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Comparison of Macro- and Microscopic Theories Describing Multicomponent Mass Transport in Microporous Media Nieck Benes* and Henk Verweij Laboratory of Inorganic Materials Science, Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received April 26, 1999. In Final Form: July 14, 1999 A detailed discussion is presented of three well-known macroscopic theories for describing the mass transport behavior of multicomponent mixtures; these include Fick’s law, the Onsager theory of irreversible thermodynamics, and the Maxwell-Stefan theory. The merits and drawbacks of these theories are discussed, together with their interrelation. These macroscopic theories and an entirely microscopic theory are applied to describe the transport of a mixture of mobile species in a microporous material obeying ideal Langmuir sorption behavior, which is an issue of considerable interest in the field of membrane technology. If a unary system, i.e., a single mobile species, is considered, the chemical diffusion coefficient appears to be independent of the concentration. In this case, mass transport can simply be described using Fick’s law. For a multicomponent mixture, the chemical diffusion coefficients become a function of composition and Fick’s law can no longer be used in a straightforward way. The two remaining macroscopic theories and the microscopic theory can adequately describe the mass transport behavior, assuming that mechanical interactions between the mobile species are negligible. When mechanical interactions are not negligible, only the Maxwell-Stefan theory yields a proper description. Applying the Onsager theory of irreversible thermodynamics in that case would lead to off-diagonal transport coefficients that are nonzero. Also, the microscopic theory used in this study is then no longer applicable, since it simply neglects interactions between mobile species. It is demonstrated that the use of virtual chemical potentials for the mobile species, by contrast with real chemical potentials of building units, complicates the description of mass transport.
Introduction Transport of multicomponent mixtures is known to play an important role in many technological fields, and numerous studies have been dedicated to describing their transport behavior. These theories can roughly be divided up into macroscopic and microscopic. This distinction is not a sharp one, since in many macroscopic theories a certain amount of microscopic information is used. In this study the following macroscopic theories are discussed: •Fick’s law, •Onsager theory of irreversible thermodynamics, and •Maxwell-Stefan theory The first two theories include essential microscopic information in the sense that a distinction is made between various distinguishable species, to which different properties are attributed. In the Maxwell-Stefan theory even more microscopic information is included, because in this theory mechanical interactions between the different species are considered explicitly. If sufficient information is available, it is also possible to describe the mass transport behavior completely on a microscopic basis. At first glance it is not always clear which theory is the most suitable for describing the mass transport behavior in a specific situation. In this paper, the merits and drawbacks of the above-mentioned theories are discussed to aid in selection of the proper theory for a specific problem. In general it can be stated that of all theories, Fick’s law has the simplest appearance and is the easiest to use for diluted systems. Using Fick’s law, it is often possible to derive analytical expressions for describing (timedependent) mass transport behavior. The chemical dif* Author to whom all correspondence should be addressed.
fusion coefficients used in Fick’s law are, however, generally not constant, given that they may depend on the composition of the mixture, thermodynamic nonidealities, and the nonequilibrium status of the system. Assuming ad hoc that they are constant may lead to erroneous results. Furthermore, for describing of the mass transport behavior of an n-component mixture, a nonsymmetric (n - 1) by (n - 1) matrix of chemical diffusion coefficients is needed.1,2 In the other two macroscopic theories, a smaller number of transport parameters is needed, as will be discussed later. In the Onsager theory of irreversible thermodynamics, the appropriate driving forces for mass transport are used, which results in a smaller number of transport coefficients compared to Fick’s law, i.e., the matrix containing the transport coefficients has the same size as in Fick’s theory but is now symmetric. Furthermore, the transport coefficients are a less complicated function of the composition, since thermodynamic nonidealities, i.e., deviations from ∇µi ) (RT/ci)∇ci are accounted for in the driving forces for mass transport. The interpretation of the transport coefficients is nevertheless still complicated, especially for systems where mechanical interactions between species are not negligible. For systems in which mechanical interactions between species are not negligible, the use of the Maxwell-Stefan theory may be advantageous, because it utilizes (microscopic) information about these interactions. The MaxwellStefan equations are often considered difficult to use because of their appearance. Another disadvantage of using the Maxwell-Stefan theory is that the information about interactions between species is not always readily available. These disadvantages seem superable, because in open literature the use of the Maxwell-Stefan theory
10.1021/la9905012 CCC: $18.00 © 1999 American Chemical Society Published on Web 10/01/1999
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The molar flux of a species is built up from a convective flux N, resulting from the overall movement of the ensemble of mobile species, and a diffusion flux Ji relative to this overall movement:
system that is in equilibrium. It is related to the mechanical mobility, b, by the Nernst-Einstein equation4 D ) b/RT. The Tracer Diffusion Coefficient, D*. The tracer diffusion coefficient follows from the average displacement of a labeled molecule or atom as a function of time, in a system that is in equilibrium. The behavior of a single molecule or atom is not necessarily identical to that of the entire ensemble, e.g., successive jumps of a molecule can be correlated. The tracer diffusion coefficient and the component diffusion coefficient are therefore, in general, not identical.4 The Maxwell-Stefan Diffusion Coefficient, ^ij. The Maxwell-Stefan diffusion coefficient is the reciprocal friction factor for mechanical interactions between a species i and a species j. Macroscopic Theories. When the transporting species are treated as continua, the kinetics of mass transport can be described without detailed knowledge of the system on a microscopic scale. Below, the three most frequently used macroscopic theories are briefly discussed. Fick’s Law. In 1855 Adolf Fick6 proposed that the flux Ji of a species i is proportional to the gradient in the concentration of a species
Ji ) ci‚(ui - u)
Ji ) -D ˜ i∇ci
for the description of multicomponent mass transport is rapidly increasing. A microscopic approach to describe mass transport has the advantage that it does not require explicit use of transport parameters such as diffusion coefficients. Instead, it is often possible to derive expressions for these parameters. On the other hand, information on a microscopic scale is needed. Definitions Used in This Paper. In what follows, a number of parameters are defined that are used throughout this paper. We define the molar flux Ni as the amount of mol transported through a unit of surface per unit of time, in a fixed laboratory system [mol‚m-2‚s-1]. It is equal to the product of the concentration ci of species i [mol‚m-3] and its velocity ui [m‚s-1] with respect to the fixed laboratory coordinate system
Ni ) ciui
(1)
(2)
where u is the average velocity [m‚s-1]. For a fluid, the mass average velocity usually is chosen as a measure for the average motion. Although this choice is connected to the momentum flux in a natural way, it is completely arbitrary,3 and other choices may be more appropriate. For instance, for mass transport in a microporous medium, the mass average velocity is not an appropriate choice, as will be discussed later in this paper. There is much confusion about nomenclature and physical meaning of the transport parameters used in the different theories. One cannot simply speak about the diffusion coefficient, because diffusion coefficients may be defined in many different ways. Common mistakes include the interchange of component diffusion coefficients, chemical diffusion coefficients, and intrinsic diffusion coefficients. In this paper we use the nomenclature described in the paragraphs below. The Chemical Diffusion Coefficient, D ˜ . The chemical diffusion coefficient is the parameter of proportionality between the diffusion flux and a gradient in composition. The Intrinsic Diffusion Coefficient, DI. When a solid material consists of multiple species with different mobilities, interdiffusion of these species may result in a net velocity of certain elements of the solid material in the fixed laboratory coordinate system.4,5 The fluxes with respect to these elements are described with the intrinsic diffusion coefficients, whereas the fluxes with respect to the fixed coordinate system are described with the chemical diffusion coefficients. The Component Diffusion Coefficient, D. This diffusion coefficient is a measure for the mobility of a species in a (1) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley and Sons: Singapore, 1960. (2) Waldram, J. R. Theory of Thermodynamics; Cambridge University Press: Cambridge, 1987; pp 238-242. (3) Lightfoot, E. N. Transport Phenomena and Living Systems, Biomedical Aspects of Momentum and Mass Transport; Wiley & Sons: New York, 1973. (4) Smalzried, H. Festko¨ rperreaktionen, Chemie des festen Zustandes; Verlag Chemie: Weinheim, 1971; pp 59, 68. (5) Darken, L. S. Am. Inst. Mining Met. Eng. Inst. Met. Div. Metals Technol. 1948, 15, Techn. Publ. 2311/2443.
(3)
where D ˜ i is the (isotropic) chemical diffusion coefficient [m2‚s-1] and ci is the concentration [mol‚m-3]. Equation 3 is referred to as the first law of Fick. Although this law is useful for diluted systems, in many practical systems the chemical diffusion coefficient may depend on the composition. In a multicomponent system, the movement of a species will also be influenced by the movement of other species present. This is accounted for in the so-called “generalized first law of Fick”:7 n
D ˜ il∇cl ∑ l)1
Ji ) -
(4)
Only (n - 1) of the diffusion fluxes and (n - 1) concentrations are independent. Hence, to describe mass transport behavior of an n-component mixture, (n - 1)2 diffusion coefficients are needed. It is emphasized that these coefficients cannot generally be assumed to be independent of mixture composition. Onsager Theory of Irreversible Thermodynamics. It was recognized many years ago that besides the presence of concentration gradients, the presence of, e.g., temperature gradients or external forces may cause transport of matter. According to the theory of irreversible thermodynamics, mass transport is due to the nonequilibrium status of the system. The entropy, S, of an isolated system determines whether the system is in thermodynamic equilibrium, i.e., any spontaneous irreversible change results in an increase in entropy. Onsager suggested that for each variable φj, which changes as the system approaches equilibrium, a thermodynamic driving force should be defined:2
Φj )
∂S ∂φj
(5)
Preserving the linear relation between transport of matter and the different thermodynamic driving forces, (6) Fick, A. Ann. Physik (Leipzig) 1855, 170, 59. (7) Taylor R.; Krishna, R. Multicomponent Mass Transfer; Wiley & Sons: New York, 1993.
