J. Phys. Chem. 1996, 100, 12569-12573
12569
Concentration-Dependent Diffusion Coefficients and Sorption Isotherms. Application to Ion Exchange Processes As an Example N. F. E. I. Nestle and R. Kimmich* UniVersita¨ t Ulm, Sektion Kernresonanzspektroskopie, D-89069 Ulm, FRG ReceiVed: February 2, 1996; In Final Form: May 8, 1996X
The physical significance of concentration-dependent diffusion coefficients is examined. It is shown that such concentration dependences cogently follow from typical sorption isotherms. As an example, NMR images of the progress of isovalent competitive ion exchange processes in alginate gels are compared with the predictions of diffusion models assuming Henry or Langmuir sorption isotherms. It is shown that the model on the basis of the Langmuir isotherm provides a much better correspondence with the experimental data than the linear absorption model based on the Henry isotherm. The better correspondence of the Langmuir model is due to the fact that saturation effects are taken into account in this model, which is based on the law of mass action.
1. Introduction In Fickian diffusion theory, the diffusion coefficient is assumed to be a constant material property. However, there are many systems such as sorption and reaction-diffusion systems in which material transport is found to be non-Fickian. A possible approach to treat such systems is to work with concentration-dependent diffusion coefficients D ) D(c). This idea has been discussed quite extensively in the literature.1-3 The most popular forms of such concentration dependences are reviewed, e.g., in ref 1. However, these functions D ) D(c) are mainly of a phenomenological nature while their physical significance remains unclear. In this contribution, we demonstrate for sorption processes that the functional relationship of the diffusion coefficient on the concentration can directly be derived from sorption isotherms such as Henry’s or Langmuir’s laws. A further purpose of this work is to demonstrate that NMR imaging is an interesting experimental tool for testing the theoretical predictions in this context. As an example, the competitive ion exchange process in alginate gels has been studied.
Figure 1. Concentration of bound sorbate vs the concentration of the unbound species as expected from the Langmuir (solid line) and Henry (dashed line) isotherms (parameters in both cases: cc ) 40 mmol/L, ca ) 50 mmol/L, K ) 10).
2. Reaction-Diffusion Systems and Sorption Isotherms A popular model for ion exchange and other reactiondiffusion systems is the linear absorption model (LAM1,4), which is based on the assumption that the concentration cb of bound molecules in a sorbent material is proportional to the concentration c of the unbound sorbate in the same material:
cb ) kc
(1)
where k is a material constant. In this case, the diffusion of unbound sorbate obeys
dcb dc dc ) D∇2c ) D∇2c - k dt dt dt
(2)
(3)
Equation 3 obviously has the conventional form of the standard X
diffusion equation, but the diffusion coefficient, albeit constant, is modified according to
Deff )
This modified diffusion equation can be simplified to
dc D 2 ) ∇c dt 1 + k
Figure 2. Effective diffusion coefficients (in arbitrary units) as a function of the concentration of the unbound sorbate in the case of a Langmuir (full line) and Henry (dashed line) sorption model using the same parameters as in Figure 1.
Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00342-5 CCC: $12.00
D 1+k
(4)
where D is the diffusion coefficient in the absence of any absorption reactions. The linear relationship (known as linear or Henry isotherm) of the concentrations of free and bound sorbate has an obvious shortcoming for many practical applications, that is, saturation © 1996 American Chemical Society
12570 J. Phys. Chem., Vol. 100, No. 30, 1996
Nestle and Kimmich
TABLE 1: Examples of Concentration-Dependent Diffusion Coefficients Derived from Physically Well-Established Sorption Isothermsa isotherm
name, ref
cb ) Acn
cb )
diffusion coefficient
Deff )
Freundlich 1906, 7
cabcn
Sips 1948, 8
Deff )
1 + bcn
cb ) A exp(-B(ln c/A)2)
Dubinin and Radushkewich 1966, 10
cb ) A ln(Bc)
Temkin 1940, 12
1 + Bc cb ) A ln 1 + B exp(-R)c
BLK, Brunauer et al. 1942, 11
Bc (1 - c/A)(1 + Kc/A)
BET, Brunauer et al. 1938, 13
cb )
Deff )
c D c + Ancn (1 + bcn)2
(1 + bcn)2 + cabncn-1
c D c - 2AB ln(c/A)exp(-B(ln(c/A))2)
Deff )
Deff )
Deff )
D
c D c+A
(1 + Be-Rc)(1 + Bc)
D
1 + BA(1 + e-R) + B(1 + e-R)c + B2e-Rc2
(1 - c/A)(1 + Kc/A) D (1 - c/A)(1 + Kc/A) + B(1 + c/(A - Ac) - Kc/(A - AKc))
a K, A, B, n, and R are constants with respect to c but may be functions of other parameters such as temperature. For a detailed discussion of the physical meaning of these constants, refer to the literature cited. D is always denoting the diffusion coefficient in the absence (or saturation) of the sorption process.
effects are not considered. This deficiency can be overcome by using a Langmuir sorption isotherm instead of Henry’s law. This isotherm was originally derived by Langmuir5 for gas adsorption on a solid surface. However, the same expression can also be deduced from the law of mass action for a reaction of the form
The Langmuir sorption term (7) leads to a reaction-diffusion equation analogous to (2):
AB + C a AC + B
Combining the terms containing the time derivative of the concentration of the free sorbate leads to
(5)
For such reactions (e.g., isovalent ion exchange), the law of mass action reads
K)
[AC][B] (ca - cb)c ) cbcc [AB][C]
(6)
where ca denotes the concentration of binding sites, cc the background concentration of the exchanger’s original counterion, and K the equilibrium constant of the exchange reaction. Resolving this expression with respect to the concentration cb of the bound sorbate, one obtains
cb )
Kcac cc + Kc
(7)
This has the same analytical form as the original Langmuir sorption isotherm. For Kc , cc, expression 7 can be approximated by
cb )
Kca c cc
(8)
From this it is obvious that Henry’s law (1) can be derived as a special limit case from the Langmuir isotherm (see also Figure 1).
Kcacc dc dc ) D∇2c dt (c + Kc)2 dt
(9)
c
(cc + Kc)2 dc D∇2c ) dt (c + K )2 + Kc c c c a c
(10)
Equation 10 can be interpreted as a diffusion equation with an effective concentration-dependent diffusion coefficient Deff (see Figure 2):
Deff )
(cc + Kc)2 (cc + Kc)2 + Kcacc
D
(11)
where D again denotes the diffusion coefficient in the absence of absorption reactions (or in the limit of full saturation of binding sites). The use of the Langmuir sorption isotherm in the theory of reaction-diffusion systems has already been suggested by Weisz and Hicks.6 However, these authors did not realize that the resulting relation leads to a diffusion equation with a concentration-dependent diffusion coefficient. The Langmuir isotherm yields an appropriate description for competitive stoichiometric ion exchange processes involving ions of the same valence. Similar approaches can also be applied to other sorption processes with isotherms explicitly resolvable with respect to cb. Typical isotherms of this sort are (1) Freundlich’s isotherm,7 (2) the Sips8 isotherm (which is essentially a synthesis between the Langmuir and Freundlich
Diffusion Coefficients and Sorption Isotherms
J. Phys. Chem., Vol. 100, No. 30, 1996 12571
Figure 3. Three “snapshots” from a NMR “movie” of the ingress of Cu into a 2% Ca-Alginate tube from a solution containing 500 µmol/L Cu and 10 mmol/L Ca. The images represent cross sections of an alginate tube in which the solution with the ions is circulating (bright area in the center). The images were recorded just after exposure (left), after 1 h 30 min (center) and after 4 h 30 min (right). Imaging parameters used are the following: field of view (9.6 mm)2; echo time, 20 ms; recovery time, 250 ms. The bright bar in the images corresponds to 1 mm.
