Department of Chemical Engineering, Thornton Hall, University of Virginia, ... generalization of the nonequilibrium molecular dynamics (NEMD) algorithm ... approach for two reasons: first, the error in NEMD using reasonably small simulation ... regime have no obvious physical significance except in the case of the NEMD ...
MOLECULAR PHYSICS, 1992, VOL. 75, NO. 6, 1345-1356
Nonequilibrium molecular dynamics study of molecular contributions to the thermal conductivity of carbon dioxide By B. Y. W A N G and P. T. C U M M I N G S Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, VA 22901, USA D. J. EVANS Research School of Chemistry, Australian National University, Canberra, ACT 2601, Australia (Received 12 June 1991; revised version accepted 18 September 1991)
We calculate the thermal conductivity of supercritical and liquid carbon dioxide using a recently developed nonequilibrium molecular dynamics (NEMD) algorithm for molecular fluids. We evaluate the translational, rotational, potential energy and force contributions to the heat flux separately. We find that at high density both the rotational contribution for a nonspherical molecule and the contribution from the total force acting on the molecule are important for predicting the thermal conductivity accurately. The NEMD results for the thermal conductivity agree well with experimental data. At a near critical state point, we observed a phase separation induced by the fictitious external field. The contribution to the thermal conductivity from the Lennard-Jones potential energy becomes important at this state point.
1. Introduction In a previous paper [1], Wang and Cummings reported calculations of the thermal conductivity of supercritical carbon dioxide along the 313K isotherm using a generalization of the nonequilibrium molecular dynamics (NEMD) algorithm developed by Evans for spherically symmetrical molecules [2]. We compared the thermal conductivity of carbon dioxide at densities corresponding to pressures of 69.latin, 197.4atm, and 493.5atm with experiment. We found that the thermal conductivity of carbon dioxide at lower densities agreed well with the experiment, whereas, at the highest density (493.5atm), the thermal conductivity was overestimated by 40% compared to experiment. As will be shown in this paper, it turns out that this overestimation was caused by our failure to account for the rotational contribution to the heat flux in the N E M D algorithm employed. Recently, Evans and Murad have developed an N E M D algorithm for the thermal conductivity of molecular fluids which can be treated as rigid bodies [3]. They added a term to the heat flux vector to account for the work performed by a rigid body rotating against an applied torque. In some situations, the rotational contribution to the thermal conductivity for nonspherical molecules can be significant. This term in the heat flux was missing from the Wang-Cummings work [1]. In this paper, we recompute the thermal conductivity of supercritical carbon dioxide at the same state points as in our prior calculation. Moreover, we calculate individually, the rotational, electrostatic, internal energy and force contributions to the thermal conductivity. By isolating these contributions, we can identify their 0026-8976/92 $3.00 9 1992 Taylor & Francis Ltd
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B.Y. Wang et al.
relative importance in determining the thermal conductivity. Understanding the relative importance of these contributions should find application in developing approximate theories for the thermal conductivity of supercritical molecular fluids. At the state point (temperature = 313 K and density p = 199.8 k g m -3) nearest to the experimental critical point (critical temperature T~ = 304.2 K, critical pressure Pc = 72.8 atm and critical density Pc = 468.2 kg/m 3) of CO2, the previously reported N E M D simulations behaved abnormally in that the predicted thermal conductivity was negative [I]. We tentatively credited this to long-time tail effects in thermal conductivity. Evans and Hanley have found that the synthetic heat field introduced in the N E M D algorithm for thermal conductivity can initiate instability in a simulated system [4]. In their case, the heat field caused a solitary wave of the molecule velocity moving parallel to the direction of the field at supersonic speed. In this paper, we examine the N E M D simulations at this state point in considerable detail. We observe an instability in our simulation system associated with two-phase separation which causes the abnormal behavior in the thermal conductivity obtained from the simulation. N E M D simulation involves simulating a system in a steady state away from equilibrium where the steady state is attained through the application of an external field. The ratio of the field induced current to the field itself gives the transport coefficient of interest. For each transport coefficient, a different external field has to be introduced. Thus, each transport coefficient requires a different N E M D algorithm corresponding to the introduced external field. For example, in the case of thermal conductivity, a fictitious external heat field Fe is used, which plays essentially the same role as the logarithmic temperature gradient V In T in an experiment. It has been proven using linear response theory that, in the linear regime, the ratio of the induced heat current to the applied fictitious field is the same as that of the real heat current induced by an actual logarithmic temperature gradient [5]. Although a Green-Kubo relation exists that would permit the calculation of the thermal conductivity from equilibrium molecular dynamics trajectories [6], we prefer to use the N E M D approach for two reasons: first, the error in N E M D using reasonably small simulation samples is quite small based on experience with simple fluids [7, 2]; second, in view of the theoretical results noted above, the field dependent properties in the linear regime can be expected to be observed experimentally in systems subjected to temperature gradients. Such field dependent properties are not accessible via equilibrium molecular dynamics. On the other hand, properties associated with the nonlinear regime have no obvious physical significance except in the case of the N E M D sllod algorithm for simulating planar Couette flow, where the algorithm is exact to all orders in the strain rate [5]. In this study, we used the three-site potential model for carbon dioxide interactions [8] which was used in our previous work. This model yielded the best overall agreement of transport properties with the experimental values for supercritical carbon dioxide among the three models evaluated in our research program [1,9]. (The other models examined were a one-site simple Lennard-Jones model and a two-site Lennard-Jones model.) The three-site model is the most realistic carbon dioxide model, since it has oxygen and carbon atoms represented explicitly and has a point quadrupole-quadrupole interaction acting between the centers of mass [8].
