ELECTRO-OSMOSIS IN CHARGED MEMBRANES THE DETERMINATION OF PRIMARY SOLVATION NUMBERS BY A. DESPI~ * AND G. J. HILLS Dept. of Chemistry, Imperial College, London, S.W.7 Received 1st February, 1956 Electro-osmosis in highly charged membranes is discussed. The two main ways of investigating electro-osmotic flow in membranes are described and the extensiveness of such measurements necessary to examine current theories of the phenomenon is emphasized. The direct and indirect study of electro-osmosis in membranes of cross- linked polymethacrylic acid in the sodium form is described and its explicit dependence . on the swelling of the membrane is deduced. The validity of the Schmid theory of electro- osmosis in membranes was investigated. From the measurement of total volume flow of solution through a membrane under- going electrolysis and a knowledge of the electro-osmotic contribution to this total flow, the difference, representing the volume of the solvated counter-ions is readily obtained.From this, the solvation number of the sodium ion in the membrane phase has been evaluated. Virtually all forms of membranes in contact with ionic solutions and subject to a potential gradient normal to the membrane exhibit electro-osmosis, The phenomenon arises from a non-uniform distribution of ions close to the walls of the capillaries of the membranes which can occur in two ways (i) by preferential adsorption of one type of ion on the walls of an uncharged membrane, or (ii) by the existence of charged groups in the structure of the membrane itself. In the first case, the distribution is that of a normal double-layer.The ions predominating in the inner or outer Helmholtz planes are considered to be relatively immobile and the electrolytic transport of the other species which predominate in the diffuse part of the double layer give rise to a flow of ions mainly in one direction. The exchange of momentum between the moving ions and the solvent then gives rise to unidirectional flow of solvent. The exact location of the slip-plane in the double layer is not known but the electrical potential in this plane is, by definition, the {-potential and it can be shown1 that the electro-osmotic velocity VE of the solution at a large distance from the walls of the capillaries is given by where e and 7, the dielectric constant and viscosity of the solution respectively are assumed to be the macroscopic values of the solvent (cf., however, ref.(2)), and E is the applied potential gradient.The corresponding volume of solution transported per second for a current density of i A cm-2 through a capillary of constant cross-section is where K is the specific conductance of solution. These equations are ideal and only apply to uncharged membranes of relatively large pore-radius in contact with very dilute solutions where a double layer of significant magnitude is obtained. Even in such systems it is necessary to make a correction for " surface conduction ", i.e. migration of ions in the Helmholtz part of the double layer. * present address : Faculty of Technology, Belgrade University, Yugoslavia.150A . D E S P I ~ AND G . J . HILLS 151 The second type of membrane, consisting generally of sheets of homogeneous or heterogeneous natural or synthetic ion-exchange substances, are characterized by a high concentration of fixed charged and a correspondingly high concentration of counter-ions. The pore radii are usually much smaller than in those of the first type, being often of molecular dimensions. All such systems approximate to the perfectly selective membrane in which there are two charged or ionic species, only one of which, the counter-ion, is mobile. In keeping with the high conductivity of these systems it is assumed that a high proportion of the counter-ions, if not all of them, are free to migrate and that the slip-plane occurs at the surface of each fixed ion or, if it is hydrated, at the surface of its primary hydration sheath. The unidirectional flow of ions will again give rise to unidirectional flow of solvent, i.e.electro-osmosis. For any applied voltage, the net current flow i will be greater than that ( i f ) flowing in a hypothetical, identical system in which there was no electro-osmosis, i.e. no net movement of solvent. Schmid 4 has derived a relationship between i and if for point charges flowing