J. Chem. SOC., Faraday Trans. 1, 1985, 81, 609-619 Concentration Dependence of Electrokinetic Transport Coefficients of Non-aqueous Binary Mixtures through Weakly Charged Porous Plugs BY ROQUE HIDALGO-ALVAREZ,* FRANCISCO JAVIER DE LAS NIEVES AND GERARDO PARDO Physics Department, Faculty of Sciences, University of Granada, Granada, Spain Received 30th April, 1984 An experimental investigation of the concentration dependence of electrokinetic transport coefficients of methanol + ethanol mixtures through a quartz plug is described. Electro-osmotic flow and streaming-potential measurements have been carried out. The Onsager reciprocity relations have been experimentally checked for all compositions of the binary mixture used. The form of the fluidity curve indicates that the ethanol+methanol mixture is a quasi-ideal system with a small negative deviation. The hydrodynamic permeability of systems comprising methanol+ethanol mixtures and a porous plug and the fluidity of the mixture vary with the composition in a similar way.The variation of the cross-phenomenological coefficient with composition is due to an analogous variation of the term Dc/r,~ with the mole fraction of methanol. On the other hand, the ratio of streaming potential to pressure difference (in the interval 0-20cmHg) decreases as the pressure drop increases. This result agrees with the theoretical findings of Rutgers and Boumans concerning the effect of turbulence on the streaming potential. The concentration dependence of the second-order coefficients has been partially explained on the basis of Jha’s equations.In recent years several papers have been published with the main objective of studying the transport phenomena of liquid mixtures through inactive porous media and ion-exchange membranes, from the point of view of the thermodynamics of irreversible processes. Special interest has been paid to the study of the dependence of the first-order phenomenological coefficients Lik on the compositions of binary and ternary mixtures.1-8 Some investigatorsly working with the same system (methanol + water mixtures and a sintered Pyrex glass membrane) have found very different trends of variation of each phenomenological coefficient with the mixture composition. For example, Srivastava et al. obtained linear relationships between the coefficients L,, and Lik and the methanol mole fraction, these dependences being explained by means of Spiegler’s frictional model.However, using a similar porous glass membrane Singh and Singh2 obtained a non-linear dependence of Lii and Lik on the mole fraction of methanol. This different behaviour of the transport coefficients has not been satisfactorily explained by these authors. Very recently Blokhra et d3 have also employed methanol + water mixtures as permeants through a chemically treated oak wood membrane. The variations of the coefficients Lki and Lkk with the mass fraction of methanol were non-linear. These trends were attributed to the structural modifications which occur in water + methanol mixtures due to hydrogen bonding. With regard to ion-exchange membranes, Rastogi et aL4 have found that with several mixtures the electro-osmotic permeability can even change sign when potential differences beyond a determined critical value are applied.The concentration depen- dence of the transport coefficient is usually more complicated in highly charged than 21 609 FAR 1610 TRANSPORT THROUGH POROUS PLUGS in weakly charged or inactive membranes. Moreover, some investigators5 have asserted that the concentration dependence of the phenomenological coefficients can be consistent with Spiegler's model when the interactions among the components of the mixture and alterations in the membrane-permeant interaction with change in composition are negligible. Hence it seems that when a liquid mixture flows through an ion-exchange membrane the above interactions are never negligible, since in these cases the dependence of the phenomenological coefficients on composition have always been non-linear. With inactive membranes and several liquid mixtures1* 5-6 some investigators have shown that Spiegler's model is adequate to describe the transport processes of those mixtures, the relation between the phenomenological coefficients and