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N E M D study o f carbon dioxide
2. NEMD algorithm The NEMD algorithm for calculating the thermal conductivity of rigid body molecular fluids is described by Evans and Murad [3]. The equations of motion for a simulated system of N molecules with a fictitious external field F e are given by dri
_
dt
dvi
Pi
m
( /
dt
-
F~ + (Ei - E)F< - l ~ Fqrij Fr +
dt
-
T,. - ~
Lp = =
9
2 j
fl =
in (-qi3
l
= 1
-dttqi3]
2
\qi4/
~ Fskr,j 2 N j,k
.
Fr
_
~Pi.
S
(l)
" F~
AlL i
/qil~
d|q,
Lrij
' _ _ v
x, y, z
-qi4
qi4 -qi3 q,,
qi2
--qi2
qil
(2) qi2 q i , ~ / o ) , P ~
-qi, qi4 q i 3 1 t ~ ) --qi3
qi41
In these equations, m is the mass of a molecule and, for molecule i, r~, Pi, oi, Li, Fi and Ti represent, respectively, the position of the centre of mass, the translational momentum, the angular velocity, the angular momentum, the force on the center of mass and the torque in the laboratory frame. E~ is the instantaneous energy of molecule i, E is the average instantaneous energy per molecule, E = Ei E,/N. The principal (or molecular) frame quantities have superscript p. The matrix Ai is the rotation matrix that converts the laboratory frame coordinates of molecule i to molecular frame coordinates, and is a function of the orientation of the molecule. The qij, J = 1. . . . , 4 are the quaternions representing the orientation of molecule i in such a way that the equations of motion are singularity-free [10, 11]. The use of quaternions in rotational equations of motion is described in detail in Chapter 3 of Allen and Tildesley [12] to which the interested reader is referred. The parameter ~, 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 ~ is obtained from application of Gauss's principle of least constraint [13]. The isokinetic requirement yields
(3) The boundary conditions imposed on the simulated system are the usual periodic boundary conditions employed in the equilibrium molecular dynamics simulation which are adequately described elsewhere [12]. The constitutive relationship between the heat flux JQ and the thermal conductivity 2 is given by Evans [2] (JQ)
=
2TF~.
(4)
1348
B. Y. Wang et al. Table 1. The CO2 state points in the simulations [16].
p/kg m- 3
Temperature/K
Pressure/atm
State
199.8 840-8 992.1 1167.4
313 313 313 220
69 197"4 493"5 9.9
supercritical supercritical supercritical liquid
In this equation, Tis the temperature o f the simulated system, and JQ, the microscopic heat flux, is given by [3]
-~
~
,.~j(p,.F,j + ~ . r?).
(5)
The only differences between this N E M D algorithm for thermal conductivity and that used by Wang and Cummings [1] are the addition in (2) of the external field term to the equation for dLi/dt and in the definition of the heat flux (5). The latter differs from the definition of the heat flux used by Wang and Cummings, who did not include the term taP - Tip to account for the rotational contribution.
3.