through cylindrical capillaries : (3) i - i f F2X2i-2 - - _ - i 8 T K ’ where F is the faraday and r is the radius of the capillaries. From this equation it might appear that (i - i’)/i -+ co as Y -+ GO but this is not so since K also depends on Y, i.e. increasing with an increasing electro-osmotic effect.A more useful form of this equation is (4) 103 F2Fi-2 A = X + 877 ’ where A and A’ are the equivalent conductances of the counter-ions with and without the electro-osmotic contribution. Both equations suffer from the limi- tations that values of Y are not directly obtainable and that in many cases they are of the same order as the-radii of the mobile ions themselves. The fact that A +co as r -tco at constant Xis in agreement with the concept of increased ionic velocity outlined above since the friction of the solvent with the walls, the sole limiting force, would tend to zero. Neither of these equations has hitherto been experimentally investigated. The direct measurement of electro-osmosis in charged membranes is usually not possible.The total volume of solution transported per faraday is readily observable,69 7 but this is only partly electro-osmotically transported solvent, the remainder, of similar magnitude, being the volume of the transported ions together with that of their solvation sheaths. Since no reliable independent values of these latter volumes are available, this method offers no absolute measurement of electro- osmosis. The contribution of electro-osmosis to ionic velocity in the membrane phase is, however, easily evaluated. The equivalent conductance and, hence, the ionic mobility of the counter ions in the membrane phase, can be determined in a number of ways.39 5.8-16 Where a suitable radioactive isotope of the counter-ion is also available its self-diffusion coefficient D can be determined in the absence of an applied potential gradient and, hence, in the absence of electro-osmosis.The corresponding equivalent ionic conductance and ionic mobility can be evaluated from the Nernst-Einstein equation : where z is the valence of the counter-ion and uf and A’ are the ionic mobility and equivalent conductance in absence of electro-osmosis. The differences between A and A’ (A > A’) in two ion-exchange resins have been recorded by Spiegler and Coryell.14/ mine AA over a wide range of X and of swelling and an 5 - / adequate variation of this last 1 - 2 . 0 / A w o - - a / - / E / - / I - 1 I I I 0 . 2 0 . 4 0 . b 0.8 I or r2) it is necessary to deter- parameter can only be obtained by removing the membrane material from contact with the 1.5 solution and partially dehy- drating it.A similar but more limited variation of swelling could be obtained by alteration of the degree of cross-linking in the membrane material but this ~ was considered undesirable in 6 view of its unknown effect on r . 0- The system, solution + JX’ membrane $ solution, was 0 . 5 only used to study the total electrolytic transport of solu- tion through the membrane. Few values of this transport have been reported 6, 7 for charged membranes and no estimate of how much of the : 1 .O I .OA . D E S P I ~ : AND G . J . HILLS 153 EXPERIMENTAL The preparation and co-polymerization of methacrylic acid with ethylene glycol di- methacrylate in the form of membranes and rods has been described elsewhere.17~ 3 Only 10 % cross-linked polymethacrylic acid was used and the neutralization of the rods with NaOH followed the same procedure as that already described.5 Each rod was equilibrated for several weeks with its appropriate dilute NaOH solution to ensure an even distribution of counter-ions.The variation of swelling of the material and of 2 with degree of neutralization is illustrated in fig. 1. The equivalent conductance and self-diffusion coefficient of sodium ions in each fully swollen rod was determined, after which it was progressively dehydrated, X and D being determined at regular intervals during this process. A section of each rod was retained for analysis 5 and from this, the sodium ion content and the water content for each rod during its dehydration was known. 