Fluid Phase Equilibria 150–151 Ž1998. 151–159
Thermal diffusion in alkane binary mixtures
A molecular dynamics approach
J.-M. Simon, D.K. Dysthe, A.H. Fuchs, B. Rousseau
)
Laboratoire de Chimie Physique des Materiaux
Amorphes, Batiment
490, UniÕersite´ Paris-Sud, 91405 Orsay Cedex, France
´
ˆ
Abstract
We have employed Direct Non-Equilibrium Molecular Dynamics ŽNEMD. simulations to compute the
thermal conductivity and the thermal diffusion coefficient a T in a methane-n-decane binary mixture in the
liquid state. We discuss the choice of a non-equilibrium molecular dynamics method rather than the equilibrium
method using Green–Kubo integrals. We emphasize the fact that using direct-NEMD, there is no need for
additional calculation of any thermodynamic property of the mixture. The n-decane molecules are realistically
modelled as flexible bodies and using the Anisotropic United Atoms ŽAUA. interaction scheme of Toxvaerd.
The statistical uncertainty for the thermal conductivity is less than 1% and that for the thermal diffusion factor is
about 5%. We show that the thermal conductivity results agree with predictions from Assael. The thermal
diffusion data are in accordance with the empirical relationships of Kramers and Broeder and experimental
results, where available. q 1998 Elsevier Science B.V. All rights reserved.
Keywords: Alkane mixture; Molecular simulation; Non-equilibrium molecular dynamics; Soret effect; Thermal conductivity;
Thermal diffusion factor
1. Introduction
When a fluid mixture is placed in a thermal gradient, one observes a separation of the components
in the mixture. This separation leads to a mole fraction gradient parallel to the thermal gradient. This
effect is known as the Soret effect or the thermal diffusion effect. Its amplitude is described by a
phenomenological coefficient, a T , the thermal diffusion factor. The separation by thermal diffusion is
usually small, but its effect can be quite important under special conditions such as in oil reservoirs
w1x. In this case, it is of considerable importance to get reliable data for a T .
Since its discovery in 1850 by Ludwig and the first systematic studies in liquid mixtures by Soret,
thermal diffusion has been the subject of both experimental and theoretical work. However, both
)
Corresponding author. Tel.: q33-1-69-15-42-02; fax: q33-1-69-15-42-00; e-mail: rousseau@cpma.u-psud.fr
0378-3812r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 2 8 6 - 6
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
152
microscopic theories based on kinetic theory of gases applied to dense fluids w2x, and macroscopic
theories based on irreversible thermodynamics are unable to give accurate results for the thermal
diffusion factor in dense fluid systems w1x. From an experimental point of view, thermal diffusion is
very difficult to measure w3,4x. Only recently, independent groups have obtained consistent results on
toluene–n-hexane w5–7x and water–methanol w5,6,8x mixtures.
During the 1980’s, several authors have attempted to compute transport coefficients using
molecular dynamics Ž MD. methods in multi-component systems w9–13x. Transport coefficients can be
obtained using the Green–Kubo ŽGK. formulae and equilibrium MD or synthetic non-equilibrium
molecular dynamics Ž NEMD. w14,15x. One can also mimic the real experiment by modifying the
conditions at the boundaries of the simulation box. This method is called boundary driÕen NEMD or
direct NEMD. In the next, we explain why direct NEMD must be used to compute the thermal
diffusion factor in organic binary mixtures in order to give ‘experimental-like’ data for the thermal
diffusion factor. Then we briefly describe the model and the potential used in our computations along
with some computational details. We finally give some data for thermal conductivity and thermal
diffusion factor obtained for the methane–n-decane binary mixture.
2. Methodology
Using the GK–MD approach, one computes the phenomenological coefficients L i j related to the
thermal diffusion factor a T . For a binary mixture, we have:
aT s
1
L1q
mw11w 1 L11
Ž1.
where w 1 is the mass fraction of the heavy component Žcomponent 1.. The sign of a T is arbitrarily
chosen so that a T is positive when the lighter component concentrates in the hot region. mw11 is the
partial derivative of the chemical potential of component 1, m 1 with respect to w 1 at constant
pressure, P, and temperature, T :
mw11 s
Em 1
ž /
E w1
Ž2.