The main difference between the original Langmuir isotherm and the first four examples of the isotherms listed above is due to a distribution of binding energies. In the Langmuir case, all binding sites are assumed to be energetically equivalent, whereas the other isotherms are based on certain distributions of binding energies (for a detailed discussion see, for example, refs 8 and 14). Application of such isotherms normally refers to the absorption of uncharged substances into porous solids either from the gas phase or from dilute solutions.15,16 However, we expect that in the case of ion exchange materials involving reaction sites with different binding energies in approximate description of the reaction behavior based on such isotherms is feasible. 3. Langmuir Sorption Model and NMR Imaging of Competitive Cu Sorption in Ca-Alginate
Figure 4. Schematic representation of image contrast for a gradient echo sequence based on partial relaxation. At low Cu concentrations, the signal intensity is increased due to lower T1. At higher concentrations the reduction of T2 leads to a lower signal intensity (assumed relaxivity of Cu in alginate is 2000 L/mol/s).
isotherms) and the Langmuir-Freundlich isotherm suggested in ref 9, (3) the Dubinin-Radushkevich isotherm,10 (4) the isotherm of Brunauer, Love, and Keenan,11 which may be approximated by Temkin’s isotherm,12 and (5) the BET isotherm,13 which accounts also for multilayer absorption effects. A Langmuir-Freundlich extension of the BET isotherm was suggested in ref 9. The concentration-dependent diffusion coefficients based on these isotherms are given in Table 1.
Ion exchange in alginate gels was studied in recent years by several researchers.4,17,18-20 The main motivation of these studies is the potential application of alginate-containing seaweed materials as biosorbents for the removal of heavy metal ions from waste waters and other dilute solutions. In some of the work, kinetic aspects of the exchange processes were studied too. In some of these publications,4 the LAM was used for interpretation of the results. In these experiments, the ion distribution within the gels was evaluated indirectly from the ion concentration in the surrounding solution. In our NMR imaging experiments, the ion distribution can be observed more directly. In our experiments, the time development of the ion distribution was monitored by relaxation-weighted NMR images (see examples in Figure 3). The contrast in the images is due to changes in the NMR relaxation behavior near paramagnetic centers such as Cu ions. The local signal intensity obeys the relationship
1 - e-(TR/T1) e-(TE/T2*) sin β (12) S(x,y) ) F0(x,y) -(TR/T1) 1-e cos β where F0(x,y) denotes the local spin density, β the excitation flip angle, TR the repetition time, and TE the echo time. T2* is the apparent transverse relaxation time determined by the “real” spin-spin relaxation time T2 and by the inhomogeneity of the magnetic field. In our case, T2 and T2* are approximately equal, since we work in a very homogenous field. The relaxation rates can be expected to be linearly dependent on the ion concentration:
12572 J. Phys. Chem., Vol. 100, No. 30, 1996
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Figure 5. Profiles of the Cu concentration in Ca alginate along the intrusion depth as a function of time. The gray shades indirectly reflect the copper concentration according to eqs 12 and 13. The experimental profiles (left) are compared with simulations according to the LAM (middle) and the Langmuir model (right). At the lowest Cu concentration, both models show a good qualitative correspondence to the experimental results. At higher concentrations, only the Langmuir model gives a good description because saturation effects are more pronounced. The stripes in the images are due to a contrast cycle using three different combinations of echo time and recovery time in order to differentiate between T1 and T2 influences in the image contrast. Parameters in the simulations are D ) 6 × 10-10 m2/s, K ) 10, and ca ) 0.05 mol/L. The relaxivity of Cu in alginate is 2000 L/mol/s.)
