Ind. Eng. Chem. Res. 1997, 36, 3927-3936
3927
Self-Diffusion in Gases and Liquids Eli Ruckenstein* and Hongqin Liu† Chemical Engineering Department, State University of New York at Buffalo, Amherst, New York 14260
A systematic study of the self-diffusion coefficient in hard-sphere fluids, Lennard-Jones fluids, and real compounds over the entire range of gaseous and liquid states is presented. First an equation is proposed for the self-diffusion coefficient in a hard-sphere fluid based on the molecular dynamics simulations of Alder et al. (J. Chem. Phys. 1970, 53, 3813) and Erpenbeck and Wood (Phys. Rev. A 1991, 43, 4254). That expression, extended to the Lennard-Jones fluids through the effective hard-sphere diameter method, represents accurately the self-diffusion coefficients obtained in the literature by molecular dynamics simulations, as well as those determined experimentally for argon, methane, and carbon dioxide. A rough Lennard-Jones expression, which contains besides the diameter σLJ and energy LJ the translational-rotational factor, AD (which could be correlated with the acentric factor), is adopted to describe the self-diffusion in nonspherical fluids. The energy parameter is estimated using a correlation obtained from viscosity data, and the molecular diameter is obtained from the diffusion data themselves. The equation represents the self-diffusion coefficients with an average absolute deviation of 7.33%, for 26 compounds (1822 data points) over wide ranges of temperature and pressure. Introduction Much effort has been made regarding the experimental determination and theoretical explanation of the selfdiffusion in fluids, especially dense fluids (Ertl et al., 1974; Tyrell and Harris, 1984; Lee and Thodos, 1988). The present available theories, such as the Enskog theory and its modifications (Chapman and Cowling, 1970; Kincaid et al., 1994), are valid only for gases at low densities. For this reason, empirical correlations have been proposed for dense fluids by Lee and Thodos (1988) and Harris (1993). Various semiempirical methods have also been suggested for correlation purposes, such as those based on free-volume models (Macedo and Litovitz, 1965; Ertl et al., 1974; Tyrell and Harris, 1984). In all these correlations, four or more adjustable parameters were used to represent the data over the whole range of gaseous and liquid states. Molecular dynamics (MD) simulations for some simple spherical fluids, such as hard-sphere (HS) and LennardJones (LJ) fluids, were also employed to generate equations (Speedy, 1987; Speedy et al., 1989) for the selfdiffusion coefficient. However, most molecules encountered in practical applications are nonspherical; a connection between real and simple model fluids can be made by using Chandler’s rough hard-sphere (RHS) theory (Chandler, 1975; Dymond, 1985). According to that theory, the diffusion coefficient of a nonspherical molecule can be described by the following equation
DRHS ) ADDHS(F,σ)
(1)
where DHS(F,σ) is the diffusivity of a hard-sphere fluid, for a number density F and an equivalent molecular diameter σ, and AD is the translational-rotational coupling factor, which accounts for the nonsphericity of the molecule. As already noted, for low densities, an expression for DHS(F,σ) could be derived theoretically (Chapman and * Author to whom correspondence should be addressed. † Permanent address: Chemical Engineering Department, Beijing University of Chemical Technology, 100029 Beijing, P. R. China. S0888-5885(97)00133-4 CCC: $14.00
Cowling, 1970). However, for high densities, molecular dynamics simulations had to be used to obtain expressions for DHS(F,σ). The coupling factor AD may depend on both temperature and density (Dymond, 1985), but for real fluids, no equation is available for this factor. A more reasonable treatment is to replace DHS(F,σ) in eq 1 by the self-diffusion coefficient DLJ ) DLJ(F,Τ,σLJ,LJ) in a Lennard-Jones fluid, where T is the temperature and σLJ and LJ are the molecular diameter and energy parameters in the LJ potential, respectively. Numerous attempts have been made to obtain expressions for DHS(F,σ) and DLJ(F,Τ,σLJ,LJ). Speedy et al. (1989) proposed the following equation for the LJ fluid:
(
DLJ ) DHS(F*,σe) exp -
1 2T*
)
(2)
where F* ) Fσe3 and T* ) kT/, with k the Boltzmann constant and σe the effective hard-sphere diameter (EHSD) which depends on temperature. On the basis of MD simulations, Speedy (1978) proposed the following equation for DHS(F*,σe):
(
DHS(F*,σe) ) D0 1 -
)
F* (1 + 0.4F*2 - 0.83F*4) 1.09
(3)
where D0 is the diffusion coefficient at very low densities. Equations 2 and 3 have been used to correlate the self-diffusion coefficients of some nonpolar molecules (Vardag et al., 1990, 1991; Enninghorst et al., 1996). Kushick and Berne (1973) and more recently Straub (1992) used the Weeks-Chandler-Andersen (WCA) (1971) fluid, based on a potential defined later in the paper, as the reference system. For example, Straub (1992) found on the basis of MD simulations that DLJ can be expressed as DLJ ) DWCA/(1 + R/T*), where R is a constant R ) 0.23, DWCA is the self-diffusion coefficient based on the WCA repulsive potential, and (1 + R/T*)-1 is a correction due to the attractive potential. DWCA can be calculated using the expression for a hard-sphere fluid in which the molecular diameter is replaced by the WCA EHSD (WCA effective hard-sphere diameter). Most recently, Enninghorst et al. (1996) adopted Straub’s © 1997 American Chemical Society
3928 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
suggestion but replaced (1 + R/T*)-1 by exp(-1/4T*) to account for the attractive contribution:
(
DLJ ) DHS(F*,σe) exp -
1 4T*
)
(4)
Equation 3 was adopted by them for DHS, and σe is the WCA EHSD proposed by Speedy et al. (1989). Equation 4 was employed to correlate their experimental data for isopentane and cyclopentane (Enninghorst et al., 1996). In this paper, a new equation for DHS is proposed based on the MD simulations carried out by Alder et al. (1970) and Erpenbeck and Wood (1991). An expression for DLJ(F,Τ,σLJ,LJ) is also obtained on the basis of a modified EHSD method and compared with numerous MD simulations over the entire range of gaseous and liquid states. A comparison between various expressions proposed for the EHSD is made. The sensitivity of DLJ to the molecular parameters, σLJ and LJ, as well as the relative contributions of the repulsive and attractive interactions are analyzed and discussed. It is found that for spherical atoms or molecules, such as argon and methane, the molecular parameters σLJ and LJ obtained by combining viscosity and second virial coefficient data can be used to predict the self-diffusion coefficient with accuracy, especially for argon. For nonspherical molecules, a rough Lennard-Jones (RLJ) type equation is used to calculate the diffusivity over wide ranges of temperature and pressure (density). The energy parameter, LJ, is estimated from a correlation obtained from viscosity data, the coupling factor, AD, is expressed in terms of the acentric factor ω, and the equivalent molecular diameter, σLJ, is determined from the diffusion data. The equation is used to correlate the self-diffusion coefficients for 26 compounds (1822 data points). Finally, a simple correlation between the molecular diameter, σLJ, and the critical volume is obtained, which provides, however, acceptable results only for the self-diffusion coefficients of some simple (nonpolar and small) molecules.
