3996
Ain A. Sonin
and Gershon Grossman
!on Transport through Layered Ion Exchange Membranes ~
Sonin" and Gershon Grossman
Department of Mechanical Engineering, Massachusetts lnstitufe of Technology, Cambridge, Massachusetfs 02 739 (Received May 30, 1972) l'ublication costs assisted by the Office of Saline Water. U. S. Deparfment of the lnterior
The steady-state transport characteristics of a family of ion exchange membranes composed of contiguous layers of anion and cation exchange materials are described in terms of a simple physical model. Membranes consisting of up to four layers are considered. Explicit analytic expressions are derived which relate the current-voltage characteristics and transport numbers t o the membrane structure and to the concentrations of the bounding solutions. The current-voltage characteristics are shown to be anisotropic with respect to current direction, showing current saturation in one or both directions, depending on membrane structure. The results are compared with available data on bipolar membranes. An analysis is also given for the general performance characteristics, including the effects of concentration polarization in the bounding solutions, of a three-layered membrane consisting of a thick central ion exchange layer sandwiched between two extremely thin ion exchange layers of opposite sign. 'This combination may serve as a model for the effects of certain types of membrane fouling in practical applications such as electrodialysis. It is shown that even very thin surface layers can reduce the limiting current to a value significantly below the diffusion-controlled one which is expected in the absence of the surface films.
1. Introduction
tended t o level off a t a saturation value which was, however, exceeded a t sufficiently high voltages where the disThis paper is concerned with ionic transport in sociation of the water began to play a dominant role. The membranes composed of contiguous layers of anion exbipolar ion exchange membrane was proposed by Mauro4 change and cation exchange materials. Such laminar as a model for the biological cell membrane. Mauro noted membranes have attracted interest for several reasons. the anisotropic nature of the current-voltage characterisMany biological membranes are thought to be composed tic, but was more directly concerned with the high capaciof, or a t least, contain, one or more layers of ion exchange tance that such a membrane would exhibit. Using a type materials, t~,ntlexplan.ations for several aspects of quasi-equilibrium analysis, he derived an expression for their behavior have been proposed in terms of models the capacitance of a bipolar membrane for small depbased on S U G : ~a structure.1-6 Synthetic layered memartures from the resting state of zero current. Following branes have been prepared in the l a b ~ r a t o r y , ~ -and l l have Mauro, the bipolar membrane was taken u p by C0ster5-~ been found to have peculiar performance characteristics who suggested it as a model for the membranes of C h a m with possible industrial agplications.8 There is also reason australis. Coster carried out an approximate analysis of to believe that in many applications where supposedly hothe steady-state response of the membrane, and showed mogeneous mjernr'sranes are used, the membranes have in that when the membrane is not too thick compared with fact thin surface Payers, caused by fouling, which give the the Debye length, there is a punch-through effect where membrane as a whole a laminar character. The occurrence the current suddenly increases over its saturation value as of s w h fouling layers has been found, for example, in the voltage is increased, much as had been observed far electrodialysis systems, where the presence o f minute the biological membrane. This punch-through resuits from quantities of certain impurities in the feed water can a breakdown of charge neutrality and is not to be conbuild up a thin ion exchange surface layer with fixed charge of opposite sign to that of the membrane i t ~ e l f . 1 2 - ~ ~fused with the similar effect which occurs even with thick synthetic membranes as a result of water dissociation, as Even extremely thin films of this nature can cause a serious degradation in !,he performance of electrodialysk sysT. Teorell, Progr. Biophys. Biophys. Chem.. 3, 335 (1953). G . Eisenman and F. Conti, J. Gen. Physiol., 48, 65 (1965). tems. K . S. Cole, "Membranes, Ions and Impulses," University of Caliiornia An interestkg property of laminar ion exchange memPress, Berkeley, Calif.. 1966. branes is that, depending on their structure, their steadyA . Maiiro, Biophys. J., 2, 179 (1962) H. G. L. Coster, Biophys. J., 5, 669 (1965) state current-voltage characteristics can be anisotropic H. G. L, Coster, Aust. J. Biol. Sci., 22, 365 (1969). with respect to current direction and can show current V . Frilette, J. Phys. Chem., 60, 435 (1956). N. lshibashi and K. Hirano, J. Eiectrochem. Soc J a p . . 