Simulation results and discussion
The simulations were performed on a system of 125 molecules with At = 10 ~Ss. The total simulation time for each state point was 100 ps. The intermolecular potential consists of Lennard-Jones potentials for the C-C, O - O and C - O interactions. The Lennard-Jones parameters are ecc/ka = 29.0, ace = 2-785A, eoo/kB = 83.1, aoo = 3.014 A (where kB is Boltzmann's constant) and the cross-interaction parameters (toc/kB and ~oc) are given by the usual Berthelot rules. The O-O bond length is 2-32A. The CO2 molecules also interact via a quadrupole-quadrupole interaction with quadrupole moment of - 3 . 8 5 DA. For the molecule as a whole, we choose e/kB = 195.1 and ~ = 2.83 A in order to dedimensionalize quantities in the simulation. The spherical cutoff of the intermolecular potential was l0 A (corresponding to 3.5a of CO2). The simulations were carried out on four computers: an IBM 3090/150E and an IBM Power Station 530 located in the Academic Comuting Center at the University of Virginia, the Cray Y-MP at Pittsburgh Supercomputing Center, and an Apollo DN10000 Workstation in the Research School of Chemistry at Australian National University. The CPU times per timestep for these four machines were, respectively, 1-12 s, 0.63 s, 0-09 s and 0.43 s. The densities and temperatures of the simulations are given in table 1 along with their pressures. The first three state points (which are the densities and temperatures used in the previous paper [1]) are supercritical. The remaining state point, which is near the triple point of C 0 2 , was chosen for the sake of comparison. We begin by reporting and discussing the different molecular contributions to the thermal conductivity at high density, namely, p = l167.4kgm -3, 992-1 kgm -3, and 840-8kgm -3. Then, we discuss the results from the simulations at the state point nearest to the critical point of carbon dioxide (p = 199-8 kg m-3). In Table 2, we present the various molecular contributions to the reduced heat flux vector J~ = tr3(m/e3)t/2JQ for the three high density states. Corresponding to
NEMD study of carbon dioxide
0
0 0 0 0
0 0 0
0 0 0
+1+1+1+1
+1+1+1
+1+1+1
0 0 0 0
~
~
+1+1+1+1
+1+1+1
+1+1+1
0 0 0 0
0 0 0
0 0 0
6o66 +1+1+1+1
oor +1+1+1
ooo +1+1+1
0 0 0 0
O0(D
0 0 0
6r
d~r
66 "
+1+1+1+1
+[+1+1
+1+1+1
0 0 0 0
0 0 0
0 0 0
III
III
0
~
~
0
~9
?
Z 0
0
0
6666 +1+1+1+1
660 +1+1+1
666 +1+1+1
6 ~
e+e
e~o
6o66 +1+1+1+1
666 +1+1+1
666 +1+1+1
~ 0 0 0
~ 0 0
~ 0 0
Ill
~q
1349
B.Y. Wang et al.
1350
equation (5), the reduced molecular contributions are identified as the following:
J~ -
1
2 ~ r~j(~o~-/7,-')
is the rotational contribution;
g~f -
1
Z riJ(Pi "Fig*)
2 u
is the contribution from the forces between molecules; J ~ and J 0"3 0"5
SN 1"7467 l'7533 1-8187 49"9090 65"6317 1'9780 64"0710 45"8622 56"4965
1354
13. Y. W a n g et al. 100
[] [] 10-
[]
[] [] []
X*
1
A
0.1
[
I
I
I
0.2
0.4
0.6
0.8
Fe z *
Figure 3.
The mean cluster size SN (in logarithmic scale) at p = 199.8 kgm 3 as a function o f F * . There are two distinct steady states between F* = 0-1 and 0-2.
