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Ind. Eng. Chem. Res. 2007, 46, 3422-3428
GENERAL RESEARCH Restrictions on Friction Coefficients for Binary and Ternary Diffusion James S. Vrentas* and Christine M. Vrentas Department of Chemical Engineering, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802
A friction coefficient formalism provides a convenient method of relating mutual diffusion coefficients and self-diffusion coefficients. However, all of the friction coefficients cannot be eventually eliminated from the analysis unless additional equations relating the friction coefficients are proposed. Restrictions on friction coefficients are useful ways of eliminating proposed possibilities for friction coefficient relationships. In this study, we consider whether the entropy inequality and the Gibbs-Duhem equation produce useful restrictions on proposed friction coefficient relationships for binary and ternary diffusion. In addition, two practical restrictions on friction coefficients are considered, and a new set of equations is derived for the four mutual diffusion coefficients for a ternary system. 1. Introduction The analysis of mass transfer problems for binary and ternary systems requires the availability of mutual diffusion coefficients, one coefficient for a binary mixture and four for a ternary mixture. The necessary mutual diffusion coefficients can be determined either from appropriate experiments or from theoretical predictions. In some cases, it is easier to carry out selfdiffusion experiments than it is to perform experiments which measure mutual diffusion coefficients. For example, nuclear magnetic resonance experiments provide a rather straightforward method of measuring self-diffusion coefficients. Also, a number of molecular theories develop expressions for self-diffusion coefficients rather than for mutual diffusion coefficients because self-diffusion coefficients describe the mobility of molecules. For example, expressions for self-diffusion coefficients can be derived using the free-volume theory of diffusion.1 It would thus be useful if equations were available which relate mutual diffusion coefficients to self-diffusion coefficients because selfdiffusion measurements or predictions could then be used to compute the required mutual diffusion coefficients. One way of relating self-diffusion coefficients and mutual diffusion coefficients is to use a friction coefficient formalism for the diffusion processes.2 Equations are written for both the self-diffusion and the mutual diffusion processes in terms of friction coefficients, component velocities, and chemical potential gradients. After mutual and self-diffusion coefficients are introduced into the equations, the goal is to eliminate completely the friction coefficients from the pertinent equations so that a particular mutual diffusion coefficient is expressed in terms of only self-diffusion coefficients. Unfortunately, it is not in general possible to eliminate all of the friction coefficients. For example, for a binary system, only two equations are available for eliminating the three friction coefficients.3 Consequently, additional information is needed in the form of an expression which relates the three friction coefficients. For a ternary system, only two equations are available for eliminating the five friction coefficients,4 and again, additional equations * To whom correspondence should be addressed. Fax: 814-8657846. E-mail:
[email protected]. Telephone: 814-863-4808.
for the friction coefficients are needed. The direct calculation of mutual diffusion coefficients from self-diffusion results clearly depends on the formulation of appropriate equations which provide additional relationships involving the friction coefficients. Because a large number of possible friction coefficient relationships can of course be proposed, it is useful to consider if there are restrictions which eliminate some of the proposed possibilities. At least three types of restrictions on friction coefficients are available. One set of restrictions is indirect, while the other two sets are applied to the friction coefficients directly. The indirect set of restrictions involves application of the entropy inequality using constitutive equations for the diffusion fluxes and introducing chemical potential derivatives and mutual diffusion coefficients.5,6 Restrictions are placed on the combined thermodynamic-diffusion behavior of binary systems (eq 79 