1018 7440-47-3; 7440-66-6;
Anal. Chem. 1988, 60, 1018-1022
Mn, 7439-96-5; N2, 7727-37-9; C, anatase, 1317-70-0.
7440-44-0;
Zn,
LITERATURE CITED (1) Skelton, I?. A.; Marston, R. E.; Painter, G. D. The Vinlend Map and the Tartar Relation; Yale Unhrersity Press: New Haven and London, 1965. (2) McCrone, W.C.; McCrone, L. B. Geol. J. 1974, 140, 212-214 Part 11. (3) McCrone, W.C. Anal. Chern. 1976, 4 8 , 676A-679A.
(4) Pigment Handbook; Patton, Temple C., Ed.; Wiley: New York, 1973:
voi. I, p 7. (5) Wallis, Helen, private communication. (6) Olin, Jacqueline S..Private communications. (7) Cahiii, T. A.; Schwab, R. N.: Kusko, B. H.; Eidred, R. A,; Moiler, G.; Dutschke, D.; Wick, D. L.; Pooiey, A. S. Anal. Chern. 1987, 59, 829-833.
RECEIVED for review August 13,1987. Accepted February 1, 1988.
Plasticized Poly(viny1 chloride) Properties and Characteristics of Valinomycin Electrodes. Current-Time Responses to Voltage Steps Michael L. Iglehart a n d Richard P. Buck* Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27514 Gyorgy Horvai and E r n o Pungor Technical University of Budapest, GellBrt t l r 4, 1111 Budapest X I , Hungary
Aoproxlmate analyds of current-tkne translento In the Kval’ Ion-carrier system can be done for short- and long-the responses. The whole transport problem Is nonlinear, but good approxlmatlons are posslble for flxed-dte membranes. A unique value of D, calculated from short- and from bng4me data, agrees wlth the steadygtate value already reported. There Is no evldence for K+ hopping; motion of Kval’ from fixed &e-to-site wlth back dlffuslon of free vallnomycln Is sufflclent to explaln translent and steady-state responses.
Current-time curves of electrochemical systems are determined in the course of defining and understanding normal behavior. Current-time curves of metal electrode/electrolyte interfaces show up two of the characteristic phenomena in conventional electrochemistry: mass transport controlled fluxes a t reversible interfaces and surface kinetic control of fluxes a t electrochemically irreversible interfaces. Although transient analysis is not as common for membrane systems, theoretical treatment (1) and an experimental example of detailed current-time curve analysis of a fully dissociated mobile site membrane system exist (2). Digital simulation methods can be used for a variety of transient response analyses ( 3 , 4 ) . Transients have also been analyzed for simple, fiied-site systems as well (5-7). Kinetic parameters of neutral carriers have been measured from current-time transients of lipid bilayers (siteless membranes) bathed in aqueous solutions (8, 9).
