Molecular Dynamics Simulation Studies of Viscosity and Diffusion of n-alkane Oligomers at High Temperatures

Transcription

1 Smulaton Studes of Vscosty and Dffuson of n-alkane Bull. Korean Chem. Soc. 2011, Vol. 32, No Molecular Dynamcs Smulaton Studes of Vscosty and Dffuson of n-alkane Olgomers at Hgh Temperatures Song H Lee Department of Chemstry, Kyungsung Unversty, Busan , Korea. * E-mal: shlee@ks.ac.kr Receved July 28, 2011, Accepted August 31, 2011 In ths paper we have carred out molecular dynamcs smulatons (MD) for model systems of lqud n-alkane olgomers (C 12 ~C 80 ) at hgh temperatures (~2300 K) n a canoncal ensemble to calculate vscosty η, selfdffuson constants D, and monomerc frcton constant ζ. We found that the long chans of these n-alkanes at hgh temperatures show an abnormalty n densty and n monomerc frcton constant. The behavor of both actvaton energes, E η and E D, and the mass and temperature dependence of η, D, and ζ are dscussed. Key Words : MD smulaton, Dffuson, Vscosty, Frcton, n-alkane Introducton A number of studes have been reported on the relatonshp between the zero shear vscosty (η 0 ) and molecular weght. For lnear polymers, η 0 ncreases lnearly wth the molecular weght M. 1 At values of molecular weght greater than a crtcal molecular weght, M c, η 0 becomes proportonal to M 3.4 proposed n 1951 by Fox and Flory 2 on the bass of measurements on narrow dstrbuton fractons of polystyrene and polysobutylene. Self-dffuson constant (D) of polymer melts have also shown a smlar trend: D ~ M 1 for M < M c and D ~ M 2 for M > M c. M c s called the entanglement couplng molecular weght. Below and above M c, the polymer chan dynamcs of unentangled and entangled chans are commonly descrbed by the Rouse model and the reptaton model. 1,3,4 At the molecular weght below the Rouse regme, D and η of n-alkanes also show power law behavors. For example, D of n-alkanes from n-octane to polyethylene of the molecular weght of several thousands was reported to follow the relaton, D ~ M α n whch the exponents α s n the range of dependng on temperature. 5-7 Although the power law dependence has the same functonal form as n the Rouse and reptaton regmes, the orgn s totally dfferent. The molecular weght dependence of D and η n the Rouse and reptaton regmes s attrbuted to the topologcal entanglement effect not to the local segmental frcton whle the exponent found below the Rouse regme reflects the molecular weght dependence of the local frcton rather than the topologcal effect. Whle the Rouse-reptaton crossover has receved consderable attenton n recent years, 8,9 the n-alkane-rouse crossover has receved relatvely lttle attenton. The equlbrum (EMD) and non-equlbrum molecular dynamcs (NEMD) smulatons on the system are useful for the study on the transport propertes of lqud n-alkanes of moderate molecular szes (C 10 -C 40 ), whch can shed lght on the polymer dynamcs n the molecular level. In ths study, we use the Green-Kubo and Ensten formulas for the calculaton of the vscostes and dffuson constants of lqud n-c 12 H 26 to n-c 80 H 162 usng EMD smulaton. The man purpose of ths study s to characterze the vscosty and self-dffuson dynamcs of small n-alkanes as a functon of chan length n (or molecular weght M) especally at hgh temperatures. Molecular Models and MD Smulaton Methods For lqud n-alkanes, we have chosen 7 systems - n- dodecane(c 12 H 26 ), n-ecosane(c 20 H 42 ), n-octaconsane(c 28 H 58 ), n-hexatracontane(c 36 H 74 ), and n-tetratetracontane(c 44 H 90 ), n-hexacontane(c 60 H 122 ), and n-octacontane(c 80 H 162 ). Each smulaton was carred out n the NpT ensemble to determne the volume of each system at gven temperatures, and after the equlbrum densty and hence the length of cubc smulaton box were obtaned, new NVT MD smulaton was performed for each system to store the confguratons of n-alkanes for later analyses. The usual perodc boundary condton n the x-, y-, and z-drectons and the mnmum mage conventon for par potental were appled. Gaussan soknetcs was used to keep the temperature of the system constant. 13,14 We used a unted atom (UA) model for n-alkanes, that s, methyl and methylene groups are consdered as sphercal nteracton stes centered at each carbon atom. Ths model have been used n the prevous smulaton studes Here, we brefly descrbe the salent features of the model. The nteracton between the stes on dfferent n-alkane molecules and between the stes separated by more than three bonds n the same n-alkane molecule s descrbed by a Lennard-Jones (LJ) potental. All the stes n a chan have the same LJ sze parameter σ σ =3.93 A, the well depth parameters were ε ε = kj/mol for nteractons between the end stes and ε = kj/mol for nteractons between the nternal stes. The Lorentz-Berthelot combnng rules [ε j (ε ε j ) 1/2, σ j (σ +σ j )/2] were used for nteractons between an end ste and an nternal ste. A cut-off dstance of 2.5σ was used for all the LJ nteractons.