The self-diffusion coefficient in a fluid can be related to the velocity autocorrelation function by (Zwanzig, 1965):
∫0∞〈ux(0) ux(τ)〉 dτ
(5)
where ux(0) and ux(τ) represent the molecular velocity at a position x at time 0 and τ, respectively, and the angle brackets indicate averaging over an ensemble. Introducing a normalized time-correlation function
φ(τ) )
〈ux(0) ux(τ)〉 〈ux2〉
(6)
where 〈ux2〉 is the average of the square of velocity, eq 5 can be rewritten as
D ) kT/ξ
(7)
where k is the Boltzmann constant, and the friction coefficient ξ is defined via the expression
ξ ) (kT/〈ux2〉)/
∫0
∞
φ(τ) dτ
For a molecule which, in addition to repulsion, has a soft attractive potential, the friction coefficient can be written as ξ ) ξR + ξS (Rice and Gray, 1965). Accordingly
D)
(8)
kT ξR + ξS
(9)
where ξR represents the friction coefficient due to the repulsive potential and ξS that due to the attractive one. For a hard-sphere fluid at sufficiently low densities, the following equation for the friction coefficient could be derived (Helfand, 1961; Rice and Gray, 1965):
8 ξR ) Fg(σ) σ2(πmkT)1/2 3
(10)
where g(σ) is the radial distribution function at contact and m the mass of a molecule. Combining eqs 9 and 10, the well-known Enskog equation for a hard-sphere fluid is obtained:
DE )
Basic Equations and the Hard-Sphere Fluid
D)
Figure 1. Ratio (DHS/DE) of the diffusion coefficient of a hardsphere fluid (MD simulation results) to that calculated with the Enskog equation against reduced density F* ) Fσ3.
D0 3 kT 1/2 1 ) 2 πm g(σ) g(σ) 8Fσ
( )
(11)
Comparing with the MD simulations, Alder et al. (1970) have noted that eq 11 is valid only at low densities, namely, F* < 0.1 (see Figure 1). A correction factor must therefore be introduced to obtain accurate results for the HS fluid in the moderate- and highdensity ranges:
DHS ) DE
( )
DHS D0f(F*) ≡ DEf(F*) ) DE g(σ)
(12)
The correction function f(F*) in the above equation can be obtained by using MD simulation results. In the density range 0.0 e F* e 0.95, several sets of MD data are available in the literature. Those obtained by Easteal et al. (1983, 1984) were used previously by Speedy et al. (1989). We prefer, however, the results published by Erpenbeck and Wood (1991), since they coincide with those obtained by Alder et al. (1970). In the high-density range 0.943 e F* e 1.08, we use the data of Woodcock (1981). The obtained equation has the form
f(F*) ) 1 + 0.94605F*1.5 + 1.4022F*3 5.6898F*5 + 2.6626F*7 (13)
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3929
The Carnahan-Starling equation (Carnahan and Starling, 1969) is adopted for the radial distribution function at contact:
g(σ) )
1 - 0.5η (1 - η)3
tive contribution can be taken into account by adding the soft friction coefficient to that due to repulsion. Combining eqs 9 and 12, one obtains the following equation for the self-diffusion coefficient in the LJ fluid:
(14) DLJ )
where
1 1 η ) πF* ) πFσ3 6 6
(15)
Figure 1 compares eq 13 to eq 3 obtained by Speedy (1987). In the density range 0.3 < F* < 0.9, the difference between the MD simulation data used by Speedy (1987) and those employed in the present paper is significant. One can also see that the Enskog equation, eq 11, is valid only within a very limited density range. As shown later, the LJ equation obtained for the self-diffusion coefficient on the basis of eq 13 can represent the experimental data for spherical molecules (argon, methane).