1 1 + krc ) Ti(c) Ti0
(13)
where Ti(c) is the relaxation time (i ) 1, 2) at concentration c, Ti0 is the relaxation time in the absence of paramagnetic centers (i.e., c ) 0), and kr is the relaxivity of the respective paramagnetic ion species. The resulting relationship between the gray shades in the NMR images and the local Cu concentrations is illustrated in Figure 4. More detailed accounts of the experiments are given in refs 21 and 22. In Figure 5, experimental results of several competitive absorption experiments are given in comparison to simulations on the basis of the LAM and the Langmuir diffusion model. While there is a good correspondence of both models to the experimental data in the case of the lowest Cu concentration, the LAM does not describe the results obtained with higher Cu concentrations any more. The main reason for this is the fact that at these higher concentrations, the saturation effects have a much stronger influence on the effective ion transport behavior. 4. Discussion We have shown that the competitive ion exchange of Cu and Ca in alginate can be described by a diffusion equation assuming
a concentration-dependent diffusion coefficient. The appropriate concentration dependence corresponding to a Langmuir sorption isotherm is a direct consequence of the law of mass action. The concept of concentration-dependent diffusion coefficients is quite a general approach for describing material transport phenomena exhibiting non-Fickian behavior. For example, many measurements with respect to this question have been performed in beds of zeolite powders (e.g., refs 23 and 2) or similar sorbent materials such as silica gel, fired clay,3 and activated carbon. However, in most of these cases, the relationship between sorption isotherms and apparent diffusion coefficients was not recognized and a systematic study of this question is not known to us. Exploiting this relationship is quite promising, since the use of concentration dependences derived from sorption isotherms allows one to obtain additional information on the respective binding processes associated with non-Fickian transport phenomena. As our results demonstrate, the use of concentration dependent diffusion coefficients based on sorption isotherms is also superior to present models for ion exchange processes under certain conditions. In the literature, data on ion exchange kinetics are usually interpreted by assuming an absorption process according to the shrinking core model (SCM17,18) or the LAM.4
Diffusion Coefficients and Sorption Isotherms The SCM24 is a simple model for reaction-diffusion systems, which was first suggested in the context of heat condition.25 The basic idea of this model is a quasistationary concentration profile between the reservoir and a sharp reaction zone, which in turn is slowly propagating upon saturation of the available binding sites. Note that this model situation cannot be represented by a concentration-dependent diffusion coefficient. In our own work, we found good agreement between the SCM assumption of a sharp reaction front and the sorption behavior of most rare earth26,27 and actinoid ions. However, the sorption behavior of Cu ions turned out be be incompatible with the SCM as well as with the LAM. Modeling of Cu absorption from a solution containing no background concentration of Ca is only possible when taking into account kinetic aspects of the inhibition by released alginate counterions (usually Ca) too. Therefore, a theory taking into account only sorption equilibria is not applicable. This is different in the case of competitive absorption: in the presence of a sufficiently high background concentration of Ca, the additional Ca ions released by the ion exchange do not perceptibly affect the sorption equilibrium anymore. This in turn allows the application of a theory based on sorption isotherms. When working with ions of the same valence, a satisfying description of the ion exchange processes can be given using a concentration-dependent diffusion coefficient derived from the Langmuir isotherm. For ions of different valence, the situation is more difficult, since the law of mass action does not provide convenient explicit expressions for cb anymore. The ion binding sites in alginate are known to have different binding energies depending on the sequence of manuronic and guluronic residues in the chain (e.g., ref 28). Therefore, one might expect that sorption isotherms more sophisticated than the Langmuir relationship should lead to an even better correspondence with the experimental data. However, in the present frame of experimental error of our data, no indication of deviations from the Langmuir model is found. In the Langmuir model, chemical and electrostatic interactions are taken into account only implicitly by the law of mass action. A more rigorous understanding of reaction-diffusion phenomena in polyelectrolyte gels could in principle be deduced from first principles, i.e., from the Nernst-Planck equation (or the Smoluchowski equation) and Debye-Hu¨ckel-type formalisms referring to the ion atmosphere around the gel chains and the activities of the free ions. However, such approaches require a complicated numerical evaluation. Our approach, based on an effective diffusion coefficient derived from a Langmuir-type binding isotherm, therefore might be considered to be a kind of compromise between rigorous but numerically expensive theories and very crude approaches such as the LAM. 5. Conclusions Concentration-dependent diffusion coefficients derived from sorption isotherms are a plausible approach for the description of certain reaction-diffusion systems such as competitive ion exchange.