2 3910 Bull. Korean Chem. Soc. 2011, Vol. 32, No. 11 Song H Lee Intally n NpT MD smulatons, the bond-stretchng was descrbed by a harmonc potental, wth an equlbrum bond dstance of 1.54 A and a force constant of kj/mol/a 2. The bond bendng nteracton was also descrbed by a harmonc potental wth an equlbrum angle of 114 o and a force constant of kj/mol/degree. 2 The torsonal nteracton was descrbed by the potental developed by Jorgensen et al.: 15 U torson (φ) = a 0 + a 1 cosφ + a 2 cos 2 φ + a 3 cos 3 φ, (1) where φ s the dhedral angle, and a 0 = kj/mol, a 1 = kj/mol, a 2 = kj/mol, and a 3 = kj/ mol. For the tme ntegraton of the equatons of moton, we adopted Gear's ffth-order predctor-corrector algorthm 16 wth a tme step of 2 femto-second for all the systems. Later n NVT MD smulatons, the bond-stretchng was swtched to a constrant force whch keeps ntramolecular nearest neghbors at a fxed dstance. The advantage for ths change s to ncrease the tme step as 5 femto-seconds wth the use of RATTLE algorthm 17 and velocty Verlet algorthm. 18 After a total of 1,000,000 tme steps (5 nano-seconds) for equlbraton, the equlbrum propertes were then averaged over 5 blocks of 200,000 tme steps (1 nano-seconds). The confguratons of all the molecules for further analyses were stored every 10 tme steps (0.05 pco second) whch s small enough for the tck of any tme auto-correlaton functons. Self-dffuson constant s obtaned from the Ensten relaton of the mean square dsplacement (MSD): 19 D = 1 -- lm ---- d < [r (t) r (0)] 2 >, (2) 6 t dt The Green-Kubo relaton for the vscosty s gven n terms of the pressure auto-correlaton functon by η = V < P αβ (0)P αβ (t)>, (3) kt dt 0 where αβ = xy, xz, yx, yz, zx, or zy and for a system of N molecules, the molecular formalsm for the pressure tensor s gven by Results and Dscusson Densty. In Fgure 1, we plot the densty of n-alkanes as a functon of temperature. The values of the densty were obtaned from a long tme of NpT MD smulatons as dscussed above. Generally the densty decreases lnearly wth ncreasng temperature as expected, but an abnormalty s found n the long n-alkanes at hgh temperatures. For the short n-alkanes, we observed that the volumes of the systems ncrease nfntely as the temperature ncreases. Ths ndcates evaporaton of the systems. However, for the long chan n- alkane, the densty loses the lnear decrement as temperature ncreases further and becomes flat at hgh temperatures. Ths may ndcates the exstence of a boundary between the small molecular melts and the Rouse regme. In ths study, the crossover s between C 44 and C 60. Self-dffuson Constant. Self-dffuson constants of n- alkanes, D, obtaned from the Ensten relaton, Eq. (2), are plotted n the nset of Fgure 1 as a functon of T. The dependence of D on T n the small molecular melts s obvously dfferent from that n the Rouse regme. The temperature dependence of the calculated dffuson constants of lqud n-alkanes over the whole temperatures consdered s sutably descrbed by an Arrhenus plot as shown n Fgure 2: D = D 0 exp( E D /RT), (6) where D 0 s the pre-exponental factor, RT has the