0.74 0.83 0.94 0-29 0.37 0.48 0.63 TABLE 1 K x 103 ohm-1 cm 4.111 4.065 3.485 0.8 18 8-55 8.13 7.67 4.56 13.81 14.16 13-68 2.182 2-17, 3.261 2.257 6.979 7.546 5.432 18.43 18.99 12.55 20.66 21-97 21.69 25-20 26.25 26-01 17.25 27-35 28.74 3.986 29-56 14.42 7.843 29.05 30.00 28.17 11-17 5.46 31.1 23-88 18-54 13.61 31.96 1 ohm-] cm2 g ion-1 10.07 8.885 6.563 3.397 1.081 14.34 11.90 8.921 3.992 1.709 16.64 14.54 12.30 1.550 0.978 3.911 18.85 15.74 3.07,- 1.935 6.195 17.92 16-20 12.8 1 16.93 13.86 12.53 5.79 0.900 16.15 13.53 11.61 3.908 1.784 15.27 13.30 9.885 2.628 1 so89 14-49 11-70 6.7 1 4.521 3.079154 ELECTRO-OSMOSIS CONDUCTANCE MEASUREMENTS.-These were carried out as described elsewhere 5 9 18 and the.results are recorded in table 1 which gives the ionic concentrations c in g ions/l.of resin phase, the volume ratio VR (see below), the interstitial ionic concentration cj in g ions/l. of interstitial solution in the resin phase, the specific conductance K and the equivalent conductance A, for each degree of neutralization and dehydration. SELF-DIFFUSION rmAsuREMENTs.-The ionic mobility and self-diffusion coefficients of sodium ions in the resin phase were determined by the method of Morgan and Kitchener,lg using 22Na. A detailed description of the diffusion experiments is given elsewhere 18 and in table 2 are given the values of c, VR, Ci, D and A' for 5 different degrees of neutralization, each at various degrees of swelling. TABLE 2 degree of neutralization 0.29 0.8049 0.9388 1.839 0-37 0.963 1.833 0.63 1.461 3.733 0.74 1.678 2.523 6.1 19 0.94 2.132 3.529 4.1 12 2.908 2.02~ VR Cf 1.030 1.234 3.480 1.192 2.893 6.944 1.762 2.656 6.628 2.01~ 3.377 2.562 4.887 15.8 6.082 D X 106 cmz sec-1 3.92 3.50 1.069 4.09 1.918 0.525 3.88 2.72 0.605 3.09 2-12 0.049 3-70 1.734 0-946 1' ohm-1 cm2 g ion-1 14-73 13-17 4.02 7.21 1.97 14-59 10.22 2-27 11.63 7.97 0.184 13-91 6-52 3.56 15-36 TOTAL VOLUME TRANSPORT MEASUREMENTS.-MembraneS of 10 % cross-linked POlY- methacrylic acid were thoroughly equilibrated with 0.01 N NaOH solution and then clamped in the silicone-greased rubber gaskets in the apparatus shown in fig.2, the flanges being I Membrane FIG. 2.-The measurement of total volume transport. ground ends of Industrial Glass Piping held together by the usual bakelite connectors. This apparatus was filled on both sides of the membrane with the same 0.01 N NaOH since under these conditions the transport number of the sodium ion in the membrane isA .DESPIC AND G . J . HILLS 155 close to unity. To prevent ordinary osmotic flow of solvent and diffusion of electrolyte across the membrane, no large concentration changes should occur during the electrolysis. Although it has been observed that neither of these processes occur readily in the mem- branes used in this work, it was decided arbitrarily to restrict the concentration changes to 5 %. Moreover, after the initial experiment the current was reversed so that the time- averaged concentration gradient across the membrane during the subsequent experiments was zero. Such a small concentration change amounts to 0.0005 equiv. or - 50 coulombs/l.of solution. Since the total volume transport was - 200 ml per faraday, the volume of each compartment had to be at least 1 litre in order that a reasonable volume change be observed. The volume increase in the cathode compartment and the volume decrease in the anode compartment were measured by observing the corresponding rise and fall of the menisci in the precision-bore capillary tubes. The electrodes were so placed that most of the gas evolved during the electrolysis escaped up the capillary tubing ; any bubbles clinging to the glass surface and to the electrodes were gently dislodged. The whole cell was immersed in a water thermostat maintained at 25 i 0.005" C and connected in series with a silver coulometer. The membranes used were soft and flexible and any difference in the levels of solution in the two tubes was partly compensated by the bulging of the membrane.It was essential therefore that when the levels of solution in the capillary tubes were being measured before and after each experiment that no net difference in hydrostatic pressure existed. The difference in levels was therefore compensated by the addition of petroleum ether to the lowest side. In table 3, are given the observed volume changes d Y in the cathode and anode compartments. TABLE 3 expt. no. no. of coulombs d Y (cathode) d V (anode) passed ml ml 1 72 0- 180 0-184 2 74 0.174 0.182 3 72 0.151 0.156 4 72 0.150 0.157 Conductance and self-diffusion measurements