Electrochemistry of capillary systems with narrow pores III. Electrical conductivity1

Journal of Membrane Science 150 (1998) 171±187

Electrochemistry of capillary systems with narrow pores III. Electrical conductivity$ Gerhard Schmid, Hans Schwarz Laboratory of Physical Chemistry and Electrochemistry, Technical University Stuttgart, Germany

Abstract The extension of the Teorell±Meyer±Sievers theory of the dialysis potential to a general theory of capillary systems with narrow pores outlined in Part I of this series of publications has been applied to electroosmotic phenomena in Part II. In this Part, the electrical conductivity, including the electrical convection conductivity, will be treated in terms of the new theory. The corresponding equations already referred to in Part I are derived. In addition, results of measurements of the electrical conductivity of collodion membranes with graded porosity and graded electrochemical activity in aqueous KCl solutions of different concentrations are reported. They are used to test the new theory. It will be shown that it is possible to determine the ®xed ion concentration A of the membranes by using electrical conductivity data. The theory predicts that the value of A should be identical with the `selectivity constant' of the Meyer±Sievers theory of the dialysis potential. This prediction will be checked in Part IV of this series of contributions. # 1998 Published by Elsevier Science B.V. All rights reserved.

1. Introduction It has been shown in Part I and Part II of this series [1,2] that the electroosmotic properties of capillary systems with narrow pores of radii smaller than about 0.05 mm can be better understood if the picture of an electrical double layer at the pore wall, introduced by Quincke, Helmholtz, and von Smoluchowski, is not used. That picture has been very successful in understanding the electrochemistry of capillary system with coarse pores. It is not applicable to capillary systems with narrow pores. The  potential of the double layer has to be replaced by ®xed charges at the wall and a homogeneous distribution of mobile counter ions in

$

Zur Elektrochemie feinporiger Kapillarsysteme, III. Elektrische LeitfaÈhigkeit, Ber. Bunsenges. Phys. Chemie (Z. Elektrochem.) 55 (1951) 295±307.

the pore ¯uid. While using this new picture we shall derive new equations for the electrokinetic phenomena. They have a structure similar to the Helmholtz± Smoluchowski equations. The equations contain the concentration A of the ®xed ions at the pore wall per unit volume of the pore ¯uid instead of the  potential. In the new model the systematic discrepancies between the predictions of the Helmholtz theory and the experimental observations disappear. Furthermore, a connection can be made between the electrokinetic phenomena and the electrochemistry of capillary systems with narrow pores. The ®xed ion concentration A is not an electrokinetic quantity like the  potential which could be used as a ®tting parameter when evaluating experimental data. It is a quantity which has been introduced by K.H. Meyer and J.-F. Sievers [3,4] in their theory of the dialysis potential. There it is called "selectivity constant".

0376-7388/98/$ ± see front matter # 1998 Published by Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00231-2

172

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

The ®xed ion concentration can be determined from measurements of the dialysis potential. Here, Part III of this series will be devoted to this subject. Results of our own measurements of the dialysis potential, made by using membranes with narrow pores, will be reported. The ®xed ion concentration can also be determined analytically by ion exchange experiments. In this communication we shall treat the electrical conductivity of a capillary system and narrow pores from the same point of view. We shall show that simple equations can be derived which describe the concentration dependence of the electrical conductivity. They contain the ®xed ion concentration A as the only unknown parameter. This gives another independent method for the determination of A. We also describe a procedure to calculate the conductivity of the pore ¯uid if the electrical conductivity of the bulk phase is known. These calculations are carried out as a function of the electrolyte concentration in the bulk phase. A knowledge of the electrical conductivity of the pore ¯uid is of interest for the analysis of electrokinetic phenomena. It is one of the parameters in the Helmholtz±Smoluchowski model as well as in the new model. In the past, it could not be measured directly and could not be calculated either. It was treated as a scapegoat for discrepancies between predictions of the Helmholtz-Smoluchowski model and the experiments. It will be seen that the agreement between the predictions of the equation for the electrical conductivity and the measured electrical conductivity of collodion membranes with narrow pores in KCl solutions is not complete. This can be ascribed to the simplifying assumptions on which the model is based. It is assumed that the ®xed ion concentration is constant and the activity coef®cient and electrical conductivity coef®cient of the mobile ions in the pore ¯uid and in the external phases have the value 1. It is preferable to accept differences between the simpli®ed theory and the experimental results rather than to introduce ®tting parameters which cannot be controlled. It will turn out that these differences warrant no basic objections for the proposed model. The model can be considered to be in agreement with the experiments. This gives us the possibility to use electrical conductivity measurements to obtain an approximate value of the ®xed ion concentration of the collodion membranes. In Part III and the forthcoming Part IV

of this series we shall compare the A-values of the same membranes with those obtained from measurements of the dialysis potential and the streaming potential. 2. Derivation of the equation of electrical conductivity As already stated in Part I [1] we shall assume that capillary systems with narrow pores are formed by a colloidal disperse matrix in an aqueous ¯uid. These systems can be treated as homogeneous electrolyte solutions. Such solutions differ from normal electrolyte solutions in one major respect: they contain immobilised ions which are ®xed to the matrix of the capillary system. T. Teorell [5] as well as K.H. Meyer and J.-F. Sievers [3,4] were the ®rst to recognise that the Donnan equilibrium distribution can be used to calculate the distribution of mobile ions between the pore ¯uid and the bulk phase in capillary systems with ®xed ionic groups. They used this approach for the theoretical treatment of the dialysis potential. We follow the arguments of K.H. Meyer and J.-F. Sievers but do not introduce activity coef®cients and electrical conductivity coef®cients. We use concentrations instead of activities and assume that the mobilities of the ions in the pore ¯uid and in the bulk phase have the same value and do not depend on concentration. We shall carry out our experiments under such conditions that the simplifying assumptions will not introduce too large an error; but some discrepancy between the predictions of the theory and the experimental results will be found. It is not possible to produce membranes with locally constant ®xed ion concentration. The pore size of the membranes used for the experiments is expected to have a more or less broad distribution around a mean value. Even if the surface concentration of the charges ®xed at the pore wall were constant the ®xed ion concentration per unit volume of the pore ¯uid would change with pore size. The assumption that the ®xed ion concentration is constant gives us the possibility to avoid the introduction of ®tting parameters and a clearer picture emerges. Eq. (1) describes the Donnan equilibrium at the membrane/bulk phase interface and Eq. (2) imposes the condition of electroneutrality within the pore ¯uid.