More complicated kinetic situations can often be diagnosed from deviations of I-t curves from expected behavior of simpler systems. One of these is the case of a reversible interfacial process combined with a high-resistance electrolyte treated by Chapman and Newman (10). The results reported in this paper bear a formal resemblance to their system, even though we are dealing with an ionic conductor (the PVC membrane) bathed on two sides by low-resistance electrolytes. The fixed site, neutral carrier PVC membrane behaves as a resistance
at short times after application of a voltage step. Subsequently, concentration polarization of the neutral carrier, valinomycin, confers diffusion control and a limiting current that is generally less, but can approach, the short-time ohmic current value. Mathematically, the two physical systems are coupled in similar ways and give similar I-t curves, although the details of interfacial potentials are quite different. In the five preceding papers of this series (11-15), the presence of membrane-trapped, anionic, fixed impurity sites in commercial poly(viny1 chloride) (PVC) has been demonstrated by a number of techniques and analysis of resulting electrochemicaleffects. Ordinarily, the conclusion to be drawn is that the membranes are simply fixed, low-site density ion exchangers. In the absence of Donnan exclusion failure conditions, current-voltage curves would be entirely ohmic at all sensible applied voltages, and current-time curves would be uninteresting constant values for all times after the interfacial double layer charging. The experimental I-V results (14)are quite different and resemble mobile-site ion exchanger curves with limiting currents at applied voltages above the ohmic region (16-18). These results have been interpreted by the steady-state carrier mechanism, Morf s “closed loop” process (19),with an amended ion budget, in which current a t high applied voltages is controlled by consumption and release of neutral carrier at the two membrane-electrolyte interfaces and the consequent linear concentration polarization profile of carrier throughout the membrane. This paper continues the observations on neutral carrier behavior in low site density, fixed-site membranes. The basic reversible interfacial potential equations and IR terms are now viewed in the transient state. The flux equation for neutral carrier is developed and the two terms are analyzed, but not solved, to suggest the expected form of the I-t response at applied voltages in the ohmic, transition, and limiting current regions as defined and illustrated in part IV (14).Unfortunately, this transport problem is nonlinear and defies analytical solution. However, the form of the I-t response at short, and separately at long times, can be determined. De-
0003-2700/88/0360-1018$01.50/0 @ 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 10, MAY 15, 1988
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the voltage is applied to two identical junction-type reference electrodes, junction potentials are equal, and IR drops in the bathing electrolytes are small and negligible. Note that the constancy of current implies that carrier concentration profiies remain parallel to each other, e.g. have equal slopes at x = 0 and x = d at each instant of time. At some later time, the surface carrier concentrations reach steady values. Redistribution of carrier occurs until the steady-state profile is reached which is linear across the membrane and has a higher value than 9 at x = 0 and a lower value at x = d . P is the average carrier concentration. The midcell value remains at In B, the surface concentrations of carriers have not reached the extreme values that correspond to the limiting current. In the limiting current region C, ohmic current at t = 0 is maintained until the surface concentration of carrier has risen at x = 0 to 29 and reached zero at x = d . Current falls while the carrier becomes distributed in a linear way across the membrane. These statements are illustrated schematically, but not precisely, in Figure 2. These deductions are based on the forms of the equations and on the experimental results. It is not clear what happens at high applied voltages in region D. The current starts at the ohmic value, but there is an almost immediate decrease that does not compare with backward running “flattened S” curves for regions A-C. It is likely that anions from the electrolytes are no longer excluded, and the total salt, as Kval+Cl-, rises in the membrane especially on the side with excess carrier. This result, in panel D, shows the Kval+ concentration line moving from the low value set by site concentration, to a higher value at each new applied voltage, corresponding to the observed higher steady-state currents above the limiting value.
v.
t -
t-
t-
t-
Flgwe 1. Composite view of carrier concentration profiles, I -t curves and steady state I-VaPp, curve: upper panels A-D, concentrationdistance plots; middle panels A-D, current-time curves; lower, current-voltage.
pendence of the current on carrier loading and on applied voltage defiie the controlling processes and the principal terms in the response function.