the composition then being linear.An investigation of electrokinetic transport in liquid mixtures is thus of interest in understanding mechanisms of transport. In view of the above, we have studied the dependence of the electrokinetic and permeation coefficients on composition for a mixture of liquids (ethanol + methanol) of similar characteristics, using a plug prepared with particles of quartz as an inactive porous medium.To obtain further insight into the phenomenon the experimental results were analysed on the basis of the classical theory of electrokinetic processes. On the other hand, in order to test the thermodynamic consistency of the electro-osmosis data, streaming-potential measurements were also carried out, and the Onsager reciprocity relations were experimentally checked. In order to gain a further understanding of electrokinetic transport in weakly charged porous plugs we have also determined the second-order phenomenological coefficients that appear in streaming-potential experiments when a high pressure difference, Ap, is applied. Likewise, we have studied the effect of mixture composition on the non-linear coefficients.This aspect has been little studied since most reports refer to the variation of the linear coefficients with compositions of binary liquid mixtures. EXPERIMENTAL The experimental apparatus used in this work has been described in earlier paper~.~-ll The quartz employed was from Cerro Muriano (Cdrdoba, Spain) and its quantitative composition, obtained by spectrophotographic methods, was 95.5% SiO,, 2% Na,O, 1% Al,O, with small impurities of CaO, K,O, Fe,O, and HgO. The quartz was crushed and sieved to obtain the fraction between 150 and 200 pm. It was washed with dilute HCl(l% ) and then with deionized water until the conductivity of the washing water remained constant. Porous plugs 0.52 cm thick were prepared with quartz particles.The liquids utilized in the experiments were analytical-grade methanol and ethanol from Carlo Erba. When the electrokinetic measurements were carried out the electrical conductivity of these alcohols increased by ca. 5% of its initial value. Consequently, the quartz particles used to prepare the porous plugs did not cause any ionic contamination of the permeant liquids (methanol and ethanol). The time taken to condition the quartz in the liquid media was 48 h. All measurements were carried out in an air thermostat maintained at 20.0k0.5 "C. RESULTS AND DISCUSSION The linear phenomenological equations for the simultaneous transport of matter and electric charge are12 J = L,, Ap+LI2 A$ I = L,,Ap+L,,A$R. HIDALGO-ALVAREZ, F. J. DE LAS NIEVES AND G. PARD0 61 1 where J and I are, respectively, the total volume flux and the electric current due to both hydrostatic pressure differences, Ap, and the electric potential difference, Ad.The coefficients L,,, L,, and L,, characterize the hydrodynamic and electro-osmotic flows and the streaming current, respectively, while the electrical conductance of the plug is given by the coefficient L,,. In order to provide a suitable interpretation of the dependence of the phenomeno- logical coefficients on the composition of the permeant it is necessary to know the relation between those coefficients and the parameters characterizing the electrohydro- dynamic properties of the system. In the capillary model a porous plug is supposed to be composed of a bundle of n capillaries entering a porous medium on one face and emerging on the opposite face. Although the actual structure of any porous plug is not as simple as described by this capillary model, it has been successfuly used by many authors.According to the classical theory we have L,, = nnr4/8qZ (3) L,, = L,, = n&r2(/4qZ (4) L,, = nnr2L/Z where n is the number of capillaries, r is the equivalent pore radius, Z is the length of the capillaries, q is the absolute viscosity, E is the dielectric constant, A is the electrical conductivity of the liquid permeant and [ is the electrokinetic or zeta potential of the solid/liquid interface. From eqn (3)-(5) it can be inferred that if the geometric parameters n, r and Z are independent of the mixture composition (x) then the dependence of L,,, L,, (= L,,) and L,, on x might be explained by means of the variations that q-l, [D/q