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

Both equations refer to the presence of a single 1±1 valent electrolyte in the bulk phase. cA cK ˆ c2

(1)

c K ˆ cA ‡ A

(2)

where cA and cK are the concentrations of the anion and cation species in the pore ¯uid. c is the electrolyte concentration in the bulk phase. Following Meyer and Sievers, the quantity A represents the concentration of the charges at the pore wall per unit volume of the pore ¯uid (Meyer±Sievers: "selectivity constant"). The value of A is de®ned in such a way that it has a positive sign when the charges at the pore wall are negative1. The dimensions of A are mol dmÿ3 pore ¯uid. Combination of Eqs. (1) and (2) leads to Eqs. (3) and (4) s  2 A A ÿ (3) cA ˆ c2 ‡ 2 2 s  2 A A 2 (4) ‡ cK ˆ c ‡ 2 2 The concentrations cA and cK within the pore ¯uid can be calculated if the value of the ®xed ion concentration A is known. Since we have assumed that the mobilities of the ions in the pore ¯uid have the same values as in the bulk phase it is possible to calculate the speci®c electrical conductivity i of the pore ¯uid. If u and v are the ionic mobilities of the cation and the anion, i is given by Eq. (5) s  2 A A ‡ …u ‡ v† i ˆ ucK ‡ vcA ˆ …u ‡ v† c2 ‡ 2 2 (5) The speci®c electrical conductivity  of the bulk phase is given by  ˆ …u ‡ v† ‡ c 1 For theoretical considerations it would have been more appropriate to define the fixed ion concentration of a matrix with negatively charged groups as A. This definition was used in Part II. In this contribution (Part II) we have followed the usual convention (fixed ion concentration of a matrix with negatively charged groups, ‡A). This basis of the usual convention is probably the fact that in nature there are more pore walls with negative charges than with positive charges.

173

In most cases we are interested in the ratio of the speci®c electrical conductivity i of the pore ¯uid and that of the bulk phase . This ratio is given by Eq. (6). s   i A uÿv A ‡ (6) ˆ 1‡ 2c u ‡ v 2c  This was given already in Part I [1] without deriving it. The electrical conductivity measurements were made with KCl solutions for which the difference (uÿv) can be set to zero to a good approximation. In this case of the second term in Eq. (6) vanishes. A generalisation and extension of the considerations to electrolytes with a higher valency leads to more complex equations which are of no interest at present. From Eq. (6) it can be seen that under the stated conditions the speci®c electrical conductivity i of the pore ¯uid is expected to be larger than that of the bulk phase. With increasing dilution of the bulk phases the ratio (i/) is expected to increase. In the limit c  A the ratio (i/) reaches large values. This explains qualitatively a phenomenon which E. Manegold and K. Solf [6] have described in detail already in 1931. But they could not give a conclusive explanation of their ®ndings. They reported that the relative increase of the electrical conductivity of the membrane at high dilution is not caused by impurities of the pore ¯uid or by a special pore geometry or by an electrical conductivity of the membrane matrix (collodion). They concluded "that the speci®c electrical conductivity of water bound to the capillaries is 10 to 20 times higher (depending on the porosity of the capillary system) than that of free water". It can be seen from Eq. (6) that in concentrated electrolyte solutions (i.e. c  A) the ratio (i/) reaches the value of 1 asymptotically. This corresponds to the observation that, for KCl concentrations above 3  10ÿ3 mol dmÿ3, the conductivity of the pore ¯uid increases proportionally to the conductivity of the solution with which the pore space is ®lled. Manegold and Solf interpreted this observation by stating that, at least in membranes with a water content larger than 80% and with coarse pores, `there is no necessity to assume any hindrance to the mobility of the solute in the water ®lled capillaries'. The observation for cA is that the ratio i/ reaches the value of 1 giving us the possibility of calculating the electrical conductivity capacity CL

174

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

[cmÿ1] of a membrane from the value of the electrical conductivity L of the membrane measured in concentrated KCl solutions. L ( ÿ1 cmÿ2] refers to a membrane area of 1 cm2. The conductivity capacity CL is given by Eq. (7) CL ˆ

L 

(7)

If the electrical conductivity capacity CL of a membrane in concentrated KCl solutions is known it is possible to determine the electrical conductivity of the pore ¯uid in dilute solution by using Eq. (8) i ˆ

L CL

(8)

L is the electrical conductivity of the membrane in the dilute solution. Combination of Eqs. (7) and (8) leads to Eq. (9)   (9) i ˆ L L conc The index conc indicates that the quantity has to be determined in concentrated electrolyte solutions (c  A). For convenience we introduce the form factor ' de®ned by Eq. (10). It is a dimensionless quantity smaller than 1. ' ˆ CL


d

(10)

Here d is the thickness of the membrane. The electrical conductivity capacity CL is reciprocally proportional to the thickness of the membrane and the form factor ' represents the conductivity of a membrane with the dimensions of 1  1  1 cm. The electrical conductivity capacity of an electrolyte solution of these dimensions in the absence of the membrane has the value 1. The form factor ' which is a quantity characteristic of each membrane is the factor by which the conductivity of a solution decreases due to only the displacement of a fraction of the solution by the non-conducting membrane matrix. It is independent of the composition of the electrolyte solution with which the pores of the capillary system are ®lled. The practical importance of the form factor is expressed by Eq. (11)   L (11) 'ˆ L0 conc

L0 is the electrical conductivity of a slice of the bulk phase having the same dimensions as the membrane. Eq. (11) is obtained from Eqs. (7) and (10) while taking into account the relations L0 ˆ (/d). In dilute electrolyte solutions (c < A) Eq. (11) does not hold. It has to be replaced by Eq. (12) which follows from Eqs. (10) and (8) on taking into account the relation L0 ˆ (/d). 'ˆ

L  L0  i

(12)

In the experimental section the resistance R of a membrane disc and the resistance R0 of a layer of the electrolyte solution displaced by this membrane will be measured. The ratios (R/R0) and (i/) and the form factor ' are related by Eq. (13).  is obtained in an independent experiment by measuring the ratio (R0/R) in concentrated electrolyte solutions (see Eq. (14); c  A)  i R0 1 ˆ  R'   R0 'ˆ R conc

(13) (14)

Eqs. (13) and (14) follow from Eqs. (12) and (11) by replacing L by (1/R). The right hand side of Eq. (13) (taking into account Eq. (14)) contains only measurable quantities. This opens up the possibility of determining the speci®c conductivity i of the pore ¯uid. We will use Eq. (13) to determine the ratio (i/) and thus to probe the validity of Eq. (6). Before beginning the experimental section we shall derive an equation for the so-called electrical surface conductivity in terms of the new theory. A more appropriate term for this phenomenon is "electrical convection conductivity" and is the one we shall use. With the equation of the convection conductivity at hand it will be possible at a later stage to see to what extent the convection conductivity contributes to the measured electrical conductivity. The term "electrical surface conductivity" has been introduced and de®ned by M. von Smoluchowski [7]. It is the electric current generated by the convection of the mobile charges in the electric double layer caused by an electroosmotic volume ¯ow. The electrical current i0 is the electrical convection current and i

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

175

is the (normal) electrical ionic current which passes the membrane and re¯ects the motion of the ions relative to the solvent molecules. Following von Smoluchowski, the ratio …i0 =i† is given by Eq. (15) which is valid only for capillary systems with coarse pores.   2 2 i0 1 "2 & 2 1 2" & 1 ˆ ˆ 2"0 (15) i 82 i  r i  r

while using Eq. (17) for the electroosmotic permeability Di. Eq. (18) is best suited for comparison with the corresponding equation given by von Smoluchowski. Our original expression for Di (see Part II [2], Eq. (20)) contains no assumption about the pore structure but only experimentally determinable quantities, except A.

