Gershon Grossman
1616
present data in Figure 7 seem to support such consideration. However, it is not clear whether the ideal, fully expanded state such as cristobalite is achieved a t Td. Taylor and Henderson assumed that leucites and sodalites were prevented from attaining the fully expanded state a t Td because of the interframework cation-framework anion bonds. Subsequently, Taylor (1972) suggested that the temperature affected not only the degree of rotation of the tetrahedra but also the amount of anisotropic thermal motion of the framework oxygens, and that consequently the ideal fully expanded state was probably achieved a t Td. Studies of this kind indicate the need for high-temperautre structure determinations. References and Notes (1)T. Y . Tien and F. A. Hummel, J. Am. Ceram. Soc., 47,584 (1964). (2)0.F. Tuttle and J. V. Smith, Am. J. Sci., 2$5, 282 (1957).
(3) D. M. Roy and R. Roy, Bull. Geol. SOC.Am., 69, 1637 (1958). (4)F. H. Gillery and E. A. Bush, J. Am. Ceram. Soc., 42, 175 (1959). (5)J. Schulz, J. Am. Ceram. Soc., 57, 313 (1974). (6)J. S.Moya. A. G. Verduck, and M. Hortal, Trans. J. Brit. Ceram. Soc.,73, 177 (1974). (7)W. Ostertag, G. R. Fisher, and J. P. Williams, J. Am. Ceram. Soc., 51, 651 (1968). (8) F. A. Hummel. J. Am. Ceram. Soc., 34, 235 (1951). (9) L. Chi-Tang and D. R. Peacor, Z.Kristaiiogr., 126, 46 (1967). (IO)L. Chi-Tang, 2.Kristaiiogr., 127, 327 (1968). (11) A. J. Majumdar and H. A. McKinstry, Phys. Chem. Solids, 25, 1487 (1964). (12)D. Taylor and C. M. B. Henderson, Am. Min., 53, 1476 (1968). (13)D. Taylor, Min. Mag., 38, 593 (1972). (14)D. Taylor, Min. Mag., 36, 761 (1968). (15)Faust, Schweiz. Mineral. Petrogr. Mitt., 43, 165 (1963). (16) R. M. Barrer and N. McCallum, J. Chem. Soc., 21,4035 (1953). (17)J. Wyart, C. R. Acad. Sci. Paris, 282, 356 (1941). (18)Naray-Szabb, Z.Kristaliogr., 99,277 (1938). (19)R. M. Barrer and L. Hinds, J. Chem. Soc.,21, 1466 (1953). (20)J. C. R. Wyart, Acad. Sci., Paris, 205, 1077 (1937). (21)R. M. Barrer and J. W. Baynham, J. Chem. Soc., 24, 2882 (1956). (22)W. H. Taylor, 2.Krisfailogr..,74,1 (1930). (23)Kopp, et al., Am. Min., 48, 100 (1963). (24)R. W. G. Wyckoff, 2.Kristallogr., 62, 189 (1925). (25)W. Johnson and K. W. Andrews, Trans. Brit. Ceram. Soc., 55, 277 (1956).
Water Dissociation Effects in Ion Transport through Composite Membrane Gershon Grossman Faculty of Mechanical Engineering, Te@nion-israel
institute of Technoiogy, Haifa, israel (Received June 25, 1975)
A model has been developed to describe the steady-state ionic transport in composite membranes consisting of alternate layers of cation and anion exchange material. Particular attention has been given to the effects of water dissociation and water ion transport which are associated with the transport of electrolyte and play an important role in many biological and engineering applications. Membranes with various layer combinations are considered. The governing equations are presented and expressions are derived for the distributions of potential and concentrations of all ionic species. Current-voltage characteristics are obtained in terms of the membrane structure and the properties of the solution. The relative portions of current carried through the membrane by electrolyte and water ions are calculated. Homogeneous anion and cation exchange membranes are shown to have a linear current voltage characteristic. Bipolar membranes exhibit an anisotropic behavior with respect to current direction, showing electrolyte current saturation in one direction. Ion-exchange membranes with thin surface films of opposite properties exhibit a similar behavior with polarization in both directions of the current.