P ,T
A direct comparison of GK–MD results of the mutual diffusion, D, and a T with experimental data
requires values of mw11. Many authors use the ideal value, but Jolly and Bearman w9x and Schoen and
Hoheisel w10x derived mw11 by integration of the partial pair correlation functions. However, Stoker and
Rowley w16x mention that it is difficult to obtain good statistics by this method and used UNIFAC w17x
to predict mw11.
According to the GK relationships w12x, L11 is given by the infinite time integral of the
autocorrelation function of the mass flux J1 and L1q is defined in terms of the cross correlation of the
mass flux and the heat flux Jq . L11 is related to the mutual diffusion coefficient D, and L1q is
directly connected to the thermal diffusion coefficient D T . J1 is the instantaneous mass flux of
component 1 and can be readily computed during an equilibrium MD run whereas Jq is not directly
accessible from a single simulation. The instantaneous heat flux, Jq , is given by the expression,
Jq s JU y Ž h1 y h 2 . J1 , where JU is the internal energy flux and h i is the specific partial enthalpy of
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
153
component i. Recently, Daivis and Evans w18x have given a microscopic expression for JU for
flexible molecular models. There exists no microscopic expression for h i for constant N ensembles.
Most of the authors use macroscopic data from experimental work or the ideal mixture expressions.
Vogelsang et al. w19x report that even the sign of a T may change when using ideal values of h i
instead of calculated ones. Some authors have attempted to compute h i using MD techniques w20x but
these simulations give poor statistics.
In order to avoid these problems, we have computed a T using the Heat EXchange algorithm
ŽHEX. proposed by Hafskjold et al. w21x. A heat flux is created in the simulation box by exchanging
kinetic energy between a hot and a cold region so that the total energy of the system is conserved.
The simulation box is replicated in a direction perpendicular to the heat flux to enable periodic
boundary conditions in three dimensions. The system develops a temperature gradient and an internal
energy flux. At the stationary state, the thermal diffusion factor is given by:
ž
aT s y
T
= w1
w 1w 2 = T
/
.
Ž3.
J1s0
The simulation box is divided into several layers perpendicular to the applied heat flux. The
composition and temperature are computed in each layer in order to compute the corresponding
gradients w21x. This method gives a direct ‘measure’ of the thermal diffusion factor in a single NEMD
run.
3. Model and computational details
We have used the anisotropic united atoms Ž AUA. model proposed by Toxvaerd w22x to model
n-decane. In the AUA model, the positions of the interaction sites are shifted from the center of mass
of the carbon atoms to the geometrical center of the methyl Ž CH 2 . and methylene Ž CH 3 . groups w23x.
In this way, the site–site interactions depend on the relative orientations of the interacting chains, as
in real molecules. The magnitude of the shift is noted d1 for CH 2 and d 2 for CH 3. This model gives
simulation results for the diffusion coefficient, the pressure, and the trans-gauche configuration in
excellent agreement with experimental data for n-pentane and n-decane in the liquid phase w23x.
Methane was modelled using a 1-center Lennard–Jones potential given by Moller
¨ et al. w24x. All the
potential parameters can be found in Table 1.
The site–site intermolecular interaction is given by the truncated Lennard–Jones potential:
°
V LJ Ž ri j . s~
4e i j
¢0
si j
12
si j
ž / ž /
ri j
y
ri j
6
, ri j - rc
Ž4.
, ri j G rc
where rc is the potential cut-off radius, ri j is the distance between sites i and j and si j and e i j are
the potential parameters. We use the Lorentz–Berthelot mixing rules e i j s Ž e ii e j j . 1r2 and si j s Ž sii
q sj j .r2 where e ii and sii are the potential parameters for like site interactions.
In addition to the intermolecular site–site interactions, the atomic units Ž CH 2 and CH 3 . are
affected by intramolecular constraints and forces. The carbon–carbon bond lengths d cc are con-
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
154
Table 1
Model and potential data
Bond length and AUA distances:
˚
˚
d cc s1.54 A
d 1 s 0.275 A
Mass of the interacting sites ŽkgPmoly1 .
m CH 2 s 0.014
m CH 3 s 0.015
Lennard–Jones potential
CH 2
e s80 K
CH 3
e s120 K
CH 4
e s150 K
Bending potential:
b 0 s114.68
kb s 520 kJ moly1
y1
Torsion potential ŽkJ mol .
a 0 s8.6279
a1 s 20.170
a 3 sy26.018
a 4 sy1.3575
˚
d 2 s 0.370 A
m CH 4 s 0.016
˚
s s 3.527 A
˚
s s 3.527 A
˚
s s 3.7327 A
a 2 s 0.67875
a5 sy2.1012
strained to their mean value using the RATTLE algorithm by Andersen w25x. The angle bending
potential is
1
2
VB Ž b . s kb Ž cos b y cos b 0 . ,
Ž5.