{} added: conversion into SI units: replace (1/4) by "0 ; "0, permittivity of vacuum; "0 ˆ 8.854 A s Vÿ1 mÿ1.  is the viscosity of the pore ¯uid [Poise] (1 Poise ˆ 10ÿ7 J s cmÿ3); " is the dielectric constant of the medium in the electric double layer;  is the electrokinetic potential [V]; r is the pore radius and  the thickness of the electric double layer. In capillary systems with narrow pores the mobile excess charges are no longer concentrated at the pore wall but are distributed homogeneously over the cross section of the pores by the thermal motion of the mobile ions. Therefore it is misleading to use the term `surface' current in these systems. But the fact that the excess electric charge formed by the mobile counter ions is carried along by the electroosmotic volume ¯ow remains equally valid. The electrical convection current generated in this way can be calculated by using the new theory in the following way: The electric convection current i0 is given by the product of the volume transported electroosmotically per unit time (Di i) and the excess charge per unit volume (F0 A) (see Eq. (16)). Here F0 is the Faraday number and Di the electroosmotic water permeability per (A s).

Di ˆ

i0 ˆ Di0 F0 Ai

(16)

The electroosmotic permeability Di is given by Eq. (17) (see Eq. (23) in Part II [2] 1 F0 A 2 r (17) Di ˆ 8 i Combination of Eqs. (16) and (17) leads to the ratio of the electrical convection current i0 and the electrical ionic current i given by Eq. (18) i0 1 F02 A2 2 ˆ r i 8 i

(18)

Eq. (18) has been given already in Part I [1] of this series and has been compared with the corresponding equation given by M.v. Smoluchowski. It contains the hypothetical pore radius which has been introduced

F0 A DH L

(19)

DH is the water permeability (mechanical permeability). Combination of Eqs. (16) and (19) leads to an equation for the ratio …i0 =i† which no longer contains the pore radius (see Eq. (20)) i0 F02 A2 ˆ DH i L

(20)

Eq. (20) is transformed into Eq. (21) on taking into account Eq. (8). i0 F02 A2 ˆ DH i i C L

(21)

This equation can be used to calculate the maximum contribution of the electrical convection conductivity to the measured electrical conductivity. This contribution is expected to be noticeable when the membrane is in contact with highly dilute electrolyte solutions. We have shown in Part I [1] i cannot become smaller than the product …uA† even in the most dilute electrolyte solutions. This follows from Eq. (6) in the limit c ! 0. The quantity u is the cation mobility in the pore ¯uid. In this limit the relation Eq. (22) holds i0 F02 A < DH : i uCL

(22)

It will be shown that the maximum contribution of the electrical convection current is a few percent under the conditions of our experiments and can therefore be neglected in most cases. Exceptions will be discussed separately. 3. Materials and membranes 3.1. Electrolyte solutions All experiments have been carried out with dilute aqueous KCl solutions (Merck, pro analysis). This

176

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

electrolyte was chosen because the K‡ and Clÿ ions have the same ionic mobility. This property simpli®ed the evaluation of the experimental results. The solutions were prepared in distilled water. Preliminary experiments had shown that the use of doubly distilled water was not necessary. The exclusion of air was not necessary for the experiments. Solutions of various concentrations were prepared by dilution of a stock solution. The ®nal determination of the concentration was carried out before or after each experiment by electrical conductivity measurements. The electrical conductivity of the water had values in the range 2±5 m ÿ1. 3.2. Membrane preparation Most of the membranes used in this study were of collodion (cellulose nitrate) prepared by following recipes given by N. Bjerrum, E. Manegold and R. Hoffmann [8,9]. A few membranes were commercial products purchased from Membran®lter-Gesellshaft, GoÈttingen. They had larger pores (UF coarse; UF middle). The collodion membranes were prepared in a closed container with the shape of a desiccator by pouring a suitable solution (see below) on to mercury. The gaseous atmosphere in the ¯ask (dry air) was mechanically stirred. A sieve plate positioned horizontally above the mercury surface was used to generate a controlled stream of air at the surface of the mercury. The parameter controlling the pore size of the membranes was the drying time in the container. Completely dried (`air dried') membranes, with a water content of about 10%, were not used. It was not possible to determine reliably the low water permeability DH of such membranes with very narrow pores. The membranes used for the experiments were transferred into water before they reached the `air dried' state. They had water contents in the range between 50% and 95%. The material from which each membrane was cut had a diameter of 10 cm. A device, like a pair of compasses with a razor blade attached to it, was used to cut from the material a membrane disk with a diameter of 75 to 85 mm. The rim of the material was used for the determination of the water content. The collodion from which the membranes were prepared was purchased from Dynamit AG, Troisdorf.

