738
Ind. Eng. Chem. Res. 1993, 32, 738-742
Negative Maxwell-Stefan Diffusion Coefficients Gerrit Kraaijeveld and Johannes A. Wesselingh' Department of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
The existence of negative Maxwell-Stefan diffusivities is investigated. For the case where the diffusion coefficients are taken to be composition dependent, it is found that the theory of irreversible thermodynamics does not require all diffusivities to be positive definite. This theoretical result is supported by literature diffusion coefficients in electrolyte solutions, which are composition dependent, and by one mixed-salt system which exhibits negative like-like ion diffusion coefficients. Further, a n approximate method of obtaining like-like ion diffusion coefficients from tracer diffusivities is used to investigate a further 35 salt systems. In most of these systems negative diffusion coefficients do occur. The only exceptions are systems containing H+ions and systems containing like ions of different valency. Finally, diffusion coefficients of a co-ion in an ion exchange membrane, which also shows negative diffusion coefficients, are provided.
Introduction In the past two decades, there has been an increasing interest in the Maxwell-Stefan description of transport processes. Lightfoot et al. (1962) showed that this theory may be used for the description of multicomponent diffusion in liquids. Graham and Dranoff (1982) applied the Maxwell-Stefan description to ion exchange resins. Thiel and Douglas (1988) applied the theory to pressuredriven membrane separation processes, Robertson and Zydney (1988)used it to describe protein transport through porous membranes, and Krishna (1990) described surface diffusion in zeolites using this theory. The Maxwell-Stefan description is consistent with the theory of irreversible thermodynamics, but has the advantage of having coefficients that show less variation with concentration than the phenomenological coefficients resulting from the standard irreversible thermodynamics approach (Graham and Dranoff, 1982). On the basis of data from Miller (1967),Graham and Dranoff (1982)found that the Maxwell-Stefan diffusivities, which represent the like-like ion interactions, were negative. This result differs from that obtained by Standart et al. (1979), who states that all the Maxwell-Stefan diffusivities should be larger than or equal to zero. It is the purpose of the present article to unify the theoretical and experimental results, to indicate which constraints are imposed on the MaxwellStefan diffusivities by the theory of irreversible thermodynamics, and to illustrate this with the experimental results which are available.
Irreversible Thermodynamics and the Maxwell-Stefan Equation
As opposed to classical thermodynamics, which can only describe processes which are reversible, irreversible thermodynamics deals with processes during which the total entropy increases. The goal of irreversible thermodynamics is to formulate a general theory for irreversible processes, using a minimum number of postulates. Using these postulates, an expression for the entropy balance (and hence for the entropy production) can be derived. This derivation can be found in most standard texts on the topic (De Groot and Mazur, 1969; Fitts, 1962; Hirschfelder et al., 1964; Lightfoot, 1974), and hence this article will only give the result of the derivation. Under isothermal conditions, the entropy production is given by (Standart et al., 1979) 08SS-5S85/93/2632-0738$04.00/0
The choice of the fluxes and their conjugated forces which occur in the entropy production equation is arbitrary (Farland et al., 1988). In this case it is convenient to chose di as the dependent variable and ( U j - V i ) as the independent variable. The phenomenological equation thus obtained is
Here the quantity di has the physical significance of being the force per unit volume of solution tending to move species i, relative to the solution (Lightfoot, 1974):
The equation which is known as the Maxwell-Stefan equation is just a combination of eqs 2 and 3:
In the systems which will be considered in this article (electrochemical systems), the only body forces that are acting are the electrical potential gradients. They may be introduced in the Maxwell-Stefan equation by defining (Lightfoot, 1974) ZiF gi = - -vf$ Mi
Under isothermal conditions, and using the assumption of electroneutrality, the Maxwell-Stefan equation thus becomes
Because the entropy production in eq 1 may not be negative, certain constraints will be imposed on the Maxwell-Stefan diffusivities, Si? In order to obtain 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32,No. 4,1993 739 Table I. Maxwell-Stefan Diffusivities for the NaCI-KCl System. c1 0.25 0.5 0.25 0.5 1.5 a
c2
&I2
0.25 0.25 0.5 0.5 1.5
-0.16 -0.17 -0.18 -0.20 -0.36
*13
&23
e14
&u
&84
0.09 0.11 0.11 0.14 0.28
0.11 0.14 0.14 0.17 0.32
1.28 1.24 1.28 1.23 0.98
1.94 1.88 1.93 1.88 1.59
2.08 2.05 2.06 2.04 1.72
812
813
B14
823
Bu
Bsr
0.91 1.89 1.83 3.72 39.5
1.82 5.82 2.73 7.54 82.1
2.48 7.12 3.98 10.0 104
1.21 1.87 3.74 4.98 51.9
1.92 3.26 5.07 7.70 79.2
3.13 7.96 6.70 13.1 142
The concentrationsare in mol/L, the diffusivities in 1 P m2/s, and the Ps in 1013 s2/m4.
