Transport Properties of Dumbbell Molecules by Equilibrium Molecular Dynamics Simulations

Transcription

1 EMD Smulatons of Dumbbell Molecules Bull. Korean Chem. Soc. 4, Vol. 5, o Transport Propertes of Dumbbell Molecules by Equlbrum Molecular Dynamcs Smulatons Song H Lee Department of Chemstry, Kyungsung Unversty, Busan , Korea Receved December 15, 3 We presents new results for transport propertes of dumbbell fluds by equlbrum molecular dynamcs (EMD) smulatons usng Green-Kubo and Ensten formulas. It s evdent that the nteracton between dumbbell molecules s less attractve than that between sphercal molecules whch leads to hgher dffuson and to lower frcton. The calculated vscosty, however, s almost ndependent on the molecular elongaton wthn statstcal error bar, whch s contradcted to the Stokes' law. The calculated thermal conductvty ncreases and then decreases as molecular elongaton ncreases. These results of vscosty and thermal conductvty for dumbbell molecules by EMD smulatons are nconsstent wth the earler results of those by non-equlbrum molecular dynamcs (EMD) smulatons. The possble lmtaton of the Green-Kubo and Ensten formulas wth regard to the calculatons of vscosty and thermal conductvty for molecular fluds such as the mssng rotatonal degree of freedom s ponted out. Key Words : Dffuson, Vscosty, Thermal conductvty, Dumbbell, Molecular dynamcs smulaton Introducton The earlest molecular dynamcs (MD) smulaton method has been wdely used to study the monatomc fluds, 1-3 consderng a Lennard-Jones (LJ) potental as the nteracton between pars of partcles, whch has lead to a determnaton of several thermodynamc and transport propertes. For argon, the calculated quanttes agree wthn a few percent wth the expermental data. The purpose of ths comparson between computed and expermental values s not to demonstrate that the real potental s a LJ potental, but to establsh that the proposed model has all the sgnfcant physcal propertes of a real system. Then the smulated system can be consdered as a reference for theoretcal studes of equlbrum and transport propertes. The earlest MD calculatons on polyatomc fluds have been carred out by Harp and Berne 4 usng a Stockmayertype potental to smulate CO and and by Rahman and Stllnger 5 to smulate H O. For datomc (dumbbell) molecules, a very smple extenson of the hard-sphere model s to consder a datomc composed of two hard spheres fused together, 6 but more realstc models nvolve contnuous potentals. Thus,, F, Cl etc. have been depcted as two Lennard-Jones atoms separated by a fxed bond length. 7-9 Recently, Lee and Cummngs reported results of nonequlbrum molecular dynamcs (EMD) smulatons for shear vscostes of pure dumbbell fluds, sphercal/dumbbell mxtures, and dumbbell/dumbbell mxtures. 1 It was found that the nteracton between dumbbell molecules s less attractve than that between sphercal molecules whch leads to lower vscostes as was observed n the expermental fact that the vscosty of normal alkanes s less than that of branched alkanes. More recently, Tokumasu et al. have studed the effect of molecular elongaton on the thermal conductvty of datomc lquds usng a EMD method. 11 It was found that the reduced thermal conductvty ncreases as molecular elongaton ncreases. Detaled analyss of the molecular contrbuton to the thermal conductvty revealed that the contrbuton of the flux caused by energy transport and by translatonal energy transfer to the thermal conductvty s ndependent of the molecular elongaton, and the contrbuton of the heat flux caused by rotatonal energy transfer to the thermal conductvty ncreases wth the ncrease n the molecular elongaton. In the present paper, the dependence of transport propertes of dumbbell molecules on molecular elongaton was analyzed by equlbrum MD smulatons. Ths paper s organzed as follows: In Secton II, we present the molecular models and MD smulaton methods. We dscuss our smulaton results n Secton III and present the concludng remarks n Secton IV. Molecular