Electrical charge conduction mechanism of a polymer membrane

Electrical Charge Conduction Mechanism of a Polymer. Membrane Electrode Selective for Hydrophobic Cations. James R. Luch, Takeru Higuchi, and Larry A...
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Anal. Chem. 1982, 5 4 , 1583-1588

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Electrical Charge Conduction Mechanism of a Polymer Membrane Electrode Selective for Hydrophobic Cations James R. Luch, Takeru Hlguchl, and Larry A. Sternson+ Department of Pharmaceutical Chemistry, The University of Kansas, Lawrence, Kansas 66045

A plastic electrode lncorporatlng a poly(vlny1 chloride)-dloctylphthalate membrane shows Nernstlan response to hydrophoblc catlons. Magnltude of response Is determlned by the thermodynarnlc actlvlty of the Ion and Its hydrophobicity (Le., partltlon coefflclent between test solutlon and membrane phase). Electrodlffuslon experlments suggest that the hydrophobic catlon Itself la capable of carrying charge and Is transmltted through the membrane by Its selectlve solvatlon by conflned negative shes In the form of the carbonyl groups of the plasticizer, dioctylphthalate. Potentlal development Is consldered to be locallzed at membrane-solution Interfaces.

Polymer membranes (“plastic”) have been used in various ion-selective electrodes involving ion association, chelation, or neutral carrier mechemisms (1-8). In many instances they have offered greater sensitivity than is provided by conventional liquid membrane electrodes and have enabled the direct potentiometric determination of thermodynamic ionic activity of both cationic (2)and anionic (3) species. The response of plastic electrodes is dependent on a variety of parameters including the nature and amount of plasticizer, temperature, nature of ion and solvent, etc. (2-5, 9). Higuchi ( I , 7,8)has described an ion-selective membrane electrode which preferentially responds to hydrophobic cations. Membrane selectivity is achieved by providing a hydrophobic membrane phase capable of solvating organic cations rather than using ion exchangers in the membrane phase to achieve the same goal. The results of a fundamental investigation on ion pair extraction by Higuchi and co-workers (10,11) have suggested that specific solvation in the organic phase enhances partitioning of organic ions from aqueous solutions. Solvating species having nucleophilic sites such as that provided by carbonyl-containing compounds are expected to solvate cations. Illian (1, 12) has shown that poly(viny1 chloride) plasticized with electron donors such as dioctylphthalate or N,N-dimethyloleamide, produce electrodes selective for hydrophobic cations. The hydrophobicity (of the plastic membrane electrode coupled with the presence of electron-rich sites within the membrane are responsible for its selectivity. Species having hydrophobic character exhibit a tendency to partition into nonpolar phases and also to associate with each other in aqueous solutions. This tendency results from the increase in entropy due to changes in the water structure accompanying the parition or association process (13). The hydrophobic character of many organic cations enables the plastic membrane electrode to distinguish them from inorganic cations. In addition, potential response to a relatively hydrophobic organic cation can be obtained in the presence of less hydrophobic organic cations (at concentrations appreciably greater than the primary cation sensed). Electrodes of this type have recently been used clinically to directly monitor methadone in urine (7) and quinolinemethanol type antimalarials in plasma and blood after separation from biological material by ion-pair extraction (8). The

purpose of this study was to elucidate the mechanism of electrical conductance of the “plastic” electrode leading to a Nernstian response to hydrophobic cations.

