Hyphenated Techniques in Dynamic Electrochemistry. 1. Mechanistic

10826

J. Phys. Chem. B 1998, 102, 10826-10835

Hyphenated Techniques in Dynamic Electrochemistry. 1. Mechanistic Diagnosis for Redox Switching of Electroactive Films Using Nonelectrochemical Population Probes Stanley Bruckenstein*,† and A. Robert Hillman‡ Department of Chemistry, State UniVersity of New York at Buffalo, Buffalo, New York 14260, and Department of Chemistry, UniVersity of Leicester, Leicester LE1 7RH, England ReceiVed: May 27, 1998

The use of hyphenated electrochemical techniques to study electroactive films has two aspects. First, there is an electrochemical control function that drives fluxes of film mobile species and may trigger film structural changes. Second, there are two responses, one characteristic of the probe and the other the conventional electrochemical response. We present a quantitative theory relating changes in electroactive film mobile species populations to different electrochemical control functions. This theory identifies the coulostatic and potentiostatic step methods as optimal for separation of the elementary steps in the redox switching of electroactive films. Combinations of the population changes to the coulostatic and potentiostatic control functions predict the population changes to all other electrochemical control functions. Although the theory is general, we discuss it specifically with regard to a model for redox switching of electroactive films that incorporates three elementary steps. These elementary steps are exemplified by coupled electron/ion transfer, solvent transfer, and film structural change. A simple procedure qualitatively identifies the rate-controlling step and quantitatively determines the rate constants. A scheme-of-cubes representation simplifies visualization of the many mechanistic possibilities for film redox switching.

1. Introduction 1.1. Overview. The use of nonelectrochemical methods to study electrochemically driven interfacial processes has been a dominant theme in electrochemistry over the past decade.1-5 These “hyphenated” techniques have provided a wealth of compositional, structural, and mechanistic information about the electrode/electrolyte interface. Careful choice of the probe technique has allowed selective monitoring of the population changes of individual species. Nevertheless, most studies have used the nonelectrochemical technique qualitatiVely, for example to identify species through their spectroscopic signatures. Quantitative correlation of the electrochemical and nonelectrochemical signatures and the exploration of this correlation in the time domain is valuable. The benefits of these correlations can be realized by an analysis of the interplay between the electrochemical control function (the “pump”) and the nonelectrochemical probe. This approach has two virtues. First, it permits optimizing experiments designed to isolate a selected kinetic process within the overall mechanistic scheme. Second, it provides the means to extract mechanistic information. The electrochemical control function causing a redox transformation has a significant influence on the way in which film populations change. Here, we consider the general transfer function relating film population responses to the electrochemical control function. Our analysis describes the changes in surface populations caused by a general electrochemical perturbation. These population changes may be measured by many probe techniques, according to the chemical problem. Possible probes include UV, visible, and IR transmission (and reflection) spectroscopies, fluorescence, ellipsometry, neutron reflectivity, and the EQCM. The experimental aspect of the problem then † ‡

State University of New York at Buffalo. University of Leicester.

becomes conversion of probe responses to surface populations. The theoretical aspect, which we address comprehensively in this paper, is obtaining the relationship of these population changes to the electrochemical parameters, E and Q (or i) with time. 1.2. Modified Electrodes: Delineating the Problem. The thermodynamics, kinetics, and mechanism of redox switching and of mediated charge transfer determine the properties of a modified electrode,6,7 by which we mean an electrode whose electrochemical response has been altered by surface treatment. A fundamental understanding of their equilibrium and kinetic behavior is central to rational optimization of modified electrode applications and the identification of new applications. For example, use of an electroactive polymer depends crucially upon detailed understanding of the dynamics of all processes accompanying its redox switching. Furthermore, besides electronic charge and other “mobile” species motions, one must consider motions of the polymeric matrix. This understanding will facilitate the logical design of films possessing specified properties. In pursuit of this objective, we present a theoretical framework based on a general model of redox switching of an electroactive polymer film. The application of our approach to experimental data provides criteria that help distinguishing among microscopic (molecular) mechanisms. The general problem we consider involves the interplay between the thermodynamics and kinetics of the system. Despite increasing awareness of the history8-15 and time scale15-19 effects germane to this issue, a means to rationalize their influence was proposed only recently.20,21 First, we recognize that the redox sites may comprise a variety of solvation and structural forms. Hence, the various initial and final film states must be identified. The film’s history defines the initial state, and the experimental time scale defines the “final” state. Ultimately, both are determined by the thermo-

10.1021/jp9823805 CCC: $15.00 © 1998 American Chemical Society Published on Web 12/03/1998

Redox Switching of Electroactive Films

J. Phys. Chem. B, Vol. 102, No. 52, 1998 10827

dynamics of the system. Second, a framework for interpreting kinetic data must be provided. The framework must incorporate all the elementary steps that connect the initial and final states. The time scales of these steps, compared with that of the experiment, will determine when global equilibrium is established. At no loss in generality, we focus on polymer-modified electrodes. As we show later, our approach is not restricted by the film material and its specific chemistry. A wealth of electrochemical data exists concerning electron-transfer rates at electroactive polymer films.6,7 These electrochemical experiments provide values of potential and charge (or current) as functions of time. Interpretation of these electrochemical data has emphasized classical electrode kinetics and diffusional processes (“electron hopping” and coupled ion motion). These data have not been used to obtain information about speciation, which we define here to include a specified redox level, degree of solvation, and polymer configurational form. Instead, these data have only been used to obtain the total populations of all oxidized (O) species (however solvated or configured) and all reduced (R) species (however solvated or configured). Thermodynamic and kinetic descriptions require populations of indiVidual forms of O and R. Considerable ingenuity has been exhibited in explaining the underlying polymer film processes using only potentiostatic and potentiodynamic current responses. However, it is now generally recognized2,4,5 that electrochemistry needs to be coupled to other physicochemical probes to detect and characterize these other processes and phenomena. In this paper, we develop a theory that relates all probe responses (through film population changes) to the electrochemical and chemical steps involved in film redox processes. This general analysis of electroactive film redox switching under permselective conditions describes the film state in terms of its potential, charge level, and species. We start by discussing the problem from a geometrical perspective, move to a quantitative analytical approach, and then select a microscopic model. 2. Geometrical Description We describe the redox and compositional state of the film at a time (t) using three normalized, time-dependent quantities, E, Q, and Λ. They are related to the instantaneous potential (E) and the charge density (qj ) and areal population density (λj) of the j-th surface species by eqs 1a-c.

