The reasons for salt rejection by hyperfiltration membranes are not understood. The subject is complicated, but it seems to us that some confusion results from the fact that explanations are sought in different directions by different authors, and sometimes several approaches are used in the same discussion.
In one approach, which for want of a better term we shall label
* 'structural/' attempts are made to correlate salt rejection with known (or more commonly, assumed) physical arrangements or chemical properties of membranes. Analogies in solution chemistry would be predictions of solubility from chemical structure or activity coefficients from various models. Such an approach may be useful qualitatively and may, for example, suggest materials from which useful membranes may be made. Quantitative predictions may even be possible in cases analogous to the concentration range treated by the Debye-Hiickel theory, where coulombic forces are so predominant that other interactions may be neglected. However, the interactions in organic-water-salt systems are beyond the reach of current theories, and even the simpler water-salt solutions are too complicated for satisfactory treatment except in dilute solutions. W e therefore believe that more progress in under
standing membrane behavior can be made in the near future by less ambitious treatments.
In the other approach, which we shall call "phenomenological," one attempts to correlate the rejection behavior of a given membrane with macroscopic properties that can be determined by measurements on it or on a suitable model solution; examples of important properties are diffusion rates and distribution of water and solute between membranes and aqueous solutions. In solution chemistry, the analogous approach would be correlation of various thermodynamic properties (e.g., electromotive force with vapor-pressure measurements), or of diffusion with free-energy gradients. Although studies of this type do not directly reveal information regarding microscopic aspects, they do correlate observations and clarify the roles of the factors affecting measurements.
In addition, when a macroscopic, phenomenological description is firmly established, one is in a better position to speculate about micro
scopic and structural mechanisms.
A. STRUCTURAL ARGUMENTS
L Distillation Mechanism
In this, the classical model of Callendar, a membrane is assumed to contain nonwettable pores, through which solvent passes as a gas.
Attempts to make a barrier operating on a similar principle will be mentioned in Section VII, D , but with the membranes usually considered for hyperfiltration, such a mechanism is obviously excluded. The water content is so high that water could exist as a gas only at high temperatures and pressure, and it is hard to imagine dry pores in these membranes.
2. Sieve Mechanism
This is the most obvious explanation for filtering action of membranes and certainly accounts in part for removal of some large species in ultrafiltration (Fe-36). For a summary of some recent developments, see Craig (Cr-64). Size also presumably is an important variable in gel filtration, or separations with molecular sieves. However, as we men-tioned in the introduction, with solutes in the size range of sodium and chloride ions, a purely steric explanation for membranes so far used becomes less attractive—the size of the ions is too near that of water for such a simple approach to be plausible. Nevertheless, there are proponents of the model, e.g., Ambard and Trautman (Am-60), who attribute the decrease of rejection by cellophane with increase of con-centration of solutes such as NaCl to changes in ionic size accompanying changes in ionic hydration with concentration.
3. Surf ace-Tension Mechanism
In a variant (Yu-58, So-63, So-64b) of the sieving mechanism, desalting is related to a decrease in salt concentration in the surface layer of the solution adjacent to the membrane; the arguments are based on the Gibbs' adsorption isotherm. Pores in the membrane are hypothe-sized to be of such a dimension that for the most part only surface layers of the solution go through. It is true that the surface tension of salt solutions is generally higher, at least at air-water interfaces, than that of water, a fact indicating that the surface layer has a lower salt concentration than the bulk of the solution. From the Gibbs' equation, the decrease is of a magnitude that possibly might explain rejection.
This decrease in surface concentration at an air-water interface is further consistent with the dielectric constant difference between the two phases (Wa-24, On-34). However, with hyperfiltration membranes, one deals with organic-water interfaces for which the dielectric-constant difference
is much less than for air-water, and one would predict much less decrease of salt concentration in the water-surface layer. Scatchard (Sc-64a) has discussed the situation of an aqueous solution in contact with a membrane, and concluded that much smaller rejection would be expected by this mechanism than has been observed. In addition, Blunk (Bl-64) has presented rejection data that show many exceptions to an attempted correlation with a surface-tension picture.
4. Hydrogen-Bonding Mechanism
This mechanism was evolved by Reid and co-workers (Br-57, Re-59a, Re-59b) specifically for cellulose acetate. It postulates that permeation occurs in the noncrystalline portions of the membrane, and is much faster for molecules which can form hydrogen bonds with the matrix.
Flow is pictured as a migration of water molecules from one hydrogen-bond site to the next. Keilin and co-workers (Ke-63b) at Aerojet have proposed what they call a "ligand" mechanism, which appears to be essentially the Reid model adapted to the active layer of the Loeb-Sourirajan membrane. Water in this layer is postulated to be mostly
"bound," by hydrogen bonds, and salt leakage is pictured as taking place at imperfections holding "capillary" water. One difference, which may be more of wording than substance, between the pictures proposed by the two groups is that Reid holds salt is rejected because it does not form hydrogen bonds, while Keilen assumes bound water does not dissolve salt.