and A, respectively, undergo when the composition varies.The equivalent pore radius can be estimated by the equation r = (~~LL,,/L,,)% (6) In an earlier work9 we have found that eqn (6) provides values of r close to the true value of the equivalent pore radius of inactive porous plugs. The values of r estimated from eqn (6) are shown in table 1. In most cases there is no significant variation of r with the composition, f = 19.2f 0.4 pm being the average pore radius in the overall interval of variation of the mole fraction of methanol in the mixture. Once the equivalent pore radius of the porous plug is known, it is possible to obtain an estimation of the number of capillaries that theoretically make up the plug.From eqn (5) n can be calculated and the values are also shown in table 1. n is also reasonably constant for the different fractions of methanol, and its average value is (7.3 f 0.3) x lo4 capillaries. Note that the thickness of the porous plug was always 0.52 cm, so that the capillary length (Z) can be taken as independent of x. Therefore it might be expected that the viscosity, q, the dielectric constant, D, the electrical conductivity of the mixture, A, and the zeta potential will be the main factors accounting for the dependence of L,,, L,, and L,, on composition. DEPENDENCE OF THE FIRST-ORDER PHENOMENOLOGICAL COEFFICIENTS ON THE COMPOSITION If the relation between (J)A9-o and Ap is linear, the hydrodynamic permeability coefficient (L,,) can be obtained from the slopes of the straight lines that result when plotting (J)Ad-o against Ap.Fig. 1 shows that for each mixture the dependence of the 21-2612 TRANSPORT THROUGH POROUS PLUGS Table 1. Values of pore equivalent radius (r), number of capillaries (n), absolute viscosity (q), dielectric constant (D) and zeta potential (c) for different values of xM (methanol mole fraction) 0.0 19.7 7.3 1.206 25.8 14.5 0.2 19.6 7.0 1.038 27.3 18.0 0.4 19.1 7.5 0.9 15 28.7 19.8 0.6 19.3 6.8 0.790 30.1 22.1 0.8 18.8 7.7 0.686 32.3 19.2 1 .o 18.7 7.3 0.582 34.4 17.4 1 . 2 1 . o - 'Y) 0.8 E m Y) 0.6 \ 0 I1 t 5 0 . 4 0 . 2 0 1 I I I I I 2 4 6 8 10 12 AplN m-2 Fig. 1. Dependence of hydrodynamic flow on pressure difference for various methanol mole fractions: (a) 1.0, (b) 0.8, (c) 0.4 and (d) 0.0.hydrodynamic flow on the pressure difference is linear for the interval of pressure used, and thus the flow regime is laminar. Fig. 2 shows the variation of L,, and q-l with the mole fraction of methanol (x,). It will be observed that the dependence of both L,, and q-l on X, is linear, but for X, 2 0.6 the straight lines fitting the experimental data undergo a sharp elevation, with greater slopes than when xM < 0.6. Taking into account the close parallel that exists between both straight lines, and the constancy of the term n&/81 [see eqn (3)] when the composition of the mixture varies, one may conclude that the dependence of L,, on X, is mainly due to an analogous variation of q-l (fluidity of the mixture) According to the form of the fluidity curve, the methanol+ethanol mixture is a quasi-ideal system with a very small negative deviation, i.e.the curve practically coincides over the whole concentration with the straight line joining the fluidities of the pure components. It is clear from the values of q (see table 1) that an increase of methanol in the mixture increases the freedom of the internal molecular motion with xM.R. HIDALGO-ALVAREZ, F. J. DE LAS NIEVES AND G. PARD0 613 12 11 7 0 0.2 0.4 0.6 0.8 1.0 XM Fig. 2. Dependence of hydrodynamic permeability (L,,, a) and fluidity (q-l, 0) on methanol mole fraction. owing to an analogous decrease in the viscosity of the mixture, which is usually interpreted as a relaxation effect of the hydrogen-bonding interactions between the polar liquids (methanol and ethanol in our case).This relaxation effect seems slightly more marked when the dominant component of the mixture is methanol (x, 2 0.6). On the other hand, eqn (4) can be written in the form K being a constant equal to 4 ~ ~ , n r ~ / 4 1 and D the dielectric constant of the liquid permeant used. If eqn (7) were valid for explaining the dependence of L,, on the composition of the ethanol +methanol mixture, a close correspondence should exist between the variations of L,, and