Thermodynamic and transport properties of carbon dioxide from molecular simulation

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1 Thermodynamic and transport properties of carbon dioxide from molecular simulation Carlos Nieto-Draghi, Theodorus de Bruin, Javier Pérez-Pellitero, Josep Bonet Avalos, and Allan D. Mackie Citation: The Journal of Chemical Physics 126, (2007); doi: / View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Transport properties of carbon dioxide and methane from molecular dynamics simulations J. Chem. Phys. 141, (2014); / Transport properties of C O 2 -expanded acetonitrile from molecular dynamics simulations J. Chem. Phys. 126, (2007); / Molecular dynamics simulations of transport and separation of carbon dioxide alkane mixtures in carbon nanopores J. Chem. Phys. 120, 8172 (2004); / Calculation of the transport properties of carbon dioxide. II. Thermal conductivity and thermomagnetic effects J. Chem. Phys. 120, 7987 (2004); / Structural and thermodynamic properties of fluid carbon dioxide from a new ab initio potential energy surface J. Chem. Phys. 109, 3153 (1998); /

2 THE JOURNAL OF CHEMICAL PHYSICS 126, Thermodynamic and transport properties of carbon dioxide from molecular simulation Carlos Nieto-Draghi Departament d Enginyeria Química, ETSEQ, Universitat Rovira i Virgili, Avinguda. dels Països Catalans 26, Tarragona, Spain and IFP, 1-4 Avenue de Bois-Préau, Rueil-Malmaison Cedex, France Theodorus de Bruin IFP, 1-4 Avenue de Bois-Préau, Rueil-Malmaison Cedex, France Javier Pérez-Pellitero, Josep Bonet Avalos, and Allan D. Mackie Departament d Enginyeria Química, ETSEQ, Universitat Rovira i Virgili, Avinguda. dels Països Catalans 26, Tarragona, Spain Received 19 September 2006; accepted 28 December 2006; published online 14 February 2007 Monte Carlo and molecular dynamics simulations have been used in order to test the ability of a three center intermolecular potential for carbon dioxide to reproduce literature experimental thermophysical values. In particular, both the shear viscosity under supercritical conditions and along the phase coexistence line, as well as the thermal conductivity under supercritical conditions, have been calculated. Together with the already reported excellent agreement for the phase coexistence densities, the authors find that the agreement with experimental values is, in general, good, except for the thermal conductivity at low density. Although extended versions of the model were employed, which include an explicit account of bending and vibrational degrees of freedom, a significant difference was still found with respect to the reported experimental value American Institute of Physics. DOI: / I. INTRODUCTION Although molecular simulation has been widely promoted as a way of predicting the physical properties of fluids, the number of articles where these techniques are employed to make quantitative calculations is fewer than could a priori be expected. This is particularly true in the case of dynamic properties where the amount of published quantitative work is surprisingly scarce. Even in the case of carbon dioxide, a widely present gas of current concern when found in high levels in the earth s atmosphere as well as a common compound in many industrial and biological processes, the number of detailed simulation studies of its dynamic and thermodynamic properties is limited. Most analyses restrict themselves to validating phase equilibrium properties of carbon dioxide, whereas dynamic properties such as the viscosity or thermal conductivity have received less attention, although specific mention can be made of a series of articles of over ten years ago by Wang and Cummings 1 5 as well as the works of Steinebrunner et al. 6 and Huber et al. 7 Knowledge of the dynamic properties of CO 2 and its mixtures with other compounds is required, for example, in the design of chemical process equipment and oil reservoir production in order to complement the insufficient experimental data available. There are two elements required to successfully predict physical properties in molecular simulation. The first part is the development of a suitable interaction potential model, usually a semiempirical classical model where the parameters are adjusted to available experimental data. The second part requires appropriate molecular simulation techniques in order to estimate the required properties as efficiently as possible. As far