3174
J . Phys. Chem. 1992, 96, 3174-3182 group ring structure. The hydrocarbon chains remain well-ordered arrays in three dimensions within SA LB films after the reaction. A structured UV-visible absorption spectrum with a exciton peak at 580 nm was obtained. The absorption edge shifted about 1.40 eV with respect to bulk PbS. PbS monolayer consists of the two-dimensional domains or lines, instead of isolated monomolecular PbS. Thus strong quantum size effects occur mainly in one dimension (quantum sheet) or in two dimensions (quantum wire). It indicates that low dimensional quantum wells can be obtained in LB films by chemical reactions. Registry No. PbS, 1314-87-0; lead stearate, 1072-35-1.
when PbS of the monolayer is linked to each other in lines. Thus it is not precise enough to determine the size of the quantum sheet or the quantum wire based on the hyperbolic band model or the particle-in-box modeL8
Conclusion PbS generated by the reaction of PbSt, LB films with H2S at a pressure of 1 Torr forms Q-state monolayers in the polar planes of S A LB films. PbS monolayer is controlled by the carboxylic group ring structure. After the reaction, the long spacing of LB films does not change because of the formation of the carboxylic
An Alternating Current Impedance Model Including Migration and Redox-Site Interactions at Polymer-Modified Electrodes M. F. Mathias*,+and 0. Haas Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland (Received: July 8, 1991)
Electron hopping in redox polymer-modified electrodes requires coincident ion movement to maintain electroneutrality, and transport effects can arise due to either process. In this paper, the equation for diffusion/migration by electron hopping is used with the classical Nernst-Planck equation for mobile ionic species to rigorously model the ac impedance response of a redox-site-containing polymer attached to an electrode and bathed by a binary aqueous electrolyte. The theory also includes the effect of polymer redox-site interactions. Donnan exclusion is assumed, permitting only one mobile ion in the film. This leads to an analytical solution to the problem because the polymer can be treated as a binary electrolyte. When ion movement is slow relative to electron hopping, the impedance spectrum contains a classical Warburg region just as when electron hopping controls the rate. Although these two types of rate control cannot be distinguished from a single ac impedance spectrum, one can determine the transport controlling process from the potential dependence of the data from a series of spectra.
Introduction About a decade ago, it was recognized that the propagation of charge through redox polymer films on electrodes frequently obeys Fick's second law. Consequently, the electron diffusion coefficient has become a characteristic parameter of charge transport through polymers on electrodes.lv2 Because of its simplicity, use of Fick's law leads to analytical models for many electrochemical experiments. Fick's law based models have been applied to chr~noamperometry,~ cyclic voltammetry: and ac impedan~e.~-~ The movement of electrons within a polymer film requires coincident transport of ions to maintain electroneutrality, and the ion-transport process can influence the overall charge propagation rate. This problem of coupled electron and ion movement has been extensively considered by Saveant and c o - w o r k e r ~who ,~~~~ first presented the modified Nernst-Planck equation that governs electron movement,I0B U C ~and, ~other^.^^,^^ ~ ~ ~ Much of the work has focused on chronoamperometry. Andrieux and Savhnt have shown that Cottrell behavior is observed even when ion movement is slow relative to electron hopping; the apparent diffusion coefficient is related to the diffusion coefficients of the ion movement and electron hopping processes.14 Buck has exploited the fact that when a single ion is mobile in the film, the ion and the hopping electron compose a binary electrolyte, and a binary diffusion coefficient can be defined.19 This binary transport is common in polymer films with a preponderance of fixed positive or negative charge; Donnan exclusion leads to low film concentrations of similarly charged mobile ions. Buck has used this approach to model the ac impedance of films sandwiched between two electrode^.^^^^' This paper establishes the analogous diffusion-migration ac impedance solution for the case where the
'
Present address: Mobil Research and Development Corp., Paulsboro Research Laboratory, Paulsboro, NJ 08066-0480.