cluster) if any o f the O - O , C - C or C - O intermolecular separations is less than 3.18 A. The calculated mean cluster sizes are reported in table 5 and plotted in a logarithmic scale in figure 3. There is a j u m p in the mean cluster size as the external heat field F* (where F* is the z c o m p o n e n t o f the external heat field vector F*) increases, indicating that there is a phase change. This suggests that the fictitious external heat field actually shifts the vapour-liquid phase b o u n d a r y so that it encompasses the p = 199.8 kg m -3, T = 313 K state point. The simulation exhibits two steady states between be* = 0-1 and 0-2 (i.e., F* = (0,0,0.1) and (0,0,0.2)). These two states are indicated in table 4 by the annotations (H) and (L), where (H) indicates that the results were obtained by performing the simulations with the initial configuration obtained f r o m a previous simulation at a higher external heat field F~* while (L) indicates using results at a lower external heat field as the initial configuration. In other words, the H runs are going in the direction o f reducing field while the L runs are going in the direction o f increasing field. The (H) steady state is more stable than the (L) ones, as is indicated by the standard deviations o f the simulation results. The reduced thermal conductivity obtained from the simulations is plotted on a logarithmic scale as a function o f reduced external heat field in Figure 4. At the lowest external heat field (F* = 0.05), the results o f thermal conductivity from N E M D simulation (table 6, figure 3, see Table 6. T 313 313 220
Comparison between pressure and thermal conductivity calculated via NEMD and experimental data [16]. P kgm -3
PNEMD atm
Pexp atm
2NEMD X 102
W m - l K -l
2~p • 10: W m - t K -1
)wc • 102 Wm tK-I
840.8 992"1 1167.4
295 616 104
197.4 493-5 9.9
8.2 12"3 19.9
9"7 12"6 17-7
8"6 17.5 --
NEMD study of carbon dioxide
1355
100 ,5 A
Su
10-
`5
1
A
,5
0
I
I
0.2
0.4
0,6
Fez*
Figure 4. The reduced thermal conductivity 2* as a function of F* at p = 199.8kgm 3 logarithmic scale on the vertical axis). 121,Results started with a configuration from a higher F*; ~x, from a lower F~*..The arrow represents the experimental value. arrow) agree very well with the experimental value. This suggests that it is possible to predict thermal conductivity at this density provided one uses a small enough external heat field, The phase separation observed at this state point (which is reasonably close to the critical point of CO2) is a nonlinear field effect because 2 is not constant as a function of F*. It evidently comes about because of a shift in the two-phase region induced by the nonlinear field, an effect which can be regarded as analogous to the shear induced shift in two-phase coexistence observed in non-Newtonian polymer mixtures. 4.
Conclusion
The rotational contribution to the heat flux in carbon dioxide makes a significant contribution to the thermal conductivity at high densities. Accurate modeling of the forces acting on the molecule at these densities are crucial for estimating the thermal conductivity. N E M D simulation results of the thermal conductivity of carbon dioxide agree well with the experiment. At the state point (T = 313 K and P = 69 atm) we considered which was nearest the experimental critical point (T c = 304-2K and Pc = 72.8 atm), a high enough fictitious external field induced a two-phase separation in the simulated system. Two steady states of thermal conductivity from N E M D simulation occur between F~* = 0-1 and 0.2. By using a low enough external field (i.e,, one wbicb does not induce the two-phase behavior), it is possible to obtain a thermal conductivity which agrees well with the experimental value, PTC and BYW gratefully acknowledge the support of this research by the National Science Foundation through grant CBT-8801213 and Pittsburgh Supercomputing Center through grant CBT-880026P. PTC acknowledges the support of the National Science Foundation through the US/Australia Cooperative Science Program (Grant INT-8913457) for the provision of travel funds to visit the Australian National
1356
B.Y. Wang et al.
University where some of the reported research was performed. BYW acknowledges financial support from the Research School of Chemistry at the Australian National University in the form of a visiting studentship. References [I] WANG, B. Y,, and CUMMINGS,P. T., 1989, Fluid Phase Equil., 53, 191. [2] EVANS,D. J., 1986, Phys. Rev. A, 34, 1449. [3] EVANS,D. J., and MURAD, S., 1989, Molec. Phys., 68, 1219. [4] EVANS,D. J., and HANLEY,H. J. M., 1989, Molec. Phys., 68, 97. [5] EVANS,D. J., and MORRISS,G. P., 1990, Statistical Mechanics of Nonequilibriurn Liquids (Academic Press). [6] MCQUARRIE,D. A., 1976, Statistical Mechanics (Harper and Row). [7] EVANS,D. J., 1982, Phys. Lett. A, 91,457. [8] MURTHu C. S., and SINGER,K., 1981, Molec. Phys., 44, 135. [9] WANG, B. Y., and CUMMINGS,P. T., 1989, Int. J. Thermophys., 10, 929. [10] EVANS,D. J., 1977, Molec. Phys., 34, 317. [11] EVANS,D. J., and MURAD, S., 1977, Molec. Phys., 34, 327. [12] ALLEN, M. P., and TILDESLI~Y,D. J., 1987, Computer Simulation of Liquids (Oxford University Press). [13] EVANS,D. J., HOOVER,W. G., FAILOR,B. H., MORAY,B., and LADD,A. J. C., 1983, Phys. Rev. A, 28, 1016. [14] VOGELSANG,R., and HOHEISEL,C., 1987, J. chem. Phys., 86, 6371. [15] SEVICK,E. M., MONSON,P. A., and OTTINO,J. M., 1987, J. chem. Phys., 88, 1198. [16] VARGAF'rlK,N. B., 1983, Handbook of Physical Properties of Liquids and Gases, 2nd Edn (Hemisphere, New York).