were also carried out on rods equilibrated irz tlze same soZution as used in the transport experiments.The results are listed in table 4. TABLE 4 C K X 103 1 D x 106 A' 2.39 32.1 13-43 2-68 10.09 2.36 31.3 13-26 2.89 10-87 2.39 32.0 13.38 2.74 10.30 2-39 31.8 13.31 2-76 10.37 2.37 31-9 13-43 2-64 9-92 mean 13-36 & 0.46 % mean 10.31 i 2.4 % DISCUSSION EVALUATION OF THE INTRINSIC DEPENDENCE OF CONDUCTANCE ON Ci AND ON SWELLING The decrease of equivalent conductance of the sodium counter-ions with inter- stitial ionic concentration was attributed to an increase in overall viscosity and a structural parameter was sought which would reflect directly the viscous effect of the resin matrix. For this system always having the same distribution of compon- ents, the same degree of tortuosity etc., the viscous effect of the matrix was taken t o be directly proportional to the fractional volume content or relative volume content of organic matter V R .~ VR values for each rod are readily evaluated 5 and are included in tables 1 and 2. Since VR and Ci change simultaneously, the intrinsic dependence of h on VR or on ci can be isolated only if one of the variables is kept constant, e.g. by interpolating from vertical lines drawn on graphs of X against ci and h against VR respectively. Fig. 3 and 4 show the variation of equivalent conductance with Cj at constant VR and the variation of X with VR at constant ci.156 ELECTRO-OSMOSIS The corresponding curves for the equivalent conductance derived from the self- diffusion coefficients are shown in fig. 5 and 6. Finally, in fig.7 and 8 are shown the corresponding relationships for AA, the electro-osmotic contribution to A, obtained by comparison of fig. 3 and 5 with fig. 4 and 6. I 2 3 4 5 c i q-ions titre-’ FIG. 3.-The variation of equivalent conductance with interstitial ionic concentration at various constant VR values. 0-2 0-3 0-4 0.5 V, FIG. 4.-The variation of equivalent conductance with VR at various constant Cj values. DEPENDENCE OF ELECTRO-OSMOSIS ON THE INTERSTITIAL CONCENTRATION The form of Schmid’s equation given by eqn. (4) suggests that the electro- osmotic contribution to h should be a function of two variables, x(= ci) and r2. The interstitial viscosity 7 is assumed to be constant and approximately equal to that of water. Keeping VR constant is equivalent to maintaining constant r values, and therefore from (4), i.e. (AA)VR should be a linear function of ci.A .D E S P I ~ : AND G . J . HILLS 157 It is evident from fig. 7 that this is not so. The inapplicability of the Schmid theory to highly cross-linked and highly charged membrane materials might be expected. If the membrane material consists of a cross-linked linear polymer the pores are probably those through the polymer spirals, the diameters of which are of the same order as that of hydrated sodium ion. The annulus of " free water " c , q-ions litre-' FIG. 5.-The variation of A' with Ci at various constant VR values. 1 I I I I I 0.10 0.20 0.30 040 0.50 vR FIG. 6.-The variation of A' with VR at various constant ci values. surrounding such an ion is therefore only a few molecular diameters in thickness and it is questionable whether the relationship between diffusion coefficient, pore radius and solvent viscosity used in the theory is admissible.For a particular Y value, the retarding influence of the walls increases with the ionic concentration, i.e. as the annulus of " free water " decreases, contrary to eqn. (4). The effect increases with the degree of primary solvation of the counter- ion and it would appear therefore that eqn. (4) and (6) are only applicable to mem- branes of large pore size.158 ELECTRO-OSMOSIS The present system could be modified by using a smaller, less hydrated counter- ion. Although corresponding conductance values are available for the potassium system,s in the absence of a convenient radio-isotope, direct determination of h’ and Ah’ for potassium has not yet been possible.It is, however, possible to deduce A’ for this system,lg and the derived Ah as a function of ci at various VR values is A X ci 9-ions litre-’ FIG. 7.-The variation of AA with ci at various constant VR values. FIG. 8.