular nteracton. In general, as the mole fracton of component A, x A, ncreases, λ tm and λ t of all the mxtures decrease whle λ pm ncreases and as a result the sum of these three thermal conductvtes decreases. λ tm of AAR, ARR, and RAR mxtures decrease wth the ncrease of A % from that of the pure repulsve system to that of the pure attractve system whle that of ARA mxture decreases wth the decrease of A % from that of the pure attractve system. Snce λ tm s the thermal conductvty by the translatonal energy transport due to molecular moton, the behavor of λ tm for all the mxtures exactly concdes wth that of dffuson constant, D, as shown n Fgure 3. In Fgure 6, we compared the behavors of D and λ tm as a functon of the mole fracton of component A, x A, for all the mxtures. It s also nterestng that λ pm of AAR, ARR, and RAR mxtures ncrease almost lnearly wth the ncrease of A % from that of the pure repulsve system whle that of ARA mxture decrease wth the decrease of A % from that of the pure attractve system. Ths s easly understood from the fact that λ pm s the thermal conductvty by the potental energy transport due to molecular moton and the LJ energy of AAR, ARR, and RAR mxtures ncreases negatvely wth the ncrease of A % from that of the pure repulsve system whle that of ARA changes rarely as shown n Fgure 2. λ t of AAR, ARR, and RAR mxtures decrease wth the ncrease of A % from that of the pure repulsve system whle that of ARA mxture decreases wth the decrease of A % from that of the pure attractve system. The behavor of λ t for all the mxtures s very smlar to that of λ tm. The translatonal energy transfer due to molecular nteracton, Eq. (11), nvolved wth two terms - velocty and nteratomc force, whch are not easly analyzed. It may be only deduced that the translatonal energy flux due to molecular moton, Eq. (9), s nvolved wth the velocty and that the nteratomc force for AAR, ARR, and RAR mxtures decrease wth the ncrease of A % from that of the pure repulsve system whle that of ARA mxture decreases wth the decrease of A % from that of the pure attractve system. The lnear decrease of λ t for ARA and ARR mxtures s notable. The sum of three thermal conductvtes, λ t, also shows a smlar behavor to λ tm and λ t. λ t of AAR, ARR, and RAR mxtures decrease wth the ncrease of A % from that of the pure repulsve system whle that of ARA mxture decreases wth the decrease of A % from that of the pure attractve system. Concluson We present new results for transport propertes of mxtures of two LJ partcles at a lqud argon state of 94.4 K and 1 atm by equlbrum molecular dynamcs (EMD) smulatons of canoncal (NVT) ensemble usng modfed Green-Kubo formulas. The mxtures are bult by consderng mxtures of two LJ partcles nteractng through the above attractve (A) or repulsve (R) potental. For two component mxture of A and B, the nteracton between A and A s chosen as the attractve (A) potental, that between A and B as the attractve potental (A), and that between B and B as the repulsve potental (R). Ths mxture s labelled as AAR. Three more mxtures - ARA, ARR, and RAR are created n the same way. The Lennard-Jones (LJ) energy of AAR, ARR, and RAR ncreases negatvely wth the ncrease of porton of attractve potental (A %) from that of the pure repulsve system whle that of ARA changes rarely. The frcton constant ncreases generally wth the ncrease of the mole fracton of A, x A. The trend of
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