Zero-current bilayer membrane potential: Part II. Diffusion potential of hydrophobic anions
J. Electroanal. Chem., 144 (1983) 373-390 373 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands ZERO-CURRENT BILAYER MEMBRANE POTENTIAL PART II. D IFFUSION POTENTIAL OF HYDROPHOBIC ANIONS GENEVIEVE AMBLARD, BERNARD ISSAURAT, BERTRAND D'EPENOUX and CLAUDE GAVACH Groupe de Recherche de Physico-Chimie des Interfaces, C.N.R.S., Boite Postale 5051, 34033 Montpellier Cedex (France) (Received 5th February 1982; in revised form 19th July 1982) ABSTRACT The diffusion potentials of hydrophobic anions through bimolecular lipid membranes (BLMs) are recorded with lecithin + cholesterol bilayers. The aqueous solutions contain a hydrophobic anion tetra- phenylborate (TPB-) or biscarbollyl cobalt (BCC-) in the presence of an excess of inorganic salt (NaCI or CsC1). The concentrations of the inorganic salt are symmetric, those of the hydrophobic anion are asymmetric. Nernstian variations of the steady-state membrane potential are obtained only in certain concentra- tion ranges of hydrophobic ions and also of inorganic salt. The deviations from Nernstian variations are accounted for by taking into consideration the transmembrane fluxes of the inorganic anions which may exist even under zero external current and a symmetry of concentration. The internal currents created by the transmembrane fluxes of inorganic ions acts as leakage currents which compensate for the diffusion current of hydrophobic anions through the membrane. Charge-pulse measurements are also carried out with symmetric systems that permit the determination of the value of some parameters which, combined with the results of membrane potential measurements, bring an estimation of the rate constant of adsorption of hydrophobic anions on to the lipid bilayer. INTRODUCTION Zero-current membrane potentials have been extensively studied in the case of thick membranes and, as a result, the basic principles of membrane ion-selective electrodes have been established [ 1 ]. Conversely, there are relatively few studies dealing with the membrane potentials of planar lipid bilayers (BLMs) which may constitute a model of biological mem- branes and are easily adaptable to the exploitation of electrical measurements. The first studies  consisted in measuring the zero-current potential of unmod- ified BLMs resulting from asymmetrical mineral ion composition of the aqueous solutions on either side of the membrane. A Nernstian response is sometimes observed. A series of monotonic and biionic membrane potential experiments have also 0022-0728/83/0000-0000/$03.00 1983 Elsevier Sequoia S.A. 374 been used to determine the permeability of the BLM to alkaline cations as induced by ionophores , while attempts have been made to correlate the initial values of the membrane potential to changes of the surface potential of lipid monolayers [4-6]. Satisfactory agreement is obtained in some cases. Our aim here is to study the zero-current membrane potential at the steady state of unmodified BLMs in contact with two aqueous solutions containing differing amounts of hydrophobic anions in the presence of an excess of an inorganic salt whose concentration is the same on both sides. The anions considered are tetraphen- ylborate (TPB-) and bis (1-2) carbollyl cobalt III (BCC-). Hydrophobic anions are known to permeate easily through BLMs which, con- versely, act as insulators towards inorganic ions. However, these latter ions do in fact go through BLMs, their passage being attributed to the formation of pores or the existence of defects [7,8]. The transfer mechanism of hydrophobic anions has been intensively studied using relaxation methods [9-12]. This transmembrane transfer is considered to be com- posed of three steps: first, the adsorption of the anion from the aqueous solution on to the first interface, then translocation between the two adsorption planes and finally, desorption to the second aqueous side. According to the classical concept of membrane potential, a theoretical Nernstian variation is to be expected when a difference of concentration of the hydrophohic anions is maintained between the two solutions. The main result of this study is the fact that Nernstian variations of the membrane potential are only recorded with hydrophobic anions within definite