8600
J. Phys. Chem. B 2008, 112, 8600–8604
Mixture Diffusion in Nanoporous Adsorbents: Equivalence of Fickian and Maxwell-Stefan Approaches Yu Wang and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt UniVersity NashVille, Tennessee 37235 ReceiVed: NoVember 2, 2007; ReVised Manuscript ReceiVed: March 28, 2008; In Final Form: April 2, 2008
Nanopore diffusion in multicomponent adsorption is described using different macroscopic theories: Onsager irreversible thermodynamics, Maxwell-Stefan, and Fickian approaches. A new equivalence between Fickian and Maxwell-Stefan formulations is described by [D] ) [ns][B]-1[Γ][ns]-1. The elements of D and B are explicitly related to the Fickian and Maxwell-Stefan diffusivities, respectively. Only when the saturation loadings nsi for different components are the same can the matrix be reduced to the generally accepted equation [D] ) [B]-1[Γ]. On the basis of the relationship between the irreversible thermodynamics and Maxwell-Stefan approaches, an equation is derived for a binary system with the symmetric form (1/Ð1 + θ2/Ð12)(1/Ð2 + θ1/Ð21) ) (L11L22)/(L12L21)(θ1θ2)/(Ð12Ð21) The Maxwell-Stefan binary exchange coefficients Ðij are shown to depend not only on the Maxwell-Stefan diffusivities, Ði, but also on the Onsager coefficients. For a strong molecular interaction, that is, Ði . Ðij, the ratio of Onsager coefficients will approach unity, giving the commonly used relation L12 ) √L11L22. In addition, the Maxwell-Stefan diffusivities, Ði, are shown to depend on the interaction effects in mixtures, and Ði in mixtures will not generally be equal to pure component values evaluated at the same total fractional loading. I. Introduction Transport of multicomponent mixtures in nanoporous materials plays an important role in many separation, biological, and heterogeneous catalysis processes. However, the understanding of mixture diffusion in these materials is relatively poor. Considerable interest exists concerning mixture diffusion in nanopores and has led to the development of both macroscopic and microscopic approaches. There is little difference between microscopic and macroscopic theories in the sense that many macroscopic theories use a certain amount of microscopic information.1 Microscopic studies, such as molecular dynamics (MD), relate the diffusivities to the microscopic behavior of the adsorbed molecules. Normally, the approaches focus on diffusion in ordered microporous material, such as zeolites and carbon nanotubes.2 Macroscopic techniques and theories can be applied to determine mass transfer mechanisms and related parameters for a wide range of materials. Generally, three macroscopic theories to describe diffusion in nanoporous materials have been used: Fick’s law, Onsager’s theory of irreversible thermodynamics (IT), and the MaxwellStefan theory (M-S). All of these theories have been applied and compared to describe mixture diffusion.1,3–12 This has clearly elucidated the driving forces for mass transport. The related transport parameters, described through the Onsager coefficients, L, are a less complicated function of composition than for other formulations because thermodynamic nonidealities are better accounted for in the driving forces for mass transport. D from Fick’s law is a function of concentration in general. Ð from the M-S approach is normally referred to as a corrected diffusivity, but recently, it has been shown to be strongly dependent on loadings for a variety of gases in zeolites.13,14 This characteristic * To whom correspondence should be addressed. Vanderbilt University, VU Station B 351604, 2301 Vanderbilt Place, Nashville, TN 37235-1604. Phone: (615) 322-2441. Fax: (615) 343-7951. E-mail: m.douglas.levan@ vanderbilt.edu.