0022-3654/92/2096-3 174$03.00/0
polymer is attached to an electrode on one side and bathed in a binary electrolyte on the other. The effect of redox center repulsion/attraction and counterion activity as well as the usually small impedance contribution from the electrolyte phase are also included in the model.
(1) Andrieux, C. P.; SavCnt, J. M. J . Electroanal. Chem. 1980,111, 377. (2) Laviron, E. J . Elecfroanal. Chem. 1980, 112, 1. (3) Shigehara, K.; Oyama, N.; Anson, F. C. J . Am. Chem. Soc. 1981,103,
2552. (4) Aoki, K.; Tokuda, K.; Matsuda, H. J . Electroanal. Chem. 1983, 146,
417.
(5) Ho, C.; Rastrick, I. D.; Huggins, R. A. J . Electrochem. Soc. 1980, 127, 343. (6) Contamin, 0.; Levart, E.; Magner, G.; Parsons, R.; Savy. M. J . Elecfroanal. Chem. 1984, 179, 41. (7) Armstrong, R. D. J . Electroanal. Chem. 1986, 198, 177. (8) Lindholm, B.; Sharp, M.; Armstrong, R. D. J . Electroanal. Chem. 1987, 235, 169. (9) Gabrielli, C.; Haas, 0.;Takenouti, H. J . Appl. Elecfrochem. 1987, 17, 82. (10) ( 1 1) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) 578.
Savtant, J. M. J . Elecfroanal. Chem. 1986,201, 211; 1987,227,299. Savtant, J. M. J . Electroanal. Chem. 1986, 238, 1. Savtant, J. M. J . Elecfroanal. Chem. 1986, 242, 1. Savtant, J. M. J . Phys. Chem. 1988, 92, 4526. Andrieux, C. P.; Savtant, J. M. J . Phys. Chem. 1988, 92, 6761. Buck, R. P. J . Electroanal. Chem. 1986, 210, 1. Buck, R. P. J . Electroanal. Chem. 1987, 219, 23. Buck, R. P. J . Phys. Chem. 1988, 92, 4196. Buck, R. P. J . Electroanal. Chem. 1988, 243, 279. Buck, R. P. J . Electroanal. Chem. 1988, 258, 1. Buck, R. P. J . Elecfroanol. Chem. 1989, 271, 1. Buck, R. P. J . Phys. Chem. 1989, 93, 6212. Lange, R.; Doblhofer, K. J . Electroanal. Chem. 1987, 237, 13. Lange, R.; Doblhofer, K. Eer. Bunsen-Ges. Phys. Chem. 1988, 92,
0 1992 American Chemical Society
An Ac Impedance Model for Polymer-Modified Electrodes
3
Polymer
I
To include the effect of activity, a model used commonly for redox polymer systems is introduced.26*2' The interfacial activity coefficients are related to interaction parameters, aij:
Electrolyte Elenmlvie Bulk
0
dmt
t
The theory will be shown to be valuable for diagnosing the controlling transport mechanism from impedance data. The potential dependence of the apparent characteristic polymer parameters determined from the spectra can indicate whether electron hopping 'or ion movement is responsible for observed Warburg behavior. A detailed application of the theory to impedance data from [ O S ( ~ ~ ~ ) ~ X ( P Vcoated P ) ] X electrodes (PVP is poly(44nylpyridine), X is the counterion) will be presented in a forthcoming paper.24
Theory Figure 1 shows a schematic of the system. We consider a redox polymer that can exist in the oxidized (0)or reduced (R) state and undergo an n electron reaction
-
0 + ne-
(1)
Here C represents the concentration of redox sites in the polymer and xc is the fraction of the redox sites that are reduced. (x, is equal to C /C of Gabrielli et al.? e of Buck,19 and CB/COEof S a ~ E a n t and ~ ~ is J ~equivalent to notation used by Chidsey and Murray.25) A completely dissociated binary electrolyte in the bathing electrolyte phase, X . J ~ , with ionic charges of z, and zy, moves within the polymer phase and in the electrolyte boundary layer to maintain electroneutrality. The polymer also may contain fixed nonredox sites, a t concentration cF with charge zF. The potential in the polymer phase is denoted by 9. Analogous notation is used in the electrolyte phase, differentiated by asterisks. The thickness of the polymer is represented by d. The thickness of the