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Analysis of ion transport with fixed site carrier membranes
Journal of Membrane Science, 56 (1991) 229-234 Elsevier Science Publishers B.V., Amsterdam
229
Short Communication
Analysis of ion transport with fixed site carrier membranes Richard D. Noble University of Colorado, Department CO 80309-0424 (USA) (Received June 6,199O;
of Chemical Engineering,
Boulder,
accepted August 27, 1990)
Abstract A morphology-dependent
ion mobility is derived for ion transport
across fixed site carrier mem-
branes. This mobility term is used in the Nernst-Planck equations along with a previously derived morphology-dependent diffusion coefficient. The rearrangement of the Nernst-Planck equations gives rise to an overall “effective” diffusion coefficient for ion transport. Setting this term equal to zero then provides a criterion for the existence of a percolation threshold in these systems. Keywords: Fixed-site carrier membranes; (coupled) facilitated transport; Nernst-Planck equation; percolation threshold
transport;
diffusion; theory;
ion
Introduction
Facilitated transport uses a chemical complexing agent, usually in a liquid film to selectively enhance solute transport. There are problems associated with this approach. The main problem is loss of the liquid film. This fluid loss is usually accompanied by the loss of the chemical complexing agent. One approach to minimize these problems is the use of ion-exchange membranes as supports [l-4]. If the complexing agent could be made ionic, it could be exchanged into the membrane and held there by electrostatic forces. This has a number of advantages which have been previously described [ 2-41. But, liquid loss is still an issue. Recently, complexing agents have been cast directly into polymer films as a means to eliminate liquid loss. Tsuchida and co-workers have published results based on various metalloporphyrins cast into polymer films for selective 0, transport [5-81 and N2 transport 191. Yoshikawa et al. [lo] described CO, transport in these fixed site carrier membranes. Two different analyses have been developed to describe transport of neutral molecules across these membranes. Cussler et al. [ 111 assumed that the reacting permeate can only be transported by the complexing agent if two complexing agents are close enough to exchange the reacting permeate between the 0376-7388/91/$03.50
0 1991-
Elsevier Science Publishers
B.V.
two. This result requires the existence of a percolation threshold, below which no facilitated transport would take place. In a previous paper, Noble [ 12 ] developed a model for facilitated transport of neutral molecules in fixed site carrier membranes. The model is analogous to facilitated transport in liquid membranes. It yields the dual-mode sorption model result in the limit of diffusion-limited transport. There are two distinct features. The effective diffusion coefficient for the solute-carrier complex is “terrain” dependent. Changes in polymer morphology cause a change in this effective diffusion coefficient. This explains why increases in the loading of the complexing agent do not automatically translate to a larger facilitation effect. Secondly, there is no percolation limit. This latter feature is confirmed for previously reported 0, transport data. There have also been reports of ion transport across fixed site carrier membranes. Yoshikawa and co-workers [ 13-161 have studied the transport of various anions across different cast polymer films. In one study, a percolation limit was observed. The purpose of this paper is to extend the previous work on neutral molecule transport to ionic transport across fixed site carrier membranes. A morphology-dependent mobility is derived for these membrane systems. This change in mobility with morphological change can explain the presence of a percolation limit. Theory The ionic transport across fixed site carrier membranes is dictated by two driving forces, concentration difference and an electrical potential gradient (induced or applied). The appropriate physical property to relate flux and driving force is the diffusion coefficient for the concentration gradient and mobility for the electrical potential gradient. In solution, these factors are usually taken to be constants except for concentrated electrolyte solutions. For cast polymer films, these factors are dependent on the morphology, or “terrain”, of the polymer films. In a previous paper, a morphology-dependent diffusion coefficient was derived [ 121. The following analysis shows the derivation of a morphology-dependent mobility. Figure 1 illustrates the ionic transport between fixed carrier sites due to the
Fig. 1. Schematic
of interactions
between an ion and three adjacent
fixed carrier sites.