EXPERIMENTAL SECTION Reagents. Dioctylphthalate, [bis(2-ethylhexyl) phthalate], obtained from Aldrich Chemical Co. (Milwaukee,WI) was vacuum distilled (bp 210 “C at 2.5 torr). Bulk polymerized Bakelite QYNL poly(viny1chloride) resin was supplied by Union Carbide (New York, NY). All tetraalkylammoniumsalts were Eastman Organic White Label (Rochester, NY). Membrane Preparation. One gram of poly(viny1chloride) and 18 mL of 1,2-dichloroethanewere added to a 50-mL glassstoppered Erlenmeyer flask and stirred at room temperature for 2 h. The flask was slowly heated to 85-90 OC and maintained at that temperature for 20-30 min while stirring was continued. The solution was then allowed to cool to 65-70 “C before 1g of dioctylphthalate was added. Stirring was continued until the solution cooled to 35-40 OC at which time the solution was poured into an 80 mm diameter flat-bottomed Petri dish. After solvent evaporation, a clear transparent membrane approximately 300 pm thick was obtained. Membranes were stored in distilled water 1-2 weeks before use. Electrode Preparation. A 1cm diameter circle of the plastic membrane was cut out with a no. 5 cork borer. The membrane section was then mounted in a Teflon electrode body which has been machined from 3/4 in. solid Teflon rod. The membrane was clamped between a silicone rubber O-ring and a Teflon washer with a Teflon screw cap. An aqueous “internal reference solution” was added to the interior chamber of the electrode body. Apparatus. A schematic diagram of the apparatus used is given in Figure 1. A current follower (amplifier I) was built to measure the current passing through the electrochemical cell. The sensitivity of the current follower covered the range of 10-13-10-9 A/mV in decade steps. A voltage follower was also constructed (amplifier 11)which provided an optional amplification factor of 2.44. The current follower and the voltage follower were interfaced to a Hewlett-Packard 2100 computer via an analog-to-digital (AID) converter. Experimental measurements were made in unstirred solutions at room temperature unless noted otherwise. The high impedence of the membrane phase (e.g., 3 X 1O1O Q) made it necessary to provide the apparatus with electrostatic shielding. Potentiometric Measurements. Standard solutions were prepared by transferring a weighed amount of a compound to a volumetric flask and diluting to the calibration mark. Lower concentrationswere prepared by successive dilution. The plastic membrane electrode was immersed in the aqueous solution to be measured and the potential output of the voltage follower was monitored with a pH meter. Current-Voltage Curves. Continuous Method. The computer was directed t o apply a specified voltage ramp to the electrochemicalcell via the D/A converter. The resulting current passed through the cell was measured by the current follower and then plotted on an oscilloscope as a function of the applied voltage. Discontinuous Method. The time-dependent electrical properties of the plastic membrane electrode were also determined by applying an external source of constant voltage to the cell and measuring the resulting current as a function of time. The experiment was repeated at several different applied voltages and a plot of the steady-state current as a function of the applied voltage was manually constructed.

0003-2700/82/0354-1583$01.25/00 1982 American Chemical Society

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ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982