usual meanng, and E D s the actvaton energy of n-alkane dffuson. The value of the actvaton energy s a drect measure of the temperature dependence of self-dffuson constant. The actvaton energes obtaned from the slope of the least square ft are 2.83, 3.41, 3.84, 4.49, 4.74, 4.87, and 4.95 kcal/mol for C 12, C 20, C 28, C 36, C 44, C 60, and C 80, respectvely. The prevous MD smulaton study 22 reported ths value as 3.98 kcal/mol for the UA model of n-tetracosane (C 24 ) whch P αβ = [ m v α v β + ( r α r jα )F jβ ] V j>, (4) where the ndces and j refer to molecules and j, and F j s the force between the centers of mass of molecules and j. Frcton constant s obtaned from the tme ntegral of the force auto-correlaton (FAC) functon 20,21 : ζ = 1 τ kT dt f ( 0) f () t 0 (5) where f (t) = F (t) <F (t)> and F (t) s the total force exerted on molecule. Molecular frcton force s defned by ζ M = 1/ σ r = ζ/m = ζ/nm wth M molecular mass of n-alkane, n number of carbons, m mass of monomer and τ r the macroscopc relaxaton tme of the FAC. 21 Fgure 1. Plot of densty (g/cc) vs. T. The nset shows plot of D (10 5 cm 2 /sec) vs. T.

3 Smulaton Studes of Vscosty and Dffuson of n-alkane Bull. Korean Chem. Soc. 2011, Vol. 32, No Fgure 2. Arrhenus plot of D (10 5 cm 2 /sec) vs. 1/T. Fgure 4. A log-log plot of D (10 5 cm 2 /sec) vs. M (g/mol). s compared wth a value of 4.04 kcal/mol obtaned from vscosty measurements 23 and a value of 4.36 from PFG NMR results. 24 The values of E D are plotted n Fgure 3. As chan length n ncreases the ncrement of E D decreases, and t s expected to approach an asymptotc value as n ncreases further. It was reported that E D ncreases lnearly wth log M from 2.32 kcal/mol for n-heptane to 5.81 kcal/mol for n- hexacontane (C 60 ). 7,25 Flesher 5 also determned E D to be about 4.8 kcal/mol for several polyethylene ndependent of molecular weght from 9 to 52.7 kg/mol, from whch we can deduce that E D of n-alkane reaches an asymptotc lmt over C 100. We also show the log-log plot of dffuson constant versus molecular mass n Fgure 4. The obtaned exponents are between 2.36 and Recent expermental study reported that D~M α, wth a changng approxmately lnearly from 2.72 to 1.85 as T ncreases. 7 Thus the apparent actvaton energy also rses lnearly wth log M. In the absence of molecular entanglements, Rouse knetcs predcts α = 1, but Cohen-Tumbull-Bueche free-volume effects due to molecular chan ends add a further nonpower-law term, 26 enhancng D ncreasngly at low M. Vscosty. Vscostes of n-alkanes, η, are obtaned from the Green-Kubo relaton. It s worth note that the pressure auto-correlaton functons of long chans of n-alkanes at low temperatures have the non-decayng long-tme tals (not shown) and that t s mpossble to get the plateau values n the runnng tme ntegrals of the pressure auto-correlaton functons. For example, n ths study, C 60 at 318 K and C 80 at 318 K and 418 K. The temperature dependence of the calculated vscostes of lqud n-alkanes over the temper- Fgure 3. Actvaton energes of D (crcle) and η (square) vs. chan length n. Fgure 5. Arrhenus plot of η (cp) vs. 1/T.