Although some observations of the University of Florida and Aerojet groups are interpreted by them as evidence for such a mechanism, and Blunk (Bl-64) has found that this picture correlated more satisfac-torily with experiment than that of surface tension, we feel that such detailed models are premature, at least until questions regarding the phenomenological description of hyperfiltration, to be discussed next, are resolved. However, those interested in a recent, more sympathetic, discussion of this mechanism are referred to Michaels et al. (Mi-64c).
B. TRANSPORT EQUATIONS AND PHENOMENOLOGICAL A N A L Y S I S
1. Solubility and Diffusion Models
Salt rejection by membranes is often attributed either to large differences in rates by which solvent and solute move through the membrane, or to the fact that the equilibrium distribution of solvent and solute between feed and membrane is such that the ratio of the concentrations of the components in the two phases is different. The
first of these has recently been stressed by Clark (Cl-62, Cl-63), by Skiens and Mahon (Sk-63), by Rickles (Ri-65), and by Lonsdale et al.
(Lo-63f); the second may be operationally classified under the "solubili-ty" or "adsorption" theory, which has been attributed (Fe-36) to a publication of THermite (He-1855) of over a century ago. A variant of the solubility model is implied in the frequent statement that salt rejection with ion-exchange membranes is related to (equilibrium) salt exclusion.
As we shall see in the next section, and as Kohlrausch implied in his argument that flux per unit force is equal to the product of mobility and concentration, these models are not contradictory but represent simply two factors in the behavior of membranes. The phenomenological equations of irreversible processes clarify the roles that distribution coefficients between phases and rates of movement through membranes play in hyperfiltration, although an exact representation of the properties of a given real membrane may be difficult.
The development of the transport equations which we shall give is specifically oriented toward hyperfiltration and differs at least super-ficially from other analyses in order to emphasize certain aspects which we feel especially important. Regarding earlier work, we have already made some references to theoretical and experimental studies of ion transport [e.g., (Sp-58), (Ke-58), and (Sc-64b)]; these cite many earlier papers, which have appeared since the application of irreversible thermodynamics to membrane behavior by Mazur and Overbeek (Ma-51) and Staverman (St-51). There is much activity in the field at present, and many papers on ion transport have appeared since Schlogl's book; a review by Li et al. (Li-65) lists a number of references. W e shall refer essentially without comment to a few others, published recently, which have come to our attention. Much pertinent work is completely neglected, particularly discussions dealing primarily with flux under thermal and electrical gradients or with "active" transport.
Eder (Ed-63), Dresner (Dr-63c), Lauger and K u h n (La-64), Dorst et al. (Do-64), and Kobatake and Fujita (Ko-64a) discuss solvent and solute flow in ion-exchange membranes. Franck (Fr-63) and Kobatake and Fujita (Ko-64b) (Ko-64c) studied electrochemical properties of porous ion-exchange membranes, with particular emphasis on periodic phenomena. Woerman and Spei (Wo-64) dealt with the effect of diffusion of one electrolyte solute component across cellulose membranes on transport of a second solute, and achieved concentration against a concentration gradient. Kedem and Katchalsky (Ke-63a) have studied permeation through composite membranes. Seaman (Se-64) gives hydraulic permeabilities of commercial ion-exchange membranes.
Lakshminarayanaiah (La-65a) has recently reviewed transport through membranes.
2. Transport Equations
Let us consider the membrane as a homogeneous, isothermal phase separating two external aqueous phases containing only one solute. The water and salt fluxes through the membrane [j\ and j2, respectively (moles c m- 2 s e c- 1) ] are then related to the gradients of chemical potentials of water and salt [μχ and μ2, respectively (liters atm m o l e- 1) , functions of both pressure and concentration] as follows4:
L12=L21. (8.25c)
Equation (8.25c) is Onsager's relation expressing the equality of the cross coefficients. Substituting Eqs. (8.25a) and (8.25c) in (8.25b) yields
This equation, which will be treated in more detail later, is particularly convenient for discussing salt flux. Note that the L-coefficients and the potential gradients are for the membrane phase; the fluxes ji also refer to the membrane. However, because of the conditions of continuity, the fluxes in the membrane must equal those leaving the exit (ø) end of the membrane; in particular y2//i = 0.018#*2 ( ω ), at steady state.
4 Note, that although the form of Eqs. (8.25) is very familiar, their derivation involves some subtlety. T h o u g h not long, this derivation is beyond the scope of this chapter. It can be found, for example, in Section 8-2 of "Non-Equilibrium T h e r m o d y n a m i c s " by D . D . Fitts (Fi-62). T h e subtlety arises from the use of the condition of mechanical equilibrium, without which Eqs. (8.25) would contain additional terms. T h e condition of mechanical equilibrium can be stated in either one of two equivalent ways: (1) the velocity of the center of mass of the components is independent of position and time, or (2) the pressure gradient equals the sum of the applied external forces. If the density of the membrane phase does not change significantly from point to point, form (1) of the condition of mechanical equilibrium follows from the requirement of continuity. T h e external forces mentioned in form (2) of the condition of mechanical equilibrium are the elastic forces applied to each element of the m e m b r a n e phase by those adjacent to it or by the clamping and supporting structure.