MODEL Normal PVC membranes contain fiied negative sites at low concentrations, =0.05-0.5 mM (15),typically 0.1 mM in our system. The normal valinomycin loading is =lo mM, and the concentration of Kval+ is the same as the negative sites, while free K+ is negligibly small. The I-V curve is easily divided into three or four typical parts and there are corresponding I-t curves expected for each I-V segment. The different I-t curves arise from the corresponding free valinomycin concentration profiles and their changes with time. The model is summarized in Figure 1 where the concentration-distance-time profiles of carrier, corresponding I-t curves, and steady-state I-V curves are illustrated schematically. A t low applied voltages, A, the I-t curve is a constant (in time) and depends only on applied voltage because current is carried by Kval+, while back diffusion of valinomycin is sufficiently rapid that only small interfacial potential changes result from the profile tipping. There must be some concentration polarization of carrier for all finite currents, but it is not until the surface concentration on the depletion side interface is nearly zero that a significant portion of applied voltage appears at the depletion interface, rather than across the membrane bulk. The length of region A, on the applied voltage axis, depends on the size of the valinomycin diffusion coefficient. In regions B and C, the initial current is the appropriate ohmic value determined by applied voltage and resistance of Kval+ motion. At each succeeding period of time, the carrier concentration rises at x = 0 and decreases at x = d for negative currents (positive applied voltages Vapplied = & - 61. The 4’s are inner potentials of the solutions on either side of the membrane. Vapplidappears as these solution potentials when
MODEL MATHEMATICS Originally by Morf, Wuhrmann, and Simon (19), and again in part I11 (13),it was shown that the total voltage across the membrane can be expressed as three terms: IR, and two interfacial potential differences (pds). From the time domain, this equation transformed is
Vappliedor Vapp,is the applied voltage, s is the Laplace transform variable, I is the current and a function of time, I(s) is the transformed current, R , is the high-frequency resistance, R is the gas constant, T i s temperature, and F is the Faraday. Valinomycin concentrations are in parentheses (val) or C,, with x the spacial membrane coordinate. The concentration of valinomycin C, is subject to two constraints: the total carrier concentration remains constant, and the fluxes at the boundaries are equal and proportional to the current. Thus the boundary concentration transforms have a familiar form
CJs) = 9 / s
f
[ I ( s ) / F A ( D , , ~ ) ’ /tanh ~] [ ( ~ / d ) * / ~ ( d / 2 ) ] (2)
where the plus applies to x = d and minus to x = 0. The concentration profile, in the steady state is C,(X) = C,(X = 0 )
+ ( 2 9 - C,(X
= O))x/d
(3)
and the limiting current, IL, in the plateau region is
IL = 2FA9DvaI/d
(4)
This result is found from eq 2 by passing to the limit of sI(s) as s = 0. A linear concentration profile of lesser slope can be found at other lower voltages, as well. The transformed I-V curve is found by substituting eq 2 into eq 1. The I-V curve in the time domain cannot be obtained by inversion because
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ANALYTICAL CHEMISTRY, VOL. 60,NO. 10, MAY 15, 1988
Table I. Transition Times for DNA Membranes“ moi/(cm3 A)
P2,sliz
VI1, mol/(cm3 A)
d2, s1I2
0.222 0.433 0.498 0.499 0.512 0.676 2.08
1.82 3.31 3.99 4.05 4.22 5.15 15.6
2.13 2.16 2.40 2.45 2.63 2.82 2.84
15.4 15.9 18.4 20.9 19.5 20.7 22.2
P/I,
X-
0
“Slope, 7.57 f 0.2; Y int, 0.1 f 0.4; correlation coefficient, 0.995; X cm2/s.
d
Dval,(1.27 A 0.04)
of the great nonlinearity. However the general expression can be formally written Va,,lied(right-left) = -IR, -
Equation 5 shows that the applied voltage is distributed over two terms. In the ohmic region, eq 5 is dominated by the ZR term because the (val) profile changes only slightly from the initial constant value across the membrane for each applied voltage. In region A, any changes in the surface (Val) are not sufficiently large to cause the second term in eq 5 to dominate the first, ohmic term. The transition region occurs when the first term dominates initially, but both terms are comparable in the steady state. This means that the surface concentrations of valinomycin are significantly perturbed by the higher currents. In the plateau region, the first term dominates at short times, but the second term controls in the steady state because the valinomycin concentration at one interface approaches zero. Relatively more complete analysis of the I-t curves can be done for applied voltages in the plateau region C. It happens that the initially ohmic current drives the surface concentrations of carrier to the steady-state conditions before the concentration profiles of carrier become sufficiently perturbed in the membrane interior to affect the slopes of valinomycin concentrations a t the interfaces. One can then treat the current-driven buildup a t one interface and depletion at the other without worry about interactions because we can assume that some of the membrane interior maintains a constant carrier concentration. During this initial time period I remains constant a t the ohmic value. Of course, this is a gross simplification because I does not remain constant over a significant fraction of the transition time. However, at the surfaces, concentrations obey approximately
C, = V f 2Vapplie~t’i2/FARm(DvaiK)li2 Defining
T
(6)
in the usual way T1i2
= F A P ( D , , ~ T ) ” ~ Vapplied/Rm) /~(
(7)
the initial period of nearly constant current should persist to about time r / 4 to r / 2 according to Z(t) = I(t = 0 ) - (RT/FR,) In ([l+ ( t / ~ ) l ’ ~ ] l / [-l ( t / ~ ) ’ / * ] )(8) This approximation is not expected to hold at all times prior to t = T because it predicts the current drops to zero. Experimentally, the initial approximation must merge smoothly into the long-term response. These initial current regions, behaving like induction periods, are more abrupt at each increasing applied voltage along
0 tSchematic version of timedependent carrierdistance profiles, carrier surface concentrations vs time, and current-time plots for different applied voltages along the plateau (limiting) current. Flgure 2.