-The variation of AA with VR at various constant Ci values. shown in fig. 9. The potassium ion is apparently small enough for the wall-effect described above to be insignificant below 3 N and for each VR value a linear section of the (Ah), against ci relation is observed. The slope of the linear section for the smallest VR value, where 7 is approximately equal to that of water, is 7.5 ohm-1 cm2 mole-2 which from eqn. (10) gives a value for Y of 7 A.A . D E S P I ~ AND G .J . HILLS 159 THE DEPENDENCE OF ELECTRO-OSMOSIS ON VR According to eqn. (4), AA is also a linear function of the square of the pore radius. The pore radius as used by Schmid was based on an earlier definition of this quantity by Bjerrum and Manegold,*o i.e. where W is the volume of solution per ml of membrane material and N is the number of pores per cm2. Schmid eliminated N in terms of r2 but, in the present work, it was found more convenient to eliminate r2 in terms of Nand to express N in terms of VR. r2 = W/rN, (7 ) c , q-ions l i t r e - ’ FIG. g.-The variation of AA for potassium counter-ions with ci at various constant VR values. The number of pores NO per cm2 of membrane organic material is a constant determined only by the structure of the matrix, and the number per cm2 of swollen material is thus where AR is the relative area of organic material, i.e.V R ~ . Since W is by definition equal to (1 - VR) N = NoAR, ( 8 ) and, therefore, eqn. (4) becomes In the absence of No values, eqn. (10) does not permit absolute values of the electro- osmotic contribution to be calculated but the dependence of (AA)cj on VR is un- ambiguously expressed. From fig. 10 it can be seen that for the potassium system eqn. (10) is obeyed. ELECTRO-OSMOTIC TRANSPORT AND PRIMARY SOLVATION NUMBERS The observed volume changes during electrolysis AVO,, must be corrected for volume changes at the electrodes. From the corrected volume change per faraday the total transport of solvent (both free and solvating) AV,,~,,,~ is obtained by subtracting from AV:E:hode and adding to AVZZde the partial molar volume of NaOH, nv,,,.160 ELECTRO-OSMOSIS The number of moles of water transported in each case is given by A Vsolvcnt/ is the partial molar volume of H20 in dilute NaOH solution.ATH20, where The mean net values of AVs,,,,,t/fiH,O are shown in table 5. TABLE 5. cathode 21 8 - 6 18 242 13.5 anode - 222 - 6 18 246 13.7 FIG. 10.-The variation of with (=)for the potassium system. VRQ The electro-osmotic contribution to this solvent transport can also be evaluated. If a current i A cm-2 flows through the membrane under a potential gradient E V cm-1, (1 1) UCjEF ciAE 1000 1000 a i - - = - The corresponding current in absence of electro-osmosis is given by ., u'c~EF - CiX'E I = - - - 1000 1000 * Eliminating E, and iji' = ulu' = X/h', Ai = i - i' = AAijA.For any time t the quantity of current passed through the two systems would be Q = it and Q' = i't respectively, where Q - Q' or AQ represents that part of Q which is transported electro-osmotically i.e., Where Q is one faraday, AQ is a number of g equiv. sodium ions transported electro-osmotically per equivalent of total transport, and, assuming all the solution AQ = Q AA/A. (15)A . D E S P I ~ AND G . J . HILLS 161 inside the membrane phase is transported, the volume of solution per faraday transported elec tro-osmo tically is and the number of moles of water transported electro-osmotically is V' = AQ/ci litres, (16) moles, AQ 1000 AA 1000 M E = - -=-- mi 18 h iiii 18 where /?li is the interstitial molality (in these experiments, 2-75).fore given by The primary solvation number of sodium ions in the membrane phase is there- This value is regarded as accurate only to & 1 and more precise work has since been carried out.22 Three criticisms of the method are (1) that possibly only part of the interstitial solvent is transported, (ii) that there may be ion-association in the membrane phase and (iii) that SNa is not a primary solvation number as such but a related parameter appertaining to the membrane phase. With respect to (i), some solvent may be primarily solvated to the fixed charges.21 The evidence for primary solvation of large anions is not convincing although Oda and Yawatawa 7 claim to have observed a degree of " trapped " solvent in their recent studies of solvent transport through cation-selective membranes, because the total concentration of the transported solution, i.e.