It was used without further puri®cation. The material was used to prepare two types of solutions: solution 1: 14 g collodion wool (air dried) 168 g ethyl alcohol (99%) 168 g ethyl ether (dried, free of peroxide) solution 2: 14 g collodion wool (air dried) 276 g ethyl alcohol (99%) 107 g ethyl ether (dried, free of peroxide) The ether had to be free of peroxide to prevent reactions with the mercury during the preparation of the membrane from producing impurities. Solution 1 was used to prepare the membranes with narrow pores labelled IV and IX. Solution 2 was used to prepare membranes labelled V, XXI, XXII, XXIII, XVI, XVII and XXIX. For comparison with the properties of membranes with large pores, membranes of that type, with a matrix similar to that used to prepare the narrow-pore membranes, were prepared. To do this, the method of casting described above was repeated with solution 3 made according to a recipe given by W.J. Elford [10,11]. To compare the properties of membranes with large pores and a matrix similar to that with narrow pores that type of membrane was produced by a similar method with solution 3 according a to recipe given by W.J. Elford [10,11]. Solution 3 : 15 g collodion wool (air dried) 25 g ethyl alcohol (99%) 115 g acetone (dried) 75 g ethyl ether (dried, free of peroxide 57 g amyl alcohol This solution was diluted with a mixture consisting of alcohol (mass fraction 0.1) and dried ethyl ether (mass fraction 0.9) in the volume ratio 1 : 1. For the preparation of the membranes on mercury a volume of 30 cm3 was used (instead of 15 cm3 of solutions 1 and 2) because solution 3 was more dilute. The evaporation period was correspondingly longer. After about 1 1 2 h the transparent solution suddenly became turbid and a pattern became visible. After a drying time of 12 h (volume ¯ow of air 500 cm2 minÿ1) the membrane was transferred into water. It was kept there for 5

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

days. The water was exchanged repeatedly until the smell of amyl alcohol had disappeared. The membrane produced in this way had an opaque appearance. The tensile strength of the product was comparable to that of commercial membranes with coarse pores. We shall refer to only one type of membrane with coarse pores (marked D). All membranes were stored in distilled water in the presence of a copper foil. A further treatment of the membranes (partial oxidation of the membrane matrix) will be described later. 4. Characterisation of the membranes The membranes were characterised by water content, by thickness and by water permeability (mechanical permeability). The methods of determining these quantities were the same as those described by N. Bjerrum, E. Manegold and coworkers. The form factor ' was an additional quantity characterising the membranes. The water content was determined by using pieces of the membrane material left over after cutting out the membrane discs. The sample was blotted with ®lter paper to remove adherent water. The membrane was weighed (wet weight), dried over concentrated sulfuric acid in a desiccator and weighed again (dry weight). The difference gave the mass of water and was used to calculate the volume fraction of voids in the wet membrane by using a value for the density of dry collodion of 1.673 g cmÿ3 (taken from Michaelis). The thickness of the membrane disk was determined with a micrometer gauge which could be read to 1 mm. Attempts to determine the membrane thickness from the volume and the area of the membrane failed. The thickness of the membrane ¯uctuated locally by up to 5 mm and in extreme cases to 15 mm. The average values listed were used for the calculations. Example : Membrane XXVII weight wet weight dry water content in g volume of the dry matrix

0.6286 g 0.0920 g 0.5366 g 0.0920/1.673ˆ0.055 cm3

volume of water total volume water content in vol.%

177

0.5366/0.9985 ˆ 0.5374 cm4 0.5924 cm3 (0.7374/0.5924)  100 ˆ90.7%

The water permeability (mechanical permeability) represents the volume of water, measured in cm3, passing per cm3 of membrane per second under the hydrostatic pressure difference of a water column 1 cm in height. Details of the apparatus used in this determination will be given in a later contribution dealing with streaming potentials. The effective area of the membrane was 4.53 cm2. The hydrostatic pressure differences had values in the ranges up to 50 cm water column and up to 64 cm Hg. The water used for the experiments was puri®ed by passing it through a membrane ®lter. The pore size can be calculated by using the following data: water content W in volume%, thickness d and water permeability DH if strongly simplifying assumptions about the membrane structure are made. On assuming an `unordered pore structure', N. Bjerrum and E. Manegold [8,9] calculated the pore radius from Eq. (23). When an `unordered' slit structure was assumed the half width of the slit was calculated from Eq. (23a). r 24DH d (23) rˆ W r 4:5DH d (23a) ˆ W  is the viscosity of the liquid passing the membrane. Finally we have determined the form factor ' to characterise the structure of the membrane. The quantity ' is the ratio of the electrical conductivity of a membrane disk with its pore structure ®lled with a concentrated KCl solution and the electrical conductivity of a disk with the same dimensions formed by KCl solution of the same concentration and in the absence of the membrane. The form factor represents the extent to which the electrical conductivity of a layer of a concentrated electrolyte solution with given dimensions decreases when this layer is replaced by a membrane with the same dimensions and containing the same electrolyte solution. The quantities used for the characterisation of the membranes are complied in Table 1.

178

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

Table 1 Characterisation of the membranes. r is the pore radius and û the half slit width. W is the water content. D is the thickness of the membrane and DH is the water permeability (mechanical permeability) at zero electrical current. ' is the form factor Membrane

W (vol.%)

d (mm)

DH107 cm ((cm WS)ÿ1 Sÿ1)

r (mm)

D M7 UF Coarse XXI XXII XXVII iV iX UF Middle XXVI V XXIII XXIX

85 78 82 95 95 91 ± ± 83 86 90 82 46

295 104 133 307 282 246 249 275 139 179 297 140 51

4000 3700 86 5.0 5.2 5.7 4.8 4.8 6.5 3.6 1.7 2.8 0.53

580 350 59 20 19 19 18 18 16 14 12 11 3.8

5. Measurements of the electrical conductivity of the membranes and calculation of the electrical conductivity of the pore fluid 5.1. Method The apparatus used for the measurements of the electrical conductivity of the collodion membranes differs from that normally used for this purpose. Instead of clamping the membranes directly between two closely ®tting platinum electrodes and measuring the conductance directly we prefer to leave a wellde®ned space between the membrane and electrodes which is ®lled with the equilibrium electrolyte solution. However we have to accept the disadvantage of obtaining the electrical conductivity of the membrane as the difference between two conductance values. This led to a larger uncertainty in the data especially when the resistance of the electrolyte solution with which the membrane was equilibrated, as well as the resistance of the membrane, was low. But this method had advantages which more than compensated for its disadvantages. The layer of electrolyte surrounding the membrane was stirred; and could be exchanged during a series of measurements. The temperature control was more precise and the in¯uence of concentration changes and fouling in the small volume between the electrodes could be avoided. It was possible to maintain the distribution equilibrium between the membrane phase and the electrolyte over

û (mm) 250 150 25 8.6 8.4 8.4 8 8 7.1 5.9 5.1 4.7 1.6

' (%) 66 50 24 77 78 74 78 74 55 64 67 58 18.5

long periods of time. In summary, we could obtain very reproducible data with this method. 5.2. Conductivity cell The conductivity cell consisted of two Plexi glass (see Fig. 1) which could be screwed together by two silver plated brass screws located at two projections A and B shown in the Fig. 1. Except for the places at which the screws were located, there existed a slit between the plates with a height of 0.4 mm. A circular brass electrode (thickness, 1 mm; diameter 25 mm) with electric connections soldered to it was melted

Fig. 1. Electrical conductivity cell.