concentration ranges of both the considered anion and the inorganic salt present in the solutions. In order to account for the observed deviation, a treatment is proposed in which the inorganic ions of the supporting electrolyte are considered as creating an electrical leakage through the lipid bilayer membrane. In Part I of this study , it was shown that the leakage due to the transfer of inorganic ions is responsible for the transient potentiometric responses induced by the dissymmetric insertion of ionic detergent into the bilayer. Within the framework of this membrane potential study, a series of charge-pulse and steady-state conductance measurements have been undertaken in order to determine the values of the translocation rate constant of the hydrophobic anion through the hydrocarbon core of the membrane and also the values of the number of adsorbed hydrophobic anions. THEORY The system composed of a BLM in contact with two aqueous solutions containing a hydrophobic anion R- with an excess of inorganic electrolyte M+X-: this system can be decomposed into several parts, as is schematically represented in Fig. 1. The central core of the BLM is composed of the hydrocarbon chains of the lipids and hydrocarbon molecules. On both sides of this central hydrocarbon layer, one can define two polar layers 375 E' 4' 3' 2' 1' c_N...c~ _ _ ,:~ / \ aqueous / solution 1" 2" 3" 4" E" C ~ ~0- c~,, ' " I2!vi BLM aqueO~oSlutton I CONCENTRATIONS t ELECTRICAL POTENTIAL Fig. 1. Schematic representation of the concentration profile and the potential profile in the BLM-aque- ous solution system under the measurement conditions: zero current and concentration difference in the bulk aqueous solutions. which are formed by the polar heads of the lipids on which some water molecules may eventually be fixed. The two adjoining aqueous layers in contact are char- acterized by space-charge distributions which exactly compensate for the electrical charges present in the bilayer, i.e. the net charge of the lipid heads and those due to the ions present in the membrane. The Gouy-Chapman theory is used to describe this spatial charge distribution. Recently, Brumleve and Buck  have elaborated a theory for the steady-state potentiometric response of site-free passive membranes. This general treatment, which is also valid for thick and thin membranes, is based on the Nernst-Planck electrodiffusion equation. Therefore, ultra-thin membranes of biological dimension are seen as hydrocarbon leaflets with a thickness smaller than the Debye length. The ion distribution inside the membrane is governed by the interracial partition of the pair of ions present in the aqueous phase and the laws of electrodiffusion of free ions moving in a non-electroneutral medium in which the diffusion coefficients are assumed to keep constant values. The more specific treatment proposed here is only valid for lipid bilayer mem- branes in the presence of strongly adsorbed hydrophobic ions. The basis is derived from the model of transport of hydrophobic ions proposed by Ketterer et al. . In this transport model, the lipid soluble ions are assumed to be located only at the membrane solution interface. A kinetic equation derived from the absolute rate theory has been written in order to analyse the ion movement through the central core of the membrane. In this transport model, the membrane is considered to be ideally non-permeable towards inorganic ions. As the mechanism of transfer of 376 inorganic ions through unmodified ultra-thin membranes is based on the formation of fluctuating hydrophilic pore  and defects  in the bilayer structure, we shall state in our treatment that inside the membrane the hydrophobic anions do not interact with the inorganic ions. The coupling between the transfers of these two kinds of ions is purely electrical. The electrical driving force of the transfer of inorganic ion is the potential difference between the two planes of closest approach created by the charging of the membrane capacitance which results from the passage of a small amount of hydrophobic ions from the concentrated to the dilute aqueous side. In the absence of inorganic ions, a Nernstian response is to be expected. As inorganic ions may create an electrical leakage, they will be able to cause a deviation from Nernst's law. The mechanism of