Dc/q with the mole fraction of methanol, for example. The L,, coefficient was obtained from electro-osmotic flow measurements. The value of this coefficient coincides with the slope of the straight line resulting when (J)A,co is plotted against At,$ (see fig.3). In the interval 0 < X, < 0.6 the coefficient L,, depends linearly on X, according to the equation When xM 2 0.6 the dependence of L,, on xM is also linear, but the straight line fitting the experimental data is then 101OL,, = (0.37+0.02)+(1.30+0.05)~~. (8) lolo L,, = (0.72f0.04)+(0.725+0.003)~~. (9)614 12 10 i s 4 2 TRANSPORT THROUGH POROUS PLUGS I 1 1 I I I 0 20 40 60 80 100 Fig. 3. Dependence of electro-osmotic flux on electrical potential difference for various methanol fractions: (a) 1.0, (b) 0.6 and (c) 0.4. A9/V In order to discover how the term D [ / q varies with composition it is first necessary to calculate the c potential. Values of the [ potential were obtained from streaming- potential data, using the following equation :14 c = CTW/AP)Z*O 1- (10) The constant CT depends on the liquid used and on the temperature.This equation allows the estimation of the 5 potential to be made independently of the capillary model used. The ratio ( A ~ / A P ) ~ - ~ was calculated for Ap --+ zero. In table 1 we show the values of 5 thus obtained, and the viscosities and dielectric constants of each mixture. The term D [ / q is then calculated, and its variation with composition is given by the following regression straight lines : lop2 Dl/V = (2.98 +0.04)+(8.765+0.010)~M (1 1) (12) for xM < 0.6 and D [ / q = (5.53 +0.08) + (4.65 + 0.05) .xM for xM 2 0.6. The values of Dc/q are given in m3C-l s-l. In all cases the linear correlation coefficient was always > 0.99. If the concentration dependence of the cross-phenomenological coefficient L,, was mainly due to the variation of the term D [ / q with xM, the product of the constant K [(l.44+_0.09) x 10-13 C V-l] by D [ / q should be equal to eqn (8) and (9).Thus KD5/1O1O = (0.43 _+ 0.03) + (1.26 0.08) XM (13)R. HIDALGO-ALVAREZ, F. J. DE LAS NIEVES AND G. PARD0 615 for 0 < xM < 0.6 and KDr/1O1O 71 = (0.80 f 0.05)+ (0.67 0.04) XM (14) for 0.6 < xM < 1. Comparing the eqn (8)-( 13) and (9)-( 14), respectively, it is inferred that, within experimental error, the concentration dependence of L,, is due to a similar variation of the term D [ / r with the composition of the methanol + ethanol mixture. To check the thermodynamic consistency of the electro-osmotic data, streaming- potential measurements were also carried out with the methanol + ethanol mixture.The phenomenological relation between and Ap was found to be non-linear for Ap up to 20 cmHg. DEPENDENCE OF THE SECOND-ORDER PHENOMENOLOGICAL COEFFICIENTS ON THE COMPOSITION For any Markoffian process, the local fluxes Ji depend on the local forces Xt, the intensive parameter ai and the structural factor G.15 Hence Ji = A X l , X,, . . ., a,, a,, . . ., G). (15) G is a factor which takes into account the characteristics of the system. If the intensive parameters are kept constant and Ji are expanded in the form of a Taylor’s series16 with equilibrium as the reference point denoted by subscript 0, we have Ji = (aJi/axJo XI+ (aJi/ax2)0 X2 + (1/2) [(a2Ji/aZ)o Z + (a2Ji/axz,), + . . . + 2(a2~~/ax, ax,), X , x, + .. .I (16) up to terms of second order. The derivates in eqn (16) are constants if the intensive parameters are kept fixed provided G is not altered. We may put (aJi/aXl), = Li1 (aJi/aX,), = Liz (17) (a2Ji/az), = Lil, (a2Ji/aP,), = Liz2 (a2Ji/aX1 ax2), = Li12. (18) Several authors16-21 have found that eqn (16) is valid for a great number of irreversible processes. For the particular case of flux of electric charge, eqn (16) is usually written as I = L,, AP + L,, A 4 + 45212 APA4 + &5211(AP)2 + iL222(A4)2 (19) where I = J,, X , = Ap and X , = A4. Taking into account the wide range of validity of Ohm’s law, the coefficient L,,, equals zero, and thus the streaming potential can be written as @#)I-, = - (L21/L22) AP - (L212/L22) AP(A4)1=0 - (L211/L22) @PI2.