mass-transfer boundary layer in the electrolyte is given by d*; this is finite for a rotating disk and infinite for a stationary disk. The distance from the polymer/electrolyte interface to the reference electrode is &[*. Electrode/Polymer and Polymer/Electrolyte Interphase Relationships. At the electrode, the current, I, can be written as - I- - kflRxe,Oc - kbyO(1 - xe,O)c (2) nFA where the forward and reverse rates are proportional to the activity of the reduced and oxidized species, respectively. The activity depends on the fraction of the redox sites that are reduced at the electrode/polymer interface, x ~ , and ~ , the interfacial activity coefficients are given by y. The forward and reverse rate constants, kf and kb are assumed to have a Butler-Volmer dependence on potential:
kj = kor eXp((1 - .)?If(v-
$0))
kb = kob exp(anf(V- $o))
= exp[-(aRRXe,O + aRO(l - xe,O))q
(5)
70
exp[-(aOO(l - XqO) + aORXe.O)q
(6)
= -GRT = -RT[(Uoo
+ aRR) - (aOR + a ~ o ) ] C
One notes that yR/yo is related to
Figure 1. Schematic of electrode/polymer electrolyte system.
R
YR
It is convenient to introduce the occupied site interaction energy, related to the interaction parameter, G,26327 both of which are constants in this model.
4
~L~++~~-A
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 3175
(3)
(4)
where ko is the potential independent rate constant and a is the cathodic transfer coefficient. The overpotential driving force is the difference between the potential of the electrode, V, and the interfacial potential in the polymer phase, &,. The potentials are both relative to the potential in the electrolyte at the tip of the reference electrode. (24) Mathias, M. F.; Haas, 0. Manuscript in preparation. (25) Chidsey, C. E. D.; Murray, R. W. J. Phys. Chem. 1986, 90, 1479.
(7)
t.
Within the polymer phase, electroneutrality is maintained. zOC - nxcC + ZFCF + z,cx + zycy= 0
(9)
where zo is the charge on the oxidized form of the redox couple. At the film/electrolyte interphase, the movement of ions across the interface required to maintain electroneutrality is assumed to be rapid and at equilibrium. Because the ion concentration inside the polymer is often different than that outside due to the electroneutrality requirement, a potential difference is established at the film/electrolyte interface to support the step change in concentration. This balance, Donnan equilibri~m,2~*~~ is governed by
where f indicates F / R T and A is a thermodynamic quantity Xi=
y*i Y i exp(
- poi RT
)
The exponential quantity is a function of the difference in the standard chemical potential for the ionic species, poi, in the electrolyte and polymer phase. In the ideal case X is equal to 1. Steady-State Distributions. At steady-state, no current flows and the concentrations and potential throughout each phase are position independent. In the electrolyte $* is 0 (the overbar indicates steady-state), and the ion concentrations are related to the electrolyte concentration c* = F * , / V , = P * , / V Y
(13)
The four unknowns in the polymer phase (x,,$, C,, and Cy) must be calculated. For I = 0, using (2)-(7) one finds = 1
+ exp(nf(P-
1 EO' - 4 - [e/(2n~)1(1- 2 x e ) ) ~
in which the formal potential, EO', is related to Eo' = Eo
(14)
c
+ [l/(nA](aRR - aoR)C + [t/(2nF)]
(15)
and the standard potential, E O , is
(26) Ikeda, T.; Leidner, C. R.; Murray, R. W. J. Elecrroanal. Chem. 1982, 138, 343. (27) Andrieux, C. P.; Haas, 0.;Savtant, J. M. J. Am. Chem. SOC.1986, 108, 8175. (28) Doblhofer, K.; Armstrong, R. D. Electrochim. Acra 1988, 33, 453. (29) Naegli, R.;Redepenning, J.; Anson, F. C. J . Phys. Chem. 1986.90, 6227.