of ref 5) and of ternary systems (eqs 21, 22, 25, and 45 of ref 6), and additional restrictions can be placed on the mutual diffusion coefficients separately for binary systems (eq 90 of ref 5) and for ternary systems (eqs 34 and 39 of ref 6). Also, for ternary mixtures, another diffusion coefficient restriction is produced if it is assumed that the Onsager reciprocal relations are valid (eq 41 of ref 6). For this case, an equation which relates the four diffusion coefficients (eq 47 of ref 6) is produced. These restrictions are indirect for the friction coefficients because, in general, mutual diffusion coefficients based on the proposed friction coefficient relationships must be first calculated before these restrictions are applied to see whether the friction coefficient relationships are acceptable. This set of restrictions is not considered here because it has been presented elsewhere5,6 and because it would be difficult to derive generally applicable restrictions on the friction coefficients themselves. However, the effect of the Onsager reciprocal relations on restrictions on the friction coefficients is discussed briefly in the next section. One of the direct sets of restrictions on friction coefficients involves utilization of the Gibbs-Duhem equation,7,8 and the other direct set of restrictions is based on the entropy inequality written directly in terms of the friction coefficients.7,8
10.1021/ie061593a CCC: $37.00 © 2007 American Chemical Society Published on Web 03/30/2007
Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3423
The principal objective of this paper is to reconsider these two direct restrictions to see just what limitations are actually placed on the choice of friction coefficients. The governing equations for the analysis are presented in the second section of the paper, and restrictions based on the entropy inequality and on the Gibbs-Duhem equation are presented in the third and fourth sections, respectively. The results of the analysis of this paper and some general comments on friction coefficient relationships are presented in the final section of the paper. Also, a new set of equations for the four mutual diffusion coefficients for a ternary system is presented in this section. 2. Governing Equations In this section, we list the equations which will be needed for the derivations presented in the paper. The friction coefficient formalism presented by Bearman2 is based on the following equation written for each component of an N-component system:
dµI dx
N
)-
∑
FJσIJ
J)1MIMJ
(uI - uJ)
] [
[
ω1
Fσ12j1 dµ2 ) dx M1M2F2
j1
d(µ1 - µ2) - tr(S‚∇v) e 0 dx
Similarly, for a ternary system, utilization of eq 1 produces the following set of equations:
] [
]
σ13(F1 + F3) σ13 σ12F2 σ12 dµ1 ) -j1 + + j2 dx F1M1M2 F1M1M3 M 1M 2 M 1 M 3
(6)
(11)
and, for a ternary system, the isothermal, one-dimensional form of the entropy inequality is an obvious extension of eq 116
d(µ1 - µ3) d(µ2 - µ3) + j2 - tr(S‚∇v) e 0 dx dx
(12)
j1
For a binary system at constant temperature, it follows from appropriate utilization of eq 1 that the self-diffusion and mutual diffusion processes are described by the following set of equations:
D1 )
RT F1σ11 F2σ12 + M1 M2
D2 )
RT F2σ22 F1σ12 + M2 M1
( )
M2Vˆ 2F1 ∂µ j1 σ12 ∂F1
(13)
(14)
(15) p
Also, eqs 13-15 can be combined to yield the following result9
D) (5)
(10)
For a binary system, the isothermal, one-dimensional form of the entropy inequality is5
D) (4)
(9)
dµ1 dµ2 dµ3 + ω2 + ω3 )0 ω1 dx dx dx
For a binary system, eq 1 leads to the equations
[
dµ1 dµ2 + ω2 )0 dx dx
(3)
Fσ12j1 dµ1 )dx M1M2F1
]
and, for a ternary system, this equation is extended in an obvious manner to give the following result:
N
jI ) 0 ∑ I)1
]
(7)
For an isothermal diffusion process, we consider the GibbsDuhem equation at constant temperature and effectively constant pressure. For a binary system, the Gibbs-Duhem equation can be written as
(2)
Bearman used a statistical mechanical analysis to establish the symmetry of the friction coefficient matrix. It can be easily shown that the Onsager reciprocal relations impose no restrictions on the symmetric friction coefficient matrix. The difference of the velocities of any two components can be expressed as the difference of the diffusion velocities of the two components relative to the mass average velocity, and, consequently, the diffusion fluxes of the components relative to the mass average velocity can be introduced into eq 1. For an N-component system, the final form of the equation set based on eq 1 contains only N - 1 of the diffusion fluxes because