E ) Eequil + ηj ) F(E - Efj + ηj)/RT

∑qj/qi,total Λ ) ∑λj/λj,total Q)

(1a) (1b) (1c)

In eqs 1a-c, the subscript j refers to the j-th film species. Eequil is the thermodynamic equilibrium potential (volts) of the system, Efj is the formal potential for a film species, j, where its half reaction is formulated as a one-electron process, ηj is its overpotential, qi,total is the areal charge density (coulombs cm-2) associated with its redox conversion on going from an equilibrated reduced film to an equilibrated oxidized film, λj,total is its areal population density (moles cm-2), and R, T, and F have their usual significance. We stress that our analysis is not dependent on the availability of any single probe technique capable of measuring the surface population of all j species. At equilibrium E, Q, and Λ define the Gibbs energy, G, of a film state through the normal definition of electrochemical

potential. Consequently, the state of the system can be described in EQΛ space by a vector, Ω(t), defined by

Ω(t) ) Q(t)i + E(t)j + Λ(t)k

(2)

i, j, and k are the unit vectors for the Q, E, and Λ axes. We define Λ(t) to be zero when all mobile species are absent from the film and E(t) to be the equilibrium potential when the overpotential is zero. The azimuthal and polar angles of Ω contain the information defining the electrochemical and population impedances. Under kinetically controlled conditions, transient variations in E, Q, and Λ will cause the system to move on a potential energy surface above the equilibrium path. For these kinetic studies, the differential form of eq 2 is useful,

dΩ/dt ) (dQ(t)/dt)i + (dE(t)/dt)j + (dΛ(t)/dt)k

(3)

Equation 3 involves charge and population fluxes and the time differential of the potential, dE/dt. Equations 2 and 3 describe the redox and compositional paths followed by Ω and dΩ/dt in 3-D EQΛ space in response to an electrochemical stimulus. The minimum-energy path defines all possible equilibrium compositions. Ideally, if the system is at a given equilibrium composition, whenever it is perturbed the compositional path it follows in returning to a new equilibrium point should be uniquely defined. However, equilibrium is often reached very slowly in electroactive polymer (and other) films. Then, film history rather than thermodynamics determines the starting point in EQΛ space at the time of an electrochemical stimulus. This starting point is probably in the vicinity of the minimum-energy (equilibrium) path. However, the complexity of the potential energy surface suggests that many local minima and local maxima may exist along the minimum-energy path. Accordingly, the system may reside in one of these local minima, rather than in the global minimum, immediately before a stimulus. Consequently, an electrochemical stimulus may inadvertently populate slightly different nonequilibrium start states that relax along different paths toward the minimum-energy equilibrium path. This situation will generate apparently “irreproducible” kinetic population responses for nonperiodic electrochemical stimuli. However, there will be an evolution of kinetic population responses with time for cyclic electrochemical perturbations. This is a common anecdotal observation, for example, for EQCM studies of both electroactive polymer22 and metal oxide11,23 films. We note that the purely electrochemical response is far less sensitive to changes in solvent population than to changes in populations of species directly involved in the redox process, since electrochemical responses are essentially “blind” when solvent transfer dominates the kinetics. Under nonequilibrium conditions, all points in EQΛ space are, in principle, accessible to Ω and dΩ/dt. An electrochemical perturbation initially moves the system from a lower to a higher energy point. It also moves the Ω and dΩ/dt vectors out of the EQ plane and creates a population flux, dΛ(t)/dt, and either a charge flux, a potential change, or both. As the system relaxes toward equilibrium, transient population and charge fluxes and/ or potential variations occur. For small perturbations, the relaxation process will consist of a weighted sum of exponential potential and/or areal population transients related to the ratecontrolling processes (see Section 5). In EQΛ space, the population flux associated with an electrochemical control function is given by

10828 J. Phys. Chem. B, Vol. 102, No. 52, 1998

dE ∂Λ dQ + (dΛdt ) ) (∂Λ ∂E ) ( dt ) (∂Q) ( dt ) Q

E

Bruckenstein and Hillman

(4)

In Figure 1, we illustrate how the choice of electrochemical technique affects the film’s population response under nonequilibrium conditions. The dashed line represents the minimumenergy (equilibrium) path for all allowed equilibrium compositions. We have drawn this line as a curve to allow for the fact that neutral species population changes are, in general, not necessarily linearly related to charge. In the special case that they are, the curve becomes a straight line. Provided the time scale of the electrochemical control function is long compared to all film processes, it is the path the film populations will follow while going from the fully reduced to the fully oxidized state. From eqs 2 and 3, as shown in Figure 1, we see that the electrochemical control function determines the direction of departure from an equilibrium starting point in EQΛ space, the open circle (E0, Q0, Λ0). The form of eq 4 immediately suggests that measurements at constant Q or constant E will have special significance. We therefore consider three ways to fully oxidize a fully reduced film: by a (1) coulostatic, (2) potentiostatic, or (3) potentiodynamic (e.g., linear voltage scan) experiment. First, a coulostatic experiment corresponds to an instantaneous translation (along line a) in the Q coordinate (film charge density ) Q100%) and in the Λ coordinate (population of counterions ) Λ1) to the triangle (E0, Q100%, Λ1). The system will then follow curve a* in an EΛ plane toward the filled circle (Eeq, Q100%, Λeq) at the end of the equilibrium path. Second, by applying a potential step, one could cause the same overall redox transformation through an instantaneous translation (along line b) in only the E coordinate to the square (Eeq, Q0, Λ0). Now the system must follow a different path (curve b*), within a QΛ plane, in its approach to the same equilibrium point (Eeq, Q100%, Λeq). Third, the general response function is illustrated by curve c and corresponds to a path in which all three coordinates change while totally oxidizing the film to a new composition (Eeq, Q100%, Λeq). The resulting path does not coincide with the equilibrium path (the dashed line). This figure highlights the fact that equilibrium is approached from different directions in EQΛ space, dependent upon the electrochemical control function. Thus, if the experimental time scale is not sufficiently long, each response will terminate at a different film composition/structure, at an arbitrary point along curve a*, b*, or c. Any control function that instantaneously establishes the equilibrium value of one of three variables reduces the dimensionality of the problem, from 3-D (EQΛ space) to 2-D (normally QΛ or EΛ space). Consequently, coulostatic and potential step experiments produce the most readily interpreted film population transients. Techniques based on a timedependent control function, e.g., cyclic voltammetry, produce more complex (less readily interpreted) response functions. These responses will be represented by paths toward equilibrium that involve changes in all three coordinates in EQΛ space. No matter what control function is used, as t f ∞ equilibrium is established and corresponds to E ) constant (η ) 0). Consequently, dE/dt ) dQ/dt ) dΛ/dt ) 0, making dΩ/dt ) 0. Also, Ω becomes a single-valued function of Q (or Λ) and E. This is shown in Figure 2, which is a 3-D plot of Ω for η ) 0 (the equilibrium path for complete redox conversion of the film). Then, the projection of the Ω locus (curve c in Figure 2) on the EQ plane is the “charge isotherm”, which may be rearranged to the form of the Nernst equation. The projection of the Ω locus on the ΛE plane (curve a in Figure 2) is the