as the intermolecular potential is concerned, most carbon dioxide models use a three center site-site Lennard-Jones interaction model with either quadrupole moments 8 or partial charges on each of the three interaction sites In general, all the models use a rigid linear structure, where the OCO angle is fixed to 180. In addition, we will also consider a semiflexible version where we study the effect of allowing the OCO angle to fluctuate, as well as a fully flexible model where, also, the CvO bond distance can vary. Perhaps the most well known model for carbon dioxide and the one that is used in this work is the elementary physical model rescaled to reproduce critical properties EPM2 due to Harris and Yung. 9 Although developed over ten years ago, this intermolecular potential is still widely used due to the excellent agreement with experimental liquid-vapor coexistence data. In Sec. III, all the details of the models employed are given, including the parameters of the EPM2 model. More recently, a detailed study of various carbon dioxide models including EPM2 was carried out by Zhang and Duan. 11 However, for the dynamic properties, only the self-diffusion coefficient was considered by these authors. With respect to molecular simulation techniques, there has been a constant evolution towards ever more efficient algorithms, although the most important benefit has come from the constant increase of computer power. In the case of Monte Carlo simulation for the estimation of phase equilibrium properties, one of the key advances has been the Gibbs ensemble technique used in this work. 12 Particular mention can also be made of the histogram reweighting /2007/126 6 /064509/8/$ , American Institute of Physics

3 Nieto-Draghi et al. J. Chem. Phys. 126, methodology, 13,14 which has certain advantages, especially for the accurate estimation of critical points. Molecular dynamics has also experienced important progress, although the nature of the methodology is less flexible and significant methodological changes are difficult to develop. The most relevant methodologies are summarized in many textbooks. 15,16 However, as far as nonequilibrium simulations are concerned, we can mention the PeX momentum exchange algorithms used in this work to obtain the thermal conductivity. With respect to approaches that fix the thermal gradient, PeX is a momentum exchange algorithm that essentially fixes the average heat flow, conserving, in addition, the overall momentum and energy. Furthermore, the histogram reweighting methodology has also been introduced for the evaluation of dynamic properties and transport coefficients for systems in thermodynamic equilibrium. 20 In Sec. II, an overview of the technical details for the calculation of the dynamic properties, the shear viscosity, as well as thermal conductivity is given. Section III describes the intermolecular potential, and Sec. IV presents the simulation details for both the Monte Carlo as well as molecular dynamics simulations. In Sec. V, we first present the equilibrium data from the Monte Carlo simulations that we then use in the molecular dynamics simulations to estimate the shear viscosity along the coexistence curve as well as the shear viscosity, thermal conductivity, and self-diffusivity at supercritical conditions. We finish with conclusions in Sec. VI. II. TECHNICAL BACKGROUND A. The Einstein relation for the shear viscosity The shear viscosity has been calculated either from NVT or NPT ensemble simulations. Due to a lack of statistics, the integral of the pressure tensor time correlation function, particularly at high densities, presents uncertainties that are reflected in the values of the viscosity obtained, through a Green-Kubo expression. However, the use of the Einstein relation 21 permits an appropriate fitting of the asymptotic behavior, strongly reducing the uncertainty in the value of the viscosity obtained in this way. Consequently, the Einstein relation has been employed as described by Smith and van Gunsteren, 21 but, contrary to the original work, we use all the elements of the stress tensor 22 to improve convergence and statistics. Thus, the viscosity coefficient is obtained from the expression = 1 20 V k B T lim t d dt P T t 2 +2 P t 2. Here, and are indices running over the three Cartesian coordinates, V is the volume, T is the temperature, and P t denotes the displacement of the elements of the pressure tensor P, according