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as in Fick’s law, the following general flux expression is obtained: n
∑l LilΦl
Ji ) -
(6)
If the system under consideration exhibits internal microscopic time-reversal symmetry, the matrix L containing the coefficients Lij is symmetric, because of the Onsager reciprocal relations Lji ) Lij.2,8 A practical example of a system that has no internal microscopic time-reversal symmetry is that of charged species moving in a magnetic field. In equilibrium, the total differential of the entropy can be calculated from the Gibbs relation.8 In a nonequilibrium system this expression does not hold. However, if the system is divided into small differential elements, it may be assumed that local equilibrium exists in these small elements. In particular, the Gibbs relation is assumed to be valid for a small element followed along its center of motion.8 This explains why in eq 6, the diffusion flux Ji, instead of the molar flux Ni, is considered. Far from equilibrium, the assumption of local equilibrium may fail, and a nonlinear dependence of fluxes on thermodynamic driving forces may be observed.9 Generalized Maxwell-Stefan Theory. Maxwell suggested that the flux of a species can be calculated from the second law of Newton: the forces acting on a system equal the rate of change of momentum of that system. The forces are evidently the same as the ones used in the Onsager theory of irreversible thermodynamics. The rate of change of momentum can be determined from the average number of collisions that occur, the average speed of the molecules, and additional (microscopic) information about the collisions. Inserting this information into Newton’s second law leads to the following equation:3,7 n
-di )
∑ l)1 l*i
xlJi - xiJl ctot\il
(7)
where x is the molar fraction [-], ctot the total concentration of the n-component mixture [mol‚m-3], and ^il the reciprocal friction factor for species i and l [m2‚s-1]. The friction between i and l or l and i is the same and therefore the matrix ^ containing all the coefficients is symmetric. di is the total force tending to move a species relative to the overall velocity [m-1]. Its derivation is analogous to that of the thermodynamic forces in the Onsager theory of irreversible thermodynamics. Transport in Microporous Media. To compare the three macroscopic theories mentioned above, they are applied to describe the transport behavior of an ncomponent mixture of mobile components in a microporous material, obeying ideal Langmuir sorption behavior. This problem is of particular relevance in the field of membrane research and catalysis,10,11 but also provides an opportunity for demonstrating the merits and drawbacks of the different theories. In addition, the mass transport is described using a microscopic approach. This makes it (8) de Groot, S. R.; Mazur, P. Nonequilibrium Thermodynamics; North-Holland Publishing Company, Amsterdam, 1962. (9) Eu, B. C. Kinetic Theory and Irrevesible Thermodynamics; Wiley & Sons: New York, 1992; p73. (10) Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. 1, The Theory of the Steady State; Clarendon Press: Oxford, 1975; p 23. (11) Qureshi, W. R.; Wei, J. J. Catal. 1990, 126, 126.