Figure 1. Schematic representation of EQΛ space for complete film redox conversion. The dashed line represents the locus of equilibrium compositions, between a fully reduced film at (E0, Q0, Λ0) (O) and a fully oxidized film at (Eeq, Q100%, Λeq) (b). Curves a and a* represent the responses to coulostatic experiments. Curves b and b* represent the responses to potentiostatic step experiments. Curves a and b are the instantaneous responses to the electrochemical control functions. Curves a* and b* are the subsequent responses as the system approaches equilibrium, in planes of constant charge and potential, respectively. Curve c represents the response in EQΛ space to a continuous electrochemical control function experiment, for example cyclic voltammetry.

Figure 2. 3-D representation of the Ω vector (full line), as defined by eq 2. The dashed lines a, b, and c, respectively, are the projections of the Ω vector on the EΛ, QΛ, and EQ planes. For presentational purposes, we show two viewing angles.

“population isotherm.” The latter two statements are only true at equilibrium, since the E axis includes a varying overpotential that becomes zero only at equilibrium. In Section 3, we show how film populations respond to specific electrochemical control functions and relate these responses to elementary kinetic steps. We also give the interrelationships among the dissimilar time-dependent population changes obtained by using different electrochemical techniques to switch electroactive films. These results allow us to select electrochemical control functions and in EQΛ



experiments to achieve mechanistic diagnosis and to obtain kinetic information about specific (electro)chemical steps. 3. Theory: Pump-Probe Transfer Function 3.1. General Considerations. We present a theoretical framework for design and interpretation of EQΛ experiments consisting of two elements. The first, developed in this section, is the mathematical analysis of the influence of the electrochemical control function on the population probe response. It guides the design of EQΛ experiments. The second element, developed in Section 4, is an interpretation of the kinetic EQΛ data that has the goal of assigning elementary kinetic steps to specific chemical processes. It is model-dependent only by virtue of the assumed chemical processes; it presupposes no particular probe technique. The results are given in terms of areal population densities of surface species. We assume that response(s) of the probe technique(s) can be converted to the total population areal density, Λ, which consists of a weighted sum of individual population densities, λj, j ) 1 to n. This approach encompasses all probe techniques, which may be selective (e.g., fluorescence or absorption/reflection spectrometry) or nonselective (e.g., gravimetry). 3.2. Analysis. We consider the most general case in which an EQΛ experiment produces responses that are sensitive to all film population changes.24 In particular, λj and thus Λ reflect the transfers of both charged and neutral species, notably solvent, between a polymer film and its bathing solution. Therefore, the total population (Λ) of all mobile species within an electroactive polymer film can be expressed as a compound function25 of potential (E), and either the charge carried by the polymer (Q) or its time differential current (i).

Λ ) Φ(Q, E) +

∑kjqj(E) ≡ Φ(i, E) + t∑kjij(E)

(5)

Φ represents the total population of net neutral species (salt, solvent, and other neutral molecules) in the film,26 and k is a constant defined by Faraday’s law; Q and E are the functions of time, Q(t) and E(t), respectively. In mechanistic studies, where the system is under kinetic control, one is commonly interested in the time rate of change of the total film population as well as the individual film populations. The time differentials of these quantities are related through eq 6, where dE/dt and di/dt represent arbitrary electrochemical forcing and response functions, or vice versa.

( ) ( ) ( )( ) ( ) ( ) ∑( )

dλj ∂Φ ∂Φ dE ) + kjij ≡ i + dt ∂qj E j ∂E qj dt ∂Φ ∂ij

E

()

dij dij ∂Φ dE + (6a) + kjij + kjt dt ∂E ij dt dt

dΛ ) dt

j

dλj dt

(6b)

The initial condition, Q(t ) 0) ) Qo (or i(t ) 0) ) io), is determined by the history of the system. 3.3. Equilibrium. When equilibrium is established, η ) 0 and Λ, Φ, and Q are uniquely defined through the Nernst equation (in terms of E). Equation 5 then reduces to

Λ ) Φ(E) +

∑kjqj(E)

(7)

However, equilibrium is not readily attained during electrochemical studies of modified electrodes. This is because some

polymer relaxation (or, more generally, film structural) processes commonly occur on time scales much longer than those of electrochemical techniques. The acquisition of a reproducible response, for example, by the frequent practice of potential cycling, is not a sufficient criterion for equilibrium. Instead, this procedure can generates a history-directed “steady state.” Consequently, before applying eq 7, we stress the need to prove equilibrium unequivocally. A simple experimental test is to “open circuit” the film to establish Q ) constant. If E does not then change with time, the film is at equilibrium. Practically, since E may show little sensitivity to neutral species populations, we recommend constant E and Λ as the experimental test for equilibrium. 3.4. Time-Dependent Control Functions. We now consider various special cases of the general eq 6 that correspond to different electrochemical techniques. We show parallel development of the method in terms of both the integral (charge) and differential (current) forms of eq 5. These are the Q formulation and the i formulation, respectively. Each has utility, depending upon the electrochemical control function used. We classify the electrochemical methods according to whether the controlled parameter is potential or charge (directly through Q or indirectly through i). Algebraic expressions (equations) for each special case of eq 6 are collected in Table 1. 3.4.1. Controlled Potential. When the potential is unchanging (dE/dt ) 0), we obtain a simple result. This happens either at constant applied potential or at potential maxima/minima, for example, at turning points in cyclic voltammetry or impedance measurements. When dE/dt is zero, the Q formulation (eq 8) is simpler than the i formulation (eq 9) in Table 1. Eq (8) gives a quantitative EQΛ, criterion for kinetic permselectiVity without solVent transfer. This is a special case, in which transient field effects cause counterion transfer to outrun all other mobile species transfers on short time scales; it is found for poly(vinylferrocene) films subject to a potential step27 or rapid potential sweep.28 Mathematically, counterion transfer alone corresponds to Q ) njFAλj ) kjλj, i.e., Faraday’s law. Then the condition for kinetic permselectivity without solvent transfer is (∂Φ/∂Q)E , kj. This condition corresponds to a plot of Λ vs Q being a straight line whose slope is determined by the molar mass of the transferring ion. We have used this criterion extensively to separate ion and neutral species (primarily solvent) transfers within a limited time range such that no polymer reconfiguration processes occurred.27,28 3.4.2. Controlled Charge or Current. As evidenced by eqs 10-13, considerable simplification results when i or (di/dt) is zero. The case where both are zero, eqs 10 and 11, applies to current interruption in a solution that does not contain an electroactive species. Any population flux under these conditions cannot involve a redox process and must arise because of slow neutral species transfer as the film approaches equilibrium. An important special case occurs when (dΛ/dt) is also zero. It arises if one factor on the right side of eq 10 or 11 is zero. When (∂Φ/∂E)Q is zero, observing a potential transient (dE/dt * 0) is then possible without a population flux. Any such potential change must be a consequence of a change in the formal potential of the polymer, i.e., a polymer structural change (reconfiguration). (We encompass in the term “polymer reconfiguration” processes such as a redistribution of charge or solvent and any intra- or interchain structural change in the polymer framework.) Consequently, in the absence of any film population changes, a transient potential response following a current interruption is diagnostic for a reconfiguration process.