231
electrical potential (CD) gradient. M; is the electrolyte concentration (moles/ cross-sectional area) at location j in the membrane. The spacing between sites is h. Therefore, jh is the distance of the sites at location j from the feed side of the membrane. The molar electrolyte concentration in the vicinity of site j is Mj/h ( = ej) . The exchange coefficients g; and g,T describe the migration of ions between sites due to changes in electrical potential. The electrical potential at site j is designated QP The charge on the ion is 2. Faraday’s constant is designated F. The net flux between sites j and j+ 1 due to the potential field is: Jj+l =g+ZFMj~j-_gj,,ZFMj,1~j+1
(1)
Some algebraic rearrangement yields: Jj+t =-y[gT
+g,
[Mj+,Oj+l-Mj@j]
+F[gT
-gsll
[Mj@j+Mj+,@j+,l
(2)
Equation (2 ) can be approximated as Jj+i z -y[gT
+g>l]
[Mjl [@j+, -@jI
+%[gT -gsll
[Mjl
[@j+@j+11
(3)
The following terms can be defined ;,g~+g,,l=C;,:
~[g~-kf~1l=U,,:
effective mobility due to the potential field function of morphology or “terrain” between sites directional preference, normally given as a convective term
For cast polymer films, there should not be any directional preference. So, the term involving Vj++ can be neglected. Equation (3 ) becomes Jj+$ =
-_ZFq+@] [@‘+;-“1
(4)
Taking the limit as h-+0 and assuming that the “terrain” between sites is not a function of j, the flux at any point in the membrane becomes J= -ZFUeE
dx
(5)
Equation (5) is analogous to the standard expression for the ion flux due to a potential field. The important distinction is the fact that U is a function of
232
morphology. So, changes in the carrier concentration in the polymer film will change the morphology, and consequently, the mobility. Next, consider a fixed site carrier membrane which separates two compartments. Compartment 1 contains an electrolyte (one cation, one anion) and compartment 2 contains a different electrolyte (again, one cation and one anion). One ion can be common to both compartments. Assume that the anions can interact with the fixed site carrier. Define: e,,
cation concentration
=2=
from compartment
1
number of cations from one molecule of electrolyte in compartment
1
The above definition preserves electro-neutrality. Writing the steady-state Nernst-Planck equation for each species O=.Z_ CJ_F-$-
(6) (7) (8)
d O=Z,+ U2+ F--e dx
(9)
Let e,_ +e,_ e,,
=e_
=e
(10)
+e2+ =e+
=e
(11)
The terms k; and k, are pseudo-first order rate constants for the rate of ion exchange with the fixed carrier sites. They are functions of the fixed site carrier concentration. The above equations can be rearranged to eliminate the potential field and ion-exchange terms. The result is: o=
(z,_
U,_
+Z,_
U2_)
[ (D,,
+D2+)
-
CD,-
+D2-
)I
[(z,_u,_+z,_u2-~-~~,+~,++~2+~2+~1
(12) The ion flux becomes zero when the term in brackets in eqn. (12) is zero (i.e. percolation limit is reached). This criterion is satisfied when
233
D,_
+a-
z,_ Ul- +zz- u,_
a+ +02+ =s+ u,++zz+G+
(13)
The important distinction here is that both D and U depend on morphology. As pointed out by Cussler et al. [ 111, the polymer film which exhibited a percolation limit had a larger, more restricted fixed carrier. The model results are analogous to Donnan dialysis with the important difference that the transport properties change with a change in morphology. So, a percolation threshold can arise due to various physical phenomena. The distance between sites can become sufficiently large that both D and U are zero. A second possibility is that both D and U change with changes in carrier loading. As the carrier loading is reduced, one reaches a point where eqn. (13) is satisfied. Conclusions An analysis is presented which provides a basis for a morphology-dependent ion mobility in fixed site carrier membranes. Use of this mobility in the NernstPlanck equations given rise to a criterion for a percolation threshold. The actual existence of a percolation threshold depends on the variation of ion mobility and diffusion coefficient with changes in morphology.
References O.H. LeBlanc, Jr., W.J. Ward, S.L. Matson and S.G. Kimura, Facilitated transport in ionexchange membranes, J. Membrane Sci., 6 (1980) 339. J.D. Way, R.D. Noble, D.L. Reed, G.M. Ginley and L.A. Jarr, Facilitated transport of CO, in ion exchange membranes, AIChE J., 33 (1984) 480. J.D. Way and R.D. Noble, Hydrogen sulfide facilitated transport in perfluorosulfonic acid membranes, in: Liquid Membranes: Theory and Applications, ACS Symp. Ser. No. 347, American Chemical Society, Washington, DC, 1987, Chap. 9. J.D. Way and R.D. Noble, Competitive facilitated transport of acid gases in perfluorosulfonic acid membranes, J. Membrane Sci., 46 (1989) 309. H. Nishide, M. Ohyanagi, 0. Okada and E. Tsuchida, Dual-mode transport of molecular oxygen in a membrane containing a cobalt porphyrin complex as a fixed carrier, Macromolecules, 20 (1987) 417. E. Tsuchida, H. Nishide, M. Ohyanagi and H. Kawakami, Facilitated transport of molecular oxygen in the membranes of polymer-coordinated cobalt Schiff base complexes, Macromolecules, 20 (1987) 1907. M. Ohyanagi, H. Nishide, K. Suenaga and E. Tsuchida, Effect of polymer matrix and metal species on facilitated oxygen transport in metalloporphyrin (oxygen carrier) fixed membranes, Macromolecules, 21 (1988) 1590. H. Nishide, M. Ohyanagi, 0. Okada and E. Tsuchida, Oxygen binding and transport in the membrane of poly[ [tetrakis(methacrylamidophenyl)porphinato] cobalt-co-hexyl methacrylate], Macromolecules, 21 (1988) 2910.