possible to extract the physical significance of the transport parameters used in the different macroscopic theories. Basic assumptions made are as follows: •The microporous medium consists of a 3D-interconnected network of qsat energetically independent voids [mol/kg], in which only one molecule of a mobile species can be present. •For reasons of simplicity, net-transport is assumed to take place in only one dimension, i.e., the z-direction. •The temperature is constant. •There are no external forces acting on the mobile species. Not all microporous materials show ideal Langmuir sorption behavior, e.g., energetic guest-guest interactions, multiple site occupancy, and different types of sites in a material might influence sorption behavior. The consequences of non-Langmuir sorption behavior for mass transport are beyond the scope of this study, but the interested reader is referred to the work of Ka¨rger and Ruthven.12 Fick’s Law. In this case, the mass transport is described by eq 4. The chemical diffusion coefficients may not be constant, and expressions for them must be obtained by rewriting other theories into a form similar to Fick’s law. Onsager Theory of Irreversible Thermodynamics. The system under consideration consists of the mixture of mobile species and the solid matrix M that makes up the microporous material. Because net-transport is assumed to take place in only one direction, the flux expressions of eq 6 for the mobile species are
Ji ) -
∑
l)1...n,M
( )
Lil
∂µ{l}
, i ) 1...n
(8)
∂z
where µ{l} is the chemical potential of the building unit13a {l} ≡ {ls - Vs}, where ls refers to the mobile species and Vs to the empty voids, i.e., vacancies. The building units denote that one cannot add a mobile species to the system without removing a vacancy, i.e., the mobile species and vacancies are conjugated. The chemical potential of a building unit {i} is the reversible work that is needed to place a molecule of species i inside the microporous material and simultaneously remove a vacancy. It is given by
µ{i} ≡ µ{is-Vs} ) µ{i},0 + RT ln
(
θi
1-
∑θl
)
(9)
where θi is the fraction of the qsat voids occupied by species i, i.e., θi ) qi/qsat. Note that eq 8 also contains the chemical potential of the solid microporous material M, which represents the reversible work needed to add a molecule of microporous material and simultaneously create a vacancy.13b
µ{M} ≡ µ{Ms+Vs} ) µ{M},0 + RT ln(1 -
∑θl)
(10)
Instead of using the chemical potentials of the building units, it is possible to describe the thermodynamic status of the system by using the “virtual” chemical potentials µls, µVs, and µMs. Here, “virtual” signifies that these chemical potentials do not reflect the fact that the species and (12) Ka¨rger, J.; Ruthven, M. Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons: New York, 1992. (13) (a) Lankhorst, M. H. R.; Bouwmeester, H. J. M.; Verweij, H. J. Am. Ceram. Soc. 1997, 80(9), 2175. (b) In fact, only a fraction f of molecules M that are added to the system result in the creation of a vacancy. This has no consequence for the discussion presented here.
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vacancies are conjugated. The implications of the use of virtual chemical potentials on the description of mass transport are discussed later in this paper. The flux obtained from eq 8 is the flux with respect to an average velocity. Because there is no viscous flow, the average velocity is assumed to be the velocity of the microporous material. This means that at any fixed position in the microporous material local equilibrium may be assumed.15a The mass transport behavior can now be completely described by considering only the flux equations of the mobile species.3,15b As was already mentioned, the entropy change accompanying linear processes is calculated from the equilibrium Gibbs relation, which implies the GibbsDuhem relation.9 If we consider the Gibbs-Duhem equation at constant temperature and pressure for the system under consideration
∑ ql l)1...n,M
( ) ∂µ{l}
)0
(11)
∂z
only n independent driving forces remain, and the flux expression reduces to
Ji ) -
∑ (Lil - θlLlM)
l)1...n
( ) ∂µ{l}
, i ) 1...n
(12)
∂z
The expression for the flux of the microporous material M is similar to eq 8,15c and after utilizing the GibbsDuhem relation it becomes
∑ (LMl - θlLMM) l)1,n
JM ) -
( ) ∂µ{l}
)0
(13)
∂z
Since the flux of the matrix is zero and the gradients in the chemical potentials of the mobile species can be varied independently, we obtain (LMl - θlLMM) ) 0. This can only be satisfied when LMl ) LMM ) 0, given that there are l independent coefficients LMl that are not directly related to LMM. The flux equations of the mobile species (eq 8) thus reduces to
Ji ) -
∑
l)1...n
( )
Lil
∂µ{l}
, i ) 1...n
∂z
In the case of a single mobile species, the flux becomes
Ji ) -Lii
( ) ∂µ{i} ∂z
(14)
The direct coefficient of a single species is the product of its concentration, Fqsatθi and mechanical mobility, bi. The mobility of a single species is determined solely by its mechanical interactions with the microporous material and will not be influenced by the nonequilibrium status (14) Benes, N. E.; Biesheuvel, M. P.; Verweij, H. AIChE J. 1999, 45(6), 1322. (15) (a) Although there may be a net flux of mobile species with respect to the microporous material, one has to realize there is also a flux of vacancies. Because the mobile species and the vacancies are conjugated, the fluxes of the mobile species and vacancies exactly oppose each other. Thus, the average motion of the mixture of all mobile species, i.e. “real” and vacancies, is zero. (b) Mason, E. A.; Malinauskas, A. P. Gas Transport in Porous Media: The Dusty Gas Model; Elsevier: New York, 1983; p 43-44. (c) To keep the microporous material fixed with respect to the laboratory coordinate system, i.e., Ji ) Ni, an additional external force should act on the material, which is directly related to the pressure difference over the material.14