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

Fig. 2. Experimental set up to measure the electrical conductivity.

into each plate. The upper surfaces of the electrodes were positioned about 0.5 mm below the upper surface of the slit. The effective diameter of the brass electrodes had a ®nal value of 19.6 mm. The surfaces of the electrodes were plated galvanically with silver and platinum and were covered by a thin layer of platinum black. After screwing together the two Plexi glass plates the two platinum plated electrodes with an area of 3.02 cm2 were separated by a distance of about 1.5 mm. The membrane was introduced between the electrodes via the slit. The third double projection C located at the lower plate (see Fig. 1) together with an enlargement at the upper plate simpli®ed the introduction under water of the membrane into the slit. Fig. 2 shows the construction of the entire experimental set up. The conductivity cell was fastened eccentrically to the axis of a stirring motor using the two cell holders Z1 and Z2 (see Fig. 1) which carried the electrical current connections. The holders consisted of copper rods isolated by paraf®n wax. They were connected to a Wheatstone bridge via the sliding contacts S1 and S2 with the electrodes (see Fig. 2). The sliding contacts consisted of two rings made of brass and two pieces of brass sheet in the form of combs. The areas of contact with the slide rings were covered with silver. They were slightly greased with oil after we convinced ourselves that this generated no electrical contact problems. The electrical resistance of the connections of the conductivity

179

cell with the Wheatstone bridge, including the sliding contacts, had a value of 0.03 measured with direct current. The lubrication with oil lowered this value of the resistance. The conductivity cell with its holder was immersed into a beaker with a volume of 2 dm3. It contained the electrolyte solution in which the measurements were carried out and was in contact with thermostated water ((temperature T ˆ (20  0.1)8C). After setting the stirring motor into motion the conductivity cell was a very effective stirrer. The centrifugal force acting on the solution within the slit of the conductivity cell ensured an immediate exchange of the liquid between the conductivity cell and the external solution. The rate of exchange could be in¯uenced by changing the rate of rotation of the stirring motor. Air bubbles in the corners of the conductivity cell were removed easily by increasing the rate of rotation of the cell for a short period of time. Located in the same beaker was a small basket K (see Fig. 2) made of glass rods. It contained the membranes to be studied. They were cut from a larger membrane disc to a shape shown in Fig. 3. They were equipped with a small silver ring. This allowed them to be moved around by using a glass hook G. With the aid of two such hooks it was possible to move a membrane within the solution from the basket into the slit of the conductivity cell and back. Hydrodynamic forces generated by the rotation of the conductivity cell moved the membrane within the cell towards the ¯anges A and B. They prevented the membrane from being swept out of the slit. 5.3. Conductivity bridge The electrical conductivity was measured with a Wheatstone bridge. Information taken from the publications of W. Grallert [12] and F. Ender [13] were

Fig. 3. Membrane with attached silver wire.

180

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

symmetric to ground. It was not necessary to take into account that the balance of the bridge changed slightly on reversing the polarity of the electrical supply to the conductivity cell. The reliability and the symmetry of the bridge was checked with resistors of known resistance while using direct and alternating current. The uncertainties were smaller than 1  10ÿ3. The slide contact on the bridge wire could be positioned within an uncertainty of a fraction of a millimeter. An uncertainty of 1 mm in the middle of the length of the wire (length 8923 mm) corresponded to a change of the resistance value of only 0.45  10ÿ3 . 5.4. Calibration of the conductivity cell

Fig. 4. Electrical circuit to measure the electrical conductivity.

helpful in putting the circuit together. It is shown schematically in Fig. 4. The bridge consisted of a wound steel wire with a resistance of 200 . A plug box resistance of 11 111

in total was used as a comparator. An adjustable capacitor with a capacity up to 2.2 mF could be connected in parallel with each branch of the bridge. Special attention was paid to the symmetry of the bridge and its grounding. The sliding contact of the wound resistor was positioned in the middle of the length of steel wire. The middle of the bridge was maintained indirectly at ground potential by using a K.W. Wagner [14,15] circuit. It consisted of a secondary bridge in parallel to the primary bridge. The secondary bridge was formed by two resistors of 150 each in combination with a potentiometer of 20 connected to ground (see Fig. 4). The thermostat was also grounded. It was not necessary to use capacitances to balance the secondary bridge. A frequency generator with a variable frequency output was used as an alternating current source. It was ®xed at a frequency of 2.1 kHz and generated about 16 V at a load of 100 . It was not completely free of overtones. The overtones at high frequencies did not disturb the measurements and it was not necessary to ®lter them out. The electric current supplied to the earphones was ampli®ed. The transformers at the entrance and exit of the bridge were not

The form factor of the conductivity cell was calibrated by using KCl solutions of eight different concentrations. The form factor had a relatively small of value of 0.0511 cmÿ1. The resistance of the most concentrated KCl solution was so low that the resistance of the connecting wires (0.03 ) had to be taken into account. The form factor was strictly concentration independent and had a constant value during the entire series of measurements. The capacitance necessary to balance the conductivity bridge corresponded to a blind resistance which was smaller by a factor of 0.002 than the ohm resistance at a frequency of 2.1 kHz. Despite the very effective temperature control of the conductivity cell the voltage applied to the conductivity bridge was set at such a value that the Joule heat generated in the conductivity cell did not exceed 0.01 W. The electrodes had an area of 3.02 cm2. Therefore it was possible to determine the effective distance between them by using the value of the form factor. It had to be taken into account that the electrical current lines were distorted by the slit in the conductivity cell. To control this in¯uence a ring (internal diameter 19.6 mm; thickness, 0.4 mm) was cut out of a foil of a non-conducting polymer so as to ®ll the slit. A few grooves cut into the foil made exchange of the electrolyte solution between the electrodes and the external solution still possible during stirring. The resistance of a 0.01 N KCl solution had a value of 57.38 in the presence of the polymer ring and a value of 57.14 in its absence. The disturbance of the electric current lines by the slit corresponded to an effective increase of the cross section of the electrodes

