THE SIGNIFICANCE OF CONVECTION IN TRANSPORT PROCESSES ACROSS POROUS MEMBRANES BY R. SCHLOGL Max-Planck-Institut fur physikalische Chemie, Gottingen Bunsenstrasse 10 Received 26th January, 1956 For the treatment of transport phenomena across membranes according to the irrever- sible thermodynamics of" continuous systems ", the particle fluxes can be separated into a convection common to all particles, and fluxes relative to the local centre of gravity. The convection velocity represents an additional unknown quantity requiring a further known relationship ; this is provided by the balance of hydrodynamical forces. The separation of the convection term has proved itself useful both in the theoretical treatment of " anomalous osmosis " and in experimental investigations on current flow across ion- exchange membranes.A procedure by Kirkwood by which he obtains linear relation- ships between fluxes and forces will be considered from this viewpoint. The current theories on the transport of charged or uncharged particles across membranes can be roughly divided into the following groups. GROUP 1 considers the membrane as a surface of discontinuity setting up different resistances to the passages of the various molecular or ionic species.1-3 The driving forces are the diferences of the general chemical potential between the two outer media. (Differences of pressure or electrical chemical potential are included in the general chemical potential). GROUP 2 considers the membrane as a quasi-homogeneous intermediate phase of finite thickness in which the local gradients of the general chemical potential act as driving forces.4-15. Convection may also contribute to particle transport within the membrane. GROUP 3 considers the membrane as a series of potential energy barriers lying one behind the other, thus forming, in contrast to group 2, an inhomogeneous intermediate phase.16-18 An (irregular) spatial lattice is formed due to the higher probability of finding a particle in the positions between the activation thresholds.The driving forces arise from the differences between the transition probabilities in opposite directions perpendicular to the membrane. This grouping attempts to classify the various mathematical approaches, according to the ideal models on which they are based. It is in fact too schematic, as many theories occupy intermediate positions.No author is likely to take the view that one of these treatments is right and the others wrong. The various descriptions supplement each other, and depending on the system under considera- tion, one of the three will prove the most suitable. It may be shown, for example, that when the number of activation thresholds becomes very large, and the distance between the lattice points sufficiently small, then groups 3 and 2 merge into each other. A transition is also to be found between groups 1 and 2, and this will be dis- cussed later. Unlike group 3, groups 1 and 2 can be classified in the general scheme of irrever- sible thermodynamics. In 1 as well as in 2, linear relationships are assumed between the particle fluxes and the driving forces.Owing to the differing characters of the driving forces, 1 is treated according to the method of " discontinuous systems ", and 2 according to that of " continuous systems ' 7 . An integration in group 2 across the membrane does not as a rule yield a linear relationship between 46R. SCHLOGL 47 the flux and the general chemical potential differences. Only for sufficientlj small differences does group 2 merge into group 1 after integration. In this sense 2 is more general than 1. Group 2 is, however, inferior to group 1 in that a number of idealizations must be assumed before an explicit integration can be effected. In a recently published work,lg the chief interest of which lies in time-dependent integrals of equations of motion, Kirkwood finds a connection between groups 2 and 1.In our present discussion we are especially interested in this transition. Kirkwood's initial flux equation differs from that normally used in the treatment of " continuous systems " in irreversible thermodynamics. For the flux ja (Le., the number of moles of the molecular or ionic species a which are transported in unit time across the membrane cross-section), he writes : j , = - p c q B dik - (a, /3= 1,2, . . . , n). D d x a and /3 are the indices for the n species which participate in the transport. P p is the general chemical potential of the species /I. dj$dx is the potential gradient at the position x. Qd is a coefficient depending on x, but not on the forces d&/dx, and in addition satisfying the symmetry relationship fzap = Qp,.However, when evaluating the production of entropy, the irreversible thenno- dynamics of " continuous systems "20 distinguishes between the movement of the various species relative to each other, and the translational movement of the common (local) centre of gravity of all mobile