J. Phys. Chem., Vol. 100, No. 30, 1996 12573 Absorption of Cu in Ca-alginate from solutions containing a substantial background concentration of Ca can be described by means of a concentration-dependent diffusion coefficient based on the Langmuir isotherm. If the ratio between Ca and Cu concentrations is especially high, saturation effects become negligible and the well-known linear absorption model (LAM) also gives an acceptable description of the data. For higher relative Cu concentrations [Ca]/[Cu] , 10, the Ca concentration cannot be considered to be a constant throughout the absorption process anymore, so neither the Langmuir model nor the LAM apply any more. References and Notes (1) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: London, 1975. (2) Hughes, P. D. M.; McDonald, P. J.; Halse, M. R.; Leone, B.; Smith, E. G. Phys. ReV. B 1995, 51, 11332. (3) Pel, L.; Kopinga, K.; Bertram, G.; Lang, G. J. Phys. D: Appl. Phys. 1995, 28, 675. (4) Chen, D.; Lewandowski, Z.; Roe, F.; Surapaneni, P. Biotechnol. Bioeng. 1993, 41, 755. (5) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (6) Weisz, P. B.; Hicks, J. S. Trans. Faraday Soc. 1967, 63, 1807. (7) Freundlich, H. Z. Phys. Chem. 1906, 57, 385. (8) Sips, R. J. Chem. Phys. 1948, 16, 490. (9) Marczewski, A. W.; Jaroniec, M. Monatsh. Chem. 1983, 114, 711. (10) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk SSSR 1966, 55, 331. (11) Brunauer, S.; Love, K. S.; Keenan, R. G. J. Am. Chem. Soc. 1942, 64, 751. (12) Temkin, M.; Pyzhev. Acta Physicochim. URSS 1940, 12, 327. (13) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (14) Jaroniec, M. AdV. Colloid Interface Sci. 1983, 18, 149. (15) Jaroniec, M.; Deryło A. J. Colloid Interface Sci. 1981, 84, 191. (16) Jaroniec, M.; Marczewski, A. W.; Einicke, W. D.; Herden, H.; Scho¨llner, R. Monatsh. Chem. 1983, 114, 857. (17) Jang, L. K.; Lopez, S. L.; Eastman, S. L.; Pryfogle, B. Biotechnol. Bioeng. 1991, 37, 266. (18) Jang, L. K. Biotechnol. Bioeng. 1994, 44, 183. (19) Lewandowski, Z.; Roe, F. Biotechnol. Bioeng. 1994, 43, 186. (20) Chaiken, R. F. Biotechnol. Bioeng. 1995, 45, 454. (21) Nestle, N.; Kimmich, R. In Proceedings of the III. International Symposium on Biochemical Engineering; Schmid, R. D., Ed.; University of Stuttgart: Stuttgart, 1995; 105-107. (22) Nestle, N.; Kimmich, R. Appl. Biochem. Biotechnol. 1966, 56, 9-17. (23) Ka¨rger, J.; Pfeifer, H. In Magnetic Resonance Microscopy; Blu¨mich, B., Kuhn, W., Eds.; VCH: Weinheim/Bergstrasse, 1991; 348-365. (24) Rao, M. G.; Gupta, A. K. Chem. Eng. J. 1982, 24, 181. (25) Stefan, J. Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A 1890, 98, 965. (26) Nestle, N.; Kimmich, R. Heat and Mass Transfer, in press. (27) Nestle, N.; Kimmich, R. Magn. Reson. Imaging, in press. (28) Stokke; B. T.; Smidsrød, O.; Zanetti, F.; Strand, W.; Skja˚k-Bræk, G. Carbohydr. Polym. 1993, 21, 39.
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