TABLE 1: Expressions for the Population Flux control variable E

condition

dE )0 dt

Q (or i)

technique

( ) (dEdt ) * 0 di i ) 0; ( ) ) 0 dt di i ) 0; ( ) * 0 dt di i * 0; ( ) ) 0 dt di i * 0; ( ) * 0 dt

potentiostatic and potential turning points potentiodynamic coulostatic

Q formulation

∂Φ i + ki ∂Q B

( )

galvanostatic and current turning points current sweep

TABLE 2: EQΛ, Responses Characteristic of Elementary Steps elementary step

response involves changes in

electron/ion transfer solvent transfer polymer reconfiguration

E, Q, and Λ Λ(primarily) and E E only

Such structural changes have been invoked to rationalize electrochemical or other responses of electroactive films,9,15,18,19 but an unambiguous criterion for their identification has not previously been stated. We have observed such behavior for poly(vinylferrocene) films29 and use this characteristic signature below (see Table 2). The alternative mathematical case, dE/dt ) 0, leading to a zero population flux, turns out to have no physical meaning for the following reason. Mathematically, it allows (∂Φ/∂E)Q * 0, and such a change in neutral species population would, according to the definition of chemical potential, lead to a change in potential (dE/dt * 0); this is a logical inconsistency. Transiently zero currents (i ) 0; di/dt * 0) are observed in many electrochemical experiments, for example, during experiments with a periodic control function when the current (as either the control or response function) is instantaneously zero. Cyclic voltammetry is a common example. Other experiments that involve current reversal, current interruption, or charge injection meet these conditions. The Q formulation of eq 12 reveals a powerful diagnostic test. Whenever i ) 0, any population flux must, by definition, be a neutral species (salt and/or solvent) flux. The full benefit of this test is realized in quantitative interpretation of EQΛ data. However, note that a simple visual inspection of raw Λ(t) and i(t) data provides an unambiguous criterion for transfer of neutral species (notably solvent). This process has been somewhat neglected in hyphenated technique studies of modified electrodes, even in those cases (e.g., the EQCM) where the probe is sensitive to solvent transfer. Our diagnostic test is illustrated neatly for electroactive polymer films by comparing the charge and population fluxes found during redox switching. See, for example, the responses of polypyrrole in chloride30 and more complex electrolytes, such as anthraquinone sulfonate30 and p-toluenesulfonate.31 EQCM data for electroactive metal oxide films may also use this test.32,33 Under such circumstances, a plot of population response vs Q (e.g., mass change vs Q found in an EQCM experiment) will have an infinite slope. In a galvanostatic EQΛ experiment (i * 0) or at turning points in the current response to any electrochemical perturbation (di/dt ) 0), the i formulation of eq 13 is simpler than the corresponding Q formulation. (∂Φ/∂E)j can be calculated as a

(8)

full eq 6

(∂Φ ∂E ) ∂Φ ( ∂E )

dE dt dE Q dt Q

zero-crossing points

i formulation

full eq 6

∂Φ ∂i

di di + kt + ki dt dt

( )( ) ( ) E

full eq 6

(10) (12)

(∂Φ ∂E )

i)0

dE dt

(11)

full eq 6

∂E + ki (∂Φ ∂E ) dt i

full eq 6

(9)

(13)

full eq 6

function of time, since dΛ/dt, dE/dt, and i are readily accessible experimental quantities. Kinetic permselectivity here is defined by [(∂Φ/∂E)j(dE/dt)/i] , k. Although the use of galvanostatic control functions in hyphenated technique studies of modified electrodes is relatively rare, kinetic permselectivity has been qualitatively identified at high current density for poly(vinylferrocene) films.34 We believe that the provision of a quantitative criterion for kinetic permselectivity will prompt the wider use of this control function with appropriate population probe techniques. Further interpretation of the data in reference34 is now possible in terms of eq 13. Qualitatively, curvature in Λ-Q (or Λ-t plots in a galvanostatic experiment34) is an unequivocal diagnostic for neutral species transfer. Quantitatively, the difference between the population flux and the ion flux yields the kinetically controlled neutral species (e.g., solvent) flux, dΦ/dt, as a function of potential, i.e., redox composition. Changing the applied current varies the time scale of this kinetic experiment. Furthermore, extraction of dE/dt and extrapolating it to i ) 0 enables one to obtain a thermodynamic quantitysthe slope (dΦ/dE)j at zero current. We note that Φ represents the concentration of water and that the actiVity of water varies logarithmically with potential (according to the definition of chemical potential). Consequently, in principle, the activity coefficient of water can also be determined as a function of potential (redox composition). 3.5. Interrelation of Derivatives. Equation 6 leads to the important conclusion that quantitative interpretation of a dynamic voltammetric EQΛ experiment is complex. However, our analysis reveals that important interrelations exist among data obtained with different electrochemical control functions. First, we note that

∂Φ dE i+( ) ( ) (dΦdt ) ) (∂Φ ∂Q ) ∂E dt E

Q

(14)

Second, (dE/dt) is nonzero in a cyclic voltammetric experiment, as is i (except momentarily after a potential reversal). Consequently, eq 14 shows that the potentiodynamic EQΛ response (described by eq 6) is a composite of the responses from a constant potential experiment (eq 8) and a coulostatic experiment (eq 10), each commencing from the same start state. Graphically, this is shown by curve c of Figure 1. A potentiostatic or a coulostatic control function moves the system within EQΛ space in only two coordinates: either a QΛ plane at fixed E or an EΛ plane at fixed Q. However, a potentiodynamic (e.g., cyclic voltammetric) experiment causes motion of the system within EQΛ space in all three coordinates (see Figure 1, curve c). That motion’s mathematical description and physical interpretation is accordingly more complex (see Section 4, also).