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TEST SCLLTICN

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Flgure 1. Schematic diagram of apparatus: I and 11, AD 523L operational amplifier (Analog Devices); SCE, saturated calomel electrode (Radiometer K4112); R,, 10.45 k 3 ; RP, 15.06 kQ R,, rotary switch with the following Vlctoreen 1% carbon film resistors, lo6, lo7, lo*, los, and 10" 3; computer: Hewlett-Packard 2100 (8K memory); oscilloscope: Hewlett-Packard 130c; A/D converter: 8 channel, 12 bit analog-todigital converter; D/A converter: 12 bit digital-to-analog converter; pH meter: Corning Model 112 research pH meter; recorder: Hewlett-Packard 7001 AMR X-Y recorder; voltage source: Fluke 335A dc voltage standard. In some cases, the transient potential response of the cell was measured immediatelyafter the applied voltage was disconnected. Conductivity Measurements. In those cases where linear i-V curves were obtained, the apparent membrane conductivity was calculated from slope of the i-V curve using eq 1,where d d i d conductivity = slope- = - A V A is the membrane thickness and A is the membrane cross-sectional area. A Randall and Stickney micrometer was used to measure the membrane thickness. The cross-sectionalmea was estimated from the dimensions of the exposed membrane surface. Electrodiffusion Experiment. Under computer control a voltage was applied to the cell to maintain a specified electric current. The computer monitored the current and made sufficient changes in the applied voltage (up to &5V) to maintain a constant current. A provision was included to record the actual current passed through the cell if the voltage necessary to maintain the current constant was outside the range of *5 V. Current (i) was integrated as a function of time to obtain the quantity of charge (4)passed through the membrane. Part 1. The test solution contained N-methylacridinium(MA') and the internal reference solution was 2.5 X M MgSO, (10 mL). A voltage was applied to the cell such that the indicator electrode was negative with respect to the reference electrode and thus MA+ migrated into the membrane. At periodic intervals the internal reference solution was sampled and analyzed by fluorescence spectroscopy (Aex = 355 nm; A,, = 482 nm) to determine the equivalents of MA+ present. The internal reference chamber was then carefully rinsed with several portions of 2.5 X M MgS04 solution before 10.0 mL of the same solution was added for the next measurement. The ratio of the moles of MA+ in the internal reference solution to the total equivalents of ions passed through the membrane (determined coulometrically) in a given measurement interval was calculated to give the fraction of the current carried by MA+ in the membrane phase during that time interval. Part 2. The internal reference solution chamber was again carefully rinsed and then refilled with 10.0 mL of 2.5 X lo-' M MgS04solution. The electrode was immersed in a second test M K2S04and a voltage again solution containing only 2.5 X applied such that the indicator electrode was negative relative to the reference electrode. Under the applied voltage the residual MA+ ions in the membrane from part 1migrated to the internal reference solution and potassium ions were transported into the membrane phase. After a sufficient number of coulombs has been passed through the membrane,the internal reference solution was withdrawn and analyzed by fluorescencespectroscopy to deter-

(

ERUIVALEIITS

IoiiS

PASSED

)

x 109

Flgure 2. The total moles of N-methylacridinium cation appearing In the internal reference solution as a function of the equivalents of ions passed through the membrane. mine the total amount of MA+ in the internal reference solution (which is equal to the equivalents of MAf and, therefore, fixed negative sites in the membrane). RESULTS AND DISCUSSION The configuration of the electrochemical cell used to elucidate the mechanism of action of the plastic electrode is given by eq 2, in which the internal reference solution is readily accessible. A saturated calomel electrode (SCE) was used (REF)

SCE]

-

int s$E:;on I membrane Isolution ref ISCE

(Ind)

(2) \

I

plastic membrane electrode

c

to make electrical contact with each aqueous solution, and all potentials were considered using the usual convention of the potential of the indicator (Ind) vs. the reference (REF) electrode ( E = EInd - E R E F ) . The selection of membrane composition containing bulk polymerized Bakelite (QYNL) poly(viny1 chloride) resin and an equivalent weight of dioctylphthalate was based on maximum observed potential response, minimum ionic contamination of membranes, and minimal physical changes to the membrane (e.g., dehydration) upon exposure to air (as discussed elsewhere (12)). Electrodiffusion Experiments. Membranes used in the preparation of these electrodes have substantial electrical conductivity as determined from current-potential plots (3 ( f l ) X lo-" 3-l cm-l) suggesting that the membrane phase contains appreciable ionic material. Electrodiffusion experiments with N-methylacridinium ion (MA+)were carried out to elucidate the nature and amount of the ionic species in the membrane phase. As the highly fluorescent MA+ is transported across the membrane, its appearance in the reference solution can be readily monitored by periodic sampling of the accessible reference compartment and fluorimetric determination of MA+. The fraction of current carried by MA+ (present in the test solution) in a given measurement interval was expressed as the tohl moles of MA+ found in the internal reference solution as a function of the total equivalents of ions passed through the membrane as determined from Coulomb's law (Figure 2). Initially, although ions passed through the membrane, current was not carried by N-methylacridinium ions, i.e., no MA+ was found in the internal reference solution during the first measurement interval even when an average current of 4.0 X lo4 A was passed through the membrane for 3.0 h. As the number of ions passed through the membrane increased, a greater share of the current was transported by MA+ ions, suggesting that the membrane contained confined negative

ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982

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. .