of the system. It can therefore be related to the component-diffusion coefficient by the Nernst-Einstein equation e.g.,4,16 Di ) bi/RT, yielding
Lii )
FqsatθiDi RT
(15)
Combining eqs 9, 14, and 15 results in a pseudo-unary system, i.e., a single mobile species, in the following flux expression:
( )
FqsatDi ∂θi Ji ) (1 - θi) ∂z
(16)
If more than one mobile species is present, the flux expressions contain cross coefficients Lil. When it is assumed that there is no coupling between the flux of a species i and the gradient in chemical potential of other species l, the cross coefficients Lil can be neglected and the flux of a species i will obey
FqsatDi
Ji ) (1 -
∑θl)
(
(1 -
∂θi
)
∂θl
+ θi∑ θl) ∑ ∂z l*i l*i ∂z
(17)
The flux of the vacancies V can be calculated from JV n ) -∑l)1 Jl. The appearance of the flux expressions obtained above will be discussed later. One should keep in mind that they are not entirely macroscopic. The expressions for the chemical potentials of the mobile components, for instance, are derived using microscopic information. If mechanical interactions between the species are not negligible, there may be a coupling between the flux of a mobile species i and gradients in the chemical potential of other mobile species. As a result, the cross-coefficients Lil cannot be neglected. Since there are no straightforward expressions for cross-coefficients, the Onsager theory of irreversible thermodynamics will not be applicable to describe mass transport properly. Maxwell-Stefan Theory. The Maxwell-Stefan theory utilizes (microscopic) information about friction between different species to explicitly account for mechanical interactions between different species. The application of the Maxwell-Stefan theory for transport in microporous materials was, as far as we know, first proposed by Krishna,17 who regarded the vacancies as an (n + 1)th pseudospecies. We find, however, that the use of vacancies as a pseudospecies may lead to confusion and an inconsistent set of diffusion equations. One cannot simply apply the Gibbs-Duhem equation to vacancies, as Krishna does, since they cannot be regarded as real particles to which a definite mass (and, hence, a chemical potential) can be assigned. The use of vacancies also complicates the determination of the reference frame of velocity. Krishna18 in 1993 defines the molar average velocity as the choice for the reference velocity and states erroneously that Qureshi and Wei,11 who use the stationary voids as the reference frame velocity, assume a vanishing vacancy flux. In another publication19 in that same year, Krishna considers the vacancies as stationary. This is clearly not possible given that a jump of a molecule has to be opposed by a jump of a vacancy. Also, combining the variable (16) Rickert, H. Electrochemstry of Solids, an Introduction; SpringerVerlag: Berlin, 1982; p 83. (17) Krishna, R. Chem. Eng. Sci. 1990, 45(6), 1779. (18) Krishna, R. Chem. Eng. Sci. 1993, 48(5), 845.
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vacancy concentration and surface diffusivity of a mobile species (^i ) ^i,n+1/θn+1)19 does not help the understanding of the physical significance of the resulting diffusion coefficient. For a proper description of the mass transport behavior of an n-component mixture in a microporous medium, the mechanical interactions between the mobile particles and the solid matrix can be accounted for by treating not the vacancies, but the qsat voids as the (n + 1)th species. This yields the following Maxwell-Stefan equation:
ui - uj
n
-di )
θiθj ∑ j)1
\ij
j*i
ui - un+1 + θi \i,n+1
(18)
The terms on the right-hand side of the equation represent the rate of change of momentum. The probability that momentum is exchanged between two species is expressed by the product of the fractions of the two species. The amount of momentum exchanged is proportional to the velocity difference of the species. The parameter of proportionality is the reciprocal friction factor ^. The left-hand side of eq 18 represents the force exerted on species i that tends to move it relative to the overall velocity, which here is zero in the laboratory coordinate system, as discussed in the previous section. Since there are no external forces acting on the mobile species3
di )
( )
θi ∂µ{i} RT ∂z
-
RT
( )
n
)
∂z
∑ j)1
e-Em,i/RT
θjJi - θiJj
Ji ) Fqsatgaνie-Em,i/RT(θi|zθV|z+dz - θi|z+dzθV|z)
^ij
+
(20)
^i
( )
∑j
J i ) ^i
θjJi - θiJj ^ij
-
Fqsat^i
(
)
with the constant single component chemical diffusion coefficient
D ˜ i ) ga2νie-Em,i/RT
(25)
For a multicomponent mixture, the flux equation becomes
(
∂θi
)
∂θl
+ θi ∑ θl) ∑ ∂z l*i l*i ∂z
(26)
From eq 26, it can easily be shown that at steady state the concentration profiles of all mobile species are linear. Discussion
(21)
In the case of a binary mixture with negligible mechanical interactions between the mobile species (^1,2 ) ∞) this equation is identical to (17) with Di ) ^i. Microscopic Approach. For the master example chosen in this paper, there is much information available on a microscopic level. Computational techniques such as Monte Carlo and Molecular Dynamics methods (e.g., ref 12, p 44) can utilize this information for a numerical simulation of the diffusion process. Within the frame of the assumptions made here, however, it is possible to derive analytical expressions for the diffusion behavior of the system and circumvent the use of such advanced methods. While vibrating around their equilibrium position in a void, the molecules of each mobile species will attempt to cross a potential barrier to reach an adjacent void. They (19) Krishna, R. Gas Sep. Purif. 1993, 7(2), 91.