these constraints, it should be noted at this point that although the velocitiesin eq 1are independent, the velocity differences are not. In fact, for an N-component system there are only (N - 1) independent velocity differences. So in order to obtain information about the constraints, eq 1 needs to be rewritten in terms of the independent velocity differences. We thus obtain the following constraint for every a and b (see the Appendix):
1.4
I
1.2
a, b = 1, ..., n a # b (7) The above equation is the only constraint which the theory of irreversible thermodynamics imposes on the MaxwellStefan diffusivities. As it turns out, there are as many constraints as there are Maxwell-Stefan diffusion coefficients.
Negative Maxwell-Stefan Diffusivities Now that the constraints for the Maxwell-Stefan diffusivities have been obtained, the question as to whether or not they can be negative may be addressed. Clearly it follows from eq 7 that for a binary system the diffusion coefficient must be positive. However for ternary and multicomponent systems more information is needed to answer the question. In general, if the Maxwell-Stefan diffusivities do not depend on the composition, then the diffusion coefficients must be positive (since eq 7 must hold for all mole fractions), and the result obtained by Standartet al. (1979) is found:
Bij1 0 (8) The only exception to this result is in electrolyte systems, where the electroneutrality condition provides an extra relation between the mol fractions. To provide a general result for these systems is difficult, but, e.g., for the mixed salt which is considered in Table I, it can be shown that B12 must satisfy eq 9 instead of 8.
However, if the diffusion coefficients are composition dependent, then in any multicomponent system the diffusion coefficients may be negative for certain ranges of the mole fractions, provided that eq 7 is satisfied. Diffusion data from Chapman (1967)clearly demonstrate that the Maxwell-Stefan diffusivities in electrolyte solutions may be composition dependent (see Figure 1). Hence in the case of electrolyte solutions, either eq 7 or eq 9 should be used instead of eq 8. So far the only area where negative diffusivities have been found is in electrolyte solutions, where the like-like ion diffusional interactions are usually negative. As an example of negative diffusivities, consider the aqueous solution of NaCl and KC1.' The required experimental
-
1.0 '1 0.0
0.6
Conc (mol/l)
Figure 1. Diffusion coefficients for Na+ in various salts based on data from the thesis by Chapman (1967).
data have been collected by Miller (19671, and the Maxwell-Stefan diffusivities can be calculated from these data using a method similar to that described by Newman (1973). Table I gives the relevant values. In the table the following numbering is used: Na+ is 1, K+ is 2,C1- is 3, and H2O is 4. To illustrate that the Maxwell-Stefan diffusivities satisfy the constraints of the irreversible thermodynamics, we define
It is clear from the table that the like-like ion diffusion coefficient is negative, and further it can be seen that all the Ps are positive, so that all the thermodynamic constraints are satisfied. Unfortunately, complete collections of phenomenological coefficients are rather scarce in the existing literature. I t is therefore interesting to note that the like-like ion diffusivities can be calculated from binary diffusion data and information concerning the tracer diffusion coefficients. Of course these calculations become less accurate as the salt concentrations increase. The equations that are required for these calculations are those expressing the Fick diffusion coefficient, the conductivity, and the transference number in terms of the binary MaxwellStefan diffusivities. The relevant expressions are (Newman, 1973)
t+ = K=--
@-O+
z+Bo+ - Z B &
(c4,+ + C + B c J B + RT c-Bo+ + ~ 3 + c@+%
Z + d C
(12) (13)
These three equations provide all the required diffusion Coefficientsfor a single salt system at a given concentration.