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

of 0.4%. This was within the range of uncertainty of the measurement of the areas of the electrodes. Nevertheless the increase of the effective cross section was taken into account. The corrected value of the electrode area (3.03 cm2) was used to calculate the distance between the electrodes on the basis of the known form factor. A value of 0.155 cm was obtained which did not change during the time in which all the experiments were carried out. 5.5. Measurement of the electric conductivity of the membrane The measurements were carried out with the conductivity cell rotating at a frequency of 150 minÿ1. At ®rst, the resistance was measured with the membrane present in the conductivity cell. Then the membrane was removed with the glass hook and the resistance was measured again. The time elapsing between the two measurements was less than 30 s. Originally the beaker was ®lled with distilled water. Known masses of KCl or known volumes of KCl solution of known composition were added to the water to change the electrolyte concentration. It was determined from the resistance value Rg measured at the end of the experiment at each electrolyte concentration. The establishment of the equilibrium distribution of the electrolyte between the membrane phase and the bulk phase after a change of the electrolyte solution in the bulk phase could be followed. It took less than 10 min under intensive stirring. To check the internal consistency of the data, the membrane was transferred from the concentrated electrolyte solution at the end of a run back into distilled water and the conductivity data were measured again with increasing electrolyte solutions in several runs. The agreement between the two data sets was always very satisfactory with the exception of the runs with "oxidised membranes". These membranes apparently lost some charges from the matrix during contact with the KCl solutions. Similar observations were reported by Sollner [16] on the basis of measurements of the dialysis potentials of oxidised collodion membranes.
R is the difference between the electrical resistance in the presence and the absence of the membrane in the slit of the conductivity cell (see Eq. (24)).
R ˆ R ÿ R0

(24)

181

R is the resistance of a given membrane and R0 the resistance of the layer of electrolyte solution displayed by the membrane. R0 can be calculated from the measured resistance Rg of the cell ®lled with the electrolyte solution in the absence of the membrane, the distance a of the electrodes (a ˆ 0.155 cm) and the thickness d of a layer of the solution which had the same thickness d as that of a given membrane. R 0 ˆ Rg

d a

(25)

The resistance of the membrane is then given by: d R ˆ Rg ‡
R a

(26)

5.6. Calculations of the specific conductivity of the pore fluid To calculate the electrical conductivity of the pore ¯uid i from the resistance R of the membrane, the conductivity capacity of the membrane CL or better the form factor ' ˆ CL d has to be known. According to Eq. (14) (i.e. ' ˆ …R0 =R†conc ) the form factor can be calculated from resistance measurements in concentrated electrolyte solutions. In the experiments, the concentration of the KCl solution was increased stepwise until the ratio (R0/R) remained constant. That value was used to calculate the form factor '. The ratio of the speci®c electrical conductivity of the pore ¯uid i and the corresponding value  for the bulk phase in equilibrium with the membrane phase in given by Eq. (13).  i R0 1 ˆ  R'

…13†

5.7. Discussion of errors The resistance values and the values of the difference
R are known accurately, with an uncertainly of less than 1%. The quantity determining the uncertainty of the data is the thickness of the membrane. It can be measured within an uncertainty of about 2% and for very thin membrane with an uncertainty of 5%. The form factor ' is accurately known. The resistance R and the electrical conductivity ratio (i/) are known with a relative uncertainty of

182

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

a few percent as long as the electrical conductivity of the membrane is lower than, equal to or only slightly larger than the electrical conductivity of the layer of solution in the absence of the membrane with the same thickness as the membrane. The disadvantage of the different methods used in the study shows up when the membrane has a higher electrical conductivity than that of the external solution. In this case the small value of the membrane resistance is in series with the relatively larger resistance of the layer of external solution between the membrane and the electrodes. In this case
R is negative. The absolute value of
R can reach values comparable to the value of the term Rg …d=a† in Eq. (26). The absolute error of R is not in¯uenced but its relative error increases proportionally to …R0 =R†. In our most extreme case …R0 =R† had a value of 14 (corresponding to …i † ˆ 18). In such cases we have to expect an increase in the uncertainty of R and (i/) by a factor of 10. There are only few cases with very large values of (i/) for which the situation is so unfavourable. 5.8. Presentation of experimental data The total set of electrical conductivity data with four membranes of series IX is compiled in Table 2.

The membranes, each with an effective membrane area of 3.03 cm2 were cut from the same sheet. The membrane thickness was determined with a micrometer. 6. Results The experimental …i =† values for each membrane studied are plotted as a function of the negative logarithm of the external electrolyte concentration (see Fig. 5). Each curve is marked by the number of the membrane to which it refers. The data on membrane IX are not shown. They are given in Table 2 and are very similar to that of membrane XXVII. The data of membrane IV are almost identical with those of membrane XXVII. There is a strong increase in the ratio …i =† with increasing dilution of the bulk phase. This is to be expected on the basis of Eq. (6). It is found that the onset of the increase in the ratio …i =† with decreasing electrolyte concentration is shifted to lower dilution (i.e. higher electrolyte concentrations) the higher the ®xed ion concentration A of the membrane. This is also to be expected on the basis of Eq. (6). The chemical composition of the matrix of all membranes

Table 2 Membrane IX1: d ˆ 278 mm; membrane IX2: d ˆ 270 mm; membrane IX3: d ˆ 274 mm; membrane IX4: d ˆ 276 mm d, membrane thickness ÿlog c

Membrane N

Rg ( )


R ( )

R0 ( )

R ( )

i/

'

3.675

IX1 IX2 IX3 IX4 IX1 IX2 IX3 IX4 IX1 IX2 IX3 IX4 IX1 IX2 IX3 IX4 IX1 IX2 IX3 IX4

1799 1801 1797 1793 982.8 985.4 985.2 983.0 234.1 234.2 234.1 234.2 20.64 20.63 20.64 20.63 10.980 10.975 10.975 10.980

ÿ179 ÿ77.5 ÿ177 ÿ176 ÿ59.1 ÿ59.4 ÿ60.0 ÿ58.5 ‡8.4 ‡8.1 ‡8.5 ‡8.4 ‡1.29 ‡1.25 ‡1.32 ‡1.30 0.667 0.667 0.688 0.678

323 314 318 319 176 172 174 175 42.0 40.8 41.4 41.7 3.70 3.60 3.65 3.67 1.97 1.91 1.94 1.96

144 136 139 143 117 113.2 114 117 50.4 48.9 49.9 50.1 4.99 4.85 4.97 4.97 2.64 2.58 2.63 2.64

3.1 3.1 3.1 3.0 2.0 2.1 2.0 2.0 1.1 1.1 1.1 1.1 1 1 1 1 1 1 1 1

0.742 0.743 0.734 0.737 0.748 0.739 0.739 0.743

3.41

2.79

1.7

1.4

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

183

Fig. 5. The ratio (i/) of the electrical conductivity of the pore fluid i and the electrical conductivity of the bulk phase  as function of the negative value of the logarithm of the molar volume concentration c of KCl (ÿlog c). Curve notation: membrane. Curve 1: calculated reference curve for a membrane with a fixed ion concentration A ˆ 1 mol dmÿ3 pore fluid.

used in this study was the same. We can assume that the surface concentration of the charges on the pore wall had about the same value in each membrane. This has the consequence that the ®xed ion concentration A per unit volume of the pore ¯uid should increase with decreasing pore size. Indeed, the sequence of the …i =† versus ÿlog c curves for the different membranes in Fig. 5 corresponds to the sequence of the 1/r values of these membranes (see Table 1). The curves for the membranes with the index `ox' should not be taken into account in this context.