Models and pt MD Smulaton Methods The dumbbell molecule s modeled as an atom-atom or ste-ste model of a datomc molecule. The total nteracton s a sum of parwse contrbutons from dstnct atoms a n molecule, at poston r a, and b n molecule j, at poston r jb : u j (r j ) = u ab (r ab ) (1) a = 1 b = 1 where r ab s the nter-ste separaton r ab = r a -r jb and u ab s the par potental actng between stes a and b: σ d u ab (r ab ) = 4 ε d () r ab σ d r ab 6 Here σ d and ε d are the Lennard-Jones (LJ) parameters for the dumbbell molecule. The nteratomc separaton n a dumbbell, l, s chosen such as the volume of the dumbbell

2 738 Bull. Korean Chem. Soc. 4, Vol. 5, o. 5 Song H Lee molecule s the same as that of a sphere of dameter σ s. Snce the volume of a dumbbell of two spheres of dameter σ d s gven by 1 3 V d = -- 1 π σ + π l πl 3, (3) 6 d -- σ 4 d for a gven nteratomc separaton l = Lσ d, σ d s determned by equatng V d = V s, and can be expressed n the form of σ d = cσ s. In ths study, we have chosen as L =, 1/1, 1/6, 1/4, and 1/3, and c s determned as 1.,.9616,.989,.91, and.877, respectvely. The other LJ parameter ε d s chosen as ε s /4. The LJ parameters, σ s and ε s, for sphercal Ar are chosen as.345 nm and.9961 kj/mol, respectvely. All MD smulatons n VT and pt ensembles were carred on 178 molecules and fully equlbrated for at least 5, tme steps of 1 15 second (l femto second). The equlbrum propertes were then averaged over 5 blocks of, tme steps for a total of 1, tme steps, and the confguratons of molecules were stored every 1 tme steps for analyses of structural and dynamc propertes. The ntermolecular potentals were subject to sphercal cutoffs as follows: the cutoff dstance was.5 σ s for pure dumbbell fluds. The equatons of translatonal moton n pt ensemble are gven by r = p /m + ε r (4a) p = F ε r α t p (4b) V = 3ε V. (4c) The equatons of rotatonal moton about the center of mass for molecular fluds n pt ensemble are derved usng quaternons 1,13 follows: T p = A T (5a) L p = T p p α r L (5b) q 1 q q 3 q 4 p = L p k , k = x, y, z ω k = 1 -- I k (5c) q 3 q 4 q q 1 ω x q 4 q 3 q 1 q p ω y, (5d) q 1 q q 4 q 3 q q 1 q 3 q 4 A 11 = q 1 q 1 + q q q 3 q 3 + q 4 q 4 A 1 = (q 3 q 4 q 1 q ) A 13 = (q q 3 + q 1 q 4 ) A 1 = (q 1 q + q 3 q 4 ) A = q 1 q 1 q q q 3 q 3 + q 4 q 4 A 3 = (q q 4 q 1 q 3 ) A 31 = (q q 3 q 1 q 4 ) A 3 = (q 1 q 3 + q q 4 ) A 33 = q 1 q 1 q q + q 3 q 3 + q 4 q 4 (5e) where L s the angular momentum of molecule, T s the p p torque on molecule n the laboratory frame, L and ω are, respectvely, the angular momentum and angular veloctes of molecule n ts prncple axs frame, A s the p rotaton matrx whch transforms vectors from the laboratory frame to the prncpal axs frame of molecule, I k s the prncpal axs of momenta of nerta of each molecule, and q k, k = 1,, 3, 4, are the quaternon parameters related to the Euler angles descrbng the orentaton of molecule n the laboratory frame e x = (q q 3 q 1 q 4 ) (6a) e y = (q 1 q 3 + q q 4 ) (6b) e z = q 1 q 1 q q + q 3 q 3 + q 4 q 4 (6c) where e α (α = x, y, z) are the components of ts orentatonal unt vector of molecule. ote that the quaternons satsfy the normalzaton q 1 + q + q 3 + q 4 = 1. The use of quaternons leads to sngularty free equatons of moton. 1,13 The translatonal and rotatonal temperature constrant parameters t and r are gven by and The dlaton rate s gven by ε = p F α t = ε p (7a) α r = [ L T ]/ L. (7b) ( r j p j )( Φ j + Φ j /r j ) j m r j ( Φ j + Φ j /r j ) + 9pV j, (7c) where Φ j s the nteracton potentals between molecules and j. Puttng ε = recovers the VT ensemble. Results and Dscusson Thermodynamc propertes for dumbbell molecules at 94.4 K n VT ensemble ( = 178 and V = nm 3 ) obtaned from our EMD smulatons are lsted n Table 1. The Lennard-Jones (LJ) energy and total energy decrease negatvely wth ncreasng nteratomc separaton n a dumbbell. Ths means that the nteracton between dumbbell molecules becomes less attractve wth the ncrease n the molecular elongaton. The calculated pressure n VT ensemble EMD smulatons, however, becomes negatve and then ncreases postvely as molecular elongaton ncreases. Snce the volume of the dumbbell molecule n varous dumbbell systems s the same as that of a sphere of dameter σ s, ths pressure change mght be due to the molecular shape of dumbbell molecules. 