hydrophobic anion transport through BLMs proposed by Ketterer et al. , and afterwards adopted by other authors is based on the existence of two deep minima in the energy barrier of the anions in the membrane, near - -but not necessarily a t - - the interface [14-18]. Adsorbed states of the hydrophobic ions correspond to these energy minima. The model for the transport of hydrophobic ions from the bulk of one of the aqueous solutions to the other is schematically represented in Fig. 2. At first there is a diffusion process of the anion from the bulk aqueous solution to the plane of closest approach 2'. This plane of closest approach, which can be the equivalent of the outer Helmholtz plane for a metal-solution interfaceis presumably located near the polar layer boundary at a distance equal to the ionic radius. The diffusion is followed by an adsorption process corresponding to the anion transfer reaction up to the adsorption plane 1' located at the minimum of the potential energy. The third step is a translocation process from one adsorption plane 1' to the other 1". The exact localization of these two adsorption planes is not exactly known. Indirect determinations of the dipolar potential at the BLM interface 4 ' 2 ' 1 ' 1" 2 '~ 4" i I t i i J Bu lk Bu lk i i t t t i ,, i i Diffusion ransiocatlo Diffusion . " Adsorption ". Desorption'. Fig. 2. Schema of the model for the transport of the hydrophobic ions R- through BLM. 377  have been based on the assumption that the two adsorption planes are located inside the central BLM hydrocarbon core but very close to the polar heads. On the other hand, certain experimental results can only be interpreted by considering that the adsorption planes are localized somewhat within the central layer . The kinetic equations for the adsorption and desorption processes do not involve an electrical term  and are J~-= kAc2,- kDN' (1) jD = kDN, , - kAc2,, (2) where JR A- and JR D represent the adsorption and desorption fluxes of the R - ions, kA and k D the rate constants, N' and N" the surface concentrations of the adsorbed R- ions in the planes 1' and 1" and c2,, cz,, the concentration of hydrophobic ions at the planes of closest approach 2' and 2". The general equations for the translocation step can be derived by applying Eyring's rate theory to the ion transfer reaction, as was done for the ion transfer reaction in the case of two non-miscible solutions  j T = kT N , exp z fa (O v -- O1,, ) -- kTU" exp - zf (1 - a ) (O v - Or, ) (3) where k T is the translocation rate constant, f = F /RT , a the transfer coefficient and Or, v, the potentials at the adsorption planes. A general expression of the steady-state membrane potential will be established in the discussion. This steady-state zero-current membrane potential, AVstmt, is the potential difference 3, - O3,,, when a concentration asymmetry of the hydrophobic ion under study is realized between the two bulk aqueous solutions; the expression of this experimental parameter can be decomposed thus: mVstn~t = O 3, - 03,, = (O 3, - 02, ) q- (02, - O1, ) + (01, - O1,, ) + (O1,,- 02,, ) + (02- - 03,, ) (4) where (03, - Oz0 , (02, - O3,, ) are the diffuse-layer potentials and (02, - Or), (Oy, - 02,, ) the potential jumps across the polar layers (see Discussion for an evaluation of these potential differences). Under our experimental conditions, i.e. at the steady state, the total flux of electrical charges through the membrane is zero. This means that the electrical current associated with the flux of hydrophobic ions JR through the membrane is exactly compensated by the electrical current due to the passage of inorganic ions X- and M +. Therefore JR = S~ = J~ = J~- (5) IJR-I = IJM+I + IJx-I (6) For simplicity we shall write IJnl = IJM+I + I/x I Non-zero values of Ju mean that a steady inorganic ion transfer takes place when a 378 potential difference occurs between the two aqueous planes. Here, Jli is estimated from the value of the membrane potential and the membrane conductance in the absence of hydrophobic ion. This assumption means that the mechanism of in- organic ion transfer is not affected by the presence of adsorbed hydrophobic anions. __ m I I m F IJI,I - AV~tat/Rstat ~ )klsltat " AVstat (7) where AVatar is the measured membrane potential and ~l)t~ the steady-state conduc- tance of a BLM