234 H. Nishide, H. Kawakami, Y. Kurimura and E. Tsuchida, Reversible coordination and facilitated transport of molecular nitrogen in poly ( (vinylcyclopentadienyl)manganese) membrane, J. Am. Chem. Sot., 111 (1989) 7175. 10 M. Yoshikawa, T. Eyaki, K. Sanui and N. Ogata, Synthetic polymer membrane with pyridine moiety for gas separation, Kobunshi Ronbunshu, 43 (11) (1986) 729. 11 E.L. Cussler, R. Aris and A. Bhown, On the limits of facilitated diffusion, J. Membrane Sci., 43 (1989) 149. 12 R.D. Noble, Analysis of facilitated transport with fixed site carrier membranes, J. Membrane Sci., 50 (1990) 207. 13 M. Yoshikawa, H. Ogata, K. Sanui and N. Ogata, Active and selective transport of anions through poly(N-propenoyl-9-acridinylamine-co-acrylonitrile), Membrane Polym. J., 15 (1983) 609. 14 M. Yoshikawa, Y. Imashiro, K. Sanui and N. Ogata, Active and selective transport of halogen ions through poly (1 -vinylimidazole-co-styrene), J. Membrane Sci., 20 ( 1984) 189. 15 M. Yoshikawa, S. Shudo, K. Sanui and N. Ogata, Active transport of organic acids through poly(4-vinylpyridine-co-acrylonitrile) membranes, J. Membrane Sci., 26 (1986) 51. 16 M. Yoshikawa, H. Ogata, K. Sanui and N. Ogata, Transport of halogen ions through synthetic polymer membranes containing pyridine moieties, Macromolecules, 19 (1986) 995. 9
229
Short Communication
Analysis of ion transport with fixed site carrier membranes Richard D. Noble University of Colorado, Department CO 80309-0424 (USA) (Received June 6,199O;
of Chemical Engineering,
Boulder,
accepted August 27, 1990)
Abstract A morphology-dependent
ion mobility is derived for ion transport
across fixed site carrier mem-
branes. This mobility term is used in the Nernst-Planck equations along with a previously derived morphology-dependent diffusion coefficient. The rearrangement of the Nernst-Planck equations gives rise to an overall “effective” diffusion coefficient for ion transport. Setting this term equal to zero then provides a criterion for the existence of a percolation threshold in these systems. Keywords: Fixed-site carrier membranes; (coupled) facilitated transport; Nernst-Planck equation; percolation threshold
transport;
diffusion; theory;
ion
Introduction
Facilitated transport uses a chemical complexing agent, usually in a liquid film to selectively enhance solute transport. There are problems associated with this approach. The main problem is loss of the liquid film. This fluid loss is usually accompanied by the loss of the chemical complexing agent. One approach to minimize these problems is the use of ion-exchange membranes as supports [l-4]. If the complexing agent could be made ionic, it could be exchanged into the membrane and held there by electrostatic forces. This has a number of advantages which have been previously described [ 2-41. But, liquid loss is still an issue. Recently, complexing agents have been cast directly into polymer films as a means to eliminate liquid loss. Tsuchida and co-workers have published results based on various metalloporphyrins cast into polymer films for selective 0, transport [5-81 and N2 transport 191. Yoshikawa et al. [lo] described CO, transport in these fixed site carrier membranes. Two different analyses have been developed to describe transport of neutral molecules across these membranes. Cussler et al. [ 111 assumed that the reacting permeate can only be transported by the complexing agent if two complexing agents are close enough to exchange the reacting permeate between the 0376-7388/91/$03.50
0 1991-
Elsevier Science Publishers
B.V.