Recent dialysis potential measurements of K. Sollner and coworkers [17±21] have shown that the ®xed ion concentration of collodion membranes increases strongly on treating them with solutions of oxidising substances like NaOBr, NaOCl and also with solutions of NaOH. The authors interpreted their ®ndings as evidence for the formation of more COOH groups on the cellulose acetate matrix of the membranes. If the increase in the electrical conductivity of the pore ¯uid is the result of an increase of the ®xed ion concentration, as predicted by Eq. (6), then the treatment of our

184

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

membranes with oxidising agents should lead to the conductivity curves starting to fan out at lower dilutions (at higher electrolyte concentrations). To check this prediction some membranes were treated with 1 N NaOH at room temperature. For example, four membranes of type IX (i.e. IX1 to IX4) cut from the same membrane sheet were treated for 10, 20, 35 and 40 min with NaOH. Thereafter they were washed with water for 4 days. The thicknesses of the membranes did not change and the form factor increased slightly, by 4%, but the membranes became more brittle. Membrane IX3 broke. The conductivity curves could be measured for the three remaining membranes. The results are shown in Fig. 5. The curves are marked by the index `ox' ox IX1 ; ox IX2 ; ox IX4 ). The increase in the electrochemical activity of the membrane matrix due to the chemical treatment, as observed by K. Sollner and coworkers2 from measurements of the membrane potential, is re¯ected by a strong increase in the electrical conductivity of the pore ¯uid. The curves obtained with the oxidised membranes lie well above the curves for each membrane before oxidation of the matrix. The electric charges on the pore wall generated by the chemical treatment are caused by low molar mass decomposition products of the matrix. They can be removed partly by an extended treatment with water and by the ®ltration of aqueous KCl solutions through the treated membrane. This also has been observed by Sollner and coworkers. The curve ox X1 0 represents data obtained with the membrane ox IX1 after washing the oxidised membrane with 2  10ÿ4 N KCl solution. After this treatment the ®xed ion concentration of the membrane 0 ox IX1 is still considerably larger than before the oxidation of the matrix (compare with the curve of membrane IX1). Membranes XXI, XXII, XXIII and XXVI have been treated in a similar manner: 10 min in contact with 1 N NaOH solution, washed with water and treated with 1 N KCl solution to remove the easily extractable reaction products of the oxidation processes. In each case an increase in the conductivity of the pore ¯uid 2

Sollner interprets the reaction of the membrane matrix with NaOH as an oxidation. In this process NaOH reacts with nitrocellulose (collodion) generating the oxidising reagent HNO2. Following Sollner, we use the index `ox' to distinguish the treated membranes from the untreated membranes.

was detected. The electrical conductivity of the pore ¯uid in the oxidised membrane was larger by a factor of 2.5 to 3 than in the non-oxidised membrane. The …i =† values of these membranes (subscript `ox') are also shown in Fig. 5. The sign (0 ) after the roman number characterising the membrane indicates that it has been used also for electrical conductivity measurements before the chemical treatment. Membrane ox XXIX is the membrane with the narrowest pores in this series. It broke before a complete series of electrical conductivity measurements could be carried out. The ratio …i =† had a value of …i =† ˆ 141 for a KCl concentration of 2  10ÿ4 N and a value of …i =† ˆ 100 for a concentration of KCl of 1  10ÿ4 N. Such high values of …i =† cannot be determined very accurately with our difference method. 7. Evaluation of the experimental results The characteristics of the curves shown in Fig. 5 are very similar except for the curves for the commercial membranes UF coarse and M700 . We shall exclude these two data sets from further consideration. Commercial membranes with narrow pores are often composite membranes composed of a thin membrane with very narrow pores and a thicker layer with coarse pores. The …i =† versus ÿlog c curve marked as "reference curve for A ˆ 1 N00 in Fig. 5 represents Eq. (6) plotted for a membrane with a ®xed ion concentration of A ˆ 1 N in contact with a KCl solution (equal mobility of the cation and the anion in the pore ¯uid, u ˆ v). Eq. (6) contains the quantities A and c as the ratio …A=c†. Consequently the …i =† versus ÿlog c curves for membranes with values of A smaller than A ˆ 1 N are shifted parallel and to the right on the log c axis. The ®xed ion concentration A of such a membrane is given by the shift on the log c axis between the reference curve with A ˆ 1 N and the curve referring to the given membrane. The curves in Fig. 5 show that the experimental data are in qualitative agreement with the theoretical predictions based on Eq. (6). However a closer analysis shows that for a given experimental curve the distance on the ÿlog c axis between the calculated reference curve and the experimental curve increases

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

185

Fig. 6. Different plot of the data shown in Fig. 5: log‰…i =†2 ÿ 1Š versus ÿlog c. Curve notation: membrane number. Curve 1: Calculated reference curve for a membrane with a fixed ion concentration A ˆ 1 mol dmÿ3 pore fluid.

with increasing dilution (i.e. decreasing electrolyte concentration). To bring out more clearly the extent of the agreement and the deviations between the theory and experiment, the reference data and the experimental data are plotted in a different way: a plot of log[…i =†2 ÿ 1] versus ÿlog c shown in Fig. 6. Straight lines with a slope of 2 are expected. Actually straight lines are found but their slopes have values in the range of 1.5 to 1.8 instead of 2. An exception is provided by the data on membrane D which has the largest pores. The deviations of the slopes of the experimental curves from the theoretical value can be attributed to the fact that the ®xed ion concentration of a given membrane is not a constant. It is not possible to quantify this interpretation; but we will show by a simple calculation that changes in A caused by a distribution of pore sizes have the effect of lowering the value of the slope of the log[…i =†2 ÿ 1] versus ÿlog c curve. For this purpose we consider two membranes of the same material and having the same form factor and thickness but different values of the ®xed ion concentration A (e.g. A ˆ 1  10ÿ3 and A ˆ 4  10ÿ4 N). The two membranes form two types of composites: a mosaic of small patches of each membranes and the two membranes placed in series. Now we calculate the

ratio …i =† for each type of composite membrane. First we consider each membrane separately. In the next step we consider the two types of composites (two electrical conductivities in parallel, and two electrical conductivities in series). Since we assume that the form factor ' of each membrane has the same value it is not necessary to use the conductivity L of each membrane for the calculations but only the ratio …i =†. The ratio (i/)parallel of the composite membrane is given by the mean value of the ratio of the two types of membrane forming the mosaic composite. The ratio (i/)series of the composite with the two membranes in series is given by the mean value of the reciprocals of the ratio (i/) of the two membranes. The calculated sets [(i/)parallel1, ÿlog c] and [(i/)series, ÿlog c] are plotted in a log[(i/)2ÿ1] versus log c diagram. For both types of composite membrane approximately straight lines are obtained but with slopes different from 2. For the mosaic composite the slope has the value 1.85 and for the layered composite the value is 1.68. The dashed line in Fig. 6 represents a curve with slope of 1.68. It may be concluded that a distribution of pore sizes and distribution in the local ®xed ion concentration of a membrane leads to a decrease of the slope of the log[(i/)2ÿ1] versus log c curve independently of the structure of the composite membrane.