2
where kb is the bending force constant, b 0 is the equilibrium bond angle, and b is the instantaneous
bond angle. The torsion potential is a fifth order polynomial in cos f , where f is the dihedral angle:
5
VT Ž f . s
i
Ý a i Žcos f . .
Ž6.
is0
The intramolecular potential also includes Lennard–Jones interactions between sites separated by
three or more centers.
The equations of motion are solved using the Verlet Õelocity algorithm with a timestep of 0.005 ps.
The site–site cut-off radius rc is 2.5 smethane . We have used a neighbor list with a site–site list radius,
r l s 1.1 rc . We used uncorrected coordinates and applied the minimum image convention based on
site–site distance. In order to obtain small statistical uncertainty, runs have been carried out for 7 to
10 ns Ž1–2 million timesteps. on a simulation box containing 1600 molecules. This system contains
10,240 centers of force and requires a large amount of computational time. A parallel version of the
code has been implemented on Cray T3D and T3E. The CPU time for one timestep is about 0.025 s
on the Cray T3E with 64 processors. Each run required 12 h. Under these conditions, the statistical
uncertainty in = x and = T is 2% and 1%, respectively. The statistical uncertainty in the mole fraction
and temperature in each layer is 0.5% and 0.1%, respectively. According to the Eq. Ž 3. , the statistical
uncertainty in a T is less than 5%.
4. Results
The methane–n-decane binary mixture can be considered as a model for gas–oil mixture and has
non-negligible excess thermodynamic properties. It has been studied for many years by several
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
155
groups. Sage and Berry have reported thermodynamic data w26x, Knapstad et al. w27x have made
systematic studies of viscosity, and Hafskjold et al. w28,29x have done measurements of inter- and
intra-diffusion coefficients, all at elevated temperature and pressure. However, there exists only one
data point for thermal diffusion in this mixture, from Lindeberg w30x.
We have studied a methane–n-decane binary mixture with a methane mole fraction x 1 s 0.4 at six
different state points in the dense liquid region, far from the critical point. The temperature and
density data used for the different runs are given in Table 2. For the highest density used in this work,
r s 0.703 grcm3, the difference between the computed and the experimental pressure w26x is less than
15%.
The HEX algorithm can be used to compute the thermal conduction k of the mixture. At the
stationary state, when the mass flux J1 s 0, k is given by the expression w31x:
ksy
² Jq :
=T
sly
D T2
D
mw11 r w 12 w 2 T ,
Ž7.
where ² Jq : is the average of the instantaneous heat flux, l is the thermal conductivity, and r is the
density of the system. The first equality of Eq. Ž 7. is used to compute k . The difference between l
and k is due to the thermal diffusion effect and is a second order term. Quantitatively, it turns out
that the difference between l and k is at the most a few percent w31x. Fig. 1 shows the thermal
conduction k computed at 307 K at four different densities. The statistical uncertainty is less than
1%. To our knowledge, there is no experimental data on thermal conductivity in methane–n-decane
mixtures. In order to check the validity of the method, we have compared the simulated data with a
prediction given by Assael et al. w32x which is reported to have an accuracy of 5%.
One observes that the two sets of data are in excellent agreement, the uncertainties are small, and
the error bars overlap. The difference between simulated and predicted data never exceed 6%. Both
the predicted and computed values increase with density, but one observes a systematic deviation
from the Assael correlation. The systematic deviation cannot be explained by the difference between
k and l which can be estimated using experimental diffusion coefficients, the simulated a T and ideal
mw11. For this system the difference is four orders of magnitude smaller than the absolute values of l
and k .
Finally, we present the results obtained for the thermal diffusion factor a T for the six different
runs. The data are plotted in Fig. 2 and reported in Table 2. First of all, we note that for the six
different runs, a T is positive. This result is in qualitative agreement with the empirical laws of
Table 2
Thermal diffusion factor a T , thermal conduction k and pressure P computed at different temperature T and density r . The
methane mole fraction is 0.4
Run
T ŽK.
r ŽgPcmy3 .