]
F2σ23 σ13(F1 + F3) dµ3 ) j1 + + dx M1M3F3 F3M2M3 σ23(F2 + F3) F1σ13 j2 + (8) F3M1M3 F3M 2M3
(1)
It suffices for the present purpose to consider only the x components of the relationships between the gradient of the chemical potential and the velocity vectors of the diffusing species. In addition, in this paper, all ordinary space derivatives are actually partial derivatives with time held constant. In the formulation proposed by Bearman, the friction coefficient matrix is symmetric:
σIJ ) σJI
[ [
dµ2 σ12F1 σ23 σ23(F2 + F3) σ12 ) j1 - j2 + dx M1M2 M2M3 F2M1M2 M2M3F2
( )
[
∂ ln a1 x1σ12 [x2D1 + x1D2] + ∂ ln x1 p x2σ22 + x1σ12 x2σ12 x1σ11 + x2σ12
]
-1
(16)
If the geometric mean relationship is used to relate the three friction coefficients
σ12 ) (σ11σ22)1/2 then eq 16 reduces to the form
(17)
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D)
( )
∂ ln a1 [x D + x1D2] ∂ ln x1 p 2 1
(18)
which has been discussed by Bearman2 and by McCall and Douglass.10 Although eq 18 is a useful result which gives reasonable predictions for D, McCall and Douglass have shown that it does not always give predictions which are in satisfactory quantitative agreement with experimental data. For a ternary system, eq 1 can be used to derive the following three equations describing the self-diffusion process:
D1 )
RT F1σ11 F2σ12 F3σ13 + + M1 M2 M3
D2 )
RT F1σ12 F2σ22 F3σ23 + + M1 M2 M3
D3 )
RT F1σ13 F2σ23 F3σ33 + + M1 M2 M3
(19)
(20)
(21)
D(ω1 ) 0) ) D1(ω1 ) 0)
(22)
D(ω2 ) 0) ) D2(ω2 ) 0)
(23)
3. Restrictions of Entropy Inequality Tyrell and Harris7 and Verros and Malamataris8 have proposed restrictions on friction coefficients based on the second law of thermodynamics and on a dissipation function proposed by Onsager.11 Here, we reconsider application of the second law of thermodynamics using the entropy inequalities stated in eqs 11 and 12. The entropy inequalities presented in these two equations are consistent with the entropy balance results of de Groot and Mazur.12 For a binary system, for v ) 0 and arbitrary j1, substitution of eqs 4 and 5 into eq 11 produces the following inequality
σ12F2j12 g0 M1M2F1F2
(24)
which leads to the following restriction on ζ12:
σ12 g 0
(25)
For a ternary system, for v ) 0 and arbitrary j1 and j2, substitution of eqs 6-8 into eq 12 gives the following inequality:
[
]
σ12F2 σ23F2 σ13(F1 + F3)2 + F1M1M2 F3M2M3 F1F3M1M3 j 1 j2
[ [
] ]
2σ13(F1 + F3) 2σ23(F2 + F3) 2σ12 + M1M2 M1M3F3 M2M3F3
j22 -
F1σ12 F1σ13 σ23(F2 + F3) e 0 (26) F2M1M2 F3M1M3 M2M3F2F3 2
Now, for j2 ) 0 and arbitrary j1, eq 26 produces the inequality
(27)
Similarly, for j1 ) 0 and arbitrary j2, the following inequality can be derived from eq 26:
σ12F1F3M3 + σ13F1F2M2 + σ23(F2 + F3)2M1 g 0
(28)
Finally, for arbitrary j1 and j2, eq 26 is a quadratic form which, after multiplication by -1, is positive semidefinite. This quadratic form will be positive semidefinite if and only if all of the eigenvalues of the coefficient matrix are nonnegative. It can be shown that this requirement is satisfied if
σ12σ13c1 + σ12σ23c2 + σ13σ23c3 g 0
Finally, for a binary system, the self-diffusion and mutual diffusion processes are physically identical at infinite dilution of either of the components of the mixture:
j12 -
σ12F2F3M3 + σ23F1F2M1 + σ13(F1 + F3)2M2 g 0
(29)
For a binary system, the entropy inequality produces a single restriction on the friction coefficients (eq 25), whereas, for a ternary system, three restrictions (eqs 27-29) are derived from the entropy inequality. All four of these inequalities are automatically satisfied if the off-diagonal friction coefficients σIJ (I * J) are taken to be positive, and, furthermore, no further restrictions can be imposed. Tyrrell and Harris7 (p 52) have noted that, if friction coefficients are defined using eq 1, then all coefficients σIJ with I * J should be positive. Consequently, our analysis of the entropy inequality effectively produces no restrictions on the σIJ. This result is in sharp contrast to the results of the investigations of Tyrrell and Harris7 and of Verros and Malamataris.8 For example, for a binary system, Tyrell and Harris derived the following restrictions (p 80 of ref 7):
c1σ12 + c2σ22 ) 0
(30)
c1σ11 + c2σ21 ) 0
(31)