particles. Thus, Here C, is the concentration of the species a, and v the velocity of the centre of gravity of all species. The coefficients Ld obey, as before, the symmetry relation- ship Lcq~ = Lp,, and are in addition linearly dependent upon one another, thus: Mu, Mp are the respective molar masses of the components a and /3. It is the aim of this paper to draw attention to the convection term C,v in (2), and to ascertain its significance in transport processes across membranes.At the same time we shall also discover under what conditions Kirkwood's eqn. (1) is equivalent to eqn. (2) for continuous systems. In one trivial case this equivalence is obvious, namely, in membrane systems with such high flow resistances that the velocity v of the centre of gravity vanishes. This is very likely the case in many biological membranes for which Kirkwood developed his theory. The equivalence is also possible, however, under more general conditions, as we shall see presently. The equations (2) contain v as an extra unknown quantity, and are therefore as such not integrable. To make the integration possible we must introduce the important relationship of the balance of hydrodynamical forces, The pressure gradient dp/dx and the electrical field d$/dx are assumed to constitute the forces driving the fluid across the membrane.,oe is the electrical space charge of a unit volume of fluid. K is a measure of the flow resistance of the membrane. An observable flow can take place especially in a membrane with porous structure. Now it can be shown that the equivalence of eqn. (I) with (2) and (4) is guaran- teed if K depends only on the membrane properties and the concentration Cp, but not on the velocities of the single components /3 relative to each other. In48 CONVECTION I N TRANSPORT PROCESSES general, this condition will not be strictly fulfilled. As is shown in the appendix, Kv takes the form IK,pp, where vp denotes the average particle velocity of the species /3, and Kp is a constant of matter (depending only on collisional cross- sections of the particles, the local concentration, and average pore cross-section). To confirm this statement we form the expression zCpdpD/dx, and with the use of the Gibbs-Duhem relationship obtain at once the equation which is, except for the signs, identical with the left-hand side of (4).Expressing v in (2) with the help of (4) and (5), we obtain Hence, we arrive at Kirkwood‘s eqn. (I), provided that K is a physical property of the system-that is, a space function which does not depend on vp. In practice this will always be a very good approximation, and any small deviations will almost certainly be insignificant compared with experimental errors. The relationship between SZll and the diffusion coefficient D, given by Kirkwood for binary systems, i.e.must, however, be re-written for a flow resistance K which is not infinite, thus forming a corresponding relationship bctween D and ,511. It is in any case only valid in this form (7) for sufficiently small concentrations C1 of the solute. With reference to the classification of membrane theories given in the introduc- tion, it would be profitable to make one more remark about the transition by which Kirkwood derives from eqn. (1) (which belongs to group 2) to an equation belonging to group 1. For this purpose he solves (1) with respect to the forces djip/dx, and integrates j, across the membrane, assuming steady-state conditions (ja independent of x). The differences of the values at either side of the membrane thus appear in place of the gradients of &.Solving again with respect to the fluxes j,, the formal linear relationships between the fluxes and the chemical potential differences are obtained, giving an equation of group 1. The coefficients now appear- ing, o c r p , again satisfy the symmetry relationships ; they are, however, dependent as Kirkwood emphasizes, not exclusively on the membrane properties, but also on the concentrations of all the components in the solutions. Generally speaking, a linear relationship of this type, which belongs to group 1, is not obtained by the integration of a group 2 equation. We can take as an example the integrals in Teorell’s theory 10 which are derived from the Nernst-Planck equations of motion. These equations are a very specialized form of the approach represented by group 2 : all coefficients LZp, excepting Lm, are assumed equal to zero, and since no convection is taken into consideration, K must be equated to CO.As is known, the integrals of these equations do not lead to a linear relationship between the fluxes and the (externally applied) chemical potential differences. This discrepancy can be explained as follows. The Kirkwood coefficients Wap depend also on the concentration profiles of all the species within the membrane. But as the profile itself depends on the electrical potential difference and the applied pressure difference, the wab