and the complicated interactions among the adsorbed molecules create a challenging task in interpreting and predicting mixture diffusion. The related models based on these macroscopic theories for mixture diffusion vary considerably in their complexity, and thus, the diffusivities obtained can differ by a large extent. In recent years, the M-S diffusion formulation has been used with considerable success to describe mixture diffusion in zeolite structures.4–9,15–18,20 Molecular interactions have been considered first with an empirical relationship of Vignes5,21 and recently with an improved logarithmic interpolation formula.9,16 The objectives of this paper are to investigate the linkage among the three general multicomponent diffusion theories and to relate the transfer coefficients from the different theories without approximation. For a general mixture system with different adsorption saturation loadings, this work gives a new relationship between the Fickian and M-S formulations. In addition, the equivalent M-S form from IT is developed for a binary system to investigate the loading dependence for M-S diffusivities. II. Equivalence of Macroscopic Mixture Diffusion Theories We consider a mixture of adsorbed molecules confined within a homogeneous nanoporous space. For many inherently heterogeneous adsorbents, the surface heterogeneity will affect mass transfer in a complicated way and can be addressed with the development of constitutive flux equations that allow for the presence of a patchwise or continuous distribution of energies of interaction.22 Here, we consider a homogeneous region to provide a basis for further studies. We recognize that our development may not apply to markedly heterogeneous surfaces, such as occur in some common zeolites. Of the three frequently used macroscopic theories to describe multicomponent diffusion of adsorbates in nanoporous materials,
10.1021/jp710570k CCC: $40.75 2008 American Chemical Society Published on Web 06/26/2008
Mixture Diffusion in Nanoporous Adsorbents
J. Phys. Chem. B, Vol. 112, No. 29, 2008 8601
the approach most commonly used in practical applications is Fick’s law, which considers the flux to be proportional to the adsorbed-phase concentration gradient,9,17,23
N ) -Fp[D] ∇ n ) -Fp[D][ns] ∇ θ
(1)
where Fp is particle density; D is the nondiagonal matrix of Fickian diffusivities, which can depend on concentrations; ns is a diagonal matrix with elements nsi , which represents the saturation loadings of components; and ∇n and ∇θ are column vectors with elements of adsorbed-phase concentrations ni and fractional loadings θi, respectively. The fractional occupancies θi are defined by θi ≡ ni/nsi . The Fickian approach is simple and convenient to use; however, the Fickian diffusivities are found to depend strongly on concentrations, and the understanding needed for theoretical evaluation of cross-terms is limited. Onsager’s irreversible thermodynamics formulation is based on chemical potential gradients’ being the driving forces for diffusion. Preserving the linear relationship between transport of matter and the different thermodynamic driving forces, the general flux expression is given by9,15,16,20
N ) -[L] ∇ µ
(2)
where ∇µ is the column vector of chemical potential gradients, and L is a symmetric matrix of Onsager phenomenological coefficients, which satisfy reciprocal relations; that is, Lij ) Lji. The chemical potential can be written in terms of the equilibrium pressure given by the adsorption isotherm using
µi ) µi0 + RT ln Pi
Both relationships are mathematically correct but have different definitions of the thermodynamic factor. We use Γ as commonly defined in M-S theory for convenient comparison among the different theories. Krishna first introduced the essential M-S concept for describing surface diffusion within a zeolite matrix.4 A detailed introduction to the M-S theory and advantages of using it have been described in review articles by Krishna and coauthors.5,6 M-S is entirely consistent with the theory of IT, and the chemical potential gradients are written as linear functions of the fluxes in the form m
-Fp
Ni θi njNi - niNj + s ∇ µi ) s s RT j)1,j*i n n Ð nÐ
∑
i j
i
i ) 1, ..., m
(8)
i
The first term on the right-hand side reflects the friction exerted by adsorbate j on the surface motion of species i, and the second term reflects the friction between species i and the surface.5 Ði is the M-S diffusivity of species i, reflecting interactions between species i and the adsorbent surface, and the Ðij′s are the binary exchange coefficients, reflecting correlation effects in mixture diffusion. To relate the M-S formulation with the Fickian approach, an m-dimensional square matrix, B, has been defined as4 m
∑
θj 1 + Ði j)1,j*i Ðij θi Bij ) Ðij
Bii )
(3)
where Pi is the partial pressure of species i and can be replaced by fugacity fi for generality. Thus, the gradients of µi at constant temperature can be written
ij
(9)
With this definition of B, eq 8 can be cast into the m-dimensional matrix form18,19
m
∑
∇µi ) RT ∇ ln Pi ) RT
∂ ln Pi ∇ nj j)1 ∂nj
Accordingly, the chemical potential gradients can be expressed in terms of the gradients of the occupancies by introduction of thermodynamic factors Γ4,5 m
∑
(5)
i
To compare the IT and Fickian approaches, it is advantageous to rewrite the Onsager form to include ∇θ by combining eqs 2 and 5 to give
N ) -RT[L][θ]-1[Γ] ∇ θ
(6)
Comparing eqs 1 and 6, we obtain the relationship between the IT and Fickian approaches without approximation as
[
RT ) [L] Fp
1 ⁄ θ1 0 0
·
0 ·· 0
0 0 1 ⁄ θm
][ [Γ]
1 ⁄ ns1 0 0
·
0 ·· 0
0 0 1 ⁄ nms
]
(7)
Chempath et al.7 have derived the relationship [D] ) [L][Γ], where Γ is defined differently as Γij ) (RT/nj)(d ln Pi/d ln nj).