In the two previous investigations,7,8 the second law was imposed by using a dissipation function proposed by Onsager,11 by assuming that all components of the mixture had the same velocity, and by setting the dissipation function equal to zero. There appear to be a number of questionable aspects of this approach. First of all, the dissipation function proposed by Onsager contains the diffusion fluxes of all N components of a mixture whereas the entropy inequalities derived elsewhere5,6,12 and used here contain only N - 1 of the diffusion fluxes. The diffusion fluxes are related by eq 3, and the forms of the entropy inequality should contain only independent fluxes. A second questionable aspect is the utilization of a dissipation function which not only includes the diffusion fluxes of all of the components of the mixture but which is applied by introducing the requirement that all of the components have the same velocity. Because the fluxes are related by eq 3, all of the N velocities for an N component system will be equal only when they are all zero. This produces a zero dissipation function and no restrictions on the σIJ. It seems reasonable to presume that the dissipation function will be nonzero when there is relative motion of the components so that a zero dissipation function is possible only when all of the component diffusion velocities are zero. Another difficulty with an approach which produces eqs 30 and 31 is concerned with the predictions based on these equations when either c1 ) 0 or c2 ) 0. For example, when c1 ) 0, σ22 ) 0 and σ12 ) 0 and, from eqs 13-15, it is evident that D, D1, and D2 are all not bounded as the limit of pure component 2 is approached. In addition from eqs 30 and 31, it follows that
Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3425
σ12 σ22 c1 ))c2 σ11 σ12
(32)
σ122 ) σ11σ22
(33)
so that
Consequently, as shown above, when eq 33 is valid, D can be calculated using eq 18, and the predictions for D should be in exact agreement with experimental data. As noted above, McCall and Douglass10 have shown that all predictions based on eq 18 are not in exact agreement with experimental data. Finally, if eqs 30 and 31 are accepted as restrictions on friction coefficients for binary systems, then eqs 13, 14, 30, and 31 constitute a set of four equations for eliminating the three friction coefficients and introducing D1 and D2. This creates the possibility that different versions of the equation relating D to D1 and/or D2 can be derived depending on which equations are used to eliminate the σIJ. An example of the possible nonuniqueness of the theory when the number of equations for the friction coefficients exceeds the number of coefficients is illustrated in the final section of the paper. From the above discussion, it can be concluded that equations like eqs 30 and 31 are not valid restrictions on the friction coefficients. Furthermore, when a correct form of the entropy inequality is utilized, the entropy inequality effectively does not impose any restrictions on the σIJ if the off-diagonal friction coefficients are taken to be positive which appears to be the accepted practice. 4. Restrictions of Gibbs-Duhem Equation Restrictions imposed on the σIJ based on the need to satisfy the Gibbs-Duhem equation can be easily determined by substituting chemical potential derivatives evaluated using eq 1 into the appropriate form of the Gibbs-Duhem equation. For a binary system, eqs 4 and 5 are substituted into eq 9 and, for a ternary system, eqs 6-8 are substituted into eq 10. In both cases, it is easy to show that the Gibbs-Duhem equation is satisfied identically for all bounded values of the σIJ. Consequently, we conclude that the Gibbs-Duhem equation does not impose any restrictions on the friction coefficients which describe the diffusion process. This result is again in sharp contrast to the results reported by Tyrell and Harris7 and by Verros and Malamataris.8 However, in both of these investigations, an incorrect form of eq 1 was used to evaluate the chemical potential derivatives. In particular, eq 1 was modified using an equation derived from an incorrect evaluation of the dissipation function. Not surprisingly, the restrictions that Tyrell and Harris derived by applying the Gibbs-Duhem equation are identical to those obtained using the dissipation function. 5. Discussion From the above results, we have shown that both the entropy inequality and the Gibbs-Duhem equation do not restrict the choice of expressions for the friction coefficients describing diffusion in binary and ternary systems. Also, no restrictions are imposed by the Onsager reciprocal relations if their validity is assumed. Restrictions based on mutual diffusion coefficients can still be utilized, but such restrictions cannot be applied directly to the friction coefficients. Consequently, as noted above, it would be difficult to formulate generally applicable restrictions on the friction coefficients themselves. In general, indirect restrictions can be used to see if proposed friction