1. Introduction
Composite ion-exchange membranes consisting of alternate layers of cation and anion exchange material have been of interest to both biochemists and engineers. Many living systems are known to contain membranes of this type and models explaining their behavior have been proposed in terms of the ion-transport phenomena associated with the layered struct ~ r e . l -Laboratory ~ experiments with synthetic composite have indicated possible industrial and other technological application^.^ I t was also found that homogeneous ion-exchange membranes in processes such as electrodialysis often exhibit a behavior peculiar to composite membranes, due to the formation of thin surface films with ionexchange properties, which give the membrane a laminar structure. The deposition of these films, known as membrane fouling, originates from small impurities present in the water, The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
and gives rise to undesirable effects which greatly reduce the efficiency of the system.1°-13 In an earlier paper by Sonin and G;aossman14a theoretical model was derived to describe ion transport through composite membranes in contact with an electrolyte solution containing a fully ionized solute in a nondissociable solvent. It was shown that currentrvoltage characteristics of composite membranes can be anisotropic with respect to current direction, depending on their structure. Moreover, the membrane can exhibit current saturation in one or both directions, even when there is little or no polarization in the external solution. These features, derived analytically from the transport equations,14 had been observed and demonstrated experimentally by numerous investigators with synthetic bipolar membra ne^,^-^ bi~logical,~ and fouled electrodialysis memb r a n e ~ . l O - ~An ~ Jadditional ~ observation reported in all these
1617
Ion Transport through Composite Membrane was the importance of water dissociation in the performance characteristics of the membrane. Dissociation becomes dominant and cannot be disregarded especially at high current densities. It results in a transport of large amounts of dissociated water ions which cause pH changes in the solution in contact with the membrane and reduce the current efficiencies with respect to electrolyte ions transport. Water dissociation and transport plays an important role in some biological membranes5 and may have some interesting technological applications.’ In the electrodialysis process, however, it reduces the efficiency of the system and has a deteriorating effect on the membranes’ life.12 An important analysis describing water dissociation effects in a membrane-electrolyte system was made by Isaacson.18 His model consists of a cell containing static electrolyte solution between two parallel homogeneous cation-selective membranes. The model has been proposed for studying water dissociation effects in an electrodialysis system. Ionic concentrations and potential distributions in the electrolyte have been calculated. Important results have been obtained in terms of evaluating the relative magnitude of various nonequilibrium effects and the deviation from chemical equilibrium of water ions in the solution. The purpose of the present analysis has been to extend the model of ref 14 to take into consideration the effect of water dissociation and water ion transport. The governing equations are presented and solved for composite membranes with a laminar structure. Current voltage characteristics and concentration distributions are calculated in terms of the properties of the membrane and the solution. Some of the results are compared with experimental data obtained in other studies.
2. Model and Basic Equations Figure 1shows a composite membrane consisting of several homogeneous layers of ion-exchangematerial in contact with an aqueous electrolyte solution. The solution is dilute and contains a fully ionized salt with one positive and one negative ionic species, as well as a certain amount of dissociated water ions H+ and OH-. A potential difference A4 is applied between the ends of the membrane, causing current at density j to flow through it. The current is due to transport of all four ionic species present in the solution, and may be divided into the part j , carried by electrolyte ions and the part j , carried by water ions, as shown. We focus our attention on one of the membrane’s layers. It is made of homogeneous ion-exchange material of thickness Am, with uniformly distributed fixed charge at concentration Cm and valency 2,. Cm is assumed to be much larger than the concentration of any of the species in the solution. The membrane thickness A, is large compared with the Debye length, which is typically a few angstroms.16 Under the above conditions, the steady-state, isothermal ion transport both in the membrane and in the solution is governed by the transport laws for ideal s01utions.l~The flux Ni of each species by mechanism of diffusion and migration is given by
electrolyte
electrolyte solution (Ut)
solution (right)
i;-
7 ( Jw
Jw
composite membrane
Figure 1. L.>deland structure of composite membrane.
write the conservation equation dNildX = Ri
(2)
where Ri is the net rate of formation of the species by chemical reaction. Note that for the electrolyte ions R+ = R- = 0 and for the water ions RH = ROH= R where R is the rate of water dissociation per unit volume. Also in effect is the law of quasielectroneutrality: in the solution c zici = [ 0-zmCm in the membrane
(3)
and the condition of local chemical equilibrium of water ions
CHCOH= K 2 (4) where K 2 is the constant of dissociation for the given temperature. We thus have ten equations (( 1)and (2) each for the four different species, (3) and (4)) to be solved for the four unknown ionic fluxes and concentrations, R and 4. The above equations are applicable in their present form everywhere in the membrane and in the solution epcept on surfaces of discontinuity, notably the interfaces between the membrane and the solution and between two layers of the membrane. The laws expressed by (3) and (4) break down in the region of the double layer near the interface.16 We have assumed that the membrane is thick compared with the Debye length and therefore this region occupies only a small part of the domain under consideration. The double layer is dealt with as part of the interface across which all ionic species are in Donnan equilibrium1’ described by the relation:
where subscripts 0 and 1apply to the left and right sides of the interface, respectively, as shown in Figure 1. Finally, the current density may be expressed in terms of the ionic fluxes by j =F
ziNi i
where Ci, Di, and zi are the concentration, diffusion coefficient, and valency of species i, respectively, and $ is the electrical potential. The subscript i may be +, -, H, or OH, applying to the positive electrolyte ion, negative electrolyte ion, H+, and OH-, respectively. For each of the species we may
j consists of j , and j , which are given by j , = FWH- NOH) j , = F(z+N+
+ z-N-)
(6a) (6b)
j , may be further divided into the part j + carried by positive The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
Gershon Grossman
1618
electrolyte ions and the part j - carried by negative electrolyte ions: j , = Fz*Nh (6c) The conditions of local chemical equilibrium of water ions (3) and of quasielectroneutrality (4) deserve further clarification. The equilibrium condition means that, locally, the rate of production of water ions by dissociation equals therate of recombination. Let us refer to eq 2 which may be rewritten explicitly for the water ion fluxes as follows:18
6 and introducing the new variables from (7), the following set of dimensionless equations is obtained:
and from (3) and (4): CHCOH =
c+
where kd and k , are the constants of dissociation and recombination. Equation 2a expresses the net rate of water ion formation R as the difference between the production by dissociation (which is independent of local concentration) and the rate of recombination (which is proportional to CH and COH).Now, if k d >> R , then eq 2a reduces to the equilibrium condition (4), which holds throughout most of the membrane and the solution.ls However, near the surfaces of discontinuity, where the changes in ionic fluxes become large, the quasiequilibrium condition (4) is no longer valid. The thickness of this nonequilibriumzone can be shown to be very small compared to that of the membrane.ls The condition of electroneutrality, likewise, is maintained throughout most of the membrane and solution and implies that an equal amount of positive and negative charges must be present a t each point. This is valid only as long as the dimensions of the domain under consideration are much larger than the Debye length, which under the presently assumed conditions of dilute solutions is of the order of a few angstroms. Electroneutrality does not hold in a thin region near the surfaces of discontinuity, known as the double layer.16Since the thickness of this layer is of the order of the Debye length, it may be dealt with as part of the interface, as may also the thin layer of nonequilibrium. To do this, the flux equation (1) for each ionic species is integrated across the interface and the two thin layers on both sides, which yields
(2)-
F 1 dX (dl - $0) = - -In RT zi o ziDiCi The rightmost term in this equation is usually very small as long as there is no resistance to diffusion through the interface. The end result is therefore the Donnan equilibrium condition ( 5 ) across the interface. We now proceed to determine the scaling laws of the problem and write the equations in a dimensionlessform. For simplicity we choose to limit the discussion to an electrolyte for which z+ = -2- = 1 and D+ = D - = D . This limitation is not essential and a more general solution may be obtained which would yield basically the same physical results. Let us define the following dimensionless variables: X x=d
.
d . .
1.z-
FDCo
d
.
J l e = -F D C o J e
. lw=-
d
FDCo
j , i*=-
d
FDC,
where d is some characteristic length and Co some characteristic concentration in the solution. By combining eq 1 and The Journal of Physical Chemistry?Vol. 80, No. 14, 1976
- c- + CH - COH =
I
k2
0 in the solution -cm in the membrane
(10) (11)
Here i+, i-, and i, are all constant with x , as follows from (2), and so are also ie and i, since
i = i,
+ i, = i, + i+ + i-
(12)
Transforming (8) and (9) with the aid of (lo),(ll),and (12) finally yields
The set of equations (13-15) may be solved for the four ionic concentrations and the potential with the appropriate boundary conditions. The potential difference applied across the membrane is usually specified, which determines the current densities. The dimensionlessrate of water dissociation can then be calculated from the following equation, obtained by combining (2) with (l),(14), and (10):
In the following sections, solutions of the equations are given for some important membrane configurations. 3. Homogeneous Membranes
Two homogeneous ion-exchange membranes consisting of a single layer of ion-exchangematerial are shown in Figure 2. One (Figure 2a) is an anion-selective membrane with high positive fixed charge concentration C , and thickness Aa, in which the diffusion coefficient of electrolyte ions is D,. The other membrane is cation selective with high concentration of negative fixed charge C,, thickness Ac, and diffusion coefficient of electrolyte ions D,. Both membranes are in contact with well-stirred solutions characterized by their electrolyte concentrations and their pH. We thus have in the solution on the left side of each membrane a positive ion concentration C+ = C L and a hydrogen ion concentration CH = 10-pHL, and similarly in the solution on the right C+ = CR, C H = 10-pHR. The interfaces between the membranes and the solutions are marked by numerals on both sides, as shown. From the equations of electroneutrality and chemical equilibrium 10 and 11 in the solution, we find for both types of membranes:
1619
Ion Transport through Composite Membrane I-
cient from D in the solution to D, in the membrane. d$ldx from (23) may be substituted in (14) with COH = k 2 / c H from (10) to give an equation in CH alone, integrable with respect to x , Once CH is found, COW can be determined from (10) and c- from (20). CH and COH are substituted back in (23) which can then be integrated to find $. Finally, c+ is determined from (13) where d$/dx is now known. Thus, the potential and concentration distributions of all ions in the membrane are obtained. The solution following the above procedure may be simplified by making use of the fact that in an anion exchange membrane with high fixed charge concentration CH