A

j

20

40

so

3;

1:

;1

t

A

A

I

-:: Flgure 3. The membrane conductivity observed during an electro( :^I _ i d .'VJLEYTS

:>;Is

:>LSED

,

diffusion experiment as a function of the equivalents of ions passed through the membrane. The call diagram given above summarizes the aqueous solution composltions durlng part 1 and part 2 of the experiment. T = 24-25 O C . sites with an equal numiber of cations or counterions to maintain electroneutrality. The confinementof these negative sites may arise from the poor mobility of the plasticizer in the highly viscous PVC membrane, rather than representing true fixed sites as is the case for the chloride specific polymer electrode described by Olra (14) in which the ion exchanger is covalently attached to the polymer matrix. Kumins and London (15)found evidence that membranes composed of only PVC contained "fixed" negative charges and mobile cations. Initially MA+ enters the membrane and a counterion in the membrane moves into the! internal reference solution. After most of the original co~interions(apparently K+) in the membrane have been replaced by MA+, then increasing amounts of MA+ appear in the internal reference solution. Eventually, all of the current passed through the membrane is accounted for by the amount of MA+ which appears in the internal reference solution. The negative sites mutrt be attached to rather high molecular weight molecules [sinceno transport of anions (even particularly hydrophobic ones) was observed, even when +5 V was applied across the membrane. Furthermore, the conductivity data taken during the electrodiffusion experiment (Figure 3) indicate that the negative sites are immobile or fiied in the membrane. If this were not the case, conductivity should decrease as the negative charges segregate in the membrane under the applied electric field. A small but significant increase in conductivity was observed as MA+ began to replace the original counterion in the membrane (Figure 3). The membrarie conductivity remained fairly constant after the replacement process was completed. Implicit in the above explanation is that MA+ has a greater effective mobility in the membrane phase than the original counterion. The concenti~itionof cations in the membrane phase is fixed by the concentration of negative sites. NMethylacridinium ion appears to have a greater apparent mobility in the membrane phase than potassium ion (Figure 3) since the oblierved condluctivity dramatically decreased as K+ replaced MA+ as coriducting species in part 2 of the electrodiffusion experiment in which K2S04solutions was used as test mixture. It is reasonable to expect that MA+ should have a greater effective mobility in the membrane phase than K+ due to its charge distribution and hydrophobicity. The apparent diffusion coefficient of N-methylacridinium ion in the membrane was estimated to be 3 X cmz/s based on the Nernst-Einstein equation (Appendix).

- V C- .._ .~ Flgure 4. Time dependence of current density through the cell after applying an external voltage of 5.0 V. The electrochemical cell contained the following aqueous solutions on both sides of the membrane: (0)0.01 M MgSO,, (0)0.01 M Na2S0,, (A)0.01 M K2S04, (0)0.01 M TBA'Br-.