measured when both aqueous solutions contain only the inorganic electrolyte at the same concentration. Combining the preceding equations, it will be possible to give a quantitative treatment of the membrane potential measurements if some parameters of the symmetrical system are known (for example, the translocation rate constant). This is the reason why we carried out charge-pulse measurements with symmetrical BLM aqueous solution systems. In charge-pulse measurements, the BLM is in contact with two aqueous solutions containing the hydrophobic anions at the same concentration. In the low concentration range of hydrophobic ions, the decrease with the time of the potential difference between the two aqueous solutions, recorded just after a sudden injection of a known number of electrical charges, is the sum of two exponentials. Benz et al.  have derived simplified expressions of the translocation rate constant k T and of the number of adsorbed ions at the equilibrium from the parameters defining the first exponential (voltage amplitude V~ and time constant '/'1 )" k T = 0.5(1/~',)(1 - al) (8a) and N v = [2a,/(1 - a,)] (Cm/f- F ) (8b) where C m is the membrane capacitance and a 1 the ratio V1/ (V l + V2), V 2 being the amplitude of the second exponential. As has been done by other authors [9,11,12], we shall again introduce P, the "partition coefficient" of R- between the adsorption planes 1' and 1" and the aqueous solutions. With p = Nv = Nv' C2' C2" which leads to P = kA/k D using eqns. (1) and (2) at the equilibrium state. It should be kept in mind that P is not a thermodynamic partition coefficient because it is dependent on the potential difference between planes 1 and 2 (see Fig. 1). EXPERIMENTAL Products Using the technique previously described , BLMs are formed, from a solution of egg lecithin (Supelco) and cholesterol (Merck) in n-decane (Fhika puriss.) in the proportions 20 mg/20 mg/1 cm 3. 379 The sodium tetraphenylborate (Fluka) was recrystallized three times from a water-acetone mixture. The caesium salt of the [bis(l.2)carbollyl]cobalt I I I anion (BCC-) was a sample obtained from Professor Koryta (the Heyrovsk2~ Institute of Physical Chemistry and Electrochemistry in Prague). Electrical measurements Membrane potential measurements The cell used for zero-current membrane potential measurements is a parallele- pipedic Teflon block with a central partition in which the membrane-supporting hole was bored. The volume of each compartment is about 8 cm 3 and the solutions can be magnetically stirred. The potential differences between the two aqueous solutions are measured using two Ag/AgC1 (area 1 cm 2) electrodes, as both aqueous solutions contain the C1- ions at the same concentration. The electrodes are connected to a potentiometric recorder (Tacussel EPL 1 + TVED) with an imput impedance > 10 tz ~. Before formation of the membrane, the two compartments of the cell are filled with the same solution of RM and MC1 (NaC1 or CsC1). In all measurements, the potential difference between the two Ag/AgC1 electrodes was always < 0.5 mV. After the BLM formation and before modifying the concentration of one of the compartments (inner compartment), it is necessary to determine the "base line", i.e. the potential difference between the two identical aqueous solutions separated by a BLM; all the results given have been obtained for symmetrical systems for which the steady-state base membrane potential was < 2 mV. The concentration asymmetry is then achieved by the addition of a small amount (a few microlitres) of a concentrated stock solution of NaTPB or CsBCC to one of the cell compartments which we shall call the inner compartment. This solution is then stirred for approximately 1 min. The variation with time of the zero-current membrane potential is now recorded until it reaches a value which remains constant for at least 20 rain; this constant value will be called the "steady-state" membrane potent ia l A Vstmt . A new aliquot of the solution containing the hydrophobic anion may now be added and the measurement repeated. Charge-pulse measurements These measurements have been carried out using the technique previously de- scribed by Benz et al. . The electronic switch was a BSV 78 (RTC) triggered by a Philips PM 5405 pulse generator and the storage oscilloscope was a Tektronix 