coefficient relationships lead to acceptable results for mutual diffusion coefficients. There are two additional practical restrictions which can be used to limit possibilities for friction coefficient relationships. First, any friction coefficient relationships should lead to correct results for binary systems when the pure component limit of either component is approached. The correct limiting behavior is given by eqs 22 and 23. One of the two restrictions can be ignored only when predictions are desired for low concentrations of one of the components. Second, the utilization of the proposed additional friction coefficient relationships should produce unique predictions for mutual diffusion coefficients. As noted previously, nonuniqueness of the theoretical predictions may result when the number of equations for the friction coefficients exceeds the number of coefficients. To illustrate the possible importance of these two practical restrictions, we consider the simplified physical model proposed by Bearman.13 For a binary mixture, Bearman assumes that the volumes of the components are additive and that the radial distribution functions are independent of composition at constant temperature and pressure. For this set of assumptions, Bearman shows that the ratios of friction coefficients are constant:
σ12 Vˆ 2M2 ) σ11 Vˆ 1M1
(34)
σ12 Vˆ 1M1 ) σ22 Vˆ 2M2
(35)
In these equations, Vˆ 1 and Vˆ 2 assume their pure component values. There are now four equations (eqs 13, 14, 34, and 35) which can be used to solve for the three friction coefficients (including σ12 which must be used in eq 15 to obtain an expression for D in terms of D1 and/or D2). Because there are more equations than friction coefficients, this can be done in several ways. For example, eq 34 can be combined with eq 13 to give
D1 )
RTVˆ 2M2 σ12
(36)
and this equation can be combined with eq 15 to produce the following equation for D:
D)
( )
j1 D1F1 ∂µ RT ∂F1
(37)
p
Another possibility is to combine eq 35 with eq 14 to yield
D2 )
RTVˆ 1M1 σ12
(38)
which can be combined with eq 15 to give a second expression for D:
D)
( )
D2 M2Vˆ 2F1 ∂µ j1 RT M1Vˆ 1 ∂F1
(39)
p
Finally, eqs 34 and 35 can be combined to give
σ11σ22 ) σ122
(40)
and this result can be used in eq 16 to give eq 18, a third expression for D. The three equations for calculating D (eqs 18, 37, and 39) give in general different predictions if
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experimental or theoretical values of both D1 and D2 are available. When the values of D1 and D2 are independent external inputs into these equations, there is a nonuniqueness in the determination of the predictive equation for D because four equations (eqs 13, 14, 34, and 35) are available to solve for the three frictions coefficients. The predictive equation for D which is derived from this equation set depends on just how these equations are used. Not only do eqs 18, 37, and 39 produce generally different values of D, but eqs 22 and 23 are not satisfied in all cases. It is evident that eq 37 does not in general satisfy eq 23 and that eq 39 does not in general satisfy eq 22, whereas eq 18 satisfies both eq 22 and eq 23. It is also possible to treat eqs 13, 14, 34, and 35 as a set of four equations in four unknowns (σ12, σ11, σ22, and D2/D1). In this case, either D1 or D2 (but not both D1 and D2) can be taken to be an independent external input because it is easy to show that D1 and D2 are necessarily related by the expression
are available for polymer-solvent systems15,16 based on the fact that the self-diffusion coefficient of the polymer is much smaller than the solvent self-diffusion coefficient. For ternary systems, there are few data sets which include both the mutual diffusion and self-diffusion data needed for data-theory comparisons. In addition, the available theories often involve either setting certain friction coefficients equal to zero or setting friction coefficient ratios equal to known constants. We believe that a reasonable approach to an approximate formulation of a ternary predictive theory is to utilize an extension of the geometric mean approximation to ternary systems so that, in addition to eq 40, we can write