In this section, we described the state of the film through the vector Ω using three control/response parameters (E, Q, and Λ). This conveniently related the film population responses to different pump techniques. No microscopic interpretation of the processes or species involved in producing these responses was made. Therefore, to relate the responses (the various partial derivatives) to rate constants and deduce a mechanism, we now model the system in terms of chemical species and possible mechanistic pathways. Our chosen model considers the redox switching of an electroactive polymer in terms of three elementary processes: electron/ion transfer, neutral species (e.g., solvent) transfer, and polymer reconfiguration. 4. Evaluation of Kinetic Parameters It has long been recognized that when a chemical system is perturbed to a small extent it relaxes back to its equilibrium state exponentially. The two experimentally accessible perturbations are charge (Q) and potential (E). They may be imposed when the system is at a global equilibrium state or a pseudoequilibrium state. There would be no obvious reason in perturbing a transient population state such as might exist during a cyclic voltammogram, since the system is already displaced from equilibrium. The issue in such a case is how to determine the nonequilibrium populations that have been created by the potential program. 4.1. Coulostatic Experiments. In the case of a charge perturbation, we obtain mass and potential relaxation transients that can yield lifetimes for interconversion between products and reactants for each reaction occurring in the system. When there is no accompanying mass transient, the potential transient will characterize structural (reconfiguration) steps. A mass transient characterizes the kinetics of neutral species transfer. Usually solvation changes have a minor effect on the potential of a film, so they may be ignored. In principle, it should be possible to deconvolute the second-order effect of a solventinduced potential transient, where a natural separation based on relaxation times does not occur. 4.2. Potentiostatic Experiments. Here, we obtain mass and charge relaxations. By using eq 5 to combine the mass and charge relaxations, the mass transient can be separated without any ambiguity into separate counterion and solvent transients. 4.3. Dynamic Potential or Current Experiments. In either case, nonequilibrium population states that vary with the time of the experiment are created. Open circuiting the filmed electrode fixes the charge state of the film, and the potential and mass transients will yield relaxation times for these processes. At equilibrium, the film composition can be determined by conventional means so that the population of the film at the instant of open circuiting can be established. In this way, by open circuiting at different times, the path followed in population space by a particular electrochemical program may be established. 5. Scheme of Cubes 5.1. Electroactive Polymer Film Model. Our model of electroactive polymer films (generalized below to other materials) assumes the following three key features. First, both the various reduced and oxidized forms of the polymer can exist as more than one solvated species; second, each of these solvated forms can have more than one polymer configuration; third, the polymer is a single phase.35 We do not discuss here the case of nonpermselectivity.21 There may be a continuum of solvation, configurational, and redox species. Considering this possible complexity, the present

Figure 3. Scheme of cubes representation for conversion of the less solvated, “a” configuration, reduced species Ra to the more solvated, “b” configuration, oxidized species OSb.

state of knowledge, and experimental limitations, we do not attempt to develop a theory that considers these possibilities. Instead, we have elected to simplify our quantitative treatment of the redox conversion of electroactive polymers by assuming that only two solvation states, two configurational states, and two redox levels (fully oxidized and fully reduced) exist. The individual partial derivatives in eqs 8-13, whose values may be determined as described in Section 4, can now be related to the rate of change of the populations of the species in this model. To clarify the chemical features of the model, we use a new pictorial representation, the scheme of cubes. This is the second element of our approach. 5.2. Visual Model. The “scheme of squares”36 has been used to great advantage to visualize the electrode processes for couples whose oxidized (O) and reduced (R) forms can each exist in two forms. Each O-R pair needs to have a very similar molecular geometry to allow electron transfer (x coordinate), and each has a unique standard potential (E°). The y coordinate in the scheme of squares represents another chemical step (e.g., isomerization, complexation, or an acid/base process). We have extended this 2-D visualization to three dimensions,20 to what we term the scheme of cubes. The third dimension (z coordinate) allows us to portray any additional chemical processsin the present instance, film restructuring (e.g. polymer reconfiguration). Figure 3 shows the cube that represents the polymer redox switching model developed here for permselective conditions. Each corner represents a species, that is, the composition at a minimum on the potential energy surface. The edges in the x, y, and z directions, respectively, connect species of a different oxidation level, solvent content, or configuration. These edges represent “valleys” connecting these local minima. As one moves from the edges to faces or the interior of the cube, the “elevation” (potential energy) generally increases, although local minima of high potential energy may exist. The symbols O and R signify the oxidized and reduced forms, respectively. A superscript “S” denotes the more solvated form. The two polymer configurations are designated as “a” and “b”. Quite generally, the elementary steps occur on different time scales. Movement of the system around the cube will be preferentially

10832 J. Phys. Chem. B, Vol. 102, No. 52, 1998 along it edges. “Diagonal” transfers (simultaneous changes in two or more coordinates) are possible, but they are energetically unfavorable. We now give a diagnostic protocol for establishing the details of the redox switching mechanism based on the interrelationships between E, Q, Λ, and t. Unlike our previous approach,27 this new protocol involves extensive use of the time domain, identified earlier16 as a stringent test of any model. Experiments carried out over a range of time scales will identify circumstances under which reaction is confined to an edge, a single face, or multiple faces of the cube. These time scales can be identified by potential and/or charge perturbation experiments. If the experimental time scale is sufficiently short, reaction will be confined to an edge of the cube (1-D), corresponding to the fastest process from the history-determined start state. Lengthening the time scale allows the reaction mechanism to include the next fastest process, i.e., over a cube face (2-D). A longer time scale requires another cube face (3-D) to describe the reaction mechanism. Shifts between edge, face, and cube mechanisms may be signaled by alternate regimes of kinetic and apparent thermodynamic control.21 5.3. Mechanistic Possibilities and Diagnosis. Our previous diagnostic procedure17,27 did not consider polymer reconfiguration; its capabilities were thus limited to restricted time regimes, similar to the scheme of squares type model employed by Peerce and Bard.16 Then we could separate only coupled electron/ion and solvent transfers, implicitly assuming a fixed configuration state and a single start and end state. The cube model described here allows a more sophisticated kinetic analysis. It includes film restructuring (e.g., polymer reconfiguration) and the possibility of multiple start and end states. 5.3.1. Single Start State. Table 2 summarizes the features of the three types of elementary steps involved in translation around the cube and the corresponding EQΛ responses. These response signatures can be exploited for mechanistic diagnosis. Initially, we consider a single predominant reduced state, for example, Ra in Figure 3. Upon oxidation, we assume it will ultimately convert to OSb through a path that depends on the selected electrochemical perturbation. We consider two cases, relating to high and low overpotential conditions, for which the six mechanistic possibilities are shown in Figure 4. First, we discuss the case of a large overpotential, charge injection technique (a coulostatic experiment). Charge injection immediately produces Oa, as shown in the first two frames of Figure 4. On open circuiting, the path to the equilibrium species OSb may go either through OSa or through Ob. In the former case, a mass change precedes a potential change,37 and in the latter case, the reverse is seen (see Table 2, relating chemical processes and experimental responses). If solvation and reconfiguration occur on similar time scales, both the mass and potential transients will overlap. The coulostatic method, with attendant mass and/or potential responses, gives the time scale of these various processes and identifies the path followed in attaining equilibrium. This mechanistic separation, based on time scales, is described in section 4. Provided an appropriately large overpotential is employed, a potential step would give the same result as the coulostatic experiment. However, when one moves to low overpotential conditions or conditions of partial redox conversion, the coulostatic and potentiostatic control functions give very different responses as we now discuss. The distinction is that high overpotential conditions (however achieved) always result in rapid electron transfer being the first step (ECC′ and EC′C mechanisms), whereas low overpotential conditions do not