549 with an high-impedance plug-in amplifier (W). Stationary conductance measurements The stationary conductance of the BLM is determined by recording the steady- 380 state current flowing through the BLM by means of a Keithley 427 current amplifier connected to an (X, t) potentiometric recorder (Tacussel EPL 1), when a small constant voltage (5 mV) is applied between the two Ag/AgC1 electrodes in the aqueous solutions. RESULTS Charge pulse measurements Table 1 gives the principal results obtained by this technique in the case of tetraphenylborate solutions 10-8-10 -6 M. The values for kT and the ratio N/c evaluated are the average of experimental values determined using eqns. (Sa) and (Sb). The concentration range is limited as the treatment can only by applied to the range of dilute solutions where the transient responses are composed of two exponentials. These results will not be discussed further as they only serve to provide numerical data for the calculation of the adsorption and desorption constants k A and kD. TABLE 1 Values of k-r and P calculated from charge-pulse measurements for differing concentrations of NaC1 and tetraphenylborate in the aqueous phase  CNacI/mo1 1- 1 108. CTpB-/mo1 1-1 kT/S- 1 lO3.P/cm 1 80 5 2 100 5.5 5 150 4 8 165 3 0.1 10 145 3 20 150 1.9 50 150 1.3 80 100 1.5 100 100 1.4 1 50 2.8 5 120 1.2 0.01 10 220 0.9 50 140 0.8 100 350 0.5 1 200 2.4 5 100 0.6 0.001 10 160 0.4 50 100 0.6 100 350 0.3 381 TABLE 2 Values of the steady-state conductance ~klsltat measured in the presence of the inert electrolyte NaC1 alone under an applied voltage of 5 mV, for different NaC1 concentrations cNacl/mOl 1- I 0.1 0.01 0.001 108. h1~t~t/9 cm 2 5 3.3 2 Stationary conductance measurements In Table 2 are given the average values, for 10 different membranes, of the steady-state conductance of the BLM when the two aqueous solutions containing NaC1 are identical, about 20 measurements being carried out for each concentration with an applied voltage of 5 inV. Membrane potential TPB - In all experiments, the Galvani potential of the solution contained in the "inner compartment" is higher than the potential of the solution less concentrated in TPB-. This means that in the diffuse layer the excess of charge is negative in the outer and positive in the inner compartment to which the TPB- has been added. Figure 3 shows the general shape of the variations with time of the membrane potential recorded just after the addition of TPB- in the inner compartment; the potential increases rapidly with the stirring, and, as soon as it is stopped, decreases down to the steady-state value mVstn~t which is shown to be constant for at least 20 min. This value is the only experimental parameter which will be taken into 15 10 5 Avm/mv addition of TPB-to the inner solution base line m AVstat 3 4 5 6 7 t/min Fig. 3. General shape of the variations with time of the membrane potential recorded just after the addition of TPB- in the inner compartment: transient and steady-state membrane potential. 382 150 12~ go 6oi / / / / / -8 I I I I I I / I ," ,S" i i i i i i / / / / 11 / ~ f ~ . / / i . . j . . - -~ / //, ' ./ ' I I -7 -6 -5 -4 Fig. 4. Variations of the steady-state membrane potential AV=~ t with the logarithm of the tetraphenyl- borate concentration C~rp B of the inner (concentrated) compartment for 10-~ M NaC1 solutions; the different curves correspond to a given TPB- concentration C~p n of the outer compartment. The dotted lines correspond to Nernstian variations according to the expression AV m ~ (RT/F) ln(c'/c"). consideration. As a general rule, the time necessary to reach the steady state is shorter when the TPB- concentration in the inner compartment is increased. In Fig. 4 are plotted the variations of the steady-state membrane potential values vs. the tetraphenylborate concentrations of the "inner" solution (semi-logarithmic plot), for NaC1 solutions 10-l M. Each curve corresponds to a given TPB- concentration of the "outer" compart- ment solution. It appears that the measured membrane potential generally deviates from the theoretical Nernstian variations, as the value of AV~mat always remains smaller than 59 mV for a 10-fold concentration ratio. Nernstian variations are recorded only in a narrow concentration range (10 -6 ext.-10 -5 M int. and 10 -5 ext.