D1 M2Vˆ 2 ) D2 M1Vˆ 1
Consequently, there are now available six equations (eqs 19-21, 40, 44, and 45) for eliminating the six friction coefficients (σ11, σ22, σ33, σ12, σ13, and σ23). Appropriate manipulation of the six equations yields the following results for the diagonal friction coefficients:
(41)
When eq 41 is used to restrict the D1/D2 ratio, it can be shown that eqs 18, 37, and 39 all give the same predictions because the D1 and D2 values are no longer independent. Consequently, there are two possibilities when eqs 34 and 35 are used to relate friction coefficients based on the simplification that the friction coefficient ratios are constant. If it is desired to utilize experimental or theoretical values for both D1 and D2, then there will be more equations than friction coefficients and hence more than one predictive equation for D. If all four equations are used in forming an equation for D, then there will be only a single equation for D but the result will be valid only if D1 and D2 are related by the expression in eq 41. Because neither of these alternatives is desirable, theories based on the assumption that certain friction coefficient ratios are constant should be avoided. We note that equations like eqs 34 and 35 are sufficient but not necessary conditions for the validity of eq 40, the geometric mean relationship. The geometric mean relationship is equivalent to the following equality of friction coefficient ratios
σ12 σ22 ) σ11 σ12
(42)
but these ratios can each be concentration dependent rather than constant. Furthermore, utilization of eq 40 adds only a single equation to the analysis so that only eqs 13, 14, and 40 are now available to determine the three friction coefficients. Finally, utilization of eq 40 in eqs 13 and 14 yields the following expression
()
σ22 D1 ) D2 σ11
1/2
(43)
This equation is very different from a similar equation, eq 41. Equation 41 does not allow an independent choice of both D1 and D2 because their ratio must assume a known value. On the other hand, eq 43 simply provides a relationship between ζ11 and ζ22, and independent values of both D1 and D2 may be utilized. Reviews of attempts to relate self-diffusion and mutual diffusion coefficients have been presented for both binary and ternary systems.2,9,10,14 For binary systems, diffusion in many systems can be adequately described by eq 18 and the geometric mean approximation. However, improved predictive equations
σ11σ33 ) σ132
(44)
σ22σ33 ) σ232
(45)
() ()
σ11 )
σ22 D1 ) D2 σ11
1/2
σ33 D1 ) D3 σ11
1/2
(46)
(47)
RT F2 D 1 F3 D1 F1 D1 + + M1 M2 D 2 M3 D 3
[
]
(48)
In addition, eqs 6 and 7 can be converted to the following forms by introducing eqs 40 and 44-47:
[
]
F2 F1 + F 3 F1 dµ1 + ) j1 D1σ11 dx D2M1M2 D3M1M3 F1 F1 (49) j2 D2M1M2 D3M1M3
[
[
]
]
F2D1 F2 dµ2 F2 + ) j1 D1σ11 dx M1M2D2 D2D3M2M3 F1 D1(F2 + F3) j2 (50) D2M1M2 D2D3M2M3
[
]
The isothermal diffusion and thermodynamic behavior of the ternary system can be described by the following set of equations when pressure gradient effects are negligible:
j1 ) -FD11
dω1 dω2 - FD12 dx dx
(51)
j2 ) -FD21
dω1 dω2 - FD22 dx dx
(52)
( )
∂µ1 dµ1 ) dx ∂ω1
( )
dω1 ∂µ1 + dx ∂ω p,ω2 2
dω2 p,ω1 dx
(53)
Ind. Eng. Chem. Res., Vol. 46, No. 10, 2007 3427
( )
dµ2 ∂µ2 ) dx ∂ω1
( )
dω1 ∂µ2 + ∂ω2 p,ω2 dx
dω2 p,ω1 dx
(54)
Inversion of eqs 49 and 50 and introduction of eqs 48 and 51-54 yield the following expressions for the four mutual diffusion coefficients:
( )
ω1 ∂µ1 D11 ) RT ∂ω1
[ω1D3M3 + ω2D1M1 + ω3 D1M1] +
p,ω2
( )
ω1ω2 ∂µ2 RT ∂ω1 D12 )
( )
ω1 ∂µ1 RT ∂ω2
[ω1D3M3 + ω2D1M1 + ω3D1M1] +
p,ω1
( )
ω1ω2 ∂µ2 RT ∂ω2
( )
ω2 ∂µ2 D21 ) RT ∂ω1
( )
( )
ω2 ∂µ2 RT ∂ω2
( )
∂µ1 ∂ω2
)
p,ω2
1+
p,ω1
ω1 ω2 ∂µ2 + ω3 ω3 ∂ω2
(60)
p,ω1
The quality of the predictions of the proposed equation set can be determined only by appropriate data-theory comparisons. It is not possible to carry out such comparisons at the present time because extensive self-diffusion, mutual diffusion, and thermodynamic data are not yet available for ternary systems. There are two principal contributions of this investigation. First, we have shown that the entropy inequality and the GibbsDuhem equation do not impose any restrictions on the friction coefficients which describe the diffusion process. Second, we have presented a new set of equations (eqs 55-58) for the four mutual diffusion coefficients for a ternary system.