3176 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
-3 .'
.-.
62
,0022 4.6
Y
,002
5 c ,0018
,001 6 ,106
Mathias and Haas of the potential in the polymer phase (10) but not the magnitude of the potential variation from complete reduction to complete oxidation of the redox sites. When the product of A, and A, >> 1 (about 100 for the case illustrated in Figure 2), Donnan exclusion collapses and E, becomes significant relative to E,. Throughout this paper although we set A, = A, = 1 for the calculations it is not a critical assumption provided that Donnan exclusion applies. Impedance Expression. To derive an expression for the electrochemical impedance, one h"around the perturbed region by replacing each of the variables with a nonfluctuating and a fluctuating component represented in general as u = zi
+ ii expuut)
(17)
where zi indicates the steady-state quantity, ii the complex fluctuating quantity, w the frequency of the perturbation, and t the time. In general, the steady-state terms and the complex exponential can be eliminated, leaving expressions in the fluctuating components. This transformation of (2) after rearrangement to solve for the impedance, 2,yields
104
&lv 102 11 -.12
I
where the charge-transfer resistance, RCT, is defined as
-.06 0 .06 12 Potential wrt Apparent Formal Potenltal (V)
Figure 2. Dependence of polymer-phase concentrations and inner potential on electrode potential (relative to and e/RT for [Os(bpy)2C1(PVP)]CI:zo = 2; n = 1; z, = -1; zy = 1; C/c* = 10; z s F / C = 4; A, = 1; A, = 1.
The four nonlinear algebraic equations, ((9)-(11) and (14)), can then be solved as a function of potential using Newton-Raphson to give the polymer phase concentrations and potential. In this paper we apply the model to a frequently studied redox polymer family, [ O S ( ~ ~ ~ ) ~ X ( P V(PVP P ) ] Xis~poly(4-vinyl~ pyridine), X is the counterion). The bathing electrolyte is assumed to be a 1:l acid (i.e. z , = -1, z, = 1) at a concentration one-tenth that of the redox sites in the polymer (Le. C/c* = 10). The polymer contains four protonated and positively charged pyridine groups for each pyridine linked to an Os complex (i.e. z + ~ / C= 4). The Os center is associated with a complexed and immobile anion and therefore has an effective charge of +2 when it oxidized (zo = 2) and +1 when it is reduced. We assume that the anion is the only mobile species and that the protons attached to the pyridines have no mobility. Figure 2 shows the steady-state results for this polymer as a function of c/RT and the electrode potential. The potential is given relative to the apparent formal potential, Eolapp (the formal potential minus the Donnan potential at the formal potential). Because the immobile sites in the polymer are positively charged, mobile anions from the electrolyte penetrate to achieve polymer electroneutrality. Donnan exclusion, (10) and (1 l), leads to small mobile cation concentrations in the polymer, an effect confirmed by isotopic tracer ~tudies.~'The anion:cation concentration ratio is approximately 3000. In order to support the concentration step changes of mobile anions and cations at the polymer/electrolyte interface, a positive potential of about 100 mV in the polymer phase is established, and this value varies only by about 5 mV from complete reduction of the active sites to complete oxidation. The effect of redox-site repulsion ( t / R T > 0) is to broaden the transition potential range from oxidation to reduction, and the effect of attraction between sites ( t / R T < 0) is to narrow it. In the calculation on which Figure 2 is based and throughout this paper, A, and X, are set equal to 1. Note that when Donnan exclusion is operative, E , is set by electroneutrality (9) and does not depend on A, or A,. The value of A, affects the absolute value (30) Forster, R. J.; Kelly, A. J.; Vos, J. G.;Lyons, M. E.G. J . Electroanal. Chem. 1990, 279, 365. (31) Kelly, A. J.; Ohsaka, T.; Oyama, N.; Forster, R. J.; Vos, J. J . Electroanal. Chem. 1990, 287, 185.