-10 -4 M int.). Finally, for the highest concentrations, the membrane potential remains almost constant, although the concentration becomes greater in the inner compartment. The curves of Fig. 5 show the influence of the inorganic electrolyte concentration (the value of which is the same on both sides of the membrane) on the measured values of the membrane potential for identical TPB- concentration differences between the two aqueous solutions. For a given TPB- concentration difference, the value of the membrane potential increases when the NaC1 concentration decreases. One notes that for 10 -3 M NaC1 solutions, almost Nernstian variations are recorded when the TPB- concentration in the outer compartment is 10 -7 M. Note that the effect of the inorganic electrolyte concentration is observed in all TPB- concentration ranges. 383 B C C - Results obtained with BCC- are different from those obtained with TPB , except concerning the sign of the membrane potential. Figure 6 shows that for BCC-, after the addition of the anion to one solution, the membrane potential presents important transient variations which may or may not i) -B nvs~at / mv (A) / / / / t -7 / / / / J I *gCC~pB- / rno l I -I) -B -5 V 7'/mV 15g 12g -7 / / / tb) 12g -6 -5 -4 -B / AVs~a t /mV / / / / (c) ~tbT -5 -4 Fig. 5. Variations of thc steady-state membrane potential AVs~at with the logarithm of the concentration of tetraphenylborate C:rp a of the inner compartment, for differing NaCl concentrations: (A) c~p a- = 10 - s M, ENaCj = 10- ~ M (a); 10 -2 M (b); 10 -3 M (c). (B) ~PB- = 10-7 M, C~acl = 10- t M (a); 10 2 M (b); 10 -3 M (c). (C) c~P B-=10 -6M,sNaC~=IM(a) ; 10 -1M(b) ; 10 3M(c). 384 /~V"/~" addition of BCC" tothe mV inner solution 20 15 10 ,i I 5 m base A Vstat line 5 10 1'5 20 t/rain Fig. 6. General shape of the variations with time of the membrane potential recorded just after the addition of the bis-carbollyl cobalt Ill anion in the inner compartment. pass through a maximum. The time necessary to reach a quasi-steady state is slow compared to the case of TPB-. When the steady state is eventually reached and for 10-~ M CsC1 as inorganic electrolyte, the value of AV~mat is approximately the same (8 mV + 2 mV) whatever the BCC- concentration gradient, when the BCC- concentration in the inner compartment is > 10 -7 M, i.e. mVstmat is no longer dependent on the concentration gradient. Nevertheless, as observed with TPB-, the deviation from Nernstian variations decreases when the concentration of the inorganic electrolyte NaCI decreases. So, in the presence of 10 -3 M CsC1 in both aqueous solutions, and for a 10 6 M con- centration of BCC- in the outer compartment, one observes almost Nernstian variations of the membrane potential with the concentration of BCC- in the inner compartment. DISCUSSION The membrane potential measured, (q53, -~y , )= AV~tmt is the difference of the Galvani potentials of the bulk of the two solutions. Considering eqns. (4), (~3' - ~3") can be related to the potential difference between the two planes 1' and 1", (~b r - ~b~,,) when both the differences between the two diffuse-layer potentials and the two potential jumps across the two polar layers are negligible. We shall attempt here to estimate these two potential differences in the concentra- tion range where it has been possible to draw numerical information from charge- pulse measurements, i.e. when the charge-pulse response is composed of two exponential transients. With TPB- this corresponds to concentrations between 10 -8 and 10 -6 M in organic anion. A good correlation has been found in several cases [4,19,22] between the surface potential change measured with lipid monolayers and lipid bilayers. Unpublished 385 results have been obtained in our group by N. Davion Van Mau for surface potential measurements with monolayers spread at the air-water interface, the monolayers being (1 : 1) mixtures of egg lecithin + cholesterol; these results show that no significant change in the value of surface potential occurs when the concentration of tetraphenylborate in the subphase increases from 0 to 10-6 m. This nearly constant value of the surface potential of a monolayer does not result from an exact compensation of variations of the potential jump across the polar layer by variations of the diffuse-layer potential. The variation of the diffuse-layer potential