4 3912 Bull. Korean Chem. Soc. 2011, Vol. 32, No. 11 Song H Lee Fgure 6. A log-log plot of η (cp) vs. M (g/mol). atures s also sutably descrbed by an Arrhenus plot except very hgh temperatures as shown n Fgure 5: η = C 0 exp( E η /RT) (7) where C 0 s the pre-exponental factor and E η s the actvaton energy of n-alkane vscosty. But the expermental result for the lnear polyethylene 6 showed that the lnes are slghtly curved, ndcatng that the temperature dependence, as n most polymers, does not follow the Arrhenus equaton. The actvaton energes obtaned from the slope of the least square ft except T over 1318 K are 3.71, 4.77, 6.29, 6.72, 7.17, 7.36, and 7.49 kcal/mol for C 12, C 20, C 28, C 36, C 44, C 60, and C 80, respectvely and are plotted n Fgure 3. It was expermentally reported for n-alkanes and lnear polyethylene 6 that the actvaton energy ncreases wth chan length and at the hghest molecular weght tested (M 4.4 kg/mol) the actvaton energy reaches 6.6 kcal/mol, whch s smlar to the average value found for NBS (6.7 kcal/mol) and the values reported by others 27,28 for hgh molecular weght lnear polyethylene ( kcal/mol). Fgure 6 shows the log-log plot of vscosty versus molecular mass. The obtaned exponents are between except C 12 (4.32). The expermental results for n-alkanes and lnear polyethylene 6 show that η s well descrbed by the power law η ~ M 1.8 at low molecular weght (M < 5 kg/mol) and η ~ M 3.64 at hgh molecular weght (M > 5 kg/mol) at 448 K. The molecular weght of the n-alkanes tested n ths study s much lower than ths expermental range of molecular weght, but the lowest exponent gves 2.41 at 818 K. Frcton Constant. It s well known that the calculaton of frcton constant from the force auto-correlaton functon s farly hard due to the non-decayng long-tme tals. As Kubo ponted out n hs fluctuaton-dsspaton theorem, 19 the correlaton functon of random force R wll decay n a tme nterval of τ c (mcroscopc tme or collson duraton tme), whereas that of the total force F has two parts, the short tme Fgure 7. Frcton constants, ζ (10 3 g/ps mol) vs. T. The nset shows plot of ζ/n vs. T. Fgure 8. Frcton constants, ζ (10 3 g/ps mol) vs. chan length n. The nset shows plot of ζ/n vs. n. From top, T = 318, 418, 518, 618, 718, 818, 918, 1118 and 1318 K, respectvely. or the fast part smlar to that of the random force and the slow part whch should just cancel the fast part n the tme ntegraton. 29 Ths means that the tme ntegral of Eq. (4) up to τ = s equal to zero. The tme ntegral n Eq. (4) attans a plateau value for τ satsfyng τ c << τ << τ r, f the upper lmt of the tme ntegral, Eq. (4), s chosen that τ c << τ << τ r because the slow tal of the correlaton functon s cut off. However, we were unable to get the plateau value n the runnng tme ntegral of the force auto-correlaton functon. Kubo suggested that the frcton constants should be obtaned from the random FAC functon not from the total FAC and that there exsts a dffculty to separate the random force part from the

5 Smulaton Studes of Vscosty and Dffuson of n-alkane Bull. Korean Chem. Soc. 2011, Vol. 32, No total force. We could obtan the frcton constants by the tme ntegral of the total FAC wth choosng the upper lmt of τ as the tme that the FAC has the frst negatve value by assumng that the fast random force correlaton ends at that tme. Fgure 7 shows the plot of ζ vs. T for gven n-alkanes. The calculated frcton constants gve a correct qualtatve trend: ζ decreases wth ncreasng T. However, the abnormalty n densty of the long n-alkanes at hgh temperatures s recovered n the frcton constant. For the short n-alkanes, ζ s nversely proportonal to T but for the long chan n-alkane, the densty loses the nverse decrement as temperature ncreases further and becomes flat at hgh temperatures. When ζ/n s plotted as a functon of T as shown n