kT 8 2 1/2 Fσ (πmkT) g(σe)f-1(F*) + ξS 3 e
In what follows, expressions for σe and ξS will be established. The Soft Friction Coefficient. The soft friction coefficient, ξS, can be expressed in terms of the intermolecular attractive potential, (r), defined below (eq 23), and the radial distribution function, g(r), through the following equation (Helfand, 1961; Rice and Gray, 1965):
σLJ r
12
-
σLJ r
6
(16)
where LJand σLJ are the molecular energy and diameter parameters, respectively. In their perturbation theory for equilibrium properties, Weeks, Chandler, and Andersen (1971) separated the LJ potential into two parts: a reference potential, which contains all the repulsive forces in the LJ fluid, and another one, which contains all the attractive forces. The repulsive potential, denoted WCA(r), is given by
{
LJ(r) + LJ, r e 21/6σLJ WCA(r) ) 0, r > 21/6σLJ
(17)
{
-LJ
G(λ) )
LJ(r), r g 21/6σLJ
(18)
Equations for the equilibrium properties of a LJ fluid were obtained by (1) replacing the expressions based on the repulsive component of the LJ potential with those for a HS fluid but using an effective hard-sphere diameter instead of the real one and (2) treating the additional soft, attractive interaction contribution as a perturbation. This approach has theoretical foundation for equilibrium properties, since, as demonstrated by the authors (Andersen et al., 1971), the hard-sphere approximation provides correct results up to the fourthorder in softness. A similar treatment was used by Barker and Henderson (1967), who justified their choice of the effective hard-sphere diameter by an analysis of the partition function. In what follows we use a similar approach for the diffusion coefficient. It should be, however, emphasized that it is not possible to provide as strong theoretical arguments in its favor as for the equilibrium properties. The first step can be achieved using the EHSD method, in which the hard-sphere diameter σ in eq 12 is replaced with σe, which generally depends on both temperature and density. The attrac-
(21)
∫[g(r) - 1] exp(-iλh‚rj) drj
(22)
Since the introduction of eq 18 in eq 20 leads to a logarithmic singularity, we replace eq 18 with the following expression, which involves a cutoff of the potential at r ) σ.
ξ(r) )
{
LJ(r), r g σ 0, r 1.12 were disregarded, since the diffusion coefficients are too small. A few data which were questionable when compared with the data of other authors were disregarded as well. From Table 2 one can see the following: (1) Among the three theoretical EHSD expressions, eqs 32-34, that of Lado (1984) (eq 34) provides the best results. (2) Among the six expressions, the Boltzmann EHSD (eq 31) with the parameters obtained by Ben-Amotz and Herschbach (1990) leads to the best agreement with the diffusion data. Equation 36, proposed by Sun et al. (1994), provides acceptable results. (3) Equation 2 overestimates the diffusivities at high densities and underestimates them at low densities, and eq 4 overestimates the diffusivities at almost all densities.
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3931 Table 1. Data Sources for the Lennard-Jones Fluid authors
T*
F*
Levesque and Verlet (1970)a Kushick and Berne (1973)b Carelli et al. (1976)c Michels and Trappeniers (1978)d Lucas and Moser (1979)e Heyes (1983)f Hammonds and Heyes (1988)g Heyes (1988)h Erpenbeck (1988)i Borgelt et al. (1990)j Heyes and Powles (1990)k Straub (1992)l Heyes et al. (1993)m Kincaid et al. (1994)n
0.760-5.09 0.740-5.13 0.687-1.90 1.300-1.50 0.788-4.02 0.710-4.58 0.720-10.0 0.720-10.0 0.722 0.673-2.54 0.722-6.00 0.750-4.00 0.707-6.02 2.00
0.30-0.85 0.30-0.85 0.30-0.84 0.00-0.3024 0.30-0.85 0.20-1.113 0.40-1.06 0.30-1.22 0.8442 0.7803-0.8839 0.50-1.40 0.30-1.050 0.50-1.00 0.05-0.50
a Levesque, D.; Verlet, L. Phys. Rev. A 1970, 2, 2514. b Kushick, J.; Berne, B. J. J. Chem. Phys. 1973, 59, 3732. c Carelli, P.; De Santis, A.; Modena, I.; Ricci, F. P. Phys. Rev. A 1976, 13, 1131. d Michels, J. P. J.; Trappeniers, N. J. Physica A 1978, 90, 179. e Lucas, K.; Moser, B. Mol. Phys. 1979, 37, 1849. f Heyes, D. M. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1741. g Hammonds, K. D.; Heyes, D. M. J. Chem. Soc., Faraday Trans. 2 1988, 84, 705. h Heyes, D. M. Phys. Rev. A 1988, 37, 5677. i Erpenbeck, J. J. Phys. Rev. A 1988, 38, 6255. j Borgelt, P.; Hoheisel, C.; Stell, G. Phys. Rev. A 1990, 42, 789. k Heyes, D. M.; Powles, J. G. Mol. Phys. 1990, 71, 781. l Straub, J. E. Mol. Phys. 1992, 76, 373. m Heys, D. M.; Powles, J. G.; Gil Montero, J. C. Mol. Phys. 1993, 78, 229. n Kincaid, J. M.; Tu, R.-F.; Lopez De Haro, M. Mol. Phys. 1994, 81, 837.
