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1 FluidPhaseEquilibria, 53 (1989) Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands 191 NON-EQUILIBRIUM MOLECULAR DYNAMICS CALCULATION OF THE TRANSPORT PROPERTIES OF CARBON DIOXIDE B. Y. WANG and P. T. CUMMINGS Department of Chemical Engineering Thornton Hall, University of Virginia Charlottesville, VA ABSTRACT We compute the thermal conductivity and diffusion coefficient of supercritical carbon dioxide along the 313 K isotherm using non-equilibrium molecular dynamics. Comparisons are made with experiment at four densities corresponding to pressures of 30, 70, 200 and 500 bar. Two intermolecular potential models for carbon dioxide are compared. INTRODUCTION In a previous paper (Wang and Cummings, 1988), we reported calculations of the shear viscosity of supercritical carbon dioxide along the 313 K isotherm using non-equilibrium molecular dynamics (NEMD). This paper continues this study of the transport properties of supercritical fluids and focuses on the thermal conductivity and the self-diffusion coefficient of carbon dioxide along the 313 K supercritical isotherm. These papers are part of our continuing research program aimed at studying the transport properties of supercritical fluids using NEMD. The NEMD algorithms for computing thermal conductivity and self-diffusion coefficient are similar to the one used for the calculation of shear viscosity. All three algorithms involve /89/$ Elsevier Science Publishers B.V.

2 192 simulating a system at steady state away from equilibrium, where the steady state is attained through the application of an external field. Th e ratio of the field-induced current to the field itself gives the transport coefficient of interest. It has been found that shear viscosity exhibits a strong dependence on the external field (which in the case of the shear viscosity is the strain field) leading to non-newtonian behavior (i.e. strain rate dependent shear viscosity) (Evans, 1983). However, little is known about the effects of the external fields employed in the NEMD algorithms for thermal conductivity and the self-diffusion coefficient. It is one of our objectives to examine these effects. One difference between the NEMD algorithm for shear viscosity and the NEMD algorithms for thermal conductivity and diffusivity is that in the former case, the strain field is physical and produces an experimentally realizable planar Couette flow. In the latter case, the applied fields are synthetic: they yield the correct hydrodynamic transport coefficients in the linear response regime, but outside the linear response regime the results obtained from simulation are not experimentally realizable (Evans and Morriss, 1984). In the simulation of the shear viscosity, we used three different intermolecular potential models: a simple spherically symmetric Lennard-Jones potential with parameters determined by fitting to the gas phase viscosity (Bird etal, 1960); a two-site Lennard-Jones model in which the oxygen centers are explicitly modeled (Gibbons and Klein, 1974); and a threesite model (in which the centers of the oxygen and carbon atoms are explicitly represented) with a quadrupole-quadrupole interaction to represent the electrostatic interaction due to the permanent quadrupole moment of CO2 (Murthy and Singer, 1981). The simulated shear viscosity using the three-site model yielded the best agreement with the experimental data. In this study, we choose the three-site intermolecular potential model. We also use the simple spherically symmetric potential model for the sake of comparison. In the next section, the NEMD algorithms are briefly described. In the section following, our results are described and compared with experimental data where available. We conclude with a summary of our findings. NEMD ALGORITHMS FOR THERMAL CONDUCTIVITY AND DIFFUSIVITY The NEMD algorithms for thermal conductivity and diffusivity for simple fluids are described by Evans (1986). For molecular fluids, one simply needs to augment these equations with the rotational equations of motion. The equations of motion for a simulated system of N molecules with a fictitious external field $, which essentially plays the same role as the temperature gradient VT in experiment, are 1 (1) where E, is the instantaneous configurational energy of molecule i, E is the average total