to and P t = 0 t P + P d 2 P T t = 0 t P 3 1 P d. The microscopic expression for the elements of the pressure tensor P appearing in the integrand of Eqs. 2 and 3 is given by P t = 1 V i p i t p i t m i + f ij t r ij t. i j In Eq. 4, p i is the component of the momentum of particle i, f ij is the component of the force exerted on particle i by particle j, and r ij is the component of the particle-particle vector, r ij r j r i. The viscosity is obtained from the slope of Eq. 1, always after some initial time, where the displacement is not a linear function of time. B. Thermal conductivity The computation of the thermal conductivity can in principle also be achieved from the Green-Kubo formalism with simulations of thermal equilibrium systems. However, for reasons of better statistical accuracy for the same computational effort, in this work we have adopted a nonequilibrium scheme based on the PeX algorithm in which an energy flow between two distinguished regions slabs in the system is imposed by means of virtual collisions between pairs of particles inside different slabs. The energy transferred per unit time due to the collision is equal, in steady state, to the total energy flow between the slabs. Thus, given the energy flow, the direct measure of the resulting thermal gradient permits the determination of the thermal conductivity from their ratio see Eq. 6. In this scheme, the total momentum and energy of the system are naturally held constant since the virtual collisions conserve these properties. Furthermore, in order to keep the average temperature of the system constant, this method can be applied together with the temperature rescaling procedure of Berendsen through a weak coupling to the thermal bath. 23 This thermostat fixes the average temperature of the system and preserves the temperature gradient. Moreover, the simulation box has periodic boundary conditions in the three dimensions of space. In particular, in the z direction, one considers two slabs of a given thickness, large enough to contain many particles on average but much smaller than the length of the box in the z direction. To maintain a temperature gradient, the particle with the highest kinetic energy in the so-called cold slab is selected with frequency to experiment a virtual elastic collision with the corresponding particle with the lowest kinetic energy in the hot slab. Consequently, after the elastic collision the particles exchange their momentum and hence their kinetic energy. Thus, the energy exchange induced by this exchange of momentum is added to the value of the accumulated energy exchanged by the two slabs. The heat flow at steady state is consequently given by the relation J z t = 1 2At transfers t,i m h 2 v new 2 old h v 2 h, where the sum runs over all the virtual collision events up to time t. In this expression, A is the cross sectional area. Fur

4 Thermodynamic and transport properties of CO 2 J. Chem. Phys. 126, TABLE I. Intermolecular potential parameters of the Harris and Yung 9 model, where the harmonic bending potential parameter is K b =1236 kj/mol rad 2 in the semiflexible version. Site Å K Charge e C O thermore, J z t is the heat flux density in the z direction i.e., the direction of the imposed temperature gradient and v old h and v new h stand, respectively, for the velocities of the particle before and after the collision. Equation 5 can be evaluated indistinctly from the cold or hot side. 17 With this special configuration, heat is delivered from the center to the right and left cold slabs. The heat flux is then half of the energy pumped by the virtual collisions per unit of time. It is required that the number of particles in both slabs is sufficiently large so that the collisions inside the slabs properly thermalize the velocity distribution before a new collision. In the case of low density systems, it is important to reduce the frequency of collision to avoid huge temperature gradients that may cause abrupt density changes in the simulation box. These requirements are fulfilled in our case. For gas densities a smaller frequency has been selected fs 1 than for the case of liquid densities fs 1. Once the thermal gradient has been stabilized and the process is stationary, the thermal conductivity can be obtained from the relation = J z t dt/dz, 6 where dt/dz is the resulting temperature gradient in the z direction. 