26, E e saturation in one or both directions, even when only one (1958). ionic species is invoived and when there is no polarization N. ishibashi, J. Electrochem. SOC Jap., 26,E 5 8 (1958). in the solution outside the membrane. These features N. lshibashi and K. Hirano, .I. Electrochem. Soc. Jap., 27, E 193 (1959). were demonstrated experimentally by Frilette7 and IshibN. Lakshminarayanaiah, "Transport Phenomena in Membranes," ashi and Hiranos-.IO with bipolar membranes consisting of Academic Press, New York, N. Y , , 1969, pp42-47. an anion and a cation exchange layer joined together. T. R. E. Kressman and F. L. l'ye, J. E/eclrochem, SCC. Eieclrochem. Sci., 116, 25 (1969). They showed tha.D t,he current rose linearly with voltage E. Koingola, F. deKorosy, R. Hahav, and M. F. Saboch. Desaiinawhen the current vector entered the membrane on the tion, 8. 195 (1970). side of the a.nion layer, while in the other direction it G . Grossman and A . A . Sonin, Desalination, 10. 157 (1972) The Joi.imal of Physica! Clwmistry. Vol. 76, No. 26, 1972
Figure 1. Four-layered membrane
is thick compared with the Dehye lengthP,SO that Poisson's equation for the electric fieid is repiaced by the simpie requirement of local charge neutrality. 'The Ssundaries between salt solution and memibrana, OT between atljacerit Payers of the membrane, are assumed t o be harp and to offer no interfacial resistance to diffusion, so that the ionic species are in Donnan equilibrium across them. The motion of the ions within a given p h a is ~ given by the flux equations
where r + and r - represent the (conskarrt) molar flux densities of the positive an llegati17e nQlI.5, C i- al'lc! C - Chedr molar concentrations, and $J
ZFrj5/p17'
(2)
is B dimensionless potential, 4 being the actual potential, 8' the Faraday constant, R the ulniversal gas const,c?nt, and T the temperature. In addition, we have the quasi-neutrality condition which couples the concentratiom of the pcsitive and negati~ieions
Mere, cm is the m o l a concentration of the fixedLcharges in the membrane and Z , their charge slumber per molecule. Solutions are given below for the ion concentration and potential distributions in anion and. cation exchange layers as well as in neutral layers. In the ion exchange layers, the solutions are obtained by expanding the equaticris in terms of the ratio of the coion concentration to the :fixed ion concentration, a:id keeping m8y terms of lowest order, consistent with the assun~ptionLhat the coion concentration is small compared with t h e fixed ion concentration in the membrane. Kt is convenient to express the solutican in eel"Ti19 of dimensionless ion fluxes, for which we choose t h e transport numbers
E. P . George and R . Simons, Aust. J. Bioi. Sci., *.99, 459 (1966). H . G , L. Coster, E. P. George, arid R . Simcns, 8ioplivs. (I., 9, 666 (1969). 0. Medem and A. Katchaisky, Trans. Faraday Soc.. 58, 1941
( i963). F. Conti and G . Eiseiiman, i%Ophys. J., 5, 51; (1$65). G. Grossman and A . A . Sonin, Fluid Mechanics L a b r a t o r y Report Nc. 72-6, Department of Mechariicai Enginewing, Massachusetts institute of Technology, Cambridge, Mass., 1972. Alsc submitted for publication i i i Desai/oat/on. The Journal of Physicai Chemistry, I'D!.
75. No. 25, 7972
Ain A. Sonin and Gershon Grossman
399
where
j = ZF(I'+ - I'-)
(5)
is the (constaat) current density. From eq 4 and 5 , we have as usual t+ -1. t.. = 1. Note that these transport numbers are defined for the general case where the ion fluxes are caused by diffusion in concentration gradients as well as by migralion in an electric field. They are dependent variables w'hose values are governed by boundary and operating conditions (see section 3). Cation Exchange Layer. Within the cation exchange layer, we set -z,c,/z
E
cc
C-
:= (c-)o h i = $0
+ t-(jx/ZFDc) -jx/ZFDccc
(7)
(8) (9)
where (c-)O and $oare the salt anion concentration and electric potential, respectively, a t x = 0, and 13, is the ion diffusion coefficient in the cation exchange layer. Equation '7 is obtained simply from eq 3, neglecting c-/c, compared with unity, Equation 8 is obtain'ed from eq 1 by first substituting eq 3 to eliminate c + , then eliminating d$/dx, and integrating the resulting equation for dc-ldx. The equation is then linearized to the form given in eq 8 by assuming that I.jt+xl /ZFD,cc is small compared with unity (see the discussion in section J.2), and keeping only terms to lowest order in c - / c c . Equation 9 can then be obtained from either of eq 1, again to lowest order in c - / cc. A n i o n Exchange Layer. Within the anion exchange layer we set
z,c,/z = c,
(10)
where c a >> c + . Proceeding as for the cation exchange layer, but now expanding in terms of c+/c, and keeping only terms of lowest order, we get
c..