-~
Table 11. Transition Times for DOS Membranes
OIL
V/I,
mol/(cm3 A)
71’2, s1i2
mol/(cm3 A)
0.30 0.32 0.35 0.36 0.39 0.53 0.69 0.70 0.76 0.77 1.05
2.17 2.33 2.96 2.79 2.88 4.24 5.48 5.74 5.57 6.09 7.35
1.25 1.91 1.98 2.02 2.08 2.20 2.22 2.30 2.40 2.80
7l/*,
s1i2
9.49 14.4 16.2 15.7 16.2 16.6 17.8 19.1 18.7 21.2
“Slope, 7.84 f 0.12; Y int, -0.08 f 0.2; correlation coefficient, cm2/s. 0.998; D,.,,(1.36 f 0.02) X
the plateau, limiting current region, as shown schematically in Figure 2. Experimentally, the shortening of the induction period is very clear for a series of applied voltages. To demonstrate the effect, Table I gives the initial ohmic current values against the apparent il/’values for dinonyl adipate (DNA) plasticized membranes. Table I1 displays similar data for dioctyl sebacate (DOS) plasticized membranes. The relationship is linear for membranes with different carrier loadings, The T values are measured by extrapolating Z to the intersection with the best straight line through the breaking region. Not only do the data obey the prediction of eq 7 , but calculated valinomycin diffusion coefficients agree with steady-state values. After the initial nearly constant current region of the I-t plot, current falls abruptly and eventually approaches, asymptotically, the steady-state limiting current ZL. This behavior has been treated before (20) and is a standard form in diffusion theory for finite systems. The basic solution follows from inversion of eq 2 for fixed concentrations at the boundaries. The solution, for long times arises from the exact 0 function solution I (long times) = IL[l+ 2 exp(-4rr2D,,lt/d2) ...I (9) The interesting observation from our experimental data is that
v.
.
ANALYTICAL CHEMISTRY, VOL. 60,
NO. 10, MAY 15, 1988 1021
- ' c
4
1
-
0
1
'++ ++
-18-
400
800
1200
1800
2000
2400
2aoo
TIME(s)
T I M E (S)
Flgure 3. Current-time transient of 1.04% valinomycin membrane -Vapp, = 15 V; bathing electrolytes, 0.001 M KCI.
Table 111. Correlation between Initial dc Resistance and ac Resistance % Val
plas
I, rA
VwPI
RDC,MQ
R,, MR
0.25 0.25 0.26 0.26 1.01 1.01 1.00 1.00 1.02 1.02
DNA DNA DNA DNA DNA DNA DNA DNA DOS DOS
10.2 12.6 1.12 1.13 20.0 17.2 14.2 4.02 18.9 4.02
55.26 55.26 4.648 4.240 54.22 55.24 55.23 17.14 55.23 10.87
5.42 5.38 4.15 3.75 2.71 3.21 3.89 4.26 2.92 2.70
5.66 4.60 4.25 3.85 2.77 3.32 3.95 4.29 2.81 2.60
Figure 4. In plot of [ I - I , ] for times from Figure 3: slope, -0.003 418;Y intercept, -12.13 correlation coefficient, 0.99978; D, from slope, 1.40 X lo-* cm2/s; D,,, from Y intercept, 2.26 X 10cm2/s.
