J. Phys. Chem. 1996, 100, 16779-16781
16779
Determination of the Pressure-Viscosity Coefficient of Decane by Molecular Simulation Christopher J. Mundy* and Michael L. Klein Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6323
J. Ilja Siepmann Department of Chemistry, UniVersity of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431 ReceiVed: June 27, 1996X
The shear viscosities of n-decane and 4-propylheptane have been calculated over a range of densities. Equilibrium molecular dynamics simulations were carried out in the canonical ensemble for an empirical force field using the united-atom representation. Quantitative agreement is obtained with the experimental shear viscosities and the pressure-viscosity coefficient.
Introduction Knowledge of how the viscosity of a given molecular fluid behaves as a function of pressure and temperature is an important parameter in the field of rheology and in the design of lubricants. However, despite the important technological issues involved, a miscroscopic level understanding on the influence of molecular structure on transport processes is presently not available. Recently, we have shown that pertaining to the estimation of critical properties of complex fluids, the field of molecular simulations has reached a state of maturity that allows the successful prediction of valuable engineering parameters.1,2 In this letter, we set out to test the accuracies of empirical force fields and molecular simulations for the prediction of transport phenomena. In particular, we have utilized the molecular dynamics simulation technique to estimate the pressure dependence of the shear viscosity for n-decane. This quantity is commonly known as the R coefficient and is defined by the Barus relation3
η(P2) ) η(P1) exp[R(P2 - P1)]
(1)
where Pi is the pressure at state point i at constant temperature T, and η is the shear viscosity, being a function of only the pressure. We have also calculated the shear viscosity of a decane isomer, 4-propylheptane, to test our approach for branched molecules. To our knowledge, thus far, only one attempt to calculate the R coefficient of a molecular fluid by simulation has been reported. Cummings and Varner5 carried out an extensive nonequilibrium molecular dynamics study for liquid water and calculated the shear viscosity, as a function of shear rate (γ), over a range of pressures.4 In agreement with experiment, their simulations yield an increase in the viscosity with increasing pressure, but the R coefficient obtained from the simulations was larger than that measured. At all densities considered, the extrapolated zero-strain viscosities deviated from experimental values by a factor of approximately 2. The main part of the disagreement between experiments and simulations is likely caused by inadequacies in the force field used to described the water molecules.8,9 In a previous paper we reported on the calculation of the shear viscosity of n-decane at one state point.10 Rather good X
Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)01919-3 CCC: $12.00
agreement with experiment (deviation of less than 10%) has been found using an empirical force field which was previously derived by fitting to thermodynamic properties. Here, we extend these calculations to different state points with the aim of testing whether or not we are able to obtain a reliable estimate of the R coefficient for this prototype of a hydrocarbon lubricant. In the next section the force field and the simulation methodology are outlined. This is followed by a discussion of the simulation results. Particular attention is given to the relationship between the calculated viscosity and the longest relaxation time in the system. We conclude that current molecular simulation methodology is able to yield qualitative resuts for properties of “real” fluids. Simulation Details The force field for the alkanes has been obtained from calculations of the coexistence curves and critical points of the homologous series of linear alkanes1,2 and of three heptane isomers11 using configurational-bias Monte Carlo in the Gibbs ensemble. The united-atom approach, in which methylene, methyl, and ternary CH groups are treated as one pseudoatom, was found to yield a very satisfactory description of the fluid phase diagrams. Details of the models can be found in refs 1, 10, and 11. The simulations for the shear viscosity were performed for 64 alkane molecules in the canonical ensemble at T ) 480 K. Four simulations at different densities (F ) 0.5937, 0.6136, 0.6251, and 0.6382 g/cm3) were carried out for n-decane, whereas 4-propylheptane was only studied at the intermediate density of 0.6136 g/cm3. The corresponding pressures ranged from 1 to 30 MPa (see Table 1). At each state point a total of three independent simulations, each of lengths 900 ps, were carried out (i.e. starting from different wellequilibrated initial configurations). For technical details of the simulations, the reader should consult ref 7. The viscosity was computed using the standard Green-Kubo (GK) formula12 that relates the fluctuations in the stress tensor to the shear viscosity by
η(t) ) (V/kBT)∫0 〈Pxy(0) Pxy(t*)〉 dt* t
(2)
where Pxy is the xy component of the pressure tensor, V the volume, kB Boltzmann’s constant, and T the temperature. Results and Discussion A plot of the viscosity of n-decane as a function of pressure is given in Figure 1 and the numerical data can be found in © 1996 American Chemical Society
16780 J. Phys. Chem., Vol. 100, No. 42, 1996
Letters
TABLE 1: Summary of the Simulation Resultsa F molecules [g/cm3] n-decane n-decane n-decane n-decane 4-propylheptane
〈P〉 [MPa]
τr τr 〈η〉 ηexp (eq 3) (eq 4) -4 Pa s] [10 Pa s] [ps] [ps]
[10-4
0.5937 1.1 ( 1.0 1.41 ( 0.12 0.6136 10.6 ( 1.0 1.90 ( 0.15 0.6251 17.2 ( 1.0 2.31 ( 0.23 0.6382 28.1 ( 1.2 2.30 ( 0.40 0.6136 1.7 ( 0.8 1.60 ( 0.15
1.60 1.92 2.12 2.43 1.40
9.2 10.2 10.7 11.6 4.5
6.5 7.3 8.0 8.9 7.3
a F denotes the density, 〈P〉 and 〈η〉 are the canonical ensemble averages of pressure (including tail corrections) and viscosity, ηexp stands for the experimental viscosity taken from refs 10 and 12, and the two relaxation times τr are calculated from the orientational autocorrelation function (eq 3) and the Rouse model (eq 4).