740 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 1.0
Table 11. Like-Like Diffusivities Calculated from Tracer Diffusivities. ~
svstem NHiCl NHiCl NHiCl BaClz BaC12 CaClz CaClz CaClz HCI Lac13 Lac13 LiCl LiCl LiCl LiCl LiN03 KBr
tracer NH4+ c1Na+ Ba2+ c1Ca*+ c1Na+ c1c1La3+
KCl KCl KCI KCl
c1K+ H+ I-
c1-
Li+ Na+ H+ Li+
BI-
KI KI
c1-
KI NaCl NaCl NaCl NaCl NaI NaI NaI NazSO4 NazSO4 ZnSO4 ZnSO4
Na+ c1-
I-
H+ INa+
INa+ c1Na+
so42 SOP ZnZ+
behavior of likelike diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity positive diffusivity positive diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity positive diffusivity negative diffusivity negative diffusivity, becomes positive above 3.0 mol/L negative diffusivity negative diffusivity positive diffusivity negative diffusivity, becomes positive above 2.8mol/L negative diffusivity negative diffusivity, becomes positive above 2.0 moVL negative diffusivity negative diffusivity positive diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity negative diffusivity
I
I
C,
0.2 0.1 0.1 1.5 1.5 1.0 1.0 0.5 4.0 0.05 0.05 2.9 3.1 2.8 2.9 5.0 3.0
0.5
n
OD
. p c
a
'p
0.0
4: -0.6
0.0
1.0
2.0
Conc (molA)
To obtain the like-like ion diffusion coefficients, tracer diffusivities are required, these are generally defined by By rearranging the Maxwell-Stefan equation, it is fairly easy to show that the tracer diffusion coefficient can be expressed in terms of Maxwell-Stefan diffusivities according to
The above equation may be used when the tracer is an anion; for cation tracers the plus and minus signs need to be interchanged. Using binary diffusion data collected by Chapman (1967) and tracer diffusion data collected by Mills and Lobo (1989), we investigated 35 systems of which 11were true ternary systems. These systems are given in Table 11. From these results, the general trend in like-like ion diffusivities appears to be that the diffusivities are usually negative at low concentrations. At high concentrations (for most systems this concentration is not reached) they may show an asymptote and become positive. Figure 2 shows an example of such general behavior. Solutions containing H+ and solutions containing like ions of different valency may deviate from the general trend, and not have any negative diffusivities. Figure 3 shows the anomalous behavior of the like-like diffusivities of various chloride solutions containing hydrogen ions. The diffusivities are positive, and the three curves all have a clear maximum.
4.0
0
0
3.0 3.0
The last column indicates the maximum concentration in mol/ L, for which the calculations were performed.
3.0
Figure 2. Like-like ion diffusivity of I- tracer in KC1, based on data from Mills and Lobo (1989)and Chapman (1967).
4.0 4.0 3.6 3.5
1.0 5.0 4.5 1.0 5.0 1.0 1.0 1.0 0.3 0.3 0.3 0.3
1
I -1.0
@$
LiCl NaCl
/!!k&zL
0.0 0.0
2.0
4.0
Conc (mol/l)
Figure 3. Likelike ion diffusivity of H+ tracer in various chloride solutions, based on data from Mills and Lobo (1989)and Chapman (1967).