An Equation for Self-Diffusion in the LJ Fluid. On the basis of the above comparison, the following equation is adopted here for the self-diffusion coefficient of a LJ fluid:
DLJ )
kT g(σe) 0.4 8 2 + Fσe (πmkT)1/2 3 f(F*) T*1.5
[
]
(37)
It can be rewritten in the following reduced form:
DLJ* ≡
DLJF* σLJ(LJ/m)1/2
)
xT* g(σe) 0.4 8xπ + [σe*(T*)]2 3 f(F*) T*1.5
[
]
(38) In eqs 37 and 38, the EHSD should be calculated using the expression
σe ) σLJσe*(T*) ) σLJσB*(T*)
deviations (percent) given by eq 38 plotted against reduced temperature and density, respectively. From these figures, one can see that the deviations are well distributed around zero and that the maximum deviations are mostly located in the low-temperature or highdensity ranges, as expected. It is interesting to compare the relative contributions of the attractive and repulsive potentials. According to eq 38, one can write that
RC ≡
(39)
where σB*(T*) is given by eq 31 and T* ) T/(LJ/k). The radial distribution function, g(σe), is given by eq 14 and f(F*) by eq 13. It should be pointed out that, in both eqs 13 and 14, the reduced density F* should be calculated by replacing σ in eq 15 by σe. Hence
F* ) Fσe3 ) FσLJ3[σB*(T*)]3
Figure 3. (a) Relative deviation (RD ) 100(Dcal - DMD)/DMD) from eq 38 against reduced temperature T* ) T/(LJ/k). (b) Relative deviation (RD ) 100(Dcal - DMD)/DMD) obtained from eq 38 against reduced density F* ) FσLJ3.
(40)
For all 411 points, eq 38 predicts the diffusion coefficients for the LJ fluid with the absolute average deviation (AAD) of 5.66%. Parts a and b of Figure 3 present the relative
0.4/T*1.5 attractive contribution (41) ) repulsive contribution g(σe)/f(F*)
Plots of RC against T* at two reduced densities, F* ) 0.2 (gaseous state) and F* ) 0.8 (liquid state), are presented in Figure 4, which shows that the attractive contribution is small compared to the repulsive one, especially in the high-density range. At low temperatures and low densities, the attractive contribution becomes significant. This conclusion is consistent with that found for thermodynamic properties (Weeks et al., 1971). Two molecular parameters, namely, the diameter, σLJ, and the energy, LJ, are involved in the expression of the LJ potential. These parameters can be, in principle,
Table 2. Calculation Results for the Lennard-Jones Fluid (T* ) 0.72-10.0, G* ) 0.00-1.12, NDP ) 411) eq 28
eq 2
EHSD expression
eq 31
eq 32
eq 33
eq 34
eq 35
eq 36
R1 AAD
0.40 5.66
0.007 17.91
0.026 13.46
0.065 10.87
0.37 10.11
0.38 7.81
a
With parameters R0 )
21/6
and T0* ) 0.5.
eq
30a
9.72
eq 4 eq 30a 13.61
3932 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
the energy parameter by 1% changes the diffusion coefficient by less than 1%. This result is consistent with the previous conclusion that the repulsive branch of the potential (eq 17) dominates the equilibrium properties (Barker and Henderson, 1967; Andersen et al., 1971; Weeks et al., 1971). Consequently, in the estimation of molecular quantities, such as the diffusion coefficient, accurate values for the diameter are very important. Applications to Real Compounds
Figure 4. Ratio of attractive and repulsive contributions to the self-diffusion in the Lennard-Jones fluid against reduced temperature T* ) T/(LJ/k). Calculated with eq 41.
The objective of this section is 2-fold: (1) to answer the question ‘which fluid can be treated as a LJ fluid?’ and also the question ‘is it possible to predict the diffusion coefficient using parameters obtained from other thermodynamic or transport properties?’; (2) to propose an equation for various real compounds. We employed data for 26 real fluids of various polarities and sizes, with 1822 data points. The compounds were chosen only when diffusion coefficients at different temperatures and pressures were available. Table 3 lists the compounds considered, data sources, and the temperature and density ranges involved. The values of the critical properties and of the acentric factors were taken from Reid et al. (1987). Which Fluid Can Be Treated as a LJ Fluid? If a real compound can be treated as a LJ fluid, its diffusion coefficient should satisfy the following equation:
D ) DLJ(σLJ,LJ) Figure 5. Sensitivities of the diffusion coefficients of the LJ fluid to changes in molecular diameter and energy parameters. Calculated using eqs 42 and 43, respectively.