Figure 4. Cube representations of the six mechanisms for redox switching of the single start state Ra to the single end state OSb. Labels e, S, and P, respectively, denote changes in redox, solvation, and (polymer) configurational states. The first two panels show the two mechanisms (ECC′ and EC′C) possible under high overpotential conditions. The remaining four panels show the four mechanisms (CEC′, CC′E, C′EC, and C′CE) possible under low overpotential conditions. (For clarity, the species labels are omitted from the cube corners; these are as shown in Figure 3.)

necessarily cause electron transfer to be the first step (CEC′, CC′E, etc. mechanisms, shown in the last four frames of Figure 4). Hence, we again consider the case of a single starting reduced species Ra that ultimately converts to OSb, but we now subject the film to an electrochemical perturbation in which the charge state of the polymer changes at a finite rate. This might be, for example, a potential sweep or a galvanostatic experiment in which the overpotential is small in at least part of the experiment. During such a perturbation, there may be sufficient time for the electron/ion transfer step to be preceded or followed by neutral species transfer and/or polymer reconfiguration steps. Under this scenario, the mass response can overlap either the current response (potential control conditions) or the potential response (current control conditions). On opening the circuit, more than one of the possible oxidation states is likely to be populated so that relaxation toward the equilibrium state may produce overlapping mass and potential responses. Consequently, we conclude that the application of a large oVerpotential immediately followed by open circuiting the filmed electrode (a coulostatic experiment) is optimum. When converting any starting reduced state to its final oxidized state (or vice versa), this kind of experiment is most likely to identify which of the two paths on the oxidized face of the cube is followed. Conversely, any low overpotential experiment, of which cyclic voltammetry is a prime example, will not allow clean separation of the various kinetic steps.

Redox Switching of Electroactive Films 5.3.2. Multiple Start States. We now consider the possibility that variously solvated or configured reduced forms exist at the instant an electrochemical experiment is begun. This situation may arise because the various reduced forms are in equilibrium with each other or the prior electrochemical history has created nonequilibrium populations of these reduced forms. The presence of multiple start states will produce a superposition of responses that are unlikely to be cleanly separated in time. As we concluded above for the case of a single start state, the best chance of achieving this will be through high overpotential experiments. Even if one is able to effect this separation, it will undoubtedly be difficult (though in principle possible) in the case of multiple start states to attribute a given elementary step to a given start state. We first assume the two constraints of a large overpotential experiment (of finite duration) and the ultimate generation of a common final state. Then, oxidation will instantaneously convert each R species to its O partner of identical solvation and configuration state. The initial instantaneous left/right translation in the scheme of cubes precludes prior interconversion of reduced start states. There is an instantaneous mass change caused by the transfer of counterions between the film and the bathing solution whose value is determined by the instantaneous charge passed. Subsequently, the four oxidized products will convert to OSb through paths that depend on the electrochemical perturbation and the rates of chemical interconversion, as discussed below in terms of Figures 3 and 4. In the case of a low overpotential experiment, interconversion of start states becomes possible because the relative time scales of the electrochemical and the nonelectrochemical steps may now overlap. Thus, using techniques with appropriately smaller overpotentials will increase the lifetimes of the start states and allow them to interconvert. Under these circumstances, this generates 11 possible parallel mechanisms for the conversion of the reduced states Ra, Rb, RaS, and RbS to the oxidized state ObS. Six of these mechanisms originate from state Ra, two from state Rb, two from state RaS, and one from state RbS. It is worth noting at this point that the continuous spectrum of overpotentials produced by a potentiodynamic sweep will generate a timedependent ratio of chemical to electrochemical time scales. Thus all the various (electro)chemical processes are not likely to be cleanly separated in the time domain. Finally, if we admit the possibility of multiple final states, analogous arguments to those of the previous paragraph lead to a total of 44 possible pathways for the redox half-cycle that involves four product states. The above considerations apply to a redox half-cycle. When one considers a complete redox cycle, the mechanistic possibilities multiply. In the general case of multiple starting and final states, probability arguments analogous to those used above result in a set of 1936 possible mechanisms. Given the experimental difficulties of reproducing film history, we are therefore not surprised that experiments on nominally “identical” films frequently (in fact, almost invariably) follow different pathways among these 1936 possible mechanisms allowed for a complete redox cycle. For example, consecutive cyclic voltammograms become identical experiments only after steadystate populations of R and O have been established. Generally, as the relationships of Table 1 show, steady-state populations are sensitive functions of electrochemical control function, film history, and experimental time scale. Associating the responses in these cases to individual mechanisms is clearly challenging under low overpotential conditions.