(20) Generally the constant factor of 3 is included in the coefficient L,,,. In fig. 4 we have plotted ( A ~ / A P ) ~ - , against Ap for some of the mixtures employed. The experimental results of ( A ~ / A P ) ~ - , were fitted by a multivariable regression method. The ratio of streaming potential to pressure difference decreases as the pressure drop increases. This suggests that the turbulent flow that arises at high pressure is responsible for the diminution in absolute value of ( A ~ ~ / A P ) ~ - , , in agreement with the theories of Rutgers et ~ 1 . ~ ~ and Bo~mans.,~ Rutgers et al. have indicated that the non-linear behaviour of the streaming potential with the pressure difference is a consequence of the great disturbance originating in the diffuse part of the electric double layer caused by the high speed of the liquid in the capillaries forming the porous plug.Likewise, Boumans theoretically found that the ratio ( A ~ ~ / A P ) ~ - ,616 W I TRANSPORT THROUGH POROUS PLUGS 0 4 8 12 16 20 24 Ap/ 1 O3 N m-z Fig. 4. Dependence of -(A#/AP)~-,, on Ap for various methanol mole fractions: (a) 1.0, (b) 0.4 and (c) 0.2. Table 2. Values of the coefficients L21, L22, 3, and L,,, for the different compositions of the binary mixture methanol +ethanol 0.0 0.35 kO.01 0.35 f0.02 1.56 0.90 0.76 0.2 0.68 f 0.03 0.66 f 0.03 1.18 0.72 2.09 0.4 0.81 f0.04 0.86 f 0.04 1.67 1 .oo 2.14 0.6 1.13 f 0.06 1.15 f0.04 1.82 1.20 3.72 0.8 1.30 f 0:07 1.32 f 0.04 2.27 1.40 4.5 1 1 .o 1.45 f 0.07 1.44 f 0.05 5.26 3.30 7.42 is lower for turbulent than for laminar flow.In an earlier we have found that with our plugs pressures over 5 cmHg cause turbulent flow. Table 2 shows the coefficient L,, and A (the electrical conductivity of the mixtures) plotted against the molar fraction of methanol. As expected from eqn (9, the changes of A explain the variation of L,, with xM. The ratio L,,/R is effectively a constant (1.67kO.07 cm) for the entire range of variation of xM. By considering the values of L,, (see table 2), L,,, L,,, and L,,, can be calculated. In table 2 are shown the values of L21, which are equal to L,, within experimental error. This resemblance between the values of both cross-phenomenological coefficients is a experimental verification of the Onsager reciprocity relation, and implies that the dependence of L,, on xM is also explained on the basis of the variation of D f [ / q .R.HIDALGO-ALVAREZ, F. J. DE LAS NIEVES AND G. PARD0 617 With the aim of analysing the dependences of the coefficients L,,, and L,,, on the composition of the mixture, we have employed the simple relations proposed by Jha et uI.,,~ which allow for the interpretation of the second-order coefficients in terms of known physical parameters of the system. In the kinetic interpretation of second-order phenomenological coefficients made by Jha et al., the basic hypothesis is that in transport processes the existence of coupling between the fluxes and thermodynamic forces leads to a decrease in the coefficients when restrictions are imposed on the fluxes, so that when J = 0 we obtain from eqn (1) and (2) According to Jha et al.this can be written as which represents the non-linear form of eqn (21) in a power series of A# (up to the second power). On substitution of [I -([2~2/2n2r2q;l)]A# in eqn (22) by the net potential [A#+([eAp/4nqJ.)] the net flow of current I is obtained as25 Comparing eqn (19) and (23) it can be seen that L,,, = (nr4p[3~3/256n2q4AP) (24) and L,,, = (nr4p[2~2/82nq3P) (25) where p is the density of the permeant and the other symbols have their habitual meaning. Taking into account that n, r, I and p are practically constant, it is to be expected that the concentration dependence of the coefficients L,,, and L,,, is due to the variation of r2c2/q3 and c3~3/q4;1 with x M , respectively. The L,,, and L,,, coefficients may be interpreted as a measure of the distortion caused by turbulence in the electric double layer, distortion that could be related to the orientation of the molecular dipoles, since the effective electrokinetic potential is made up of the charge distribution and dipole potential,,' and it seems that the dipole orientation in the electrical double layer is affected by the streaming pressure.According to eqn (24) and (25) the distortion effect will be more pronounced as the [ potential increases. As pointed out by Gur and Ravina,26 non-linearity of the electrokinetic processes is detected only for wide ranges of driving forces and in systems with high surface potentials. Therefore the values of L,,, and L,,, are conditioned not only by the turbulent flow in the capillaries, but also by some specific characteristics of the electric double layer (i.e.