with the number of electrical charges borne by hydrophobic ions adsorbed at the planes 1' and 1" can be estimated from the Gouy-Chapman theory. This variation is < 1 mV when the number of adsorbed hydrophobic ions is < 6 10 -12, 2 10 12 and 6 10 13 mol cm -2 respectively for concentrations of the inorganic electrolyte NaC1 or CsC1 10-1, 10-2 and 10 -3 M. Further, it would appear from Table 1 that the actual surface concentrations of adsorbed ions given by charge-pulse measure- ments, i.e. the values of N = P . Cvp B- are lower than the maximum values indicated above. One will notice that the so-called partition coefficient P of the individual ion R - between the interface and the aqueous solution does not maintain constant values. From the thermodynamic view point, this fact means that the electrochemical potential of R- at the interface is not related to the interfacial concentration by mean of a simple expression. One can hence assume that, at the steady state, the value of the potential difference ~b v - b v, will be equal to the value of the membrane potential AVs~at for the concentration range considered. In the proposed transfer model of hydrophobic and inorganic ions through BLM, the non-zero flux of R - ions crossing the membrane at the steady state is exactly compensated by a flux of inorganic ions, so that the net transmembrane current is zero; the flux of inorganic ions can be evaluated, making a simplifying assumption, as a function of the measured steady-state conductance of the BLM (X It) measured in the presence of the inorganic electrolyte alone and of the membrane potential (~V~t~t). Thus, using eqns. (6) and (7), the expression of the flux of R - ( JR ) and of the flux of inorganic ions (Jl i) at the steady state are given by I JR I = IJ,,I = AVst~t~tII/r (9) From eqns. (1)-(3) and (5) the numbers N' and N" of hydrophobic ions adsorbed at the steady state on the planes 1' and 1" can be expressed by the following equations: N' = ]J"[ "1- kTP(C2" ''}- c2'') exp( -0 .5 (zF /RT) AVsmat) (10) k y [exp( - 0 .5 (zF /RT) A Vsm, t ) + exp(0.5(zF/RT) A v~tmat )] U" - - I J ' l ] + kTP(C2" + c2'') exp(+0.5(zF /RT) AV~tmat) (11) k-r [exp((0.5zF/RT) AVatar) + exp(( - 0.5zF /RT) AV~tmt )] where [Jill can be evaluated from eqn. (9); k T and P have been evaluated in the range 10-s_ 10-6 M from charge-pulse measurements; AVstmt is the measured mem- brane potential. 386 Using the numerical values, one records that, in the concentration range consid- ered, eqns. (10) and (11) can be simplified as follows: 1 N' = P(c 2, + c2,, ) (12) 1 + exp( (zF /RT) AV~t~t ) 1 U" = P(c 2, + c2,, ) (13) 1 + exp(( - zF /RT) AVatar) And from these two equations, one obtains for AVstmt AV~ t = (RT /F ) I n (N ' /N" ) (14) The expression of the membrane potential therefore shows a Nernstian variation when expressed in terms of the surface concentrations of hydrophobic ions adsorbed on the planes 1' and 1" at the steady state. This simple relation is only valid when the flux of hydrophobic ions through the membrane is very small and eqn. (14) can then be directly deduced from eqn. (3) in which JR- is considered to be zero. The Nernstian variation is therefore a limiting law in the absence of the leak of inorganic ions through the membrane. In the low concentration range, the deviation from the Nernst law increases with the ratio of the concentration of the inorganic salt over the concentration of the hydrophobic anion. This result means that at the steady state the fluxes of inorganic ions can no longer be neglected with respect to the flux of R - whicl'iis diffusion limited. The adsorption rate constant k A is now given, using eqns. (1) and (16), by = IJR-I (15) C2'- (C2' "~ c2"')/exp((zF/RT) AVatar) To evaluate k A it is necessary to know the values of JR- through the BLM under zero-current conditions, and of the concentration c 2, at the plane of closest ap- proach. The value of JR- can be deduced, using eqn. (9), from the values of the membrane potential and of ;kI~tat. The estimation of c~, is somewhat cumbersome as it is necessary to know the exact value of the thickness of the diffusion layer. Several authors have examined the transport of hydrophobic ions in the aqueous diffusion layers in contact with a BLM [18,23-26]. The thickness of the diffusion layer is a parameter which depends