terms are also functions of the applied “forces ”. The integrated equations are thus only apparently linear with respect to the forces, that is, they only appear to belong to group 1.However, the deductions drawn by Kirkwood in his paper remain valid for sufficiently small differences App, so that within this range, his theory belongs to group 1. We now return to the “ convection term ” in eqn. (2) and ask when the termR. SCHLOGL 49 C,CB/K in (6) becomes comparable with the term Lap. To my knowledge, no measurements have as yet been presented on the “ mixed ” coefficients L,p, for ct $; /3. All theories of group 2 which have attempted to make an explicit integra- tion of the equation of ionic motion have, for this purpose, equated the coefficients Lap to zero. This should in fact cause no serious error if the solvation shells are treated as belonging to the particles they surround.One can, however, arrive at an estimate of Lm with the help of eqn. (7), in which SZ, should be substituted by La,, if D is known. Investigations which we have made on membranes of ion-exchange material (phenolsulphonic acid) have revealed that the term Cz/K outweighs the term L,. It may be said in general that for porous membranes in concentrated solutions (over 0.1 M), the two terms are already comparable for pore sizes of 10 A. To illustrate the influence of convection on transport phenomena, I should like here to quote two examples taken from other papers.21.22 Fig. 1 shows the c.11 If c+ It c * (4 (b) FIG. 1 .-Calculated cation concentration profiles in a cation exchange membrane under the influence of electric current ; (a) positive current, (b) negative.DIDO is a measure of the flow resistance of the membrane. Boundary concentration C; = 1.5 (inside left) and C’; = 4 (inside right). (Z. plzysik. Chern. N.F., 1955, 5, 384.) steady-state concentration profiles within a cation exchange membrane when an electric current is applied ; each curve is calculated for a different flow resistance. The electrolyte is NaCl, the concentration of which at the left side of the membrane is 1 M and at the right side 3.5 M. D/& is a (dimensionless) measure of the flow resistance K. The profiles are calculated by means of the Nernst-Planck equations of motion, in which a convection term, corresponding to that in eqn. (2), is inserted. The diagram on the left is distinguished from that on the right by the direction of the applied current.Equal current densities are chosen for the two diagrams. The curves in which DID0 = 03 correspond to the theory of Teorell in which the convection vanishes, With diminishing flow resistance the curvature is inverted. The electro-osmotic liquid flow, whose direction in a cation exchanger coincides with that of the electric current, finally reverses the distribution completely. The contribution of convection to electrolyte transfer is, as the calculation shows, greater at points of higher concentration. S-shaped curves with an inflection point will be formed for membranes with average flow resistances. Fig. 2 shows concentration profiles measured by Schodel21 for phenolsulphonic acid membranes. The membranes used in the upper curves were strongly cross- linked, those in the lower curves weakly cross-linked.Comparison with the cal- culated curve indicates that the contribution of convection, rather than being insignificant, is actually quite considerable. The value of the flow resistance estimated from the measured concentration profile was in good agreement with the water permeability measured directly on the same membrane material. The contribution of convection to electrical conductivity lay roughly between 10 and 45 %.50 CONVECTION I N TRANSPORT PROCESSES As a further example22 we consider the same system, but with no electrical current. We assume a Donnan equilibrium at the two membrane surfaces. At these phase boundaries not only does there exist an electrical potential difference, but also, due to the different concentrations of the osmotically effective particles, the hydrostatic pressure within the membrane exceeds that in the adjacent solutions.Now the hydrostatic pressure within the membrane, as can be easily demonstrated, is higher on the more dilute side (left) than on the more concentrated side (right). FIG. 2.