obtained from any thermodynamic or transport property of the fluid. There are, however, two limitations: (1) As noted in the next section, only a few substances can be treated as LJ fluids. (2) Accurate models or equations which contain these molecular parameters are rarely available over the whole range of gaseous and liquid states. For this reason, the molecular parameters are mostly estimated from experimental data regarding the property described by the theoretical equation, the diffusion coefficient in the present case. For parameter estimation, it will be helpful to have some information regarding the sensitivity to the changes in the values of the parameters. Here we define two sensitivity quantities as follows:
∆σ ≡ ∆ ≡
|
|
| | |/| | | | |/| |
∂ ln DLJ ∂DLJ ) ∂ ln σLJ DLJ
∂ ln DLJ
∂ ln(LJ/k)
)
∂DLJ DLJ
∂σLJ σLJ
(42)
∂(LJ/k) LJ/k
(43)
Figure 5 illustrates the behavior of ∆σ and ∆ at two densities corresponding to gaseous and liquid states, respectively. It is found that the diffusion coefficient is very sensitive to the value selected for the diameter, especially at high densities; the change of the diameter by 1% may lead to a change of the diffusion coefficient by more than 10%. The diffusion coefficient is much less sensitive to the energy parameter, since changing
(44)
where DLJ(σLJ,LJ) is given by eq 37. The parameters σLJ and LJ can be obtained either from data sources other than diffusion or from diffusion data themselves. Regarding the first method, we employed the parameters estimated by Tee et al. (1966) from viscosity and second virial coefficient data. The compounds tested are some simple molecules, argon, methane, carbon dioxide, ethylene, and ethane, for which a large number of reliable data points are available over the entire range of gas and liquid states. The parameters used and the predictions are listed in Table 4, along with those predicted by the equation of Speedy et al. (1989) (eq 2). From the predictions listed in Table 4, one can conclude that (1) only argon can be strictly considered a LJ fluid, since its diffusion coefficient can be predicted with an accuracy comparable to the experimental one, using parameters obtained from viscosity and/or second virial coefficient data; (2) methane and carbon dioxide can be approximately treated as LJ fluids and acceptable predictions can be made using proper molecular parameters (combining viscosity and second virial coefficient data); (3) the nonspherical molecules, such as ethylene and ethane, cannot be treated as LJ fluids; (4) generally the parameters obtained by combining viscosity and second virial coefficient data provide better results than those obtained from the second virial coefficient alone. The parameters σLJ and LJ in eq 44 have also been obtained from diffusion data themselves by minimizing an objective function. It was found that the equation can correlate the data for 26 compounds with acceptable accuracies (within an AAD from 5 to 12%). However, most of these compounds cannot be considered LJ fluids, since the values of the parameters estimated are not comparable to those obtained from other properties. The Rough Lennard-Jones Model for Real Compounds. Since most compounds are nonspherical, we
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3933 Table 3. Properties of Compounds and Data Sourcesa no.
substance
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
Tr
Fr
data sourcesb
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
argon methane ethane isopentane cyclopentane cyclohexane n-hexane n-octane n-hexadecane ethylene carbon dioxide carbon disulfide benzene toluene chloromethane dichloromethane chloroform carbon tetrachloride fluoroform carbon tetrafluoride perfluorocyclobutane chlorotrifluoromethane sulfur hexafluoride tetramethylsilane pyridine acetonitrile
150.80 190.56 305.40 460.40 511.70 554.15 507.40 568.80 717.00 282.40 304.20 552.00 562.09 591.80 416.30 510.00 536.40 556.40 299.30 227.60 388.50 302.00 318.70 448.61 620.00 545.50
48.70 45.39 48.80 33.90 45.10 40.40 30.10 24.90 14.00 50.40 73.80 78.00 48.34 41.00 67.00 63.00 53.70 45.60 48.60 36.90 26.70 38.70 37.10 27.84 55.60 48.30
74.90 100.00 148.30 306.00 260.00 311.70 370.00 492.00 944.00 129.00 94.04 170.00 258.66 316.00 138.90 138.00 238.90 276.00 132.70 140.00 326.80 180.40 198.80 362.00 254.00 173.00
0.001 0.011 0.099 0.227 0.196 0.212 0.299 0.398 0.742 0.089 0.239 0.109 0.212 0.263 0.153 0.199 0.218 0.193 0.260 0.000 0.000 0.198 0.286 0.600 0.243 0.327
0.5139-2.3422 0.4760-2.3825 0.4816-1.6076 0.6473-0.7124 0.5824-0.6410 0.5648-0.6911 0.4398-0.6566 0.4174-0.5854 0.4160-0.4856 0.4361-1.4099 0.8979-1.2267 0.4859-0.5674 0.5127-0.9308 0.3710-0.5503 0.4468-0.9753 0.3647-0.7961 0.4045-0.7401 0.5090-0.5988 0.4744-0.8353 1.0683-1.5297 0.8314-1.2175 0.4404-1.1528 0.7531-1.2488 0.6643-0.8315 0.4890-0.6825 0.4642-0.6291
0.0012-0.8112 0.0499-3.1000 1.1767-3.2973 2.5077-3.0401 2.6572-3.0862 2.5896-3.0840 2.6696-3.5461 3.1552-4.0003 3.0678-3.5559 0.0997-3.1562 0.0097-2.3197 2.7522-3.3707 1.7550-3.2864 2.8683-3.4509 2.1092-3.3538 1.8332-2.6043 2.7040-3.4472 2.7219-3.1041 2.1198-3.0571 0.1806-2.1314 1.7644-2.9407 0.5159-3.3958 0.1193-16.268 2.4634-3.4415 2.7679-3.6574 3.0843-3.8358
1-5 6-11, 12c 11, 12c 13 13 14 15 16 17 18-20 21-24, 25c 26 8, 27, 28 16 29, 30c 29, 30c 8, 29, 30,c 31 8, 32, 33 34 35 36 30,c 37, 38 39, 40 27 41 42