(24)
Transport Parameters in the Case of Negligible Mechanical Interactions Between Mobile Species. For an n-component mixture with negligible mechanical interactions between the species, the flux expression can be presented in Fickian notation:
×
∑θl) ∂θi ∂θl + θ i∑ (1 - ∑θl) ∂z l*i l*i ∂z
(1 -
∂θ1 ∂z
˜1 Ji ) -FqsatD
with ^i ≡ ^i,n+1. Substitution of eq 9 into eq 20 and rewriting yields n
(23)
which, for only a single mobile species, can be rewritten to
˜ i (1 Ji ) -FqsatD
Ji
(22)
where Em,i is height of the potential barrier. If mechanical interactions between the mobile species are negligible, a jump will be successful only if an adjacent void is empty. The probability that the adjacent void is empty is proportional to the fraction of empty voids, i.e., vacancies V. The average distance traveled during a jump is denoted a. The fact that a jump is not exactly in the direction of mass transport is accounted for by a geometrical factor g. The flux between two adjacent imaginary planes, normal to the z direction, can be calculated from
(19)
where the chemical potential is given by eq 9. After substitution of eqs 2 and 19 into eq 18, keeping in mind that un+1 ) 0, the following expression is obtained for species i:
Fqsatθi ∂µ{i}
do so with an attempt frequency νi. Only molecules with sufficient energy are able to surpass the potential barrier between the voids. This is reflected by the Boltzmann term
[∂θ∂zh]
J h ) -Fqsat[D]
[
(27)
]
The diffusion matrix obtained using the microscopic theory is n
D ˜ 1 (1 -
[D] )
· · ·
θl) ∑ l)2
D ˜ nθn
· · ·
D ˜ 1θ1
· · · · · · D ˜ n(1 ·
·
·
l)1
∑ θl)
n-1
(28)
where the constant single-component chemical diffusion coefficient, D ˜ i, is given by eq 25. It is clear that there can be a net flux of a species i while there is no gradient in the concentration of the species. This effect is caused by a gradient in the concentration of vacancies, due to the presence of mobile species other than i.
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Using the theory of irreversible thermodynamics, the following flux expression is obtained for a unary system, i.e., single mobile species in a microporous material:
J1 ) -
( )
FqsatD1 ∂θ1 (1 - θ1) ∂z
(29)
When eqs 28 and 29 are compared, the following relation between the single-component component diffusion and chemical diffusion coefficient is found:
˜1 D1/(1 - θ1) ) D
(30)
From this relation it is clear that the component diffusion coefficient must be proportional to the vacancy concentration, i.e., (1-θ). This can be explained by a decreased mobility of a molecule when the sites adjacent to it become increasingly occupied. In that case, a smaller number of jumps will be successful. When mechanical interactions between different species are negligible, the mobility of a species i thus obeys
bi ) b0i (1 -
∑ θl)
(31)
where bi0 is the mechanical mobility of species i in the microporous material for an infinitesimally low occupancy. For a unary system, the decrease in mobility with occupancy is balanced by an increase in the so-called thermodynamic factor Γii ) 1/(1 - θi), obtained from the following definition for a multicomponent mixture
θi RT
n
∇µi ≡
Γil ∇θl ∑ l)1
(32)
However, for a multicomponent mixture the following diffusion matrix is obtained:
[D] )
[
n
D1(1 -
1 n
(1 -
θl) ∑ l)1
· · ·
θl) ∑ l)2
Dnθn
· · ·
D1θ1
· · · · · n-1 · · · Dn(1 θl) ·
∑ l)1
]
using this theory. In the microscopic theory, presented here, mechanical interactions between the mobile species are simply not taken into account. Thus, this theory also will not give a proper description of the mass transport. Of the theories discussed in this study, only the Maxwell-Stefan theory can account quantitatively for the mechanical interactions in multicomponent systems in a straightforward manner. In eq 21 the interactions are accounted for by the term containing the reciprocal friction factor ^ij. One should keep in mind that there may also be mechanical interactions between molecules of a single species, which will influence the mobility of the species. The component diffusion coefficient, Di, and the MaxwellStefan diffusion coefficient, ^i, in that case may no longer be simply linearly dependent on the vacancy concentration. Vacancies as Pseudospecies. In the description of the mass transport using the Maxwell-Stefan theory, we have accounted for the presence of the porous medium by treating it as an (n + 1)th species. This is not a new idea; in fact, in 1860 Maxwell accounted for the presence of a macroporous medium in gas transport by treating the medium as a huge species, held motionless by some external force.20 The influence of the microporous medium