[ potential). Moreover, eqn (24) indicates that the L,,, depends on the electrical conductivity of the liquid permeant too. For all cases studied, the sign of L,,, was found to be opposite to that of L,,,. This fact is clearly explained by the eqn (24) and (25) if we keep in mind that the potential was negative (see table 1) for all compositions of the methanol+ethanol mixture. It is also noticeable that the contribution of the coefficient L,,, is greater than that In fig. 5 we have represented L,,, and c2O2/q3 as functions of the composition of the mixture. Both L,,, and C2D2/q3 show a monotonic increase as the mole fraction of L211, i*e- I L,l& I ' I L,,, AP I.618 7 - 6 - 5 - - I > z d 4 - E 2 2 ---- 3 - - I N 2 4 c.2 '- TRANSPORT THROUGH POROUS PLUGS m M X - 1 2 m E > 2 -88 ,? r m. 4 s CI Q - ' 4 - 0 0 0.2 0.4 0.6 0.8 1 .o XM Fig. 5. Variation of the coefficient L,,, (0) and r2D2/q3 (0) with the methanol mole fraction. of methanol in the mixture increases. The value of the coefficient L212, however, undergoes a sharp rise when the liquid permeant is pure methanol, whereas the corresponding increase of c2D2/q3 is less sharp. Concerning the coefficient L211, we have found that it decreases monotonicaly as the methanol mole fraction decreases (see table 2), while the term C3D3/q43L does not vary with xM in a similar way. Thus eqn (24) does not explain the behaviour of L,,, with xM sufficiently well. In conclusion, we have found that the variation of the phenomenological coefficients with the composition of the permeant is mainly determined by the variation of the electrohydrodynamic parameters of the liquid mixture and the c potential of the solid/liquid interface.Although this conclusion should not be generalized to all porous media, it seems clear that those parameters must always be taken into account when explaining the concentration dependence of the first- and second-order phenomeno- logical coefficients. When ion-exchange membranes are used, permeant-membrane interactions should probably also be considered. R. C. Srivastava, M. G. Abraham and A. K. Jain, J . Phys. Chem., 1977,81, 906. K. Singh and J. Singh, Colloid Polym. Sci., 1977, 255, 379. R. L. Blokhra, M.L. Parmar and S. Chand, Indian J . Chem., 1982, 21A, 341. R. P. Rastogi, K. Singh and J. Singh, J . Phys. Chem., 1975, 79, 2574.R. HIDALGO-ALVAREZ, F. J. DE LAS NIEVES AND G. PARD0 619 R. C. Srivastava and M. G. Abraham, J. Chem. SOC., Faraday Trans. 1, 1977,81,906. R. L. Blokhra, M. L. Parmar and S. C. Chauhan, J. Membr. Sci., 1983,14, 67. R. L. Blokhra, S. K. Agarwal and N. Arora, J. Colloid Interface Sci., 1980, 73, 88. M. L. Srivastava and S. N. Lal, Colloid Polym. Sci., 1980, 258, 877. R. Hidalgo-Alvarez, F. Gonzalez-Caballero, J. M. Bruque and G. Pardo, J. Colloid Interface Sci., 198 1, 82, 45. lo F. Gonzalez-Caballero, R. Hidalgo-Alvarez, J. M. Bruque and G. Pardo, Physicochem. Hydrodyn., 1982, 3, 15. l1 R. Hidalgo-Alvarez, F. Gonzalez-Caballero, F. J. de las Nieves, J. Non-equilibrium Thermodyn., 1982, 7, 269. l2 I. Prigogine, Introduction to the Thermodynamics of Irreversible Processes (Wiley, New York, 1968). l 3 I. Gyarmati and J. Sandor, Colloid J. USSR, 1966, 18, 305. l4 V. Smoluchowski, Handbuch der Elektrizitat und des Magnetismus (Graetz, Leipzig, 1921), vol. 2. l6 R. P. Rastogi and M. L. Srivastava, Physica, 1961, 27, 265. J. C. M. Li, J . Chem. Phys., 1958, 29, 747. l7 J. C. M. Li, J. Chem. Phys., 1962, 37, 1592. J. C. M. Li, J. Appl. Phys., 1962, 33, 616. lB I. Gyarmati, Period. Politech., 1961, 5, 219. 2o I. Gyarmati, Period. Politech., 1961, 5, 321. 21 R. P. Rastogi and R. Shabd, J. Phys. Chem., 1977,81, 1953. 22 A. J. Rutgers, M. de Smet and G. de Myer, Trans. Faraday SOC., 1957, 53, 393. 23 A. A. Boumans, Physica, 1957, 23, 1007. 24 R. Hidalgo- Alvarez, F. Gonzalez-Caballero, J. M. Bruque and G. Pardo, J. Non-equilibrium 25 K. M. Jha, M. D. Zaharia and S. P. Jha, J. Indian Chem. SOC., 1976,58, 745. 26 Y. Gur and 1. Ravina, J. Colloid Interface Sci., 1977, 72, 272. Thermodyn., 1981, 6, 295. (PAPER 4/698)