3 193 energy, E = CiEi/N. The rotational equations of motion are the same singularity-free equations (Evans and Murad, 1977) used in our previous paper (Wang and Cummings, 1988). The parameter (Y, the thermostating multiplier, is used to constrain the translational motion so that the translational kinetic energy is fixed at the required temperature. The functional form of the equation containing LY is obtained from application of Gauss s principle of least constraint (Evans et al., 1983). Th e isokinetic requirement yields The boundary conditions imposed on the simulated system are the usual periodic boundary conditions employed in equilibrium molecular dynamics simulations which are described adequated elsewhere (Allen and Tildesley, 1987). The constitutive relation between the heat current JQ and the thermal conductivity X is given by Evans (1986) + A=& (3) e In this equation, T is the temperature of the simulated system, and & is the microscopic heat current given by (4) The NEMD algorithm for computing the diffusion coefficient is based on applying a synthetic color field to the simulated system. This color field acts in the same way as a chemical potential gradient which drives diffusion. The translational equations of motion are given by Evans and Morriss (1984) dc/dt = p7 Jm (5) dpzi/dt = Fzi - XdCi dpgildt = Fyi - Aspy; (6) dpzi Jdt = Fzi - Xspzi I where c, is the color charge of molecule i, c; = (-l)i, The parameter Xd is the driving constraint of the color field. The function form of Xd follows from the application of Gauss s principle of least constraint where the constraint of color field is T,E-Jc=O (7) In the equation, J, is the color current applied to the simulated system. The thermostating constraint parameter, X,, is obtained in similar fashion to the thermostating constraint a in the NEMD algorithm for thermal conductivit,y. Thus, we have the following equations for Xd snd X,: Xd = k Ci CjF,; As = C;(pigFig + PizFiz)/ Ci(PTy + Pfz) (8)

4 194 Table 1: NEMD results of thermal conductivity for the one-site model of CO2 along the 313 K isotherm. F, Timesteps u f/(nk*t) x x lo3 (ky?m3) i,b/ (W/m I<) x lo x 10s x 10s x lo x x Table 2: NEMD results of self-diffusion coefficient for the one-site model of CO* along the 313 K isotherm. (ky/4n3) J,* Timesteps u /(NkBT) D, x 10 (cm2/s) The work, Pd, generated by color constraint Ad, is given by: pd = Ad c &hi (9) I The self-diffusion coefficient, D,, is then obtained from D =_(N-l) J, k T.9 7-B < Pd > (10) where kg is Boltzmann s constant and < Pd > is the ensemble average of Pd. RESULTS AND CONCLUSIONS In Tables 1 and 3, we report the pressure, configurational internal energy and thermal conductivity for the one-site and two-site model of COs; in Tables 2 and 4, we report the configurational internal energy and self-diffusion coefficient. The simulations of the one-site model were performed on systems of 108 molecules with time step 1.32 x lo-l4 set; the threesite model simulations were performed on systems of 125 molecules with time step lo-l5 sec.

5 195 rable 3: NEMD results of thermal conductivity for the three-site model of CO2 along the 313 K isotherm. (ky;m ) F, Timesteps p ucon /(NkgT) x x lo3 (atm) (W/m. K) Table 4: NEMD results of self-diffusion coefficient for the three-site model of CO2 along the 313 K isotherm. (ks/4n3) J, Timesteps u f/(nk,t) D, x lo4 (cm /s) The spherical cutoff of the intermolecular potential is 12Afor the one-site simulations and loafor the three-site model. The diffusion coefficient simulations for the three-site model were computed on a CSPI bit word array processor (attached to a VAX 11/750 located in the Center for Computer-Aided Engineering at the University of Virginia) and the rest of the simulations were carried out on a IBM 3090/100 (located in the Academic Computing Center at the University of Virginia). At the state conditions studied, the onesite model is apparently more supercritical than the three-site model as is evidenced by the much higher pressures for the one-site model. From Table 5, the diffusion coefficient of the three-site model yields the best agreement with experimental data (which was not available at the highest pressure). The thermal con- ductivity of the three-site model given in Table 5 agrees well with experiment at low pressure

6 196 A : p=840.8 kg VI-~ I: p=992.1 kg ni UW~~) a n-- a o,o~~ ~---.._*.._~ I OO I I I I Fe Figure 1: The thermal conductivity X for the one-site model (dashed lines) and the three-site model (solid line) of CO2 as a function of F,*. The symbols are simulation results (to which the straight lines are least squared fits) and the arrows indicate the experimental data. I p= k.g mm _ = --- _& p=199.8 kg me 0 0 D,x104 (cm2/s) Figure 2: The self-diffusion coefficient D, for the one-site model (on the left) and three-site model (on the right) of COs as a function of Jz. The symbols, lines, and arrows have the same significance as in Figure 1.