not only on the stirring of the solutions but also on the area of the BLM. In the absence of stirring and for a period of time long enough to allow the thickness of the diffusion layer to be determined by natural convection as in our steady-state measurements of the membrane potential, we shall state that the thickness of the diffusion layer is 200 #m. The value 200 /~m for the thickness of the diffusion layer of BLM has been already proposed by several other authors [18,23]. Considering the transport of hydrophobic anions in the diffusion layers, the flux of R - at the steady state can be expressed by the classical diffusion equation 387 JR -= D AC/~ (16) where D is the diffusion coefficient of R- . For TPB- its value is 5 10 -6 cm 2 s - 1. Here, Ac is the R - concentration difference between the two extreme planes of the diffusion layers me = C4 ' - C3" = C3" -- C4" (17) where c a, and Ca,, designate the R - concentrations of the bulk solutions. In Fig. 1 they are also designated C~-pB- and C~pB-. The potential drop of the diffuse layers being negligible, the values of c 3, and c3,, will be equivalent to those of c 2, and c2,,. Considering eqn. (17), the expression c 2, + c2,,, which appears in eqns. (10)-(13) and (15), is strictly equal to the sum of the bulk concentrations, C~p~-+C~pa . Therefore, combining eqns. (15) and (16), the value of the adsorption rate constant k A can be evaluated and, consequently, also the desorption rate constant k o = kA/P . For the calculation of k A and kD, we chose the value of P obtained with 10 -7 M TPB- (see Table 1), because k A and k D are obtained from measurements involving two different concentrations of TPB- in the [10-8,10-6 M] concentration range and P does not maintain a constant value (although the differences are not very large). In Table 3 are gathered the calculated values of N', N", k A and k o corresponding to the systems giving the experimental points from number 1 to number 18 in Fig. 5. It is only possible to draw positive values of the adsorption and desorption rate constants when the NaC1 concentration is 10-1 M. With 10 -2 and 10 -3 M NaC1 the calculated values of k A and k D are negative; this point will be examined further. With 10 - l M NaC1, the average value of k a is about 4 10 -5 cm s -1, with the exception of the experimental point number 4, which gives a higher value of k A and which also corresponds to the highest ratio of TPB- concentrations between the two aqueous solutions. This value of 4 10 -5 cm s i for k A is in accordance with the values calculated from steady-state conductance measurements under a very low applied voltage . Nevertheless the critical point of the estimation of the adsorption rate constant from the values of the steady state membrane potential lays in the fact that the values of the leakage flux of inorganic ions are calculated from conductance measurements carried on in the absence of hydrophobic anions and for a small applied voltage. We have no mean to check that the presence of hydrophobic anions in the bilayers does not affect the molecular process at the basis of the transfer of inorganic ions. This consideration may explain the overestimation of JH which in the case of 10 -2 and 10 -3 M NaC1 leads to negative values of k A. In the highest concentration range, the membrane potenial tends to keep a constant value. In this case, differences between the potential jumps through the polar layers are no longer negligible. Surface potential measurements of lipid monolayers show that, in the [10-5_ 10-3 M] TPB- concentration range, the values of the surface potential decrease appreciably with the TPB- concentration . In 388 .1 ..'-'7 \T , < \T > I l l xXXXXX V V V V V V V V V V V V I I 1 ~ ~ xXXXXX V V V V V V V V V V V V . . . . 7~ . . . . I I I ~ ~ ~ ~ ~ ~ ~ 389 this highest TPB- concentration range, where the number of free adsorbed anions cannot be evaluated from charge pulse measurements, the only conclusion which can be drawn, is that the potential drop through the central hydrocarbon layer tends to compensate for the difference between the potential jumps across the two polar layers of the membrane. This conclusion may also be stated considering the results obtained with BCC- in the case of 10- t M CsC1. Considering eqn. 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