(10)
We obtain the general relationship between the Fickian diffusivities and M-S diffusivities from eqs 1 and 10 as
[ ] [
[D] ) [ns][B]-1[Γ][ns]-1 ns1
θi Γij ∇ θj ∇ µi ) RT j)1 njs ∂ ln Pi θi ∂ ln Pi Γij ≡ s ni ) ∂nj θj ∂ ln θj n
RT [D] ) [L][θ]-1[Γ][ns]-1 Fp
N ) -Fp[ns][B]-1[Γ] ∇ θ
(4)
) 0
0 · ··
0
0
0
1 ⁄ ns1
-1 0 [B] [Γ] nms
·
0 0
0 ·· 0
0 0 1 ⁄ nms
]
(11)
It should be noted that the matrix ns and its inverse should not be canceled out for most systems because saturation capacities are typically different. Only when the saturation loadings for different adsorbates are the same can the above equation be further simplified to the generally accepted equation
[D] ) [B]-1[Γ]
(12)
which has been applied in previous publications for predicting Fickian diffusivities from M-S diffusivities. The nonequivalence of the two relations for D for the general case of differing nsj ′s is shown by the following simple reasoning. For simplicity, let the result of the matrix multiplication in eq 12 be represented by
[B]-1[Γ] )
[
a11 a12 a21 a22
]
(13)
This can be compared with the matrix multiplication in eq 11, which gives
[
8602 J. Phys. Chem. B, Vol. 112, No. 29, 2008
[ns][B]-1[Γ][ns]-1 )
[
ns1 0 0
ns2
[ ] a11
)
][
a11 a12 a21 a22
ns2 a21 ns1
][
1 ⁄ ns1
0
0
1 ⁄ ns2
ns1 a12 ns2
]
a22
[ ] [ ]
Fp ) 0 RT 0
·
0 ·· 0
θ1
0
-1
0 [B] nms
0
0
·
0 ·· 0
0
0 θm
(15)
This relationship is consistent with the expression derived by Krishna and Baten,15,16
[∆] ) [B ′ ]-1
B′ii )
∑
θj 1 + Ði j)1,j*i Ðij
-L21ns1θ2
-L12ns2θ1 L11n2
(16)
Ð1 )
|L| RT RT (L11L22 - L12L21) ) Fp L22n1 - L12n2 Fp L22n1 - L12n2
Ð2 )
|L| RT RT (L11L22 - L12L21) ) Fp L11n2 - L21n1 Fp L11n2 - L21n1
Ð12 )
RT |L| RT (L11L22 - L12L21) ) Fp L ns Fp L12ns2 12 2
Ð21 )
RT |L| RT (L11L22 - L12L21) ) s Fp L n Fp L21ns1 21 1
]
(20)
Θis B′ij ) - s Θj
θ ) θ1 + θ2 ) θi Ðij
(17)
(21)
It is clear from eq 21 that the M-S diffusivities, Ði, for a binary mixture are dependent on the loading, ni, and the values of the Onsager coefficients. In the M-S approach, the value of Ði in a mixture is commonly assumed to be equal to the corresponding pure component diffusivity,7,13,15 evaluated at the occupancy of the mixture
where B′ is a square matrix with elements m
[
L22n1
The M-S diffusivities are explicitly expressed using Onsager coefficients as
Fp s [n ][B]-1[θ] RT ns1
θ2 θ1 1 + Fp 1 Ð1 Ð12 Ð12 [B] ) ) × θ2 θ1 RT |L| 1 + Ð21 Ð2 Ð21
(14)
Clearly, the cross-term diffusion coefficients calculated by eqs 11 and 12 differ unless the saturation capacities are identical. Mathematically, the matrices [ns] and [ns]-1 are referred to as similar matrices, and the transformation in eq 11 is a similarity transformation.24 The relationship between IT and M-S formulations can be obtained on the basis of eqs 7 and 11 as
[L] )
]
Wang and LeVan
n1 ns1
+
n2 ns2
(22)
From eqs 9 and 19, the pure component Ð1 is given by
Ð1 )
and ∆ is related to the Onsager matrix L and is defined conveniently by
RT L1 Fpns1θ
(23)
Introducing an occupancy fraction parameter,
[∆] N ) -Fp [Θ] ∇ µ kBT
(18)
where kB is Boltzmann’s constant and Θ is the matrix of loadings Θi expressed in molecules per unit cell. For a binary system, the general relationship shown in eq 15 reduces to a form previously developed by Chempath et al.7 The equivalence among the transfer coefficients from the three approaches is shown in Figure 1. The M-S diffusivities are embedded in the B matrix, as indicated in eq 9. It is clear that the mass transfer coefficients from the three different approaches can be directly related. The three formalisms are all mathematically