7 197 Table 5: Comparison between thermal conductivity and diffusivity calculated via NEMD and experimental data (Vargaftik, 1983; McHugh and Krukonis, 1986). (JVk3) Model ANEMD x lo3 &,, X lo3 DJNEMD X lo4 Dsezp x lo4 (at4 (at4 (W/m I<) (Wlm. K) One-site Three-site 9.4 One-site Three-site One-site unavailable Three-site but overestimates the experimental value at the higher pressure. The simulation results and experimental values are also shown in Figures 1 and 2. In our previous study of the shear viscosity of COs, the shear viscosity exhibits the same behavior (slight underestimate at low density, overestimate at high density). The similarity in results is likely due to the similarity in collisional mechanisms for momentum and kinetic energy transfer. At the lowest density state point, which is nearest of those considered to the experimental critical point of CO2 where the critical temperature T = K and critical pressure P = 72.8 atm, the thermal conductivities obtained from NEMD for the three-site model are negative, while for the one-site model they are very large (see Tables 1 and 3). Evans (1986) noted this behavior in simple fluid systems and proposed that it is due to long-time tail effects in the conductivity with respect to the external field. In order to verify that this behavior only occurs near the critical point, we performed a simulation for which the pressure (30 atm) is much lower than the critical pressure. As shown in Table 3, at this state point the NEMD thermal conductivity is positive and agrees well with experiment. One final note is that all properties appear to behave nearly linearly as a function of applied synthetic field. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of this research by the National Science Foundation through grant CBT and through equipment grant CPE which provided matching fundings for the purchase of the CSPI 6420 array processor used in this research. We are grateful to Professor Sohail Murad for helpful discussions. LIST OF SYMBOLS color charge in NEMD algorithm for diffusivity >, self-diffusion coefficient E average configurational energy & instantaneous configurational energy of molecule i Fc fictitious field in NEMD algorithm for thermal conductivity

8 198 pi 43 JC JQ k.b In N p pd q(t) ; VT t V * x Ad x, P force on the center of mass of molecule i force on molecule i due to molecule j color field in NEMD algorithm for diffusivity heat current in NEMD algorithm for thermal conductivity Boltzmann s constant mass of a molecule number of molecule linear momentum vector work generated by color constraint in NEMD algorithm for diffusivity spatial coordinates position vector temperature temperature gradient time volume thermostatting constraint in NEMD algorithm for thermal conductivity thermal conductivity driving constraint of the color field thermostatting constraint in NEMD algorithm for diffusivity density REFERENCES Allen, M. P. and Tildesley, D. J., Computer Simulation of Liquids, First Edition, Clarendon Press, Oxford, p. 24. Bird, R. B., Stewart, W. E. and Lightfoot, E. N., Transport Phenomena, Wiley and Sons, New York, Appendix B. Evans, D. J., Computer experiment for nonlinear thermodynamics of Couette flow. J. Chem. Phys., 78: Evans, D. J., Thermal conductivity of the Lennard-Jones fluids. Phys. Rev. A, 34: Evans, D. J., Hoover, W. G., Failor, B. H., Moran B. and Ladd, A. J. C., Nonequilib- rium molecular dynamics via Gauss s principle of least constraint. Phys. Rev. A, 28: Evans, D. J. and Morriss, G. P., Non-Newtonian molecular dynamics. Comp. Phys. Rep., 1: Evans, D. J. and Murad, S., Singularity-free algorithm for molecular dynamics simulation of rigid polyatomics. Mol. Phys., 34: Gibbons, T. G. and Klein, M. L., J. Chem. Phys., 60: 112. McHugh, M. A. and Krukonis, V. J., Supercritical Fluid Extraction, Principles and Practice. Butterworths, Boston. Murthy, C. S. and Singer, K., Interaction site models for Carbon Dioxide. Mol. Phys.. 44, 1: Vargaftik, N. B., Handbook of Physical Properties of Liquids and Gases. Second Edition, Hemisphere, New York. Wang, B. Y. and Cummings, P. T., Non-equilibrium Molecular Dynamics CalcuIa- tion of the Shear Viscosity of Carbon Dioxide. Int. J. Thermophysics, accepted for publication.

Shear viscosity of liquid rubidium at the triple point

J. Phys. F: Met. Phys. 18 (1988) 1439-1447. Printed in the UK Shear viscosity of liquid rubidium at the triple point P T Cummingsi- and G P Morriss-l: i Department of Chemical Engineering, Thornton Hall,