19 III. INTERMOLECULAR POTENTIAL The thermodynamic and transport properties of pure CO 2 have been computed using Monte Carlo and molecular dynamics simulations, respectively. As mentioned in the Introduction, the EPM2 potential from the Harris and Yung 9 model has been employed to simulate the carbon dioxide molecules. According to this model, each carbon dioxide molecule consists of three Lennard-Jones centers and three electrostatic charges centered at each atom. The site to site intermolecular energy is described as follows: U ij =4 ii 12 r ij r ij 6 + q iq j, r ij where q i is the partial charge on site i, ij and ij are the Lennard-Jones interactions between sites i and j on different molecules, and r ij = r j r i is the separation distance between the corresponding sites. In Table I the values of the Lennard- Jones parameters and charges are given. Cross interactions were computed by the Lorentz-Berthelot rules. In the rigid version the carbon-oxygen bond lengths are fixed and equal to Å, and the molecule has a fixed OCO angle of 180. We have also tested two other versions of the EPM2 model: a semiflexible one with a harmonic bending potential K b =1236 kj/mol rad 2 as given by Harris and Yung 9 and a fully flexible version that includes the vibrational modes of the CvO bond distance. In the fully flexible case, we have described the bond distance using a harmonic potential with K v = kj/mol Å 2 on a per mole of oscillators basis which we obtained with quantum mechanical calculations performed with the GAUSSIAN 03 suite of programs. 24 The B3LYP functional 25 has been applied in combination with the 6-311G d all-electron basis set 26 with spinrestricted singlet wave functions. The geometry relaxation has been carried out in a symmetry constrained linear optimization using the default convergence criteria for the energy, forces, and atomic displacements, followed by the analytical evaluation of the Hessian matrix to derive the force constants of the vibrational modes. IV. SIMULATION DETAILS The saturation pressure and coexistence density of carbon dioxide were calculated using the Gibbs ensemble Monte Carlo method. 12 The simulations were performed at different temperatures, imposing constant global volume between the two simulation cells. The implemented Monte Carlo moves were translations, rigid body rotations, molecular transfers, and volume interchanges between the two simulation cells of the Gibbs ensemble. The probabilities were set to 0.2 for translations, 0.1 for rotations, for transfers, and for volume interchanges. The total number of molecules used in the simulations was 333, and the cutoff distance for all the site-to-site interactions was set equal to half the box length where standard long-range corrections were included for the Lennard-Jones interactions. In addition, we have performed molecular dynamics simulations with two different ensembles. The viscosity coefficient has been obtained using equilibrium molecular dynamic simulations either in the NVT ensemble or in the NPT ensemble, using a weak coupling bath 23 with long-range corrections for pressure and energy. 15 The thermal conductivity coefficient has been computed, as previously mentioned, using a constant temperature PeX nonequilibrium molecular dynamics scheme. 17 All simulations have been carried out with 300 or 500 depending on the density molecules except for the case of the thermal conductivity at supercritical conditions and low density, where 1000 molecules have been required to ensure that there are sufficient particles in the system. The size of the simulation box for the viscosity computation in the NVT ensemble under coexistence conditions has been adjusted to reproduce the density obtained from the Gibbs ensemble calculations. For supercritical conditions, we employed the same density as reported in the experimental thermal conductivity data. 27 A cubic box is used for the viscosity computation and a parallelepiped to determine the thermal conductivity coefficient via nonequilibrium simulations. In the latter, we have chosen l z =2l x =2l y, where l x,l y,l z are the dimensions of the box. The equations of motion have been integrated using the leapfrog 15 algorithm with a time step of 2 fs 0.5 fs for the model with internal vibration, while leapfrog implicit quaternions 28,29 have been used to integrate the rotational part of the equations of motions. All simulations have been performed with periodic boundary