1 Especally note that the modfcaton from the sphere of dameter σ s to the dumbbell system of L=1/1, of whch the nteratomc dstance s.81 σ s, brngs a huge pressure change from 4 atm to the negatve pressure of -17 atm. In the cases of L=1/4 and 1/3, the calculated pressures are postve due to the repulsve nteracton

3 EMD Smulatons of Dumbbell Molecules Bull. Korean Chem. Soc. 4, Vol. 5, o Table 1. LJ energy (E LJ n kj/mol), total energy (E tot n kj/mol), and pressure (p n atm) of dumbbell molecules at 94.4 K n VT ensembles (V = nm 3 ) Propertes L= L=1/1 L=1/6 L=1/4 L=1/3 E LJ ± ± ± ± ±.13 E tot -4.6 ± ± ± ± ±.14 p ± ± ± ± ± 3.5 Table. LJ energy (E LJ n kj/mol), total energy (E tot n kj/mol), pressure (p n atm), volume (V n nm 3 ) and of dumbbell molecules at 94.4 K n pt ensembles Propertes L= L=1/1 L=1/6 L=1/4 L=1/3 E LJ ± ± ± ± ±.53 E tot ± ± ± ± ±.53 p 1.18 ± ± ± ± ± 1.1 V 84.1 ± ± ± ± ± 3.5 between dumbbell molecules caused by the ncrease n the molecular elongaton. The correspondng pt ensemble (p = 1 atm) EMD smulatons mmedately carred out wth a careful consderaton of the long-range correcton due to the sphercal cut-off the potental, whch s a tal correcton estmatng the contrbuton from pars of partcles whose dstance apart s greater than the cut-off dstance. 14,15 Table lsts the resultng thermodynamc propertes for dumbbell molecules at 94.4 K obtaned from our EMD smulatons. The calculated pressures n pt ensemble EMD smulatons are much close to 1 atm. Agan the modfcaton from the sphere of dameter σ s to the dumbbell system of L=1/1 brngs a huge volume change from 84 nm 3 to 79 nm 3. Also note that beyond the elongaton of L=1/3, for example, n the dumbbell systems of L=5/1 and 1/, the system becomes nfnte dluton, keepng the pressure of the system as 1 atm. Fgures 1 and show the center of mass radal dstrbuton functons of dumbbell molecules at 94.4 K n VT and pt Fgure 1. Center of mass radal dstrbuton functon of dumbbell molecules at 94.4 K n VT ensembles wth r * =r/σ s. Fgure. Center of mass radal dstrbuton functon of dumbbell molecules at 94.4 K n pt ensembles wth r * =r/σ s. ensembles, respectvely. The radal dstrbuton functon, g(r), s defned as g(r) = r, r ( ) (8) ρ Vr, r ( ) where ρ o s the bulk densty, (r, r) s the number of molecules n a shell whch s between r r/ and r+ r/ from the center of a molecule wth r =. nm, V(r, r) s the volume of the shell, and <Î> ndcates the correspondng ensemble (VT or pt) average. In both VT and pt ensemble EMD smulatons, as the molecular elongaton ncreases, the frst and second peaks n the center of mass g(r) dmnsh gradually and the mnma ncrease. It s also observed that the nearest dstance between centers of dumbbell molecules becomes shorter wth the ncrease n the molecular elongaton. Ths s because the centers of mass of dumbbell molecules come closer as the molecular elongaton ncreases. When the center of mass g(r) n VT ensemble EMD s compared wth that n pt ensemble EMD for the same value of L, the dfference n the

4 74 Bull. Korean Chem. Soc. 4, Vol. 5, o. 5 Song H Lee frst peaks of g(r) ndcates the volume change effect from VT to pt. At L= the center of mass g(r) s almost the same snce volume change s so small, but n the cases of L=1/1 and 1/6, the decreasng volume makes the frst peak of g(r) hgher, whle n the cases of L=1/3 and 1/4, the ncreasng volume makes the frst peak of g(r) lower. As dynamc propertes, we consder dffuson constant (D), vscosty (η), thermal conductvty (λ), and frcton constant (ζ) of dumbbell systems. Dffuson constant can be obtaned through two routes: the Green-Kubo formula from velocty auto-correlaton functons (VAC): D s = v () t