Nomenclature
[D3M3 - D1M1] (57)
[D3M3 - D1M1] (58)
p,ω1
This set of four mutual diffusion coefficients is based on eqs 51 and 52 and hence can be used to describe the mass diffusion fluxes of components 1 and 2 relative to the mass average velocity. For some problems, it is preferable to use the volume average velocity rather than the mass average velocity to describe the convective terms in the species continuity equations. The proper constitutive equation for the diffusion fluxes relative to the volume average velocity is given by eq 5 of ref 6, and the mutual diffusion coefficients for this case can be evaluated by substituting eqs 55-58 into eq 6 of ref 6. For a binary system composed of components 1 and 3, it can be shown that eqs 55-58 yield the result
D11 ) D )
p,ω2
ω2 ω1 ∂µ1 + ω3 ω3 ∂ω1
This study was supported by funds provided by the Dow Chemical Company.
p,ω2
( )
ω1ω2 ∂µ1 RT ∂ω2
1+
Acknowledgment
[ω2D3M3 + ω1D2M2 + ω3D2M2] +
p,ω1
∂µ2 ∂ω1
[D3M3 - D2M2] (56)
p,ω1
[ω2D3M3 + ω1D2M2 + ω3D2M2] +
p,ω2
ω1ω2 ∂µ1 RT ∂ω1 D22 )
[D3M3 - D2M2] (55)
p,ω2
( ) [ ] ( ) ( ) [ ] ( )
ω1 ∂µ1 [ω D M + ω3D1M1] RT ∂ω1 p 1 3 3
(59)
and a similar equation can be derived from eqs 55-58 for a binary system with components 2 and 3. It is easy to show that an equation of the form of eq 59 is equivalent to the previous binary result given by eq 18. For any ternary system, the four mutual diffusion coefficients can be determined if equations or data are available for the thermodynamic properties of the system (µ1 and µ2) and for the self-diffusion coefficients of the system (D1, D2, and D3). For example, for a ternary system composed of two solvents and a polymer, the solvent chemical potentials can be calculated using the Flory-Huggins theory of polymer solutions,17 and the three self-diffusion coefficients can be calculated using the free-volume theory of transport.1,3 It should be noted that the four chemical potential derivatives in eqs 55-58 are related6
a1 ) activity of component 1 cI ) molar density of component I (mol/m3) D ) binary mutual diffusion coefficient (m2/s) DI ) self-diffusion coefficient of component I (m2/s) DIJ ) diffusion coefficients used for mass diffusion fluxes relative to mass average velocity (m2/s) jI ) x component of mass diffusion flux of component I relative to mass average velocity (kg m-2 s-1) MI ) molecular weight of component I (g/mol) p ) pressure (kg m-1 s-2) R ) gas constant (J mol-1 K-1) S ) extra stress tensor (kg m-1 s-2) T ) temperature (K) uI ) x component of velocity of component I (m/s) v ) mass average velocity (m/s) Vˆ I ) partial specific volume of component I (m3/kg) x ) spatial variable (m) xI ) mole fraction of component I Greek Letters σIJ ) friction coefficient between components I and J (J m s mol-2) µI ) chemical potential of component I per unit mass (J/kg) µ j I ) chemical potential of component I per mole (J/mol) F ) mass density of mixture (kg/m3) FI ) mass density of component I (kg/m3) ωI ) mass fraction of component I Literature Cited (1) Vrentas, J. S.; Vrentas, C. M. Predictive Methods for Self-Diffusion and Mutual Diffusion Coefficients in Polymer-Solvent Systems. Eur. Polym. J. 1998, 34, 797. (2) Bearman, R. J. On the Molecular Basis of Some Current Theories of Diffusion. J. Phys. Chem. 1961, 65, 1961. (3) Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free-Volume Theory. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 403. (4) Vrentas, J. S.; Duda, J. L.; Ling, H.-C. Enhancement of Impurity Removal from Polymer Films. J. Appl. Polym. Sci. 1985, 30, 4499. (5) Vrentas, J. S.; Vrentas, C. M. A General Theory for Diffusion in Purely Viscous Binary Fluid Mixtures. Chem. Eng. Sci. 2001, 56, 4571.
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ReceiVed for reView December 11, 2006 ReVised manuscript receiVed February 7, 2007 Accepted February 27, 2007 IE061593A