-3
(c+)lJ
- t+(jz/ZFD,)
c - = c,
LJ= #O
- jx/ZFD,c,
(11)
(12) (13)
where (c+)o rind $0 are the cation Concentration and the potential at x 0, and D , is the ion diffusion coefficient in the anion exchange layer. In deriving eq 11 we assume that l j t - x ] /ZFD,c, is small compared with unity. Neutral Layer. In a phase without fixed charge, C, = 0, and we have from eq 3 c+ = c- = c
From these equations it follows that
(6)
where cc >> e.-. From eq 1, 3, and 4, we obtain to lowest order in c - IC, c, = cc
Again, $0 is the potential a t x = 0. Condition Across Phase Boundaries. If a sharp phase transition occurs between point 1 and point 2 which are infinitely close together, and if the interface offers no barrier to diffusion, then, regardless of the magnitude of the ion fluxes across the boundary, the positive and negative ion species must be in Boltzmann equilibrium across the transition. That is
(14)
Equation 17 is the familiar Donnan boundary condition. I t follows directly from eq 1 once one makes use of the fact that because of the sharp transition region, each of the two gradient terms on the right is very large in magnitude compared with the flux term on the Ieft. 3. Performance Characteristics of Layered Ion Exchange Membranes We are concerned with membranes consisting of two or three layers of ion exchange material, with anion and cation exchange materials alternating in order. The various possible two or three membrane combinations can all be viewed as subcases of a four-layer membrane, shown in Figure 1, which consists of two anion and two cation exchange membranes in alternating order. Once the performance of the four-layer system is described in analytic form, the characteristics of membranes consisting of all the various combinations of one to three layers can be derived by letting one or more of the four membrane thicknesses go to zero. The four-layer membrane shown in Figure 1 is in contact on its left side with a salt solution whose concentration and potential a t the membrane surface are C J and $ 1 ~ respectively, and on its right side with a solution whose concentration and potential a t the surface are c r and $r. The layers have di€ferent thicknesses, but for simplicity the two anion exchange layers are assumed to have the same fixed ion concentration C, and ion diffusion coefficient D,, and the two cation exchange layers are also assumed t o have the same properties. Points 1 to 10 represent stations a t the interfaces between the phases. The performance of the composite membrane is described by expressing the transport numbers t + and iand the total dimensionless potential drop across the membrane
*=
$1
- $10 = 91 - $ r
(19)
as a function of the current density, the exterior salt concentrations, and the properties of the membrane. The potential drop is given by
Eliminating d$/dx between eq 1 and integrating, we obtain the well-known result
where co is the salt concentration a t x = 0 and D , is the ion diffusion coefficient in the neutral layer. Using this in either of eq 1, we ob1;ain
The Journal of Physical Chemistry, Voi. 76, No. 26, 1972
Here, the first term on the right represents the potential drop across the interface 1-2 (eq 17b, with ( c - ) ~ = cl and ( c - ) ~ = c,), the second the ohmic potential drop across the interior of the left-hand anion exchange membrane (eq 13), the third the drop across the interface 3-4 (eq 17a,
where A a 3 A,] -B-
622)
&I.
& = ACl -I" Acr reprebent t he to1 al thii:knesses of the anion change Payers in $.liesystem, respectively. The concentriitions wlxch appear in eq be obt&ncd from the equations derived Carrying the soli~it~tn rrom point 1 to point we obtain ((+) (c
-)z (C+),
=
(e-),
= (e-)3
and cation ex21 can readily in section 2. 10 in Figure 1,
=
e,
(23)
=i
ea
(24)
cc
(27)
= c,'/c,
(e.,.), = (&
=
(39)
(28) (29)
e P.1) t. CQa, + pCJ -k (F&! __._~--_____ - BcPd73 (a) I- a2(1- Pal? - pcpa~jl[c~2Pcl f crZ(l-. &) ,&!2cr/!
CICr[Cr~(pal
Here, eq 24 follows froin eq 42, eq 25 is obtained by applying eq 38 to the transition 1-2, eq 26 then follows from eq 11, and so on. ?.'iie results are obtained in a straightforward " m e r , using ecl 18 for the transitions across phase boundaries, eq 7-8 for the concentration distributions within the cation exchange membranes, and eq 11-12 for the distributions within the anion exchange membranes. The requirement expressed by eq 36 determines the transport numbers. Solving eq 36 and 35 with t.k -I- t - = I., we get
We note that although the transport numbers are usually positive arid less than unity (as they invariably are when the only transport ine