v ( ) dt (9a) and the Ensten formula from mean square dsplacements (MSD): D s = 1. (9b) 6 -- lm d< r() t r( ) t dt Vscosty and thermal conductvty are calculated by the modfed Green-Kubo and Ensten formulas for better statstcal accuracy 16 : η = V -----, (1a) kt dt P αβ ( ) P αβ () t where P αβ () t = 1 wth V -- [ mv α () t v β () t + r α () t f β () t ] αβ = xy, xz, yx, yz, zx, zy, and the modfed Ensten formula for shear vscosty s η = 1, (1b) V lm ---- d [ Q ktt dt αβ () Q t αβ ( ) ] where Q αβ (t) Q αβ () = dτ P αβ (τ). The modfed Green-Kubo formula for thermal conductvty 16 s V λ = , (11a) kt dt J Qα ( ) J Qα () t where J Qα (t) = -- 1 {ε (t) ½ v α (t) + r α (t) ½ [v (t) ½ f (t)]} wth α V = x, y, z, and ε () t = p () t The modfed m -- φ[ r j () t ] j Ensten formula for thermal conductvty 16 s λ = 1, (11b) V kt lm ---- d [ ε dt α () ε t α ( ) ] t t where ( ε α () ε t α ( ) ) = dτj. Frcton constant s Qα( τ) obtaned from the tme ntegral of the force auto-correlaton functon 17,18 : ζ = τ = , (1) τ r 3kT dt f () t f ( ) where f (t) = F (t) <F (t)>, F (t) s the total force exerted on molecule, and τ r s the macroscopc relaxaton tme of the FAC. 18 The frcton constants were obtaned by the tme ntegral of the total FAC wth choosng the upper lmt of τ as the tme whch the FAC has the frst negatve value by assumng that the fast random force correlaton ends at that tme. Dffuson constant, vscosty, and thermal conductvty of dumbbell molecules through the Green-Kubo and Ensten formulas and frcton constant at 94.4 K n VT and pt ensembles obtaned from our EMD smulatons are lsted n Tables 3 and 4, respectvely. In both VT and pt ensemble t Table 3. Dffuson constant (D n 1 5 cm /sec), vscosty (η n mp), and thermal conductvty (λ n 1 4 cal/k cm sec) by the Green-Kubo and Ensten formulas and frcton constant (ζ n g/mol ps) of dumbbell molecules at 94.4 K n VT ensembles Propertes L= L=1/1 L=1/6 L=1/4 L=1/3 D G-K.433 ± ± ± ± ±.18 E.49 ± ± ± ± ±.118 η G-K.98 ± ± ± ± ±.171 E 3. ± ± ± ± ±.64 λ G-K ± ± ± ± ±.4 E 3.13 ± ± ± ± ±.181 ζ Eq. (1) 18.6 ± ± ± ± ±.4 Eq. (13) Table 4. Dffuson constant (D n 1 5 cm /sec), vscosty (η n mp), and thermal conductvty (λ n 1 4 cal/k cm sec) by the Green-Kubo and Ensten formulas and frcton constant (ζ n g/mol ps) of dumbbell molecules at 94.4 K n pt ensembles Propertes L= L=1/1 L=1/6 L=1/4 L=1/3 D G-K.59 ± ± ± ± ±.73 E.516 ± ± ± ± ±.4 η G-K.93 ± ± ± ± ±.68 E 3.18 ± ± ±..874 ±.6.66 ±.15 λ G-K ± ± ± ± ±.387 E ± ± ± ± ±.8 ζ Eq. (1) 179. ± ± ± ± ± 1.8 Eq. (13)

5 EMD Smulatons of Dumbbell Molecules Bull. Korean Chem. Soc. 4, Vol. 5, o EMD smulatons, t s evdent that as the molecular elongaton ncreases, the dffuson constant ncreases and the frcton constant decreases. The frcton constant ζ s related to the dffuson constant D by Stoke- Ensten relaton: ζ = kt/d. (13) Table contans the frcton constants obtaned from Eq. (13) usng D obtaned from the Green-Kubo formula n Table. Both the frcton constants gve a correct qualtatve trend: decrease wth the ncrease n the molecular elongaton. The calculated frcton constants from the FAC's usng Eq. (1) are always less than those obtaned from Eq. (13) whch reflects that the random FAC decays slower than the total FAC as descrbed by Kubo (see Fg. n Ref. 18), but the overall agreement s qute good n the same magntude of order. When dffuson and frcton constants n VT ensemble EMD are compared wth those n pt ensemble EMD for the same value of L, the dfferences n dffuson and frcton constants also ndcate the volume change effect from VT to pt. In the cases of L=1/1 and 1/6, the decreasng volume makes dffuson constant lower and frcton constant hgher, whle n the cases of L=, 1/3 and 1/4, the ncreasng volume makes dffuson constant hgher and frcton constant lower. On the other hand, the vscosty (η) obtaned from our VT and pt ensemble EMD smulatons does not show any clear trend of dependency on the molecular elongaton. Accordng to the Stokes' relaton for a sphercal partcle of radus σ (ζ = 6πση), η should be proportonal to ζ. But the calculated vscosty n VT ensemble EMD s almost ndependent on the molecular elongaton wthn statstcal error bar, whle the calculated vscosty n pt ensemble EMD decreases wth the ncrease n the molecular elongaton except L=. Ths result s nconsstent wth the VT ensemble EMD smulaton result for dumbbell molecules at T=135 K and ρ = 1.34 g/cm 3, whch reported the decrease of the vscosty wth the ncrease n the molecular elongaton for L =, 1/5, 1/3, and 1/. 