J. Phys. Chem. B, Vol. 102, No. 52, 1998 10833 In the case of multiple starting states, interconversion between starting states may occur during the time required to complete the redox switching process. In one case that has been studied,29 which started with a nonequilibrium partially oxidized film, “ebb and flow” of charge between oxidized and reduced species occurred on open circuiting. This situation was the result of interconversion between species having the same nominal charge state. It will occur whenever interconversion of species of a given oxidation level is faster than reaction paths involving electron transfer to or from the other level. History effects correspond to nonequilibrium occupation of states within the cube. In the specific context developed here, this corresponds to nonequilibrium solvation and polymer configuration states within a given redox state. They are equivalent to local minima in the population space. They represent pseudoequilibrium states that are slow to convert to the true equilibrium state. Consequently, they may be difficult to detect through a short term electrochemical response.22 However, the EQCM is ideally suited to detecting these effects, i.e., Λ ) M, typically through evolution of the EQM relationship. This has been observed under certain conditions for films of Os(bpy)22+-poly(vinylpyridine),22 Nafion-Os(bpy)32+,38 and polypyrrole.39 The EQCM technique also provides evidence for progressive partition of species into the film from solution. The evolution of mass-potential curves is characteristic of this situation and is illustrated by the partition of Ru(NH3)63+ between aqueous solutions and polypyrrole films.40 These examples illustrate the central role that solvent plays in film dynamics. Although our purpose here is not focused on experimental methodology, we note the implication that the probe technique must be capable of quantitating solvent transfer dynamics. 5.4. Generalization to Other Processes. We have illustrated our approach by considering (i) a specific material (electroactive polymers), (ii) an unspecified population probe, and (iii) specific elementary chemical steps (solvation and reconfiguration). However, our approach is intrinsically much more general. First, we could have chosen materials other than electroactive polymers so that “reconfiguration” could correspond to some other structural change. For other molecular species, notably organic materials, one chemical step could involve protonation and the structural change might be an isomerization. For redox conversion of electroactive metal oxide films, such as nickel oxide41 and lead oxide42 that are very important in battery technology, the structural change might include a phase change. As an aside, we note that phase separation may occur under kinetically controlled conditions in a dynamic electrochemical experiment and equilibrium may not be reached rapidly. For example, transient phase separation has been observed during galvanostatically controlled cyclic redox switching of poly(vinylferrocene) in aqueous LiClO4 solution.43 In our model, phase separation is, in principle, possible along each coordinate of the cube. We do not explicitly discuss possible situations in which various species segregate into different phases; phase separation does not preclude the cube equilibria from being established between the phases, subject to the phase rule. For example, consider the case that all oxidized species are miscible and all reduced species are miscible but no oxidized species is miscible with a reduced species. This yields a reduced phase and an oxidized phase, between which electron exchange occurs. We do not dwell on the phase diagrams that would result from a situation in which one or more of the eight species in the cube segregate into a different phase. It might be signaled by


Figure 5. Cube representation for redox switching of poly(thionine) in aqueous acetic acid. L and T denote the reduced (leucothionine) and oxidized (thionine) states, respectively. Superscript s designates the more solvated state. Subscript a denotes acetic acid coordinated state. The mechanistic pathway for the redox cycle, based on reference 45, is illustrated by the heavy arrows.

the E-Q behavior (invariant potential or activity42 with extent of conversion) or by discontinuous E(t) or Λ(t). Second, we could have considered redox processes involving any two chemical processes without a structural change. This case is nicely illustrated by the poly(thionine) system in aqueous acetic acid medium, whose redox switching mechanism we studied using the EQCM.44,45 Oxidation involves coupled electron/proton removal, solvation change (water entry), and coordination of acetic acid to the redox sites; reduction involves the reverse processes. Our previous description45 of the mechanism was based on a scheme of squares visual approach. Accordingly, we were obliged to employ a set of cartoonlike pictures to visualize the mechanism. Here, with the aid of the scheme of cubes, we use the x, y, and z axes, respectively, to represent coupled electron/proton transfer, solvent transfer, and acetic acid coordination. Four equilibrium constants describe the coordination reactions for the four pairs of species on the left and right faces of the cube. The mechanism is shown clearly by a single figure. Thus, Figures 6 and 7 of reference45 can now be encapsulated within Figure 5. Another important case involves the redox switching of electroactive films under nonpermselective conditions. We previously represented this situation via a 4-D hypercube21 in which we added salt transfer to the set of three elementary steps chosen here. We speculate that there are time regimes in which salt transfer may occur with the film “frozen” in one (historydetermined) structure. For these time regimes, a scheme of cubes model describes systems in which the z axis represents salt transfer, rather than reconfiguration. Now, a set of four equilibrium constants describes salt partitioning between the corresponding pairs of species on the upper and lower faces of the cube. 5.5. Population Probes. One of the most frequently used in situ probe techniques for studying the switching of electroactive polymers is the EQCM,46-48 i.e., gravimetry. It provides straightforward and sensitive quantitation of mobile species in a surface-immobilized film through crystal resonant frequency changes. The underlying assumption in converting EQCM frequency data to mass change data using the Sauerbrey equation is that the film behaves as an elastic solid; this can be confirmed by making crystal impedance measurements. The obvious advantage of the EQCM is the facility to monitor solvent transfer in situ. Under permselective conditions, which may be readily engineered by control of electrolyte composition, this is unequivocally derived from film mass changes. Also, it

Bruckenstein and Hillman is straightforward, from correlation of mass and charge data, to determine which ion (anion or cation) satisfies electroneutrality within the film as a function of time and electrochemical conditions. The general capability of the EQCM is not shared by other techniques. Despite the wide qualitative application of the EQCM, little appreciation seems to exist for the quantitative information it can yield. In particular, we point out that since molar mass is environmentally invariant, EQCM responses directly yield population changes. In terms of eq 7, this corresponds to absolute values of kj directly yielding λj. In cases where more than one species transfers, transient experiments can separate the component processes. Normally, correlation of mass and charge data allows the species to be identified, e.g., counterion and solvent.28 On this basis, we have used the EQCM to study overall changes in solvent49 and salt50 populations under thermodynamic conditions. Under kinetically controlled conditions, the EQCM has been used to probe phenomena such as ebb and flow of solvent17 and kinetic permselectivity.34 A significant limitation of the EQCM is the difficulty of deconvoluting mass changes accompanying multiple species transfers, e.g., in mixed electrolytes. Accordingly, techniques offering specificity of detection are attractive. A wide range of spectroscopic methods,1,2,5 particularly UV/visible and infrared spectroscopies in both transmission and reflection modes, have been used to study electrochemically controlled interfaces. Although these methods have generally been employed qualitatively (usually to elicit structurally related information), on occasion they51 have been employed quantitatively. Their quantitative use requires a calibration experiment to establish the relationship between the spectroscopic response and film population changes. For example, the measurement of absorbance change as a function of electrochemical charge gives the extinction coefficient () for the species of interest in the interfacial region. Thus, the constant “k” in eq 5 for the measured absorbance (A) would be /F. Extension of this notion to other population probes, such as fluorescence, ellipsometry,3 or neutron reflectivity52 is straightforward. It would generate expressions for k in terms of extinction coefficient and quantum efficiency, complex refractive index, or scattering length density, respectively. This approach assumes that the proportionality constant between probe response and areal population does not change with total population, an assumption that may be difficult to confirm. For example, are there solvatochromic effects in absorption measurement, or are fluorescent lifetimes environmentally sensitive? We note that techniques based on interactions with the nucleus, rather than valence electrons, will be less vulnerable in this respect. In general, use of any in situ technique requires knowing the values of kj that relate the probe response to λj and using the latter values in the definition of Λ, (eq 1c). 6. Conclusions A general mathematical description has been developed for electrode film population responses to different electrochemical control functions. A 3-D vector describes the compositional state (chemical potential) of the system. The analysis gives the film population response to potential, current, or electrochemical charge. Compositional space can be explored under transient conditions using a variety of electrochemical control functions. Coulostatic and potentiostatic step experiments yield responses that allow the most straightforward qualitative separation of the elementary steps involved in film redox switching and allow the quantitative determination of the relevant kinetic parameters.