To calculate the above-mentioned Maxwell-Stefan diffusivities, experimental data on Fick diffusion coefficients, tracer diffusion coefficients,transference numbers, conductivities, and activity coefficients have been used. In all these data inaccuracies are to be expected. These inaccuracies might cause the (rather unexpected) negative values of the diffusivities reported here. To establish whether or not this is the case, the effect of an error of * 5 % in the experimental values was investigated. Even for this relatively large error in the experimental data, it was found that the systems in Table I1still showed negative like-like diffusivities. Hence it can be concluded that the negative diffusivities reported here are not due to experimental inaccuracies. Recent work by Narebska (1987) indicates that the Maxwell-Stefan diffusivities in ion exchange membranes may also be negative. Table I11 contains the MaxwellStefan diffusivitieswhich were calculated using Narebska's results. Again it can be seen that all the j3's are positive, and further as is illustrated in Figure 4, the diffusion coefficients representing the anion-cation and anionmembrane interactions are negative over certain concentration ranges. Both diffusivities also appear to have an asymptote similar to that in Figure 2. Conclusions This article has illustrated that the Maxwell-Stefan diffusion coefficients may be negative. This is consistent
Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993 741 Table 111. Maxwell-Stefan Diffusivities for the NaOH-Nafion 120 Systema msait B+- B+w B+, B-w B..m BmW P + 0.1 0.25 0.5 1.0 2.0 4.0
-0.27 -0.43 -0.85 3.11 1.68 0.60
3.24 3.41 3.49 3.54 2.72 1.85
0.23 0.22 0.22 0.24 0.25 0.30
3.94 12.9 14.1 12.0 9.07 8.44
0.63 0.34 0.48 1.17 7.35 -1.03
4.17 4.25 3.86 3.15 2.09 1.51
0.08 0.24 0.70 2.51 8.14 34.5
R+w
B+m
40.7 43.9 49.9 61.1 116. 264.
40.6 44.1 50.2 60.5 111. 224.
4-w 0.08 0.23 0.68 2.68 9.86 43.6
R-IT
RW, ~
0.07 40.9 0.21 44.5 0.66 51.0 2.52 61.6 8.08 114. 26.3 215. + represents the cation, - represents the anion, w represents the water, and m represents the membrane. The molality is in mol/kg, the diffusivities are in 10-lom2/s, and the ps are in 1015s2/m4.
Greek Symbols = as defined in eq 10, s4/m2 4 = electrical potential, V y = activity coefficient of the salt K = conductivity, S/m pi = chemical potential of species i, J/mol rS = entropy production per unit volume in J/(K.m3.s) p = density, kg/m3 oi = mass fraction of component i ,
pj
I
Subscripts
+, - = cation and anion
" . l
I
0.0
io
~
4:O
Conc (mol/l) Figure 4. Maxwell-Stefan diffusivities in an ion exchange membrane, based on data from Narebska (1987).
0 = water i, j = species i and j m = membrane tr = tracer
Appendix: Constraints for the Maxwell-Stefan Diffusivities Defining
with the theory of irreversible thermodynamics. Examples of negative diffusion coefficients in aqueous electrolyte solutions and in ion exchange systems have been provided. Further, a method of obtaining information about likelike diffusion coefficienb from tracer diffusion coefficients has been provided. A total of 35 electrolyte systems have been analyzed with this method, of which most exhibit the behavior shown in Figure 2. The only exceptions are systems containing hydrogen ions and systems containing like ions of different valencies.
Acknowledgment The investigations were supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for Scientific Research (NWO).