-Measured cation concentration profile in a phenolsulphonic acid membrane ; (a) and (b) cross-linked more strongly than (c) and ( d ) ; arrows show direction of electric current. Boundary concentrations differ somewhat from the values chosen in fig. 1. (2. physik. Chem.N.F., 1955, 5, 376.) The pressure difference at the phase boundary does not act as a driving force, for here we have assumed equilibrium. The pressure difference within the membrane, however, acts as a driving pump on the pore liquid, forcing it from the more dilute to the more concentrated solution. In other words if the contribution of the electric field vanishes in (4), a positive osmosis results due solely to the inner osmotic pressure difference of the membrane.In the example with an electric current investigated previously, the contribu- tion of the pressure term does not vanish, but the electrical term begins to pre- dominate at a rather low current density. In the case investigated above where there is no current, the pressure makes the chief contribution. Of course electrical fields also share in the process, their magnitude being calculable from the cation and anion mobilities. Without going into details, we should like to state the following consequences of the theory.If the cation is more mobile than, or as mobile as the anion, we find a positive osmosis. If the anion is much more mobile than the cation, we find a fluid transport from the concentrated to the dilute solu- tion, in other words a negative osmosis Taking the latter case and referring to eqn. (4), the contribution of the diffusion potential outweighs that of the innerK .F s CHLO GL 51 Osmotic pressure difference. Thus we arrive at a natural interpretation of the phenomenon of “ anomalous osmosis’’ and find that it yields to quantitative explanation. To summarize, it may be said that for membranes whose water permeability is not too low, among which can be reckoned ion exchangers with more than 65 % water content, convection effects can play a considerable role.When such systems are subjected to theoretical treatment, it is advantageous to split the ionic fluxes according to eqn. (2) and make use of the balance of hydrodynamical forces (eqn. (4)); this method is likely to lead to the clearest mathematical separation of the essential physical processes involved. APPENDIX THE BALANCE OF MECHANICAL FORCES FOR SEVERAL COMPONENTS In the molecular theory of viscosity it is shown that between two adjacent fluid1 ayer, moving in the x-direction with slightly different velocities v and v + dv, the exchangc of momentum in the direction normal to the surface (z-coordinate) results in an inter action given by the force f per unit surface area : dv 1 d V f = q - = -rnIwN - dz 3 dz’ where rn is the mass, I the mean free path, w the mean molecular velocity, and N thc number of molecules per unit volume.A complete analogue is found for various moleculat species /3 having various translational velocities vb. It should be noted that in this case vp = +rnplgwgNg depends, on account of Zg and wp on the concentrations of all species. When the concentrations Cg are very low, Zg and wi are for physical reasons independent of Cp Thus, because of the factor Ng, 70 becomes proportional to Cg as Cp-t 0. For the pressure gradient in a cylindrical tube, a perfect analogue to the Hagen-Poiseuille equation is found : where r = radial distance from cylinder axis. If we now assume that the particle concentration does not depend on Y, the fluid being homogeneously mixed over the cylinder cross-section, we find after integrating twice in the radial direction, where 5 p is the mean value of vg over the cylinder cross-section, and a the cylinder radius.The capillaries in a membrane deviate considerably from the cylindrical form (having statistically twisting and branching axes, and statistically changing cross-sectional forms), Taking the average over many pores yields, in place of the factor of 8/&, another numerical factor depending only on the membrane properties. If we now leave out the averaging bar over VB, we can write the general expression - dn L- = 1 K p g . dx B Here Kp depends only on the concentrations C, and on the membrane properties. over, for small concentrations Cp, Kp will be proportional to Cs. More-52 MEMBRANB POTENTIALS 1 Wiebenga, Rec. trao. chim., 1946, 65, 273. 2 Mazur and Overbeek, Rec. trav. chim., 1951,70, 83. 3 Staverman, Trans. Faraday SOC., 1952, 48, 176. 4 Teorell, Proc. SOC. Expt. Biol. Med., 1935, 33, 282. 5 Meyer and Sievers, Helv. chim. Acta, 1936, 19, 649, 665, 987 ; 1937, 20, 634. 6 Goldman, J. Gen. Physiol., 1943, 27, 37. 7 Hodgkin and Katz, J. Physiol., 1949, 108, 37. 8 Eriksson, Ann. Roy. Agric. Coll., Sweden, 1949, 16, 420. 9 Keynes, J. Physiol., 1951, 114, 119. 10 Teorell, Z. Elektrochem., 1951, 55, 445. 11 Schmid, 2. Elecktrochem., 1951, 55, 295 ; 1952, 56, 181. 12 Lorenz, J. Physic. Chem., 1952, 56, 775. 13 Scatchard, J. Amer. Chem. SOC., 1953, 75, 2883. 14 Schlogl, Z. physik. Chem. N.F., 1954, 1, 305. 15 Mackie and Meares, Proc. Roy. SOC. A., 1955, 232, 498. 16 Davison and Danielli, The Permeability of Natural Membranes (Cambridge, 1943, 17 Laidler and Shuler, J . Chem. Physic., 1949, 17, 851. 18 Zwolinski, Eyring and Reese, J. Physic. Clzem., 1949, 53, 1426. 19 Kirkwood, Ion Transport Across Membranes (Acad. Press, New York, 1954), p. 119. 20 De Groot, Thermodynamics of Irreversible Processes (Amsterdam, 1951), p. 94. 21 Schlogl and Schodel, 2. physik. Chem.N.F., 1955, 5, 372. 22 Schlogl, Z. physik. Chem. N.F., 1955, 3, 73. chapter 21 and appendix A).