n2FAfqkf(l - ( Y ) x ~ + T ~ k p ( 1 - x,)T0} (19) which can be simplified using (2) to RCT-' = n2FAfCkfX,TR (20) For fast electrode kinetics, Rcr = 0. In this paper, the effect of the electrode double layer capacitance, CDL,has not been included. The data reduction approach to be introduced relies on data a t frequencies smaller than 1/ (RCrCDL)and outside the range of the high-frequency semicircle. Electron/Charge Transport in Polymer. Solution of the impedance requires and &, as functions of the current, 1. Using the flux expression for hopping electrons with the effect of sitesite interactions (ref 25, eq 2 5 ) , one derives the following electron conservation equation
in which the time derivative has been linearized using (17). When t = 0, this is equivalent to the modified Nernst-Planck equation derived by Sav6ant.Io Assuming by Donnan exclusion that E, >> E, and therefore that the transference of ion x is much greater than that of ion y, one only need consider the movement of ion x. The linearized conservation equation is
Linearizing the electroneutrality expression, (9),yields -nj$ + zZ,, = 0
(23)
At the electrode/polymer interface, x = 0, the current is carried exclusively by the electrons giving the boundary conditions
and
At the polymer/electrolyte interface, x = d, the current is carried exclusively by the mobile ions to give
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 3177
An Ac Impedance Model for Polymer-Modified Electrodes
Jo= J*o
E*o ++ -dK1A + zfcf
+
t , cosh ((md) t,
and
Electroneutrality, (23), also applies at each boundary. The polymer is a binary electrolyte (electrons and one mobile ion species),I9 and the potential can consequently be eliminated from (21)-(23) to yield
where the definition of the binary diffusion coefficient, D, is
L, ( d m i ) cjwDFAf (38)
(t, - tx)lfi tanh n28C z,Zzx
where the conductivity of the polymer, K
= (n2BCD,
K,
is defined as
+ z,Z~?,D,)fl
(39)
Charge Trausprt in Binary Bathing Electrolyte. The first two terms in (38) require expressions for quantities in the elecJrolyte just outside the polymer, E*o and $*o, as a function of I . The conservation equation for ion x in the bathing electrolyte phase is
+ zJz*x-d2a* dx*2 An analogous expression holds for ion y. Electroneutrality in the electrolyte also applies: E* = E*,/v, = f*y/vy (41)
and ,8 = 1
+ (Bt/Rr)
At x = 0, (23)-(25) can be combined to give the boundary condition
For a 1:l electrolyte, E* = E*, = f* . The boundary condition at x* = 0 for the ion mobile in the film is
7 = -D*,[ z,FA in which the transference number for the mobile ion, t,, is defined as t, =
ZXQxDx z,ZZ,D, n2BCD,
+
-
At the outside of the electrolyte mass-transfer boundary layer, for a stationary disk, one applies QJ
E*, = 0 in which the transference number for the hopping electrons, re, is defined as 1,
=
-
t,
cash
L[ nFAC
X*
= d*,,f
(~W + t ,X cosh ) (d-(x sinh
(4-4
J*
(35)
e
(45)
0
The binary electrolyte conservation equations allow elimination of the potential, 4*, using (40) and (41) to yield
The solution to this problem is
ii, =
(44)
At the potential datum, the tip of the reference electrode where
n29CD, z,2ZxDx+ n28CDe
(42)
0 = -D*
x* = d* for a rotating disk or x*
(34)
+ z J ? *dJ* ,~l0]
For the ion excluded from the film, the boundary condition is
(33)
and a t x = d, one combines (23), (26), and (27) to obtain
$lo
I
- d))
where the binary diffusion coefficient in the electrolyte, D*, is defined as
(36)
providing Re as a function o f f . Equation 36 e n be inserted into (23) to provide E,. To obtain equations for 4, E, is substituted into (22) and (25), producing a differential equation in 3 and a boundary condition at x = 0, respectively. At the polymer/ electrolyte interface, this approach yields a redundant boundary condition. Instead (IO), the Donnan equilibrium expression, can be linearized at x = d, x* = 0 to give
(37) where we have used the fact that Z*x,o/c*x= E*o/c*. Notice that A, does not appear in the linearized form of (10). The following solution for 4 is obtained at x = 0
D*
e
D*xD*y(z,- zy) z,D*, - z ~ D * ~
(47)
The transformed boundary condition at x* = 0 is obtained from (4 1)-(43).