Nomenclature c = total concentration, m0l/m3 ci = concentration of species i, m0l/m3 d , = as defined by eq 3, l/m D = Fick diffusion coefficient, m2/s Sij= Maxwell-Stefan diffusion coefficient for species i and j , m2/s F = Faraday constant, 96 485 C/mol gi = total body force acting on species i, m/s2 Ji = flux of species i, mol/(m2.s) Mi = molecular weight of species i, kg/mol n = number of components p = pressure, Pa R = universal gas constant, J/(mol.K) t+ = transference number of cation T = temperature, K ui = average velocity of species i, m/s Vi = partial molal volume, m3/m01 xi = mole fraction of species i zi = charge number of species i
uij = ui- uj and
= xixj/Gij For a ternary system, we obtain from eq 1 (cRT/2)(f12u212
(All
fij
+ f13u312 + f21u122 + f23u232 + f3lU13' + f32U23')
2 0 (A21
Which reduces to
+ f13u132 + f 2 3 u 2 2
'
(A31 For a three-component system, there are two independent velocity differences. Hence one of the velocity differences may be eliminated from the above equation. f12U122
'
(A41 f12u122 + f13u132 + f 2 3 ( u 1 2 - U 1 3 ) 2 This may be rearranged to give an inequality which is quadratic in the ratio of the independent velocity differences:
This inequality will be satisfied if the equality has no solutions, and hence the following constraint is found:
+f23)(f13 +f23) - f 2 2
'
(A61 Since the analysis should be indifferent to the choice of the independent velocity differences, it follows that the above equation must hold for every combination of components: (f12
The proof for a four-component system is similar; only the number of terms in the equations increase dramatically. The general constraint for a N-component system is then
742 Ind. Eng. Chem. Res., Vol. 32, No. 4, 1993
obtained by induction: n
. -
iza
n
i#L
There are as many constraints asthere are Maxwell-Stefan diffusivities. The constraints obtained in this appendix were checked numerically for a thee-, four-, and five-component system, using a small computer program which calculated the entropy production for every possible combination of velocity differences and diffusion coefficients at various mole fractions. Literature Cited Chapman, T. W. The transport properties of concentrated electrolytic solutions. Ph.D. Thesis, University of California, Berkeley, 1967. De Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland: Amsterdam, 1969. Denbigh, K. The principles of Chemical Equilibrium; Cambridge University Press: London, 1964. Fitta, D. D. Non-Equilibrium Thermodynamics; McGraw-Hilk New York, 1962. Forland, K. S.; Ferland, T.; Ratkje, S. K. Irreversible Thermodynamics, theory and applications; Wiley: New York, 1988. Graham, E. E.; Dranoff, J. S. Application of the Stefan-Maxwell Equations to Diffusion in Ion Exchangers. Znd. Eng. Chem. Fundam. 1982,21, 360. Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964. Krishna, R. Multi-component Surface Diffusion of Adsorbed Species: A Description based on the Generalized Maxwell-Stefan Equations. Chem. Eng. Sci. 1990,45 (7),1779.
Lightfoot, E. N. Transport Phenomena and Living System; Wiley: New York, 1974. Lightfoot, E. N.; Cussler, E. L.; Rettig, R. L. Applicability of the Stefan-Maxwell Equations to Multi-component Diffusion in Liquids. MChE J. 1962, 8 (5),708. Miller, D. G. Application of Irreversible Thermodynamics to Electrolyte Solutions 11. Ionic Coefficients for Isothermal Vector Transport Processes in Ternary Systems. J. Phys. Chem. 1967, 71 (3), 616. Mills, R.; Lobo, V. M. M. Self-diffusion in electrolyte solutions; Elsevier: Amsterdam, 1989. Narebska, A.; Kujawski, W.; Koter, S. Irreversible Thermodynamics of Transport across Charged Membranes. Part I1 Ion-Water Interactions in Permeation of Alkali. J. Membr. Sci. 1987, 30, 125. Newman, J. Electrochemical Systems; Prentice-Hall: Englewood Cliffs, NJ, 1973. Robertson, B. C.; Zydney,A. L. A Stefan-MaxwellAnalysia of Protein Transport in Porous Membranes. Sep. Sci. Technol. 1988, 23 (12&13), 1799. Standart, G. L.; Taylor, R.; Krishna, R. The Maxwell-Stefan formulation of irreversible thermodynamics for simultaneous heat and ma88 transfer. Chem. Eng. Commun. 1979, 3, 277. Thiel, S. W.; Douglas, R. L. Application of the Stefan-Maxwell Equations to the Pressure-Driven Membrane Separation of Dilute Multi-component Solutions of Nonelectrolytes. J. Membr. Sci. 1988, 37, 233. Received for review July 30, 1992 Revised manuscript received November 25, 1992 Accepted December 15,1992