186

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

Now we shall determine the effective values of the ®xed ion concentration A of the membranes by using the experimental electrical conductivity data. As already mentioned, the quantity log A is the distance between the calculated reference curve for a ®xed ion concentration A ˆ 1 N and the experimental curve measured along the log c axis in a (i/) versus log c plot. This distance is concentration dependent because the slope of the curves is smaller than theoretically expected. A appears to increase with increasing dilution. To obtain an effective value of A for each membrane, the log A value is determined for the membrane at the same electrolyte concentration, namely c ˆ 2  10ÿ4 mol dmÿ3 (i.e. ÿlog c ˆ 3.7). This concentration has been selected because the streaming potentials with a KCl solution have been measured at that concentration. Those data will be reported in part V of this series of publications. Values of A determined in this way are complied in Table 3. It can be seen that the effective value of A for the non-oxidised membranes increase with decreasing values of the pore radius, as expected. The A values of the oxidised membranes are larger than those of the non-oxidised membranes. The increase in A depends Table 3 Fixed ion concentration A for different membranes in 2  10ÿ4 N KCl Membrane N

Pore radius (mm)

A  103 (mol d-3)

…i0 =i†max

D M7 UF coarse XXI1 XXII1 XXVII1 IX UF medium XXVI1 V2 XXIII1 XXIX OX XXI2 OX XXII1 OX XXVI1 OX XXIII2 OX IX1 OX IX1 OX IX2 OX IX4

580 350 58 20 19 19 18 16 14 12 11 3.8 20 19 14 11 18 18 18 18

(0.33) (0.8) (0.9) 0.95 1.1 1.3 1.3 1.5 1.7 2.3 2.1 4.0 2.8 3.0 4.0 5.2 6.5 5.5 7.2 8.3

± ± ± 3 3 4 3 4 2 2 2 1 8 8 6 5 16 14 18 20

on the extent of the oxidation. No value of A can be given for the ®rst three membranes, D M7 and UF coarse, because they do not meet the criterion of a membrane with narrow pores. The values given in brackets have been calculated in the same way as those for the membranes meeting the requirement of a membrane with narrow pores. The values of A given in Table 3 will be used again in the forthcoming Parts IV and V of this series to calculate the dialysis potentials and the streaming potentials of the same membranes. We shall now turn to the question as to what extent the electrical convection conductivity, which we have neglected so far, might in¯uence the evaluation of the electrical conductivity data. For this purpose we calculate the maximum value of the ratio …i0 =i† by using Eq. (22). The values of CL of the membranes are calculated with Eq. (10) from the known values of ' and d. They are given together with the values of DH in Table 1. The values of A are taken from Table 3 and for u a value of u ˆ 66 ÿ1 cmÿ2 is used. The calculated maximum values of …i0 =i† are given in the fourth column of Table 3. It can be seen that with the nonoxidised membrane the value of …i0 =i† is within the limits of uncertainty of the measurements. With the oxidised membranes the contribution of the convection conductivity is larger and can reach values up to about 20%. These are maximum values which will be reached only in very dilute solutions. This is the range of electrolyte concentrations in which the ratio (i/) reaches large values. We have mentioned that the uncertainty in the (i/) values increases with increasing dilution. This is caused by our method of determining (i/). We conclude that the contribution of the electrical convection conductivity to the measured electrical conductivities of the membranes studied is still small enough to be considered negligible. Acknowledgements This study has been carried out in 1947/1948 in the temporary Laboratory of the Technische Hochschule Stuttgart in SchwaÈbisch-GmuÈnd. We thank the Director of the Research Institute of Nobel Metals in SchwaÈbisch-GmuÈnd, Professor Raub, for this support. We thank the Dynamit-Nobel Company for supplying us with collodion wool.

G. Schmid, H. Schwarz / Journal of Membrane Science 150 (1998) 171±187

References [1] G. Schmid, Ber. Bunsenges, Physik. Chem. Z. Elecktrochem (Part I) 54 (1950) 424. [2] G. Schmid, Ber Bunsenges, Physik. Chem. Z. Elecktrochem (Part II) 55 (1951) 229. [3] K.H. Meyer, J.-F. Sievers, Helv. Chim. Acta 19 (1936) 649, 665, 987. [4] K.H. Meyer, J.-F. Sievers, Helv. Chim. Acta 20 (1937) 634. [5] T. Teorell, Proc. Soc. Exp. Biol. 33 (1935) 282. [6] E. Manegold, K. Solf, Koll. Z. 55 (1931) 273. [7] M.v. Smoluchowski, Phys. Z. 6 (1905) 530. [8] N. Bjerrum, E. Manegold, Koll. Z. 42 (1927) 97. [9] E. Manegold, R. Hoffmann, Koll. Z. 50 (1930) 22. [10] W.J. Elford, Trans. Faraday Soc. 33 (1937) 1094. [11] J.H. Bauer, T.P. Hughes, J. Gen. Physiol. 18 (1921) 143.

187

[12] W. Grallert, Ber. Bunsenges, Phys. Chem. Z. Elektrochem. 42 (1936) 330. [13] F. Ender, Ber Bunsenges, Phys. Chem. Z. Elektrochem. 43 (1937) 217. [14] K.W. Wagner, Ber. Bunsenges, Phys. Chem. Z. Elektrochem. 32 (1911) 1001. [15] K.W. Wagner, Ber. Bunsenges, Phys. Chem. Z. Elektrochem. 33 (1912) 635. [16] K. Sollner, J. Phys. Chem. 50 (1946) 473. [17] K. Sollner, J. Abrams, J. Gen. Physiol. 24 (1940) 1. [18] K.J. Sollner, J. Abrams, C.W. Carr, J. Gen. Physiol. 25 (1941) 7. [19] K. Sollner, J. Phys. Chem. 49 (1945) 47. [20] K. Sollner, J. Phys. Chem. 50 (1946) 53, 473. [21] K. Sollner, H.P. Gregor, J. Phys. Chem. 51 (1947) 29922.