5 Nieto-Draghi et al. J. Chem. Phys. 126, conditions and the reaction field methodology 30 with the choice RF 0 to account for the long-range electrostatic interactions. The reaction field and Lennard-Jones cutoff length are set to times the characteristic size of the oxygen atom of the CO 2 molecule depending on the density. A nearest neighbor list technique, 15 with a cutoff radius of 1.1 times the Lennard-Jones cutoff, was also employed. For the viscosity computations, an equilibration run of 500 ps was carried out prior to each 5 ns production run in order to eliminate any memory of the initial conditions. Similarly, a 5 ns simulation was run for the thermal conductivity computations. V. RESULTS The simulation conditions have been chosen in order to be able to compare against experimental data. In the case of the shear viscosity, data exist at coexistence conditions for both the liquid and vapor phases. In order to carry out the molecular dynamics simulations, it is necessary to provide either the coexistence densities or saturation pressures. Rather than using the experimental coexistence data, we decided to independently estimate the thermodynamic properties obtained by the model. In this way we avoid the possibility of entering the two phase region during the molecular dynamics simulations in the NVT ensemble and, in addition, both the equilibrium and dynamic properties are predicted in a consistent manner. In Fig. 1 we plot the simulation results for the liquidvapor coexistence densities versus temperature from the Gibbs ensemble and the experimental values. As expected, excellent agreement is found, although the model values are found to be slightly within the reported experimental values, see Table II. Together with the coexistence densities, the saturation pressures have also been calculated during the same simulations. In Fig. 2 the resulting model values are plotted against the experimental data. Although the agreement is still reasonable, there is a larger deviation than with the coexistence densities and the model tends to overpredict the saturation pressure by more than 10%. The shear viscosity has been estimated using two different approaches. In the first case, NPT simulations have been employed using the saturated pressures, as given in Fig. 2 from the Gibbs ensemble simulations. In the second, NVT simulations have been carried out using the saturated liquid and vapor densities from the Gibbs ensemble simulations see Fig. 1. In Fig. 3 and Table II a comparison between the experimental variation of the shear viscosity of CO 2 along the liquid vapor equilibrium coexistence curve and our simulation results is given. In general, we observe good agreement between our simulation results and the experimental data for both the liquid and the vapor phases. However, the liquid viscosity is underestimated by up to 15% in the lowtemperature and high density liquid region. For the NPT simulations, it is difficult to maintain the simulations in one phase because we are working precisely at the saturation pressure. At these conditions, the fluctuations of the density in the system can affect the viscosity calculations causing the viscosity of the liquid phase to be lower than expected. In FIG. 1. Comparison of our simulation data for the EPM2 model with experimental liquid-vapor coexistence densities Ref. 32 of CO 2. order to check the reliability of these calculations, we have carried out simulations in the NVT ensemble using the coexistence densities. As can be seen from Fig. 3, the NVT results are in agreement with the NPT results. Hence, we expect that the deviation for the low-temperature liquid phase is due to the fact that the model liquid coexistence densities are slightly less than the experimental ones. In Fig. 4, the shear viscosity along the saturated liquid line is plotted as a function of density. As can be seen, there is an excellent agreement between the simulation values and the experimental results. There is also an excellent agreement between the results from the NVT and NPT ensembles. As before, the only significant difference is found at the highest density, which corresponds to the lowest temperature, where the simulations underpredict the experimental data. In addition to the simulations along the coexistence curve, simulations have also been carried out under supercritical conditions, where experimental data are available for the density variation of the shear viscosity at K, 31 see