equivalent.12 The Fickian diffusivity matrix, D, is explicitly related to the product of the thermodynamic factor Γ and the matrix of Onsager coefficients or M-S diffusivities. The manner of dependence of D on Γ demonstrates that the Fickian diffusivities have a more complicated concentration dependence than the IT and M-S transfer coefficients. The equivalence among theories provides a theoretical basis for studying the relationship between the M-S diffusivities, Ði, and the loading dependence. Starting from the relationship between IT and M-S given by eq 15, we obtain
[B] )
Fp [θ][L]-1[ns] RT
For a binary system, eqs 9 and 19 give
(19)
f1 ≡
θ1 n1 ⁄ ns1 ) θ n ⁄ ns + n ⁄ ns 1 1 2 2
(24)
eq 21 gives for a binary mixture
Ð1 )
2 RT 1 (L11 - L12 ⁄ L22) L12ns2(1 - f1) Fpns1θ f1 1L22ns1 f1
(25)
Thus, from eqs 23–25, evaluated at the same occupancy θ, only when
L1 )
(L11 - L122 ⁄ L22) 1 f1 1 - L ns (1 - f ) ⁄ (L ns f ) 12 2 1 22 1 1
(26)
will Ði for mixtures have the same value as for pure components. On the basis of the complicated nature of this relationship relating loadings and different host-guest combinations for Onsager coefficients, Ði will not generally have the same values for a pure component or as a component in a mixture at the same occupancy. This point is consistent with results of previous molecular simulations, which show that the values of Lij vary with loadings for different mixture systems.7,9,15 For the limiting case of negligible binary interaction, i.e., L12 ) L21 ) 0, the exchange coefficient Ðij approaches infinity, and the M-S
Mixture Diffusion in Nanoporous Adsorbents
J. Phys. Chem. B, Vol. 112, No. 29, 2008 8603
Figure 1. Relationships among macroscopic diffusion theories. New equivalence is shown between the Fickian and Maxwell-Stefan approaches.
diffusivities, Ði, for a mixture will reduce to Ði ) (RT/Fpnsi θ)(Lii/ fi), which is similar to eq 23 for a pure component. Thus, at a constant occupancy θ, if the relationship between Lii and fi in this equation is nonlinear, the assumption that the M-S diffusivity, Ði, in a mixture is the same as that for a pure component will be violated. It is also notable that the M-S diffusivity matrix loses the symmetry property because of the appearance of nsi . As shown previously,9 the exchange coefficients satisfy
ns2Ð12 ) ns1Ð21
(27)
Similarily, the matrix elements of eq 19 give a relationship between the M-S diffusivities Ði and the binary exchange coefficients Ðij in the form
(
)(
)( )
θ2 1 θ1 L11L22 θ1θ2 1 + + ) Ð1 Ð12 Ð2 Ð21 L12L21 Ð12Ð21
(28)
The binary exchange coefficients, Ðij, are found to depend not only on the M-S diffusivities, Ði, but also on the ratio of Onsager coefficients. For a limiting case when the molecular interaction is very strong, that is, when Ði . Ðij, the ratio of Onsager coefficients will approach unity. This means that the common assumption for the cross-term Onsager coefficients of L12 ) L21 ) √L11L22 is only appropriate for the very strong molecular interaction case. However, considering another limiting case when the molecular interaction is very weak, that is, when Ðij f ∞, eq 20 will require Lij f 0. On the basis of eq 28, for a general system (with Ði positive), the Onsager coefficients will satisfy L11L22 > L12L21. Thus, allowing for the limiting cases, the cross-term Onsager coefficient for a binary system will obey 0 e L12 e √L11L22. III. Example Sanborn and Snurr25 have performed molecular simulations to investigate the Fickian transport diffusivities of binary mixtures of methane (component 1) and CF 4 (component 2) in the zeolite faujasite at 300 K. Krishna26 verified the M-S formulation by comparing Sanborn and Snurr’s simulation results with Fickian diffusivities calculated from M-S diffusivities using eq 12. However, we have shown that the traditional relationship between Fickian diffusivities and M-S diffusivities given by eq 12 applies only if the saturation loadings for the