two. This result requires the existence of a percolation threshold, below which no facilitated transport would take place. In a previous paper, Noble [ 12 ] developed a model for facilitated transport of neutral molecules in fixed site carrier membranes. The model is analogous to facilitated transport in liquid membranes. It yields the dual-mode sorption model result in the limit of diffusion-limited transport. There are two distinct features. The effective diffusion coefficient for the solute-carrier complex is “terrain” dependent. Changes in polymer morphology cause a change in this effective diffusion coefficient. This explains why increases in the loading of the complexing agent do not automatically translate to a larger facilitation effect. Secondly, there is no percolation limit. This latter feature is confirmed for previously reported 0, transport data. There have also been reports of ion transport across fixed site carrier membranes. Yoshikawa and co-workers [ 13-161 have studied the transport of various anions across different cast polymer films. In one study, a percolation limit was observed. The purpose of this paper is to extend the previous work on neutral molecule transport to ionic transport across fixed site carrier membranes. A morphology-dependent mobility is derived for these membrane systems. This change in mobility with morphological change can explain the presence of a percolation limit. Theory The ionic transport across fixed site carrier membranes is dictated by two driving forces, concentration difference and an electrical potential gradient (induced or applied). The appropriate physical property to relate flux and driving force is the diffusion coefficient for the concentration gradient and mobility for the electrical potential gradient. In solution, these factors are usually taken to be constants except for concentrated electrolyte solutions. For cast polymer films, these factors are dependent on the morphology, or “terrain”, of the polymer films. In a previous paper, a morphology-dependent diffusion coefficient was derived [ 121. The following analysis shows the derivation of a morphology-dependent mobility. Figure 1 illustrates the ionic transport between fixed carrier sites due to the
Fig. 1. Schematic
of interactions
between an ion and three adjacent
fixed carrier sites.
231
electrical potential (CD) gradient. M; is the electrolyte concentration (moles/ cross-sectional area) at location j in the membrane. The spacing between sites is h. Therefore, jh is the distance of the sites at location j from the feed side of the membrane. The molar electrolyte concentration in the vicinity of site j is Mj/h ( = ej) . The exchange coefficients g; and g,T describe the migration of ions between sites due to changes in electrical potential. The electrical potential at site j is designated QP The charge on the ion is 2. Faraday’s constant is designated F. The net flux between sites j and j+ 1 due to the potential field is: Jj+l =g+ZFMj~j-_gj,,ZFMj,1~j+1
(1)
Some algebraic rearrangement yields: Jj+t =-y[gT
+g,
[Mj+,Oj+l-Mj@j]
+F[gT
-gsll
[Mj@j+Mj+,@j+,l
(2)
Equation (2 ) can be approximated as Jj+i z -y[gT
+g>l]
[Mjl [@j+, -@jI
+%[gT -gsll
[Mjl
[@j+@j+11
(3)
The following terms can be defined ;,g~+g,,l=C;,:
~[g~-kf~1l=U,,:
effective mobility due to the potential field function of morphology or “terrain” between sites directional preference, normally given as a convective term
For cast polymer films, there should not be any directional preference. So, the term involving Vj++ can be neglected. Equation (3 ) becomes Jj+$ =
-_ZFq+@] [@‘+;-“1
(4)
Taking the limit as h-+0 and assuming that the “terrain” between sites is not a function of j, the flux at any point in the membrane becomes J= -ZFUeE
dx
(5)
Equation (5) is analogous to the standard expression for the ion flux due to a potential field. The important distinction is the fact that U is a function of
232
morphology. So, changes in the carrier concentration in the polymer film will change the morphology, and consequently, the mobility. Next, consider a fixed site carrier membrane which separates two compartments. Compartment 1 contains an electrolyte (one cation, one anion) and compartment 2 contains a different electrolyte (again, one cation and one anion). One ion can be common to both compartments. Assume that the anions can interact with the fixed site carrier. Define: e,,
cation concentration
=2=
from compartment
1
number of cations from one molecule of electrolyte in compartment
1
The above definition preserves electro-neutrality. Writing the steady-state Nernst-Planck equation for each species O=.Z_ CJ_F-$-
(6) (7) (8)
d O=Z,+ U2+ F--e dx
(9)
Let e,_ +e,_ e,,
=e_
=e
(10)
+e2+ =e+
=e
(11)
The terms k; and k, are pseudo-first order rate constants for the rate of ion exchange with the fixed carrier sites. They are functions of the fixed site carrier concentration. The above equations can be rearranged to eliminate the potential field and ion-exchange terms. The result is: o=
(z,_
U,_
+Z,_
U2_)
[ (D,,
+D2+)
-
CD,-
+D2-
)I
[(z,_u,_+z,_u2-~-~~,+~,++~2+~2+~1
(12) The ion flux becomes zero when the term in brackets in eqn. (12) is zero (i.e. percolation limit is reached). This criterion is satisfied when
233
D,_
+a-
z,_ Ul- +zz- u,_
a+ +02+ =s+ u,++zz+G+
(13)
The important distinction here is that both D and U depend on morphology. As pointed out by Cussler et al. [ 111, the polymer film which exhibited a percolation limit had a larger, more restricted fixed carrier. The model results are analogous to Donnan dialysis with the important difference that the transport properties change with a change in morphology. So, a percolation threshold can arise due to various physical phenomena. The distance between sites can become sufficiently large that both D and U are zero. A second possibility is that both D and U change with changes in carrier loading. As the carrier loading is reduced, one reaches a point where eqn. (13) is satisfied. Conclusions An analysis is presented which provides a basis for a morphology-dependent ion mobility in fixed site carrier membranes. Use of this mobility in the NernstPlanck equations given rise to a criterion for a percolation threshold. The actual existence of a percolation threshold depends on the variation of ion mobility and diffusion coefficient with changes in morphology.