Figure 2. Orientational autocorrelation function, c(t), for liquid n-decane (DEC) and 4-propylheptane (4PH) at F ) 0.6136 g/cm-3.
Figure 1. Shear viscosity, η, for liquid n-decane as a function of pressure at T ) 480 K.
Table 1. Over the entire pressure (density) range, agreement with experiment13 is quantitative (deviations of approximately 10%). With one exception (F ) 0.5937 g/cm3) the experimental results fall within the statistical accuracy of the simulations. From weighted linear fits using eq 1 the R coefficient for our model n-decane was determined as 21 ( 5 GPa-1 compared to the experimental value of 14 GPa-1. Most of the disagreement stems from the simulation at the lowest density which is very close to the coexistence density of our model n-decane at T ) 480 K.2 As is evident from the error bars in Figure 1, for a given temperature and the same MD simulation length, the precision associated with the calculated viscosity worsens as the pressure increases. This is to be expected, since at higher pressures the longest molecular relaxation time will, in general, increase. From calculations of various relaxation modes, we found that for n-decane, at the state points studied here, the slowest relaxation is associated with the orientational relaxation of a decane molecule, i.e., the time for a molecule to complete an end-overend rotation. This time can be monitored via the orientational autocorrelation function (OACF) of the normalized molecular end-to-end vector, denoted by u, e.g., c(t) ) 〈u(0)‚u(t)〉. The OACFs are shown in Figure 2, and the relaxation times, τr, can be extracted by fitting the decay to a simple exponential:
c(t) ) exp(-t/τr)
(3)
Knowledge of both the viscosity and the longest relaxation time allows for an interesting comparison with the modified Rouse model, which predicts for the relation of above quantities7,14
τr ) 6ηM/π FkT 2
(4)
where M is the molecular mass, F the density, and η the bulk viscosity. Values of τr calculated from eqs 3 and 4 are compared in Table 1. At all four densities the Rouse model predicts relaxation times approximately 30% smaller than those calculated directly from the OACFs, but this can be considered as reasonable agreement since the validity of the Rouse model assumes no excluded volume. The effects of entanglement clearly can be neglected for such a short molecule. The shear viscosity of 4-propylheptane is 1.6 × 10-4 Pa s at the state point under investigation (see Table 1). For this condition, we were not able to find an experimental result. For comparison, we have extrapolated experimental results at lower temperatures15 to the state point of interest using the Arrhenius relation. The so-estimated experimental shear viscosity is 1.4 × 10-4 Pa s and thus just outside of the certainty range of the simulation result. The OACF of 4-propylheptane is also shown in Figure 2. The deviation from the simulated and the calculated relaxation time estimated from eq 4 is most likely attributed to the fact that the side branches of 4-propylheptane are too short to behave like a polymer. In summary, equilibrium molecular dynamics calculations have been employed to determine the shear viscosities of n-decane and 4-propylheptane and the pressure-viscosity coefficient of the former. We have demonstrated that an alkane model fitted to the critical parameters can be used to obtain satisfactory values for the shear viscosity and the R-coefficient. However, the prediction of transport properties still lacks some of the accuracy that can be presently achieved for experimental thermodynamic and structural properties. Nevertheless, the agreement in the macroscopically measurable transport quantities is likely to be sufficient to allow insight into microscopic-level processes occurring in lubricating fluids. The methodology employed herein can readily be extended to more complex systems and hence should be a useful aid in the design of lubricants. Acknowledgment. The work described herein was supported by the National Science Foundation and Albemarle Corp. J.I.S. acknowledges additional support by the Camille and Henry Dreyfus New Faculty Awards Program. We thank Dixie Goins, Bill Moehle, Burney Lee, and Paul Kung for their encouragement. References and Notes (1) Siepmann, J. I.; Karaborni, S.; Smit, B. Nature 1993, 365, 330. (2) Smit, B.; Karaborni, S.; Siepmann, J. I. J. Chem. Phys. 1995, 102, 2126.
Letters (3) Barus, C. Am. J. Sci. 1893, 45, 87. (4) The viscosity at γ ) 0 has to be obtained from an extrapolation of the values obtained at finite (high) γ. However, the scaling relation which connects the finite-shear viscosity to the γ ) 0 value has not been determined unequivocally and is subject to considerable uncertainty.6,7 Cummings and Varner5 used the relation η ∝ γ1/2 to fit their results. (5) Cummings, P. T.; Varner, T. L. J. Chem. Phys. 1988, 89, 6391. (6) Chynoweth, S.; Michopoulos, Y. Mol. Phys. 1994, 81, 133. (7) Daivis, P. J.; Evans, D. J., J. Chem. Phys. 1994, 100, 541. (8) Smith, P. E.; van Gunsteren, W. F. Chem. Phys. Lett. 1993, 215, 315. (9) Balasubramanian, S.; Mundy, C. J.; Klein, M. L. J. Chem. Phys., submitted. (10) Mundy, C. J.; Siepmann, J. I.; Klein, M. L. J. Chem. Phys. 1995, 102, 3376. The pressure of the simulation reported in ref 7 is wrong due
J. Phys. Chem., Vol. 100, No. 42, 1996 16781 to a mistake in the evaluation of the tail corrections, the correct pressures are reported in this letter. (11) Siepmann, J. I.; Mundy, C. J.; Klein, M. L. Mol. Phys., submitted. (12) Evans, D. J.; Morriss, G. P. Statistical Mechanics of Nonequilibrium Liquids; Academic Press: New York, 1990. (13) Stephan, K., and Lucas, K., Viscosity of Dense Fluids; Plenum Press: New York, 1979. (14) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (15) Properties of Hydrocarbons of High Molecular Weight; American Petroleum Institute, Research Project 42, Washington, 1966.
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