components are the same. For this example, the estimated saturation capacities for CH 4 and CF 4 are ns1 ) 10 molecules/ supercage and n2s ) 6.1 molecules/supercage, respectively.26 Thus, our new equivalence given by eq 11, which relates the Fickian matrix D and the M-S matrix B for differing saturation capacities, will give a different result as compared to Krishna’s calculation using eq 12. We adopt the same procedure and parameters in Krishna’s paper to predict Fickian diffusivities. Comparisons will be given for a total mixture loading of six molecules per supercage. Mixture isotherms are described by a multicomponent Langmuir isotherm. The definition of thermodynamic factor shown in eq 5 in Krishna’s paper is the same as in eq 5 here. Accordingly, the elements of Γ for the multicomponent Langmuir isotherm are given in eq 6 in Krishna’s paper as
Γij ) δij +
θi 1 - θ1 - θ2
i, j ) 1, 2
(29)
where δij is the Kronecker delta. In Γ, ∂ ln fi/∂ ln θj can be calculated as a function of mole fraction of CH4. We have recalculated ∂ ln f2/∂ ln θ1 and ∂ ln f1/∂ ln θ2 for Krishna’s example because of an apparent error in his Figure 1.26 Following Krishna, the M-S diffusivity for CH4 is considered to be independent of loading, with the value Ð1 ) 35 × 10-9 m2/s, whereas Ð2 for CF4 is assumed to have a loading dependence given by Ð2 ) Ð02(1-θ1-θ2), where Ð02 ) 20 × 10-9 m2/s. The interchange coefficient Ðij is estimated by the logarithmic interpolation formula, Ðij ) [Ði]θi/(θi+θj)[Ðj]θj/(θi+θj). Figure 2 shows the comparison between the old and new relationships for calculation of the Fickian diffusivities from M-S diffusivities. It is clear that for main-term diffusivities, the new and old relationship give the same results, but for crossterm diffusivities, the simulation results differ by the ratio of the saturation loadings, as indicated in eq 14. An important point to notice in the theory and example is that the method used to calculate the Maxwell-Stefan parameters does not in any way impact our conclusion that the Fickian coefficients calculated from eqs 11 and 12 will differ if the adsorbates have different saturation capacities. Note that both of these equations contain [B]-1 in an equivalent way, and this matrix is the only place the Maxwell-Stefan parameters appear. The equations differ only in that the second has the inner product
8604 J. Phys. Chem. B, Vol. 112, No. 29, 2008
Wang and LeVan exhibits negligible binary interactions, the assumption will be violated when Lii varies nonlinearly with the occupancy fraction parameter fi given by eq 24. Appendix Nomenclature Ð ) M-S diffusivity of component i Ð ) binary exchange diffusivity Ðii ) self-exchange diffusivity D ) main-term Fickian diffusivity Dij ) cross-term Fickian diffusivity D0 ) corrected diffusivity fi ) occupancy fraction parameter N ) flux of component L ) Onsager phenomenological coefficient kB ) Boltzmann constant n ) adsorbate concentration P ) pressure R ) ideal gas constant T ) temperature Greek Letters
Figure 2. Comparison of Fickian diffusivities calculated from M-S diffusivities using the new relationship (eq 11) with those using the traditional relationship (eq 12).