References O.H. LeBlanc, Jr., W.J. Ward, S.L. Matson and S.G. Kimura, Facilitated transport in ionexchange membranes, J. Membrane Sci., 6 (1980) 339. J.D. Way, R.D. Noble, D.L. Reed, G.M. Ginley and L.A. Jarr, Facilitated transport of CO, in ion exchange membranes, AIChE J., 33 (1984) 480. J.D. Way and R.D. Noble, Hydrogen sulfide facilitated transport in perfluorosulfonic acid membranes, in: Liquid Membranes: Theory and Applications, ACS Symp. Ser. No. 347, American Chemical Society, Washington, DC, 1987, Chap. 9. J.D. Way and R.D. Noble, Competitive facilitated transport of acid gases in perfluorosulfonic acid membranes, J. Membrane Sci., 46 (1989) 309. H. Nishide, M. Ohyanagi, 0. Okada and E. Tsuchida, Dual-mode transport of molecular oxygen in a membrane containing a cobalt porphyrin complex as a fixed carrier, Macromolecules, 20 (1987) 417. E. Tsuchida, H. Nishide, M. Ohyanagi and H. Kawakami, Facilitated transport of molecular oxygen in the membranes of polymer-coordinated cobalt Schiff base complexes, Macromolecules, 20 (1987) 1907. M. Ohyanagi, H. Nishide, K. Suenaga and E. Tsuchida, Effect of polymer matrix and metal species on facilitated oxygen transport in metalloporphyrin (oxygen carrier) fixed membranes, Macromolecules, 21 (1988) 1590. H. Nishide, M. Ohyanagi, 0. Okada and E. Tsuchida, Oxygen binding and transport in the membrane of poly[ [tetrakis(methacrylamidophenyl)porphinato] cobalt-co-hexyl methacrylate], Macromolecules, 21 (1988) 2910.
234 H. Nishide, H. Kawakami, Y. Kurimura and E. Tsuchida, Reversible coordination and facilitated transport of molecular nitrogen in poly ( (vinylcyclopentadienyl)manganese) membrane, J. Am. Chem. Sot., 111 (1989) 7175. 10 M. Yoshikawa, T. Eyaki, K. Sanui and N. Ogata, Synthetic polymer membrane with pyridine moiety for gas separation, Kobunshi Ronbunshu, 43 (11) (1986) 729. 11 E.L. Cussler, R. Aris and A. Bhown, On the limits of facilitated diffusion, J. Membrane Sci., 43 (1989) 149. 12 R.D. Noble, Analysis of facilitated transport with fixed site carrier membranes, J. Membrane Sci., 50 (1990) 207. 13 M. Yoshikawa, H. Ogata, K. Sanui and N. Ogata, Active and selective transport of anions through poly(N-propenoyl-9-acridinylamine-co-acrylonitrile), Membrane Polym. J., 15 (1983) 609. 14 M. Yoshikawa, Y. Imashiro, K. Sanui and N. Ogata, Active and selective transport of halogen ions through poly (1 -vinylimidazole-co-styrene), J. Membrane Sci., 20 ( 1984) 189. 15 M. Yoshikawa, S. Shudo, K. Sanui and N. Ogata, Active transport of organic acids through poly(4-vinylpyridine-co-acrylonitrile) membranes, J. Membrane Sci., 26 (1986) 51. 16 M. Yoshikawa, H. Ogata, K. Sanui and N. Ogata, Transport of halogen ions through synthetic polymer membranes containing pyridine moieties, Macromolecules, 19 (1986) 995. 9