a T ) T/T , F ) F/F . b [1] Hutchinson, F. J. Chem. Phys. 1949, 17, 1081. [2] Corbett, J. W.; Wang, J. H. J. Chem. Phys. 1956, 25, 422. r c r c [3] Mifflin, T. R.; Bennett, C. O. J. Chem. Phys. 1958, 29, 975. [4] Cini-Castagnoli, G.; Ricci, F. P. J. Chem. Phys. 1960, 32, 19. [5] De Paz, M.; Turi, B.; Klein, M. L. Physica 1967, 36, 127. [6] Dawson, R.; Khoury, F.; Kobayashi, R. AIChE J. 1970, 16, 725. [7] Oosting, P. H.; Trappeniers, N. J. Physica 1971, 51, 418. [8] Dullien, F. A. L. AIChE J. 1972, 18, 62. [9] Harris, K. R. Physica A 1978, 94, 448. [10] Harris, K. R.; Trappeniers, N. J. Physica A 1980, 104, 262. [11] Schmid, A.; Wappmann, S.; Has, M.; Ludemann, H.-D. J. Chem. Phys. 1991, 94, 5643. [12] Younglove, B. A.; Ely, J. F. J. Phys. Chem. Ref. Data 1987, 16, 577. [13] Enninghorst, A.; Wayne, F. D.; Zeidler, M. D. Mol. Phys. 1996, 88, 437. [14] Jonas, J.; Hasha, D.; Huang, S. G. J. Phys. Chem. 1980, 84, 109. [15] Harris, K. R. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2265. [16] Harris, R. K.; Alexander, J. J.; Goscinska, T.; Malhotra, R.; Woolf, L. A.; Dymond, J. H. Mol. Phys. 1993, 78, 235. [17] Dymond, J. H.; Harris, R. K. Mol. Phys. 1992, 75, 461. [18] Takahashi, S. J. Chem. Eng. Jpn. 1977, 10, 339. [19] Arends, B.; Prins, K. O.; Trappeniers, N. J. Physica A 1981, 107, 307. [20] Baker, E. S.; Brown, D. R.; Jonas, J. J. Phys. Chem. 1984, 88, 5425. [21] O’Hern, H.; Martin, J. J. Ind. Eng. Chem. 1955, 47, 2081. [22] Takahashi, S.; Iwasaki, H. Bull. Chem. Soc. Jpn. 1966, 39, 2105. [23] Robinson, R. C.; Stewart, W. E. Ind. Eng. Chem. Fundam. 1968, 7, 90. [24] Etesse, P.; Zega, J. A.; Kobayashi, R. J. Chem. Phys. 1992, 97, 2022. [25] Pitzer, K. S.; Schreiber, D. R. Fluid Phase Equilib. 1988, 41, 1. [26] Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1982, 78, 583. [27] Parkhurst, H. J., Jr.; Jonas, J. J. Chem. Phys. 1975, 63, 2698. [28] Collins, A. F.; Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1975, 71, 2296. [29] Prielmeier, F. X.; Ludemann, H.-D. Mol. Phys. 1986, 58, 593. [30] Densities were estimated by the HankinsonBrobst-Thomson method; see Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. [31] Harris, K. R.; Lam, H. N.; Raedt, E.; Easteal, A. J.; Price, W. E.; Woolf, L. A. Mol. Phys. 1990, 71, 1205. [32] Collins, A. F.; Mills, R. Trans. Faraday Soc. 1970, 66, 2761. [33] McCool, M. A.; Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1972, 68, 1971. [34] Prielmeier, F. X.; Lang, E. W.; Ludemann, H.-D. Mol. Phys. 1984, 52, 1105. [35] Khoury, F.; Kobayashi, R. J. Chem. Phys. 1971, 55, 2439. [36] Finney, R. J.; Fury, M.; Jonas, J. J. Chem. Phys. 1977, 66, 760. [37] Harris, K. R. Physica A, 1978, 93, 593. [38] Has, M.; Ludemann, H.-D. Z. Naturforsch. A, 1989, 44, 1210. [39] Tison, J. K.; Hunt, E. R. J. Chem. Phys. 1971, 54, 1526. [40] DeZwaan, J.; Jonas, J. J. Chem. Phys. 1975, 63, 4606. [41] Fury, M.; Munie, G.; Jonas, J. J. Chem. Phys. 1979, 70, 1260. [42] Hurle, R. L.; Woolf, L. A. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2233. c References from which density data were obtained when they are not available from the same reference as the diffusion coefficient.
adopted the following rough Lennard-Jones (RLJ) expression (which is similar to the RHS eq 1), to describe them:
shape, it is reasonable to correlate this parameter with the acentric factor, ω. The following correlation was obtained:
D ) ADDLJ(σLJ,LJ)
AD ) 0.9673 - 0.2527ω - 0.70ω2
(45)
where DLJ(σLJ,LJ) is given by eq 37, and the translational-rotational coupling factor, AD, is taken as a temperature- and density-independent parameter. Equation 45 contains three parameters: AD, σLJ, and LJ. Because for the LJ fluid the diffusion coefficient is much more sensitive to the diameter σLJ than to the energy parameter LJ, the relation between the energy parameter and the critical temperature, obtained from viscosity data (Reid et al., 1987),
Tc LJ ) k 1.2593
(46)
is employed. The other two parameters were determined from diffusion data. Since the coupling factor AD is expected to depend on the molecular size and
(47)
As shown below, σLJ can be correlated with the critical volume, but the error in the value of diffusivity can be as large as 50%. Consequently, only one adjustable parameter, σLJ, remains in eq 45 to be determined from diffusivity data. Of course, this parameter represents an equivalent molecular size of the nonspherical molecules. Table 5 lists the calculation results for 26 compounds with 1822 data points. From this table one can see that eq 45 provides accurate correlations for small and nonpolar molecules and acceptable results for long chain-molecules and polar ones. For most compounds, the AAD is comparable to the experimental uncertainty. Higher deviations are found for long-chain or polar molecules. The grand AAD for all points is 7.33%. Considering the wide ranges of temperature and pres-
3934 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 4. Prediction Results AAD substance
sourcea
σLJ (Å)
LJ/k (K)
NDP
eq 37
eq 2
argon
µ B µ+B µ B µ+B µ B µ+B µ B µ+B µ B µ+B
3.434 3.499 3.429 3.774 4.010 3.678 3.881 4.416 3.832 4.257 4.433 4.200 4.480 5.220 4.221
120.72 118.13 121.85 143.81 142.87 166.78 216.06 192.25 230.56 201.93 202.52 219.01 208.46 194.14 274.48
72
4.88 6.16 4.94 16.88 38.43 11.68 15.47 42.04 14.39 40.81 49.99 38.02 69.73 79.06 33.26
11.11 13.02 11.31 23.86 42.17 21.88 22.27 45.41 21.93 44.00 55.78 41.48 66.53 405.15 36.41
methane CO2 ethylene ethane
369 191 118 65
a
µ, parameters from viscosity data; B, parameters from second virial coefficient data; µ + B, parameters from both viscosity and second virial coefficient data. All parameters are taken from Tee et al. (1966).