4
i
a Y
-16
rt vi
s
-17-
Y
-18
the exponential form applies over a very long time period, almost from the transition time T .
-I0
-
t
EXPERIMENTAL SECTION The chemicals, apparatus, electrodes and cells, and the procedure were described in part IV (14).
RESULTS AND DISCUSSION The principal results of this study are the I-t curves for normal membranes a t applied voltages in the plateau or limiting current range. An example is shown in Figure 3. There is an initial current that declines slowly for about 250 s. The current then breaks into a sharp monotonic decrease until the steady state is reached. The initial current is exactly given by the applied voltage divided by the high-frequency resistance. Comparison of R , calculated from Z(t = 0) with the width of the impedance plane semicircle is given in Table 111. The width of the initial constant current region, reported as T , is inversely proportional to the square of the initial average current. This observation is critical to establishment of the model and is illustrated by the data in Tables I and 11. The sharpness of the break-over depends on the precision of surface concentration change dependence on root t. For longer T ' S , and therefore lower currents, concentration profile interactions across the membrane may cause some lowering of carrier slopes at the interfaces and corresponding rounding of the I-t curves. There is an inflection in the I-t curves, but no center of symmetry as one sees for reversible polarographic waves. From the inflection point time, to the steady state, a plot of log (I - ZL)vs time is linear. The slope is accurately -47?Dvd/d2 because the values of Dvd, which can be independently determined, agree with other values from T and from IL,itself. A log plot of Figure 3 is shown in Figure 4 where recurrent positive deviations are apparent in the vicinity of t = T . Extrapolation of the log plot to t = 0 gives the value 3ZL.
'TIME(s)
Flgure 5. In plot of [ I - IsleadylltaW]for time transient from Donnan failure region, -Vapp,= 55 V.
Table IV. Diffusion Coefficients from Current Decay (In [Z - ZL]vs t ) of DNA Plasticized Membranes % Val
slope
intercept
corr coef
Dvd
0.24 0.26 1.00 1.00 1.03 1.47 1.47
-0.00357 -0.00427 -0.00292 -0.00363 -0.00337 -0.00282 -0.00260
-14.0 -13.3 -12.7 -12.0 -11.9 -11.9
0.9952 0.9991 0.9990 0.9950 0.9970 0.9998 0.9976
1.70 1.83 1.25 1.56 1.44 1.24 1.26
0.99" 0.99' 1.00'
-0.00492 -0.00454 -0.00420
-11.0 -11.4 -11.8
0.9927 0.9977 0.9994
2.10 1.94 1.80
-12.1
DOS plasticized membranes.
However, the intercept does not give an accurate indication of Dvd because eq 9 does not apply at short times. Calculating Dvd at the transition time hs the intercept time (when diffusion control dominates) gives a value that is close to, but not in complete agreement, with other diffusion coefficients from limiting currents (14). Table IV gives an analysis of the monotonic decay of currents for membranes with 0.25-1.5% valinomycin. The average diffusion coefficient of valinomycin computed from the slope of these plots is 1.5 X cm2/s while steady-state measurements give also 1.5 X cmz/s. This result agrees with Dvalmeasured by Thoma (21) for a similar system. Linear plots of In (Z- Zskadystate) vs time, after
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 10, MAY 15, 1988 ,
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LITERATURE CITED + I
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of time, to the steady state. This characteristic is routinely found over a 20-V range for plateau, limiting current in the corresponding steady I-Vappli,d curves. Analysis of the I-t curves by the constant current theory for short-time behavior, and total membrane carrier polarization theory at long times, predict four different methods for determining Dvd. Computed values by all methods must be the same if the model, with its mathematical approximations, is correct. Close agreement is found. Registry NO.PVC, 9002-86-2;DNA, 151-32-6;DOS, 2432-87-3; Kval+, 27073-68-3; K, 7440-09-7;valinomycin, 2001-95-8.
T -:
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