P "2% ŽMPa.
k "1% ŽWPmy1PKy1 .
a T "5%
1
2
3
4
5
6
307
307
307
307
350
350
0.66
0.67
0.68
0.703
0.66
0.67
16.1
27.1
38.5
69.8
49.7
62.3
0.133
0.138
0.144
0.158
–
–
1.34
1.21
1.26
1.05
1.12
1.08
156
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
Fig. 1. Thermal conduction k vs. density obtained by NEMD Ž`. and thermal conductivity l predicted from the Assael
correlation Žv ..
Kramers and Broeder w33x which states that for a homologous series, the component enriching at the
cold side is the one having the greatest number of carbon atoms. The experimental result obtained by
Lindeberg at 307 K and 0.67 g P cmy3 is 0.82 " 0.08. Under identical conditions, we found a value of
1.21 " 0.06, about 30% higher than the experimental value. However, this difference is not very large
compared to the general discrepancies often observed between experimental thermal diffusion data
from different authors.
There is a very small number of systematic studies of the thermal diffusion factor with temperature,
pressure or composition. It is, therefore, very difficult to predict the correct trend for the methane–ndecane in the liquid state. Zhang et al. w6x, have observed a decrease of the Soret coefficient ST
Ž ST s a TrT . with increasing temperature in toluene–n-hexane liquid mixtures. We observed the same
trend in our system. We also observed a decrease of a T with increasing density at constant
Fig. 2. Thermal diffusion factor a T vs. density obtained by NEMD at 307 K Ž`. and 350 K Že.; Experimental point from
Lindeberg Žv . at 307 K.
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
157
temperature. Although this behavior is different from what is observed in gas mixtures w34x, it is in
qualitative agreement with Dougherty and Drickamer results in the CS 2 –n-octane mixture in the
liquid state w35x. Hence, we note that the general trends of a T from our simulations in the
methane–n-decane system with temperature and density are in qualitative agreement with experimental results on binary mixtures in the liquid state.
5. Conclusion
We have presented a general methodology to compute the thermal conductivity and thermal
diffusion factor in a ‘realistic’ binary n-alkane mixture. Using the HEX algorithm, we avoid the
computation of quantities difficult to obtain using molecular dynamics, namely the partial derivatives
mw11 and the partial enthalpies h i . Parallelising the NEMD code has enabled us to obtain k and a T
with a statistical uncertainty of less than 1% and 5%, respectively.
The thermal conduction data computed from direct NEMD are in excellent agreement with
predictions from Assael. The thermal diffusion factor data are in good agreement with previous
experimental results, and the general behaviour with temperature and density is consistent with
experimental results on binary mixtures in the liquid state. These results show that direct NEMD can
be a powerful tool for the study of thermal diffusion in molecular liquid mixtures.
6. Nomenclature
ai
d1
d2
d cc
D
DT
h
hi
Ji
Jq
JU
kb
Li j
P
rc
ri j
rl
ST
T
VLJ
VB
Parameters for the intramolecular torsion potential
AUA distance for CH2
AUA distance for CH3
Carbon–carbon bond length
Mutual diffusion coefficient
Thermal diffusion coefficient
Integration time step
Specific partial enthalpy of component i
Instantaneous mass flux of component i
Instantaneous heat flux
Instantaneous internal energy flux
Bending force constant
Phenomenological coefficient
Pressure
Intermolecular potential cut-off distance
Distance between sites i and j
Neighbour list radius
Soret coefficient
Temperature
Lennard–Jones potential
Intramolecular bending potential
J.-M. Simon et al.r Fluid Phase Equilibria 150–151 (1998) 151–159
158
VT
wi
xi
Intramolecular torsion potential
Mass fraction of component i
Mole fraction of component i
Greek symbols:
aT
Thermal diffusion factor
b
Instantaneous bond angle
b0
Equilibrium bond angle
ei j
Lennard–Jones potential parameter for type i–type j interaction
k
Thermal conduction
l
Thermal conductivity
mi
Chemical potential of component i
mw11
Partial derivative of the chemical potential of component 1 with respect to the mass
fraction of component 1
f
Dihedral angle
r
Mass density
si j
Lennard–Jones potential parameter for type i–type j interaction
Acknowledgements
We are very grateful to François Montel, for having initiated this work and for fruitful discussions.
We wish to thank Elf Aquitaine Production for a grant for one of us Ž JMS. . We thank the Institut du
Developpement
et des Ressources en Informatique Scientifique Ž IDRIS. for technical support and a
´
generous allocation of Cray T3E computer time.
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