Table III. In this case, the experimental density has been used as input for the NVT ensemble simulations. As can be seen from Fig. 5, there is an excellent agreement between the simulation and experimental values over the full range of densities including the high density region. This fact corroborates our previous observation that the deviations found for the simulations along the coexistence line for the high density liquid phase are due to the underprediction of the liquid coexistence density by the model. For the thermal conductivity, experimental data are available along the supercritical isotherm at 470 K from Johns et al. 27 as well as from correlations from NIST Ref. 32 and from Rah and Eu. 33 Molecular dynamics simulations have been carried out both at high and low densities at this temperature. The results are given in Fig. 6 and Table III.At the high or liquidlike density, reasonable agreement between the simulation and the extrapolation of the experimental

6 Thermodynamic and transport properties of CO 2 J. Chem. Phys. 126, TABLE II. Comparison of simulation results for the liquid-vapor equilibrium obtained with the EPM2 model with NIST data Ref. 32. Density in kg/m 3, vapor pressure P sat in kpa, and shear viscosity in Pa s. Statistical uncertainty of the simulation results is 0.2% for the density, 0.5% for saturation pressure, and 0.5% for the shear viscosity. T K Ensemble Sim. Expt. % Dev. Ensemble Sim. Expt. %Dev l GEMC v GEMC l NPT NVT P sat GEMC v NPT v v NPT l GEMC l NPT NVT v GEMC v NPT P sat GEMC v v NPT l GEMC l NPT NVT v GEMC P sat GEMC v v NPT l GEMC l NPT v GEMC P sat GEMC Avg. Dev. l 0.9 l NPT 8 v 15 NVT 8 P sat 11 v NPT 7 value is found; although a slight underprediction of 9% is observed. However, at the low density conditions at =300 kg/m 3 the simulation yields a value 35% below the experimental value, in a system with 300 molecules and a rigid model potential. This discrepancy deserves further analysis. In order for the nonequilibrium molecular dynamics simulations to be reliable, the number of particles in each of the two slabs used to provide the temperature gradient must be large enough to allow for the velocity to be properly thermalized before the next hypothetical collision takes place. To check this, the low density simulation has been repeated with FIG. 2. Comparison of our simulation data for the EPM2 model with the experimental saturation pressure Ref. 32 of CO 2 along the liquid-vapor coexistence curve. FIG. 3. Comparison of our NVT and NPT ensemble simulations with the experimental shear viscosity Ref. 32 of CO 2 along the liquid-vapor coexistence curve.

7 Nieto-Draghi et al. J. Chem. Phys. 126, FIG. 4. Comparison of our NVT and NPT ensemble simulations with the experimental shear viscosity Ref. 32 of CO 2 along the saturated liquid line. a system of 1000 carbon dioxide molecules; however, the resulting thermal conductivity was found to be practically the same as for the smaller system see Fig. 6. Another important check on the correctness of the simulations is obtained by observing the temperature gradient along the z axis of the simulation cell. Figure 7 shows the temperature for the original rigid EPM2 model as an example. A reasonably linear behavior is observed as expected for the steady state transfer of heat in our simulation cell. Furthermore, the local variations of the temperature gradient with respect to the overall temperature gradient are much smaller than those required to produce the observed deviation of the thermal conductivity from the experimental value. Thus, provided that the simulations are carried out properly inside the range of validity of the overall approach, we have turned our attention to other possible sources of discrepancy between simulated and experimental data, namely, the validity of the intermolecular potential, especially in the determination of the thermal conductivity at low densities. Since the degrees of freedom associated with the flexibility of the molecule are important in the transport of heat, we decided to implement flexible versions of the model and analyze the effect of these changes on the thermal conductivity. In particular, two versions of the model have been tested, one due to Harris and Yung 9 where the molecule is allowed to bend, and a second, developed in this work, where, in addition to the O C O bending, the C O distance is also allowed to fluctuate. As can be seen from Fig. 