[B]-1[Γ] premultiplied by [ns] and postmultiplied by [ns]-1, giving the similarity transformation. Our main point is that with differing saturation capacities, the Fickian cross coefficients are calculated incorrectly from Maxwell-Stefan parameters by published methods. This has nothing to do with the Maxwell-Stefan parameters themselves or with the matrix [B]-1, which contains them. Rather, it is due to the former omission of the matricies [ns] and [ns]-1 in the equation used to calculate D. IV. Conclusions Three macroscopic mixture diffusion theories have been compared to show their equivalence. The Fickian, IT, and M-S approaches are mathematically equivalent and can be chosen on the basis of convenience. The Fickian diffusivity matrix can be obtained from Onsager coefficients or M-S diffusivities as [D] ) [ns][B]-1[Γ][ns]-1 ) (RT/Fp)[L][θ]-1[Γ][ns]-1. For the simple case of equal saturation loadings for all adsorbates, the relationship reduces to the commonly accepted form [D] ) [B]-1[Γ]. On the basis of the relationship between the IT and M-S approaches, an equation has been derived for a binary system having the symmetrical form (1/Ð1 + θ2/Ð12)(1/Ð2 + θ1/Ð21) ) (L11L22/L212)[(θ1θ2)/(Ð12Ð21)]. The binary exchange coefficients Ðij are shown to depend not only on the M-S diffusivities Ði but also on the Onsager coefficients. For a general system, the cross-term Onsager coefficient for a binary system satisfies 0 e L12 e √L11L22. The M-S diffusivities Ði generally depend on the loading, ni, and mixture interaction effects. This is in contrast to the assumption commonly used in the M-S approach that the mixture Ði is equal to the pure component Ði at the same total fractional loading. For the Ði values to be truly equal, eq 26 must be satisfied. As a simple limiting case, when the system
Γ ) thermodynamic correction factor µ ) chemical potential θ ) fractional coverage of adsorption sites Θ ) adsorbate loading in molecules per unit cell Fp ) particle density ∆ ) matrix of M-S diffusivities Subscripts s ) saturation loading References and Notes (1) Benes, N.; Verweij, H. Langmuir 1999, 15, 8292. (2) Beerdsen, E.; Dubbeldam, D.; Smit, B. Phy. ReV. Lett. 2005, 95, 164505. (3) Curtiss, C. F.; Bird, R. B. Ind. Eng. Chem. Res. 1999, 38, 2515. (4) Krishna, R. Chem. Eng. Sci. 1990, 45, 1779. (5) Krishna, R.; Wesselingh, J. A. Chem. Eng. Sci. 1997, 52, 861. (6) Krishna, R. Sep. Purif. Technol. 2003, 33, 213. (7) Chempath, S.; Krishna, R.; Snurr, R. Q. J. Phys. Chem. B 2004, 108, 13481. (8) Krishna, R. Chem. Eng. Sci. 2001, 84, 207. (9) Skoulidas, A. I.; Sholl, D. S.; Krishna, R. Langmuir 2003, 19, 7977. (10) Wang, Y.; LeVan, M. D. Ind. Eng. Chem. Res. 2005, 44, 4745. (11) Wang, Y.; LeVan, M. D. Ind. Eng. Chem. Res. 2007, 46, 2141. (12) Sholl, D. S. Acc. Chem. Res. 2006, 39, 403. (13) Krishna, R.; Van Baten, J. M. Microporous Mesoporous Mater. 2008, 109, 91. (14) Li, S.; Falconer, J. L.; Nobel, R. D.; Krishna, R. J. Phys. Chem. C 2007, 111, 5075. (15) Krishna, R.; Van Baten, J. M. J. Phys. Chem. B 2005, 109, 6386. (16) Krishna, R.; Van Baten, J. M. Ind. Eng. Chem. Res. 2006, 45, 2084. (17) Krishna, R.; Vlugt, T. J. H.; Smit, B. Chem. Eng. Sci. 1999, 54, 1751. (18) Paschek, D.; Krishna, R. Langmuir 2001, 17, 247. (19) Kapteijn, F.; Moulijn, J. A.; Krishna, R. Chem. Eng. Sci. 2000, 55, 2923. (20) Krishna, R.; Paschek, D. Chem. Eng. J. 2002, 87, 1. (21) Vignes, A. Ind. Eng. Chem. Fundam. 1966, 5, 189. (22) Do, D. D. Adsorption analysis: equilibria and kinetics, 2nd ed.; Imperial College Press: London,1998, p 679. (23) Krishna, R.; Van Baten, J. M. Chem. Phys. Lett. 2006, 420, 545. (24) Golub, G. H. ; Van Loan, C. F. Matrix Computations, 3rd ed.; Johns Hopkins University Press: Baltimore, MD, 1996, p 311. (25) Sanborn, M.; Snurr, R. Q. AIChE J. 2001, 47, 2032. (26) Krishna, R. Chem. Phys. Lett. 2002, 355, 483.
JP710570K