The molar concentration of the ionic material in the membrane was also estimated from the electrodiffusion data. The equivalents of the original mobile counterion can be obtained from the X intercept of Figure 2. Alternatively, after purging the membrane of MA+ with K+ under an applied electric field (in part 2 of the electrodiffusion experiment) the equivalents of mobile counterion in the membrane phase can be estimated as the equivalents of MA+ in the fresh internal reference solution, since electroneutrality requires that the equivalents of counterionsbe equal to the equivalentsof occupied negative sites. The concentration of the ionic species in the membrane phase, obtained by dividing the number of equivalents observed by the volume of the exposed membrane phase, was 5 (f0.4)X M for three membranes. Current-Voltage Curves. The current-voltage curves obtained for the plastic membrane electrode were linear when the aqueous solution on both sides of the membrane contained a hydrophobic species a t moderate concentration. The potential of the membrane electrode under zero current flow (obtained from the X intercept of the linear i-V curve) corresponded with the value obtained from potentiometric measurement. This value was not appreciably affected by applied voltage when the electrode was immersed in aqueous solutions containing 0.01 M tetrabutylammonium bromide (TBA') and remained relatively constant 10 s after the external voltage source had been disconnected from the cell. The linearity of the i-V curves indicate that (1)the ohmic resistance of the membrane phase must remain relatively constant and (2) if the potential of the membrane changes under current flow, it must do so as a linear function of the current density. At extremely low analyte concentrations or in systems in which only hydrophilic cations were present, current-voltage curves were nonlinear. When only hydrophilic cations are available to carry charge, activation polarization is observed (16). Since partitioning of these ions into the membrane is unfavorable, a buildup of charge results at the interface, which in turn causes the development of an EMF in opposition to the applied potential. Thus, current density, in this case, decreases with time (Figure 4) resulting in a nonlinear response, unlike that observed with more hydrophobic cations, e.g., tetrabutylammonium ion, where current density remain essentially constant with time.

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Table I. Potential Response of the Hydrophobic Electrode to Selected Quaternary Ammonium Ionsa potential [ CH, sensitivity,b response at (CH,),],N+ mV/decade CR I@= X = concn R,N+ 10-5 fi ( m v ) 2

3

4 5

51.5 58.5 59.5 58.5

t 5.6 t 113.5 t 240.1 t 349.6

a Measurements made in aqueous solutions containing 0.1 N KBr and 0.1 N KH,PO,. &I Potential measurements made over the analyte concentration range 10-’-10-2 M with sensitivity determined from slopes of plots of emf vs. R,N+ concentration. All plots were linear with correlation coefficient > 0.99.

At low aqueous concentrations of all ions responding to the electrode (e.g., 51 X lo4 M tetrahexylammonium bromide), sufficient charge carrying species cannot be delivered to the membrane interface to support the current flow through the membrane phase, resulting in a separation of charge (producing a change in potential (16)). Potentiometric Properties. Potential Response. The plastic membrane electrode gives Nernstian potential response to a wide variety of relatively hydrophobic cations requiring that only the analyte cation, I+, be reversibly transported across the interface. The membrane potential which developed appeared to be directly related to the tendency of the cation to partition into the membrane phase. The potential developed increases linearly as a function of cation thermodynamic activity and a t a particular activity increased in the order of increasing hydrophobicity for a series of tetraalkylammonium salts (Table I). The free energy of transfer of a methylene (CH,) group from the aqueous to the membrane phase was calculated to be -680 cal/mol from these data (AG = -nFE) (12)which compared reasonably well with values (ca. -800 cal/mol) obtained by other investigators (17)using alternative techniques. The plastic membrane electrode has been observed to discriminate against hydrophilic cations such as sodium ion, potassium ion, and hydrogen ion (7). At higher concentration or with more hydrophobic cations larger initial gradients in the electrochemicalpotential of the analyte across the interface result in a greater separation of charge when the steady state is reached, and consequently,a larger EMF is developed. The poor response of the plastic membrane electrode to inorganic ions results from their inability to effectively partition into the membrane phase. Membrane Phase Counterion. The identity of the original counterion in the membrane was not determined, although the conductivity data of Figure 3 suggest that the apparent mobility of the original counterion in the membrane is close to that of potassium ion. The plastic membrane electrode continued to give Nernstian response to N-methylacridinium ion even when the counterion in the membrane was Nmethylacridinium. Approximately 1 X lo-’ equiv of MA+ was passed through the membrane by applying 6 V across the cell for 16 h. When additional equivalents of ions were passed through the membrane, it was verified that all of the current was transported by MA+ (see electrodiffusion experiments). The membrane potential was then recorded for a series of MA+ solutions and Nernstian response was found. The potentiometric response of the same electrode (having MA+ as the counterion for the negative sites in the membrane) to tetrabutylammonium bromide solutions also yielded Nernstian response. Internal Reference Solution Composition. The cqmposition of the internal reference solution did not influence the ability