6, although these two flexible models yield a thermal conductivity significantly closer to the experimental value than the rigid model, there still remains a large difference that is not due to the simplicity of the latter, 22% and 23%, respectively. Once again, we have checked that the temperature gradient in these simulations is reasonably linear see Fig. 7. Hence, we face the following facts: on the one hand, even when all the possible degrees of freedom are included in the model, the prediction of the thermal conductivity at low density is still not in agreement with the experimental value, while excellent agreement is found for viscosity as well as for the thermal conductivity at higher densities. On the other hand, the energy transport due to the motion of the particles themselves carrying energy in its degrees of freedom at low densities is significant compared with the collisional energy transport which dominates at high densities. Thus, the details of the intermolecular potentials must be less relevant in the determination of collective properties in low density than in high density phases. However, Bock et al. 34 report excellent agreement from kinetic theory calculations at the very low density limit of the thermal conductivity from an optimized quantum mechanical intermolecular potential for carbon dioxide. We thus believe that an increase in the accuracy of the thermal conductivity at low density might require a quantum treatment of the atomic vibrations inside the molecule. An empirical check of this hypothesis lies unfortunately beyond the scope of this work, since it would require a completely different approach incorporating the molecular dynamics simulation of the quantum motions of, at least, the three TABLE III. Comparison of the shear viscosity and thermal conductivity obtained with the EPM2 model with the experiments of Johns et al. Ref. 27 and the correlation of Rah and Eu Ref. 33. Density in kg/m 3, shear viscosity in Pa s, and thermal conductivity in W/m K. Statistical uncertainty of the simulation results is 0.5% for the shear viscosity and less than 1% for the thermal conductivity. T K N m Model Sim. Expt. % Dev. Sim. Expt. %Dev EPM2-rigid EPM2-bending 29.4 Corr EPM2-rigid 27.7 Corr EPM2-rigid EPM2-flexible EPM2-rigid EPM2-rigid EPM2-rigid 86.3 Expt. a Corr EPM2-rigid EPM2-rigid EPM2-rigid Avg. Dev. 9 a Reference 38.

8 Thermodynamic and transport properties of CO 2 J. Chem. Phys. 126, FIG. 5. Density variation of the shear viscosity of CO 2 at K compared with experimental data Ref. 31. FIG. 7. Temperature gradient of CO 2 at 470 K obtained in the computation of thermal conductivity for our fully flexible version of the EPM2 model and the original rigid model. atoms inside each molecule. One more argument in favor of this hypothesis comes from recent quantum simulations of water using classical effective potentials These simulations are consistent with the fact that a quantum treatment of the atomic motions reproduces the results obtained with a classical treatment of the system using the same intermolecular potential but from 20 to 50 K above the temperature of the classical simulations. We should add that such a quantum mechanical treatment would require molecular interactions that account for the subtle details of the potential energy hypersurface, which could also contribute significantly to the correct description of the system at the densities considered in this work. FIG. 6. Variation of the thermal conductivity of CO 2 at 470 K with density. Simulation data with different models and different system sizes. Experimental data from Johns et al. Ref. 27 and NIST Ref. 32 ; correlations: Rah and Eu Ref. 33. The dotted line is an extrapolation of the NIST data using a cubic spline at high density. VI. CONCLUSIONS The aim of this paper has been to study the reliability of semiempirical carbon dioxide intermolecular potentials for the prediction of viscosity and thermal conductivity. In particular, the well known three site EPM2 model of Harris and Yung 9 has been used, since it is known to give excellent agreement with experiment for the liquid-vapor coexistence densities. In the case of the shear viscosity, we have compared against experimental data at supercritical conditions as well as along the coexistence curve. For phase coexistence, and in order to be consistent, the model liquid and vapor coexistence densities have been calculated using the Gibbs