Figure 6. Temperature dependence of the effective molecular diameter of some compounds (calculated with eqs 31 and 39) against reduced temperature Tr ) T/Tc. The values of σe(T) are in angstroms.
Table 5. Calculation Results from Eq 45a compounds
AD
argon methane ethane isopentane cyclopentane cyclohexane n-hexane n-octane n-hexadecane ethylene carbon dioxide carbon disulfide benzene toluene chloromethane dichloromethane chloroform carbon tetrachloride fluoroform carbon tetrafluoride perfluorocyclobutane chlorotrifluoromethane sulfur hexafluoride tetramethylsilane pyridine acetonitrile total
1.0 1.0 0.9354 0.8739 0.8909 0.8823 0.8292 0.7558 0.3944 0.9393 0.8669 0.9314 0.8823 0.8524 0.9123 0.8893 0.8789 0.8925 0.8543 0.9673 0.9673 0.8898 0.8378 0.5637 0.8646 0.8098
σLJ (Å) LJ/k (K) NDP
AAD
3.4236 3.6387 4.1336 5.3987 5.1088 5.4805 5.5731 6.1092 7.5920 3.9658 3.4482 4.2476 5.0506 5.3176 4.0294 4.3936 4.7763 5.2335 4.0968 4.4646 5.6569 4.4789 4.7229 5.6111 4.8610 4.0970
4.81 5.96 5.11 5.73 4.42 2.64 8.31 5.22 11.15 7.73 7.52 2.92 4.98 6.68 8.62 9.46 10.16 5.34 9.61 7.49 4.66 13.13 13.50 7.88 11.45 6.12 7.33
119.75 151.32 242.52 365.60 406.34 440.05 402.92 451.68 573.33 224.25 241.56 438.34 446.35 469.94 330.58 404.99 425.95 441.83 237.67 180.74 308.50 239.82 253.08 356.24 492.34 433.18
72 369 65 22 21 39 59 43 25 118 191 29 65 54 46 43 64 37 38 53 59 115 38 42 55 60 1822
a The values of A are calculated from eq 47, except for argon D and methane; the values of LJ/k are calculated from eq 46; σLJ is an adjustable parameter determined from diffusion coefficient data.
sure involved and the large number of data points used, the correlation results can be considered very good. Besides the equation itself, there are other sources of errors: (1) the experimental errors, especially the differences between different authors (for instance, those for chloroform and sulfur hexafluoride); (2) the errors from estimating the densities (the densities of chloromethane, dichloromethane, and chlorotrifluoromethane were estimated using the HankinsonBrobst-Thomson method (Reid et al., 1987)). Equation 45 with two parameters, AD and σLJ, was also used to correlate the self-diffusion in liquid water and alcohols (methanol, ethanol, and n-propanol). Higher deviations, around 30%, were found. For these compounds, molecular clusters rather than single molecules diffuse in the liquid. Figure 6 presents the temperature dependence of the effective molecular diameter σe of some compounds,
Figure 7. Correlation of the equivalent molecular diameter parameter (in angstroms) with the critical volume (in cm3/mol). The numbers represent the compounds listed in Table 3.
calculated by combining eqs 39 and 31. In these equations, σLJ is the equivalent molecular size determined from diffusion data. As shown in the figure, the effective molecular diameter decreases slowly as the temperature increases. Although the changes in diameter are small, the resulting changes in the diffusion coefficient are significant. This demonstrates again the extreme sensitivity of the diffusivity to the molecular diameter. For real substances, this sensitivity is even higher than that for the LJ fluid. A change of the molecular diameter by 1% may lead to a change in the diffusion coefficient by 10-20%. In Figure 7 the equivalent diameter parameter, listed in Table 5, is plotted against the critical volume. This parameter can be correlated with the critical volume using the following equation:
σLJ ) 0.7889Vc1/3
(48)
where the critical volume is expressed in cm3/mol and σLJ in angstroms. This result is comparable to that obtained from viscosity data (Reid et al., 1987): σLJ ) 0.809Vc1/3. A somewhat less accurate correlation is provided by the expression:
( )
σLJ ) 0.5058
RTc Pc
1/3
(49)
which may be useful when critical volume data are not available.
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3935
Finally, combining eqs 46-48 with eqs 37 and 45 yields the following corresponding-state expression for the self-diffusion coefficient:
Dr ) Dr(Tr,Fr,ω)
Nomenclature AAD ) average absolute deviation,
(50)
AAD )
cal exp 100 NDP Di - Di
∑|
NDP i)1
[m/(kTc)]1/2(D/Vc1/3),
where Dr ) Tr ) T/Tc, and Fr ) F/Fc. It was found that, for the compounds for which the equivalent diameter provided by the diffusion data deviated from eq 48 by less than 1% (see Figure 7), the diffusion coefficient was predicted by eq 50 with an AAD of less than 15%. For the other compounds, the deviations in diffusion coefficients can be higher than 30%. Therefore, the completely predictive equation, eq 50, is recommended only for small, nonpolar molecules.