of the membrane electrode to give a Nernstian response. Aqueous solutions of hydrophilic salts such as KC1 and MgS04,as well as hydrophobic salts like tetrabutylammonium bromide were examined as the internal reference solution. In all cases, Nernstian response was obtained. It was qualitatively observed, however, that the magnitude of the membrane potential of a given ionic species, at a particular activity, became less productive as the ionic species in the internal reference solution became more concentrated and/or more hydrophobic. The only requirement for the internal reference solution is that ita composition remain effectively invariant with time. Response Time. The response time of the plastic membrane electrode to concentrationchanges was found to be quite rapid and in most cases a stable potential was observed in 0, the hydrophobic cation selectively partitions into the membrane generating a membrane potential across the interface. Eventually, a steady nonequilibrium state of “local equilibrium”is reached in which the electrochemical potential of I+in the aqueous solution (a)is equal to the electrochemical potential of I+ in the membrane phase (a*),at least in the vicinity of the interface. When the steady state is reached, the rate of the forward transport of I+ from the aqueous solution is exactly equal to rate of the reversed reaction. Interfacial Potentials and Charge Transport. Ciani, Eisenman, and Szabo (18)described the membrane potential determined by neutral carrier complexes of ions in thin membranes (approximately 50-100 A) where it was assumed that violations from electroneutrality could exist throughout the entire membrane phase. This analysis has been adapted to the thick plastic membrane barrier where each cation in the membrane is considered to be solvated by the plasticizer dioctylphthalate (but not in the sense of a specific complex

ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982 I+ 0 J+:

t e s t solutions caticne

M‘:

c r i g i n a l counterion i n membrane phase

F+:

internal reference solution cation

R-

I

f i x e d n e g a t i v e s i t e in membrane

x-

:

t e s t a i d l n t e r n z l referbnc? solution a n i o n

6

r

,

Figure 5. Sketch of Ionic speicles concentration proflles: I+, J+, test solutions cations: M+, origintall counterion in membrane phase; P’, Internal reference solution catiton; R-, flxed negatlve site in membrane: X-, test and internal reference solution anion: X, membrane thickness. All concentration profiles are highly exaggerated for clarhy. In (A) only I+ Is assumed to enter the membrane, whlle In (6)both I+ and J+ are able to enter the membrane.

having long term stability) and potential development is considered to be localized at the membrane-solution interfaces. In most mechanisms of potential response, the membrane potential is considered to be generated across the entire membrane, although the relative potential contribution as a function of distance is generally not specified; i.e., the EMF is considered to be associated with the free energy change for the process of transferring I+ from one aqueous solution to the other, I+(a? I+(a”). Several experimentalobservations support the view that tho potential response of the plastic membrane electrode is more realistically described as being composed of potential calntributions generated at each interface: (1)The plastic membrane electrode responds rapidly to changes in the aqueous solution activity of I+ even though the electrodiffusion results indicate that approximately 3 h are required for MA+ to cross the membrane (even when 5 V are applied to the membrane). (2) The magnitude of the EMF generated appears t o be directly related to the ability of the cation to partition into the membrane phase. More specifically, the free energy of transfer of a -(CH2)- group calculated from the potentiometric results (Table I) agrees reasonably well with values obtained by other methods. (3) The composition of the internal reference solution does not affect the potential response as long as the composition remains effectively constant. The most reasonable explanation for these results is to postulate that the short-term EMF response is observed only at the test solution-membaane interface. The concentration of I+ in the membrane is considered to decrease rapidly as the distance from the interhce increases until a constant value of 5 X lowsM is reached. .Although the actual shape of the concentration profiie of I+ in the membrane is unknown, there are probably very few “e:wess” cations 1000 1\ beyond the interface. As shown in Figure 5, a concentration gradient of anions also exists in the aqueous solution adjacent to the