[total Na+]/[total H20], is greater than the interstitial ionic concentration. This must always be so when the counter-ions move faster than the water but the implication that the instantaneous concentration of moving solution is also greater is fallacious, since the latter remains constant throughout the electrolysis. Although therefore there is no direct experimental evidence for trapped water in cation-selective membranes, it still remains a possibility. It would, however, lead to higher values of ci and mi in eqn. (16) and (17), to lower of VE and ME and hence to higher values for the primary solvation numbers. If ion-pair forma- tion were appreciable in the membrane phase it would increase VE and ME and hence reduce SNa but, contrary to earlier suggestions, there is no evidence of ion association in cross-linked polymethacrylic acid + alkali metal ion systems.' 8 The third criticism has two aspects : (a) the special effects the membrane phase has on solvation number and its determination and (b) the significance of primary solvation numbers.The special effects could involve (i) the high ionic concentration in the membrane phase, (ii) "pumping action" of ions in close fitting pores, or (iii) frictional stripping of part of the solvation sheath. There is no evidence that the degree of primary solvation is concentration dependent ; (ii) and (iii) can be investigated by studying SNa as a function of current density and extrapolating it to zero current density.This has been done22 and a more precise value of SN~(= 8) has been obtained. A discussion of primary solvation numbers is outside the scope of this work. There is no doubt that, in water, small cations at least have associated with them significant number of solvent molecules which move with them whether they migrate in an electric field or simply by diffusion. It is this number which migrates in this sense which has been determined in the present work. It is distinct from the additional solvent transported by an ion in an electric field. The steady migration of an ion, solvated or not, gives rise to movement of the '' free " solvent about it, the magnitude of which will be proportional to the viscosity of the solvent, the radius of the ion and to the velocity of the ion.This movement is not generally observed in solutions because the resultant of the motion in opposite directions of cations and anions is small and would invariably be balanced by an opposing F162 SPECIFIC TRANSPORT ACROSS MEMBRANES hydrostatic pressure. It is observed in any system where one type of ion is im- mobilized and is called electro-osmosis. In any such system therefore the solvent is divided into that moving with the velocity of the ion, i.e. as its solvation sheath, and that moving with a velocity limited by the frictional resistance offered by the membrane structure and considerably smaller than that of the ion. 1 Kruyt, Colloid Science, vol. 1, pp. 198 et seq. (Elsevier, 1951). 2 Elton, Proc. Roy. SOC. A, 1948,194,259,275 ; 1949,197,568. Elton and Hirschler, 3 Hills, Kitchener and Ovenden, Trans. Faraday SOC., 1955, 51, 719. 4 Schmid, 2. Elektrochem., 1950, 54, 424. Schmid and Schwarz, 2. Elektroclzem., 1951,55,245,684. Schmid, Z. Elektrochem., 1952, 56, 181. Schmid and Schwarz, 2. Elektrochem., 1952, 56, 35. 5 Despib and Hills, Trans. Faruday SOC., 1955, 51, 1260. 6 Despik, Thesis (London), 1955. 7 Oda and Yawatawa, Bull. SOC. Chem. Japan. 1955, 28, 263. 8 Manecke and Bonhoeffer, 2. Elektrochem., 1951,55,475. 9 Sober and Gregor, J. Colloid Sci., 1952, 7, 37. 10 Moulton, Diss., Abstr., 1952, 12, 321. 11 Ishibashi, Seyama and Sakai, J , Electrochem. SOC., Japan, 1954,22, 684. 12 Manecke and Otto-Laupenmuhler, 2. physik. Chem., 1954,2, 336. 13 Juda, Rosenberg, Marinsky and Kasper, J. Amer. Chem. SOC., 1952, 74, 3736. Clarke, Marinsky, Juda, Rosenberg and Alexander, J. Physic. Chem., 1952, 56, 100. 14 Speigler and Coryell, J. Physic. Chem., 1953, 57, 687. 15 Wyllie and Kanan, J. Physic. Chem., 1954, 58, 73. 16 Hills, Kitchener and Jakubovic, J. Polymer SOC., 1956, 19, 382. 17 Howe and Kitchener, J. Chem. SOC., 1955, 2143. 18 Despid and Hills, (in preparation). 19 Morgan and Kitchener, Trans. Furaday SOC., 1954, 50, 51. 20 Bjerrum and Manegold, Kolloid-Z., 1927, 43, 5. 21 Glueckauf and Kitt, Proc. Roy. SOC. A , 1955,228,322. 22 Hills, Jacobs and Lakshminaryan (in preparation). Proc. Roy. SOC. A, 1949, 198, 581.