1 The man reason for ths nconsstency may be due to the mssng of the contrbuton of rotatonal degree of freedom to the vscosty of dumbbell molecules n EMD smulatons through Green-Kubo and Ensten relatons. The thermal conductvty (λ) obtaned from our VT and pt ensemble EMD smulatons ncreases and then decreases as molecular elongaton ncreases. Ths result s also nconsstent wth the EMD smulaton result for two-center LJ molecules, 11 whch reported that the reduced thermal conductvty ncreases as molecular elongaton ncreases. It was found that the contrbuton of the flux caused by energy transport and by translatonal energy transfer to the thermal conductvty s ndependent of the molecular elongaton, and the contrbuton of the heat flux caused by rotatonal energy transfer to the thermal conductvty ncreases wth the ncrease n the molecular elongaton. In the EMD smulaton, 11 the rotatonal degree of freedom was taken nto consderaton n the heat flux of each lqud, but the heat flux, J Qα(t) n Eq. (11a), s mssng the term ω (t) T (t), where ω and T are the angular velocty and the torque vector of molecule, respectvely. Also, the total energy ε (t) s mssng the rotatonal knetc energy term, 1/ω I ω, where I s the tensor of nerta of molecule. Ths rotatonal degree of freedom mght be taken nto consderaton n the vscous moton for dumbbell molecules. Ths should be ponted out as a possble lmtaton of the Green-Kubo and Ensten formulas wth regard to the calculatons of vscosty and thermal conductvty for molecular fluds. Recalculatons of the vscosty and thermal conductvty, takng nto consderaton the rotatonal degree of freedom, for dumbbell molecules are presently under study. Concluson The effect of molecular elongaton on the transport propertes of dumbbell fluds has been studed by VT and pt ensemble equlbrum molecular dynamcs (EMD) smulatons usng Green-Kubo and Ensten formulas. The nteracton between dumbbell molecules becomes less attractve wth the ncrease n the molecular elongaton, whch leads to hgher dffuson and to lower frcton. The calculated dffuson and frcton constants, D and ζ, are well descrbed by Stoke- Ensten relaton. The results of vscosty and thermal conductvty for dumbbell molecules obtaned from our EMD smulatons do not show any clear trend of dependency on the molecular elongaton. But the earler results usng EMD smulatons reported that the vscosty decreases and the thermal conductvty ncreases as molecular elongaton ncreases. It s ponted out that the early Green-Kubo and Ensten formulas for monatomc molecules s mssng the rotatonal degree of freedom whch should be taken nto consderaton for the transport propertes of molecular fluds. Acknowledgment. Ths research was supported by Kyungsung Unversty Research Grants n 4. References 1. Alder, B. J.; Wanwrght, T. E. J. Chem. Phys. 1959, 31, Rahman, A. Phys. Rev. 1964, 136A, Berne, B. J.; Harp, G. D. Advan. Chem. Phys. 197, 17, Harp, G. D.; Berne, B. J. Phys. Rev. 197, A, Rahman, A.; Stlllnger, F. H. J. Chem. Phys. 1971, 55, Streett, W. B.; Tldesley, D. J. Proc. R. Soc. Lond. 1976, A348, Barojas, J.; Levcsque, D.; Quentrec, B. Phys. Rev. 1973, A7, Cheung, P. S. Y.; Powles, J. G. Mol. Phys. 1975, 3, Snger, K.; Taylor, A.; Snger, J. V. L. Mol. Phys. 1977, 33, Lee, S. H.; Cummngs, P. T. Mol. Sm. 1, 7, Tokumasu, T.; Ohara, T.; Kamjo, K. J. Chem. Phys. 3, 118, Evans, D. J. Mol. Phys. 1977, 34, Evans, D. J.; Murad, S. Mol. Phys. 1977, 34, Allen, M. P.; Tldesley, D. J. 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