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interface which should help to compensate for the electrostatic repulsion (space charge) between the excess cations in the membrane (or the excess anions in the aqueous solution). It is expected that the membrane interior is electroneutral and no potential difference develops in the bulk of the membrane phase. In a similar fashion, a potential difference will develop at the internal reference solution-membrane interface. As long as the activity of I+ in the internal reference solution is constant, the electric potential developed at the internal reference solution-membrane interface (E”)should be constant once a local equilibrium is established, and the total membrane potential (E)is given by the difference of the two interfacial potentials E’and E”(i.e., E = E’- E‘?. Thus, the emf essentially corresponds to the free energy for the transfer of I+ from aqueous solution to a point in the membrane phase; its magnitude should be indicative of the partitioning ability of the cationic species. To simplify the derivation of membrane potential, the test solution-membrane interface is considered first, and ideal behavior is assumed for I+in the membrane phase. In contrast to the Ciani-Eisenman-Szabo (18)theory, electroneutrality is maintained in the bulk of the plastic membrane by the presence of fixed negative sites. The flux (J)of the cation I+ in the membrane can be expressed by the Nernst-Planck flux equation (eq 4)which assumes that the driving force for

diffusion of an ionic species (of concentration C* in the membrane phase) is the gradient of its electrochemical potential in the x direction (19). A steady state is considered to exist which is characterized by the flux of all permeable ionic species being zero. The apparent diffusional mobility (m)of the cation I+ in the membrane phase is assumed to be constant. Setting eq 3 equal to zero, rearranging, and integrating the resulting equation from x = 0 to x = y yield eq 5 which describes the diffusion potential within the membrane phase from 0 to y.

If it is assumed to a first approximation that the activity of I+ at the interface (a? is equal to its activity in the bulk solution, [a(O)],then an expression for the interfacial potential can be written (eq 6) by combining eq 3 and 5. The distance

RT m*Ka’ E’ = $*(y) - $’ = - In zF m*C*(y)

(6)

x = y is considered to be far enough removed from the interface so that the concentration of “excess” I+ is negligible. Under these circumstances C*(y) is a constant, fixed by the concentration of negative sites and the requirement of electroneutrality in the bulk of the membrane. A similar expression can be derived for the potential contribution generated at the internal reference solution-membrane interface (E”) (eq 7). Thus, by combining eq 6 and

7 an expremion for the membrane potential (eq 8) is generated.

If the membrane potential is considered to be composed of two separate interfacial contributions and the internal

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ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982

reference solution composition is held constant, the interfacial potential contribution, E“,should be a constant, and eq 8 simplified to the Nernst equation for the free energy difference of I+ between the test solution and the membrane phase at x = y (eq 9). In this case, the magnitude of the emf is seen

RT E = E ’ - E ” = - In [m*Kal+ constant ZF

(9)

to depend explicitly on the partition coefficient, K , and ionic activity, a/, of the analyte. This model can be extended to the more general case (Figure 5 ) in which the test solution contains two cations I+ and J+,the internal reference solution contains the cation P+, and M+ is the original counterion in the membrane phase (eq lo), where Kijis the selectivity coefficient (Kij= mj*Kj/mTKi).

E = -RT In [a{ F

+ Kijaj’]

Thus, it seems resonable that the cation I+ which partitions into the membrane can rapidly displace some of the original counterion in the membrane (M+) resulting in a constant concentration of I+ developed for some small distance into the membrane. The charge separation across the interface is assumed to be the major contribution to the membrane potential, although a liquid junction potential exists in the membrane where the M+, P+,and I+concentrations overlap. The liquid junction concentration to the overall membrane potential is expected to be s m d and changing only very slowly. Under conditions where significant amounts of both I+and J+ enter the membrane, eq 10 and Figure 5 show that the potential response arises from the separation of charge due to both I+and J+. Equation 10 emphasizes that the selectivity of the plastic membrane electrode depends on the relative activities of the responding cations and on the ratio of mobility and partition coefficient terms for I+ and J+. It is expected, however, that the partition coefficient ratio will be the dominating factor in the electrode selectivity.