on mass transport can also be accounted for by treating the vacancies as an (n + 1)th pseudospecies. In this case, one should realize that mass transport is described in terms of rather abstract interactions between vacancies and mobile species. Furthermore, the thermodynamic status of the system should also be described in terms of mobile species and vacancies. The vacancies formally have no chemical potential, since they have no mass. However, since the mobile species and microporous material are conjugated with the vacancies, it is possible to ascribe a virtual potential to the vacancies. The use of virtual chemical potentials is a mathematical device. For the system under consideration, the virtual chemical potentials are obtained by splitting up the building units {i} and {M} in is, Ms, and Vs:
[
µ{i} ) µi0s + RT ln
( )] [ qi
sat
q
- µV0 s + RT ln
From this it is clear that, for multicomponent mixtures, the change in mobility with occupancy is not exactly balanced by the changes in the thermodynamic factors. Even in the case that the mechanical interactions are negligible, the movements of different species will be mutually influenced. The diffusion matrix obtained using the MaxwellStefan theory is identical to eq 33, when mechanical interactions are negligible. Thus, the Maxwell-Stefan diffusion coefficient and the component diffusion coefficient are identical here: ^i ) Di. Transport Parameters in the Case of Mechanical Interactions Between Mobile Species. When mechanical interactions between mobile species are not negligible, a description of mass transport behavior becomes more complicated. Due to the mechanical interactions, the cross-coefficients Lij in the Onsager theory of irreversible thermodynamics will no longer be zero. Since there are no straightforward expressions for these coefficients, the mass transport cannot be properly described
[
( )] [
)]
∑ ql
sat
q ) µis - µVs (34)
(33) µ{M} ) µ0Ms + RT ln
(
qsat -
qsat 0 + µ{V + RT ln s} qsat qsat ql
(
∑
sat
q
)]
) µ0Ms + µVs (35)
0 equals where µV0 s may be chosen in such a way that µM s zero. The thermodynamic status of the system can thus be described by
µis ) µi0s + RT ln(θi) µVs ) µ0Vs + RT ln(1 -
∑ θi)
(36)
and it can be easily verified that these chemical potentials obey the Gibbs-Duhem relation. (20) Maxwell, J. C. Philos. Mag. 1860, 20, 21; reprinted in Scientific Papers, 1; Dover: New York, 1962; 392.
8298
Langmuir, Vol. 15, No. 23, 1999
Benes and Verweij
The Maxwell-Stefan equation, in terms of mobile species and vacancies, becomes
-
( )
Fqsatθi ∂µis RT
∂z
n
)
∑ j)1
θjJi - θiJi
+
(1 -
∑ θl)Ji - θiJV
^ij
j)1
(37)
It is clear that summation of eq 37 over all mobile species and the vacancies corresponds with the Gibbs-Duhem equation. For a pseudobinary system, i.e., mobile species and vacancies, eq 37 reduces to
( ) ∂θi ∂z
[
( )]
( )
These expressions contain two Maxwell-Stefan diffusion coefficients, as opposed to one if for the same system the real chemical potentials are used. This is due to the fact that the mobile species and vacancies are now introduced in eq 37 via the flux of the vacancies, which contains the parameters of interaction of the vacancies with all mobile species. The flux expression of a species i will thus contain the Maxwell-Stefan diffusion coefficients ^j,V of all mobile species present. When mechanical interactions between species are not neglected, the manipulation of the flux equations is further complicated. The use of the Onsager theory of irreversible thermodynamics also becomes more complicated when virtual chemical potentials are used. If we consider a pseudobinary mixture, i.e., one mobile species and the vacancies, the flux expression of the mobile species becomes:
( ) ∂µis ∂z
- LiV
( ) ∂µVs ∂z
(40)
After the Gibbs-Duhem equation is applied, the expression reduces to
Ji ) - L/ii
( ) ∂µis ∂z
) - FqsatDiV
Fick, microscopic eq 3 eq 16 eq 21 for n)1 microscopic eq 24
ON, MS, microscopic
MS
eq 4 eq 17 eq 21
eq 4 eq 21
eq 26
-
ficient” DiV. When more than one mobile species is present, the flux expression becomes n
∂θi ∂θj Fqsat ^i,V(1 - θi) - ^j,Vθi ∂z ∂z (1 - θ1 - θ2) (39)
Ji ) - Lii
preferred theory Fick ON MS
(38)
Note that the Maxwell-Stefan diffusion coefficient ^i,V is independent of the concentration. It represents the rather abstract reciprocal friction between the mobile species and the vacancies. In light of the assumed ideal Langmuir sorption behavior and the negligible mechanical interactions between the molecules of the mobile species, the friction between a mobile species and the vacancies is independent of concentration. For a multicomponent mixture, the use of virtual chemical potentials complicates the use of the MaxwellStefan theory, since the fact that the mobile species and vacancies are conjugated is no longer accounted for in the driving forces for mass transport. Consider, for example, the transport of two mobile species and the vacancies. When the mechanical interactions between the mobile species have been neglected, the following flux expression is obtained:
Ji ) -
n>1 n>1 negligible mechanical mechanical interactions interactions
n)1
^i,V
Ji ) -Fqsat ^i,V
Table 1. Preferred Theory, Based on the Number of Mobile Species Present and Whether Mechanical Interactions between These Species Are Eminent
( ) ∂θi ∂z
(41)
with the concentration-independent “interdiffusion coef-
Ji ) -
∑ l)1
Lil
( ) ( ) ∂µls
- LiV
∂z
∂µVs
(42)
∂z
If mechanical interactions between the mobile species are neglected, the corresponding cross coefficients are zero. Applying the Gibbs-Duhem equation now results in
Ji ) -
L/ii
( ) ∑( ∂µis ∂z
n
-
Lil - LiV
l*i
)( )
θl ∂µls θV
(43)
∂z
From the equation it is clear that the flux of a species i is influenced by the gradient in the virtual chemical potential of any other species present, even when mechanical interactions are negligible. Because there are no straightforward expressions for the cross-coefficients LiV, it is not possible to rewrite the flux expressions in terms of diffusion coefficients and gradients in the concentrations of the different species. Conclusions Three famous macroscopic theories and an entirely microscopic theory have been used for the description of the mass transport behavior of a multicomponent mixture in a microporous material that obeys ideal Langmuir sorption behavior. Which theory is the most appropriate for the description of mass transport in the specific problem addressed here depends on the number of mobile species present and whether mechanical interactions between these species are significant (see Table 1). When mechanical interactions between the molecules of mobile species are negligible and only a single mobile species is present in the microporous material, the singlecomponent chemical diffusion coefficient D ˜ i of this species appears to be independent of concentration. In this case, Fick’s law gives a proper description of the mass transport behavior. The fact that D ˜ i is constant results from the decrease in mobility, which is exactly opposed by an increase of the thermodynamic factor, as defined in eq 32. When more then one mobile species is present, the multicomponent diffusion coefficients become complicated functions of concentration and the use of Fick’s law is no longer straightforward. The mass transport behavior can now more easily be described using the Onsager theory of irreversible thermodynamics (ON), the Maxwell-Stefan theory (MS), and an entirely microscopic theory. In both the ON and the MS theories, the net rate of movement of each mobile species is related to the thermodynamic driving forces acting on the system and the mobility of the species. The mobility of each species can be related to the component diffusion coefficient by the Nernst-Einstein
Mass Transport in Microporous Media: Theories
equation and appears proportional to the concentration of empty voids 1 - ∑ θl. The presence of a mobile species influences both the mobility and the chemical potential of other mobile species present. The change in the chemical potential of such a species is not exactly opposed by the change in its mobility. As a result, there is a net influence of the presence of one species on the movement of another, and the chemical diffusion coefficients D ˜ ij therefore become functions of concentration. When mechanical interactions between mobile species become significant, the microscopic theory presented in this paper can no longer be used, given that it simply neglects mechanical interactions between molecules of mobile species. The Onsager theory of irreversible thermodynamics also can no longer be used in a straightfor-
Langmuir, Vol. 15, No. 23, 1999 8299
ward manner, since the matrix containing the transport coefficients will contain nonzero off-diagonal elements. The MS is then the preferred theory for a proper description of the mass transport behavior since it explicitly accounts for such interactions. The thermodynamic status of the system has been described by using the chemical potentials of building units. It is shown that the thermodynamic status of the system also can be described using the virtual chemical potentials of the n mobile species and the vacancies. However, the use of these virtual chemical potentials complicates the description of multicomponent mass transport. LA9905012