ensemble Monte Carlo method. The results show that the agreement with experimental data for the viscosity is, in general, excellent, except for a slight deviation at high densities that can be explained by a small underprediction of the liquid density by the intermolecular potential used. The thermal conductivity has been compared against experimental data at supercritical conditions both at high density and low density conditions. In the case of high density, although there is a slight deviation with respect to an extrapolation of experimental data, the agreement is good. However, at low density a much larger relative error of 35% is observed. A semiflexible version of the model has been implemented, where the bond angle is allowed to change, as well as a fully flexible model, where in addition the bond lengths are allowed to fluctuate. The parameter of the fully flexible model has been obtained by quantum mechanical calculations, namely, the spring constant of the harmonic potential used in this work. Although these extra degrees of freedom displace the predicted thermal conductivity towards the experimental value, the remaining deviation of 22% is still significant. Given the previous success of the model, both for equilibrium values such as the phase coexistence densities and dynamic properties such as the shear viscosity as well as,

9 Nieto-Draghi et al. J. Chem. Phys. 126, indeed, thermal conductivity at higher densities and that all of the degrees of freedom of the model have been included; it is tempting to speculate on the possibility that we are reaching the limits of what can be expected from a classical treatment of dynamic properties under the conditions discussed in this work. In fact, speculation on the limits of classical models and the need to include quantum effects has been the subject of several recent papers in the literature. A detailed quantitative analysis, however, lies beyond this article. ACKNOWLEDGMENTS The authors would like to acknowledge Dr. Philippe Ungerer for his encouragement and advice. This study has been supported by the Spanish Government Ministerio de Educación, Ciencia y Deporte Grant No. CTQ / PPQ as well as by the IFP. 1 B. Y. Wang and P. T. Cummings, Fluid Phase Equilib. 53, B. Y. Wang and P. T. Cummings, Int. J. Thermophys. 10, B. Y. Wang, P. T. Cummings, and D. J. Evans, Mol. Phys. 75, B. Y. Wang and P. T. Cummings, Mol. Simul. 75, P. T. Cummings, Fluid Phase Equilib. 116, G. Steinebrunner, A. J. Dyson, B. Kirchner, and H. Huber, Collect. Czech. Chem. Commun. 63, H. Huber, A. J. Dyson, and B. Kirchner, Chem. Soc. Rev. 28, J. Vrabec, J. Stoll, and H. Hasse, J. Phys. Chem. B 105, J. Harris and K. Yung, J. Phys. Chem. 99, J. J. Potoff and J. I. Siepmann, AIChE J. 47, Z. Zhang and Z. Duan, J. Chem. Phys. 122, A. Z. Panagiotopoulos, Mol. Phys. 61, R. H. Swendsen and A. M. Ferrenberg, Phys. Rev. Lett. 61, R. H. Swendsen and A. M. Ferrenberg, Phys. Rev. Lett. 63, M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, 1st ed. Clarendon, Oxford, D. Frenkel and B. Smit, Understanding Molecular Simulation, from Algorithms to Applications, 2nd ed. Academic, San Diego, C. Nieto-Draghi and J. B. Avalos, Mol. Phys. 101, D. Reith and F. Muller-Plathe, J. Chem. Phys. 112, F. Muller-Plathe, J. Chem. Phys. 106, C. Nieto-Draghi, J. Pérez-Pellitero, and J. B. Avalos, Phys. Rev. Lett. 95, P. E. Smith and W. F. van Gunsteren, Chem. Phys. Lett. 215, D. K. Dysthe, A. H. Fuchs, and B. Rousseau, J. Chem. Phys. 110, H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteden, A. DiNola, and J. R. Haak, J. Phys. Chem. 81, M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 03, Revision C.02, Gaussian Inc., Wallingford, CT, A. Becke, J. Chem. Phys. 98, A. McLean and G. Chandler, J. Chem. Phys. 72, A. I. Johns, S. Rashid, J. T. R. Watson, and A. A. Clifford, J. Chem. Soc., Faraday Trans. 1 82, D. J. Evans, Mol. Phys. 34, M. Svanberg, Mol. Phys. 92, M. Neuman, J. Chem. Phys. 85, A. Fenghour, W. Wakehan, and V. Vesovic, J. Phys. Chem. Ref. Data 27, Data taken from the Saturation Properties of Carbon Dioxide at the webbook.nist.gov 33 K. Rah and B. C. Eu, J. Chem. Phys. 117, S. Bock, E. Bich, E. Vogel, A. S. Dickinson, and V. Vesovic, J. Chem. Phys. 120, C. D. Wick and G. K. Schenter, J. Chem. Phys. 124, L. H. de la Peña and P. G. Kusalik, J. Am. Chem. Soc. 127, L. H. de la Peña and P. G. Kusalik, J. Chem. Phys. 125, Value extrapolated from the experimental data.