APPENDIX Estimation of a Typical Diffusion Coefficient for MA+ in the Membrane Phase. Conductivity ( K ) is related to the concentration (ci*) of charged species i and their diffusional (mi*)by eq Al, where F is the Faraday constant and zi is the i

K = FCz:mi*Ci*

valence. The use of eq A1 presumes that the only driving force for migration is the electric field. The Nernst-Einstein equation (eq A2) provides a relationship between the diffusion

Di = RTmi* (A2) coefficient (Di)and the mobility of species i, where R is the gas constant and T i s the absolute temperature, and can be substituted into eq A1 to yield a relationship between the observed membrane conductivity and the diffusion coefficient (eq A3). A typical value for the conductivity of N-methyl-

acridinium ion in the membrane phase at 25 “C was 6 X Q-l cm-l (Figure 3). On the basis of the presence of 5 X M fiied negative charges in the membrane phase an “effective” diffusion coefficient of MA+ was calculated from eq A3 as 3 x cm2/s,

LITERATURE CITED (1) (1) Hlguchl, T.; Illlan, C. R.; Tossounlan, J. L. Anal. Chem. 1970, 42, 1674. Cattrall, R. W.; Frelser, H. Anal. Chem. 1071, 4 3 , 1905. James, H.; Carmack, G.; Frelser, H. Anal. Chem. 1072, 4 4 , 856. Cattrall, R. W.; Tribuzld, S.; Frelser, H. Anal. Chem. 1974, 4 6 , 2223. Frlfflths, G. H.;Moody, 0. J.; Thomas, J. D. R. Analyst (London)1973, 97, 420. (6) Carmack. G. D.:Freiser. H. Anal. Chem. 1975. 47. 2249. (7) Srlanujata, S.; White, W. R.; Hlguchl, T.; Sternson, L. A. Anal. Chem. 1078, 50, 232. (8) Mendenhall, D.;Hlguchl, T.; Sternson, L. A. J . Pharm. Sci. 1979, 68. (2) (3) (4) (5)

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(9) Lienado, R. A. Anal. Chem. 1075, 4 7 , 2243. ( I O ) Hlguchl, T.; Michaelis, A.; Tan, T.; Hurwltz, H. Anal. Chem. 1067, 3 9 , 974. (11) Hlguchl, T.; Michaelis, A.; Ryttlng, J. H. Anal. Chem. 1971, 43, 287. (12) Illlan, C. R. Ph.D. Thesis, The University of Kansas, Lawrence, KS, 1971. (13) Tanford, C. “The Hydrophobic Effect: Formation of Micelles and Blological Membranes”; Wlley: New York, 1973. (14) Oka, S.; Slbazakl, Y.; Tahara, S. Anal. Chem. 1981, 53, 588. (15) Kumlns, C. A.; London, A. J. Polym. Sci. 1960, 46, 395. (16) Ruetschl, P. I n “The Encyclopedia of Electrochemlstry”; Hampel, C. A.; Ed.; Relnhold: New York, 1964; p 939. (17) Harris, M. J.; Hlguchl, T.; Ryttlng, J. H. J . fhys. Chem. 1973, 7 7 , 2694. (18) Cianl, S.; Elsenman, G.; Szabo, G. J. Membr. Blol. 1080, 1 , 1. (19) Bockrls, J. O M ; Reddy, A. K. N. “Modern Electrochemlstry”; Plenum Press: New York, 1970; Vol. 1, p 398.

RECEIVED for review February 26, 1982. Accepted April 23, 1982. This work was supported in part by Training Grant CA-09242 from the National Cancer Institute, National Institutes of Health.