the nset of Fgure 7, the overall behavor of ζ/n for all n-alkanes s rather concded except at hgh temperatures. We plot ζ vs. n n Fgure 8 n whch ζ ncreases lnearly wth ncreasng n for a gven T. The slope of ζ vs. n decreases wth ncreasng T. An atomstc MD smulaton of lnear polyethylene melts 30 reported that the frcton constant extracted from D by applyng the Rouse model s seen to ncrease from value of short n-alkanes to a chan-lengthndependent plateau one, reached n a regon of n = The lnear ncrease of ζ wth ncreasng n as shown n Fgure 8 mght ndcate that the lqud n-alkanes over the whole temperatures consdered are far away from the Rouse regme. When ζ/n s plotted as a functon of n as shown n the nset of Fgure 8, ζ/n s almost ndependent of n for gven temperatures, whch ndcates the frcton s governed by the local segmental frcton not by the topologcal entanglement effect. Concluson We have presented the results for vscosty and selfdffuson constants of model systems for lqud n-alkane olgomers (C 12 ~C 80 ) at hgh temperatures (~2300 K) usng molecular dynamcs (MD) smulatons n a canoncal ensemble. The long chans of these n-alkanes at hgh temperatures show an abnormalty n densty. Calculated vscosty η and self-dffuson constants D are comparable wth expermental results and the temperature dependence of both η and D s sutably descrbed by the Arrhenus plot. The behavor of both actvaton energes, E n and E D, ndcates that the actvaton energes approach asymptotc values wth ncreasng chan length, whch s consstent wth the expermental observaton. The calculated monomerc frcton constant ζ shows an abnormalty agan as n densty. The slope of ζ vs. n decreases wth ncreasng T but does not provde a chan-length-ndependent plateau. Acknowledgments. Ths research was supported by a Kyungsung Unversty Research Grant n References 1. Berry, G. C.; Fox, T. G. Adv. Polym. Sc. 1968, 5, Fox, T. G.; Flory, P. J. J. Phys. Collod Chem. 1951, 55, Lodge, T. P.; Rotsten, N. A.; Prager, S. Adv. Chem. Phys. 1990, 9, Ferry, J. D. Vscoelastc Propertes of Polymers; 3rd ed.; Wley: New York, Flesher, G. Polym. Bull. (Berln) 1983, 9, Pearson, D. S.; Ver Strate, G.; von Meerwall, E.; Schllng, F. C. Macromolecules 1987, 20, Von Meerwall, E.; Beckman, S.; Jang, J.; Mattce, W. L. J. Chem. Phys. 1998, 108, Trrell, M. Rubber Chem. Technol. 1984, 57, De Gennes, P.-G. Scalng Concepts n Polymer Physcs; Cornell Unversty Press: Ithaca, New York, Mundy, C. J.; Sepmann, J. I.; Klen, M. L. J. Chem. Phys. 1995, 102, Cu, S. T.; Cummngs, P. T.; Cochran, H. D. J. Chem. Phys. 1996, 104, Cu, S. T.; Gupta, S. A.; Cummngs, P. T.; Cochran, H. D. J. Chem. Phys. 1996, 105, Evans, D. J. J. Chem. Phys. 1983, 78, Brown, D.; Clarke, J. H. R. Mol. Phys. 1984, 51, Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J. Am. Chem. Soc. 1984, 106, Gear, C. W. Numercal Intal Value Problems n Ordnary Dfferental Equaton; Prentce-Hall, Englewood Clffs, NJ, Andersen, H. J. Comput. Phys. 1984, 52, Allen, M. P.; Tldesley, D. J. Computer Smulaton of Lquds; Oxford: Oxford Unv. Press, 1987; p McQuarre, D. A. Statstcal Mechancs; Harper and Row: New York, Cccott, G.; Ferraro, M.; Hynes, J. T.; Kapral, R. J. Chem. Phys. 1990, 93, Kubo, R. Rep. Prog. Phys. 1966, 29, Mondello, M.; Grest, G. S. J. Chem. Phys. 1995, 103, Nederbragt, G. W.; Boelhouwer, J. W. M. Physca 1947, 13, See page 7161 n Reference Ertl, H.; Dullen, F. A. L. AIChE J. 1973, 19, Cohen, M. H.; Tumbull, D. J. Chem. Phys. 1959, 31, Mendelson, R. A.; Bowles, W. A.; Fner, F. L. J. Polym. Sc., Part A-2, 1970, 8, Raju, V. R.; Smth, G. G.; Marn, G.; Knox, J. R.; Graessley, W. W. J. Polym. Sc., Polym. Phys. Ed. 1979, 17, Kubo approxmately descrbed these two force auto-correlaton functons n hs orgnal papers, see Fg. 2 n Ref Harandars, V. A.; Mavrantzas, V. G.; Theodorou, D. N. Macromolecules 1998, 31, 7934.