--[ -i.]
f = D* dE* v,z,FA
t*,
The transference numbers are defined as t*, = z,D*,/(z,D*, - zYD*,)
(49)
and t*, = -z,D*,/(z,D*,
-z~D*~)
(50)
3178 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
Mathias and Haas control ( t , = 0.99, t , = 0.01). The binary diffusion coefficient, D, was assumed to be cm2/s in all cases. The values of D* and t*y were based on dissociated HC1, and the reference electrode was assumed to be sufficiently close to the stationary polymercoated electrode so that ohmic resistance of the bathing electrolyte could be ignored. The curves depend on only two parameters: t , and (w/D)I/*d. Because the transference number expressions in (51) ( 2 t , t , and te2 + t X 2 )are symmetric about t , equal to 0.5, the response for the ion movement control case is identical to that obtained when electron hopping controls. Thus ion movement control produces a classical Warburg impedance. This is analogous to what Sav6ant has shown for the potential step e~periment;'~ Cottrell behavior is observed even when ion movement controls. For mixed control ( t , = t , = O S ) , Figure 3 shows that the conductivity of the polymer phase becomes small, and a large pure resistance due to the polymer phase is observed. Finally, the low-frequency resistance, RLF, is seen to be independent of t,. Simplified Impedance Expressions. Low-Frequency Behavior. In the purely capacitive region where w 0 and at a stagnant electrode (d* m), (54) simplifies to
-
-
0
.2
.4
.6
Z(w) =
Real Impedance (Z') Figure 3. Impedance spectra calculated for the polymer with various values of I ,and t,: zo = 2; n =: 1; z , = 1; zy = -1; C/c* = 10; z + ~ / C = 4; A, = I ; A, = 1; D = cm2/s; D* = 3.34 x 10-5 cm*/s; t,* = 0.82; RCT = 0; d* m; V = E"'app,The impedance has been made The frequency parameter dimensionless by dividing by d/(FAfn20CD). is (w/D)'I2d.
-
(55)
and the boundary condition based on (44) also holds. The solution to the problem is t*, sinh
Ignoring the contribution from the electrolyte, one can determine CLF, the characteristic low-frequency capacitance, from
( d G ( x * - d*))
m
z"= -1 / ( O c L F )
cosh ( d w d * )
(56)
and ( 5 5 ) to be
Combining this with-(40), (41), (43), and (45) yields the boundary value problem for 4*. The solution at x* = 0 is
(57)
The characteristic low-frequency resistance, RLF, from (55) is
RLF=
The electrolyte conductivity, K*
K*,
= (z,Zu,D*,
is defined as
+z~u,D*,)~c*
(58)
(53)
Results and Discussion The overall impedance can be calculated by inserting (36), (38), (51), and (52) into (18). The solution is
When Ra and the electrolyte resistance are ignored, RLF is made up of the sum of the resistance of the polymer and a contribution from the Warburg region. These last two terms in (58) can then be manipulated to yield
\
showing, as seen in Figure 3, that RLF does not depend on t, but only on the binary diffusion coefficient, D. High-Frequency Behavior. In the Warburg region where w -, a,(54) simplifies to
The first three terms represent the pure resistance (the sum of the charge transfer, polymer, and electrolyte contributions), and the final two terms represent the complex impedance of the polymer and electrolyte phase, respectively. Typically D