Conclusions A systematic study of the self-diffusion in hardsphere, Lennard-Jones, and real fluids was carried out. A new expression for the hard-sphere fluid was obtained using the molecular dynamics simulations of Alder et al. (1970), Erpenbeck and Wood (1991), and Woodcock (1981). Combining the expression for the hard-sphere fluid with the effective hard-sphere diameter method and with an expression for the soft friction coefficient, an equation for the self-diffusion in a Lennard-Jones fluid was derived and compared with numerous molecular dynamics simulations available in the literature. Several expressions for the effective hard-sphere diameter were compared, and the Boltzmann diameter with the parameters obtained by Ben-Amotz and Herschbach (1990) was adopted. The relative contributions of the repulsive and attractive branches of the interaction potential and the sensitivities of the diffusion coefficient to the parameters have been analyzed for the equation derived for a Lennard-Jones fluid. The repulsive branch dominates the diffusion in a dense fluid, and the diffusion coefficient is extremely sensitive to the molecular diameter. The equation based on the Lennard-Jones potential, which contains the diameter σLJ and the energy LJ as parameters, was tested with a large number of diffusion data for 26 real compounds over wide temperature and density ranges. The predictions show that only argon can be treated as a Lennard-Jones fluid and that methane and carbon dioxide can be approximately treated as Lennard-Jones fluids. For these fluids, the parameters obtained from viscosity data or from a combination of viscosity and second virial coefficient data can be used to predict the diffusion coefficients. For all other fluids, except water and alcohols, a rough Lennard-Jones type equation, which contains, in addition to σLJ and LJ, the translational-rotational coupling factor (AD), was used to correlate the diffusion coefficient. A correlation between the coupling factor and the acentric factor was proposed. It was found that the diffusion data for all compounds, including long-chain and polar molecules, can be represented accurately with an equation in which only one parameter, the equivalent molecular diameter σLJ, had to be obtained from diffusion data. Finally, a correlation between the molecular diameter parameter and the critical volume was derived, which is comparable to that obtained from viscosity data. This correlation allows one to make acceptable predictions regarding the self-diffusion of some simple (small, nonpolar, and weakly polar) fluids.
Dexp i
|
AD ) translational-rotational coupling factor CO2 ) carbon dioxide D ) self-diffusion coefficient, cm2/s D0 ) self-diffusion coefficient at very low gas densities, cm2/s;
(FD0) ) lim (FD) ) Ff0
3 kT 1/2 8σ2 πm
( )
Dr ) [m/(kTc)]1/2(D/Vc1/3) DLJ* ) reduced diffusivity defined by eq 38 EHSD ) effective hard-sphere diameter f0 ) constant in eq 29 f(F*) ) correction function defined by eq 12 g(r) ) radial distribution function g(σ) ) radial distribution function at contact G(λ) ) quantity defined by eq 22 HS ) hard sphere hn(z) ) quantity defined by eq 26 I ) quantity defined by eq 25 k ) Boltzmann constant LJ ) Lennard-Jones m ) mass of a molecule, kg MD ) molecular dynamics NDP ) number of data points P ) pressure, bar r ) radial coordinate R ) gas constant RD ) relative deviation, RD ) 100(Dcal - DMD)/DMD RHS ) rough hard sphere RLJ ) rough Lennard-Jones RC ) relative contribution defined by eq 41 T ) temperature, K Tr ) T/Tc T*) kT/ T0*) parameter in eq 30 ux(τ) ) molecular velocity at time τ and position x U(λ) ) quantity defined by eq 21 V ) molar volume, cm3/mol WCA ) Weeks-Chandler-Andersen y(r) ) background correlation function, y(r) ≡ g(r) exp[β(r)] z ) variable in eq 25 Greek Letters R ) constant R0 ) parameter in eq 30 R1 ) parameter in eq 28 β ) 1/kT, J-1 ) molecular energy, J (r) ) molecular potential, J φ(τ) ) normalized time-correlation function defined by eq 6 ∆σ ) sensitivity of diffusivity to the molecular diameter parameter defined by eq 42 ∆ ) sensitivity of diffusivity to the molecular energy parameter defined by eq 43 η ) πFσ3/6 for HS fluid, ) πFσe3/6 for LJ and real fluids λ ) parameter in Fourier transform, eqs 21 and 22 ω ) acentric factor F ) number density Fr ) F/Fc F* ) Fσ3 (for HS fluid), Fσe3 (for LJ and real fluids) σ ) molecular diameter, m σe ) effective hard-sphere diameter
3936 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 σe* ) σe(T)/σLJ σLJ ) molecular diameter of a LJ fluid or equivalent molecular diameter of nonspherical molecules, parameter in eq 45 τ ) time, s ξ ) friction coefficient Subscripts att ) attractive potential B ) Boltzmann effective hard-sphere diameter BH ) Barker-Henderson c ) critical point e ) effective hard-sphere diameter E ) Enskog HH ) Hammond-Heyes LJ ) Lennard-Jones LWCA ) Lado-Weeks-Chandler-Andersen r ) reduced quantity R ) repulsive contribution RHS ) rough hard sphere S ) soft attractive contribution SBT ) Sun-Bleazard-Teja t ) triple point WCA ) Weeks-Chandler-Andersen Superscripts * ) reduced quantity cal ) calculated value exp ) experimental value
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Received for review February 7, 1997 Revised manuscript received June 10, 1997 Accepted June 13, 1997X IE9701332
Abstract published in Advance ACS Abstracts, August 15, 1997. X