Redox Switching of Electroactive Films Other electrochemical control functions are combinations of these two. Utilizing this approach requires an in situ probe technique that has a single-valued relationship between the probe response and population. Although the theory is general, we discuss it specifically with regard to a model for redox switching of electroactive films that incorporates three elementary steps. These elementary steps are exemplified by coupled electron/ ion transfer, solvent transfer, and film structural change. The theory can be used to interpret data from any probe technique that yields information about the population(s) of one or more mobile species in the film. A scheme of cubes representation visualizes the mechanistic possibilities for redox switching of an electroactive polymer film. The three cube axes in the model are coupled electron/ ion transfer, solvent transfer, and polymer reconfiguration. The basic model or simple extensions can rationalize a variety of phenomena that might otherwise be ascribed to “irreproducibility”. In particular, the model highlights the importance of previous film history and experimental time scale, which determine the populations of the initial state(s) and “final” state(s), respectively. The combined 3-D vector/scheme of cubes approach applies generally to the redox switching of surface films accompanied by any pair of chemical steps, represented by all permutations of ECC′ mechanisms. For example, this pair could be selected from solvent transfer, salt transfer, transfer or coordination of a neutral molecule, polymer reconfiguration, or isomerization. Acknowledgment. We thank the National Science Foundation (Grant CHE 9616641) for support of this work. References and Notes (1) Heineman, W. R.; Hawkridge, F. M.; Blount, H. N. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1984; Vol. 13, p 1. (2) Gale, R. G. Spectroelectrochemistry: Theory and Practice; Plenum: New York, 1988. (3) Gottesfeld, S. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1989; Vol. 15, p 143. (4) Hubbard, A. T. The Handbook of Surface Imaging and Visualization; CRC Press: Boca Raton, 1995. (5) Christensen, P. A.; Hamnett, A. Techniques and Mechanisms in Electrochemistry; Blackie: Glasgow, 1994. (6) Murray, R. W. Molecular Design of Electrode Surfaces; Wiley: New York, 1992. (7) Hillman, A. R. In Electrochemical Technology of Polymers; Linford, R., Ed.; Elsevier Applied Science Publishers: London, 1987; p 103. (8) Li, H. S.; Garnier, F.; Roncali, J. Solid State Commun. 1991, 77, 811. (9) Odin, C.; Nechtstein, M. Phys. ReV. Lett. 1991, 67, 1114. (10) Sleigh, A. K.; Murray, J. J.; McKinnon, W. R. Electrochim. Acta 1991, 36, 1469. (11) Skaarup, S.; West, K.; Zachauchristiansen, B.; Jacobsen, T. Synth. Met. 1992, 51, 267. (12) Zhang, W. B.; Dong, S. J. Electrochim. Acta 1993, 38, 441. (13) Fan, F.-R. F.; Mirkin, M. V.; Bard, A. J. J. Phys. Chem. 1994, 98, 1475. (14) Pyo, M. H.; Reynolds, J. R.; Warren, L. F.; Marcy, H. O. Synth. Met. 1994, 68, 71. (15) Pohjakallio, M.; Sundholm, G.; Talonen, P.; Lopez, C.; Vieil, E. J. Electroanal. Chem. 1995, 396, 339. (16) Peerce, P. J.; Bard, A. J. J. Electroanal. Chem. 1980, 114, 89. (17) Hillman, A. R.; Swann, M. J.; Bruckenstein, S. J. Phys. Chem. 1991, 95, 3271. Ebb and flow is a kinetic situation that arises when the

J. Phys. Chem. B, Vol. 102, No. 52, 1998 10835 electroneutrality condition within the film is transiently satisfied by transfers of a very mobile ion, which is then displaced from the film by a less mobile ion that is needed to satisfy thermodynamic equilibrium. (18) Chinn, D.; DuBow, J.; Liess, M.; Josowicz, M.; Janata, J. Chem. Mater. 1995, 7, 1504. (19) Chinn, D.; DuBow, J.; Li, J.; Janata, J.; Josowicz, M. Chem. Mater. 1995, 7, 1510. (20) Hillman, A. R.; Bruckenstein, S. Faraday Trans. 1993, 89, 339. (21) Hillman, A. R.; Bruckenstein, S. Faraday Trans. 1993, 89, 3779. (22) Clarke, A. P.; Vos, J. G.; Hillman, A. R. J. Electroanal. Chem. 1993, 356, 287. (23) Mo, Y. B.; Hwang, E.; Scherson, D. A. J. Electrochem. Soc. 1996, 143, 37. (24) The issue of rigidity does not compromise our description either in general terms (of film populations) or in the specific terms employed here (mass changes); in the latter case, crystal impedance measurements provide the capability to separate mass changes from film viscoelastic phenomena. (25) Courant, R.; John, F. Introduction to Calculus and Analysis; Wiley: New York, 1974; Vol. 2, Chapter 3. (26) Hillman, A. R.; Loveday, D. C.; Swann, M. J.; Bruckenstein, S.; Wilde, C. P. Faraday Trans. 1991, 87, 2047. (27) Hillman, A. R.; Loveday, D. C.; Bruckenstein, S. J. Electroanal. Chem. 1991, 300, 67. (28) Hillman, A. R.; Loveday, D. C.; Bruckenstein, S.; Wilde, C. P. Faraday Trans. 1990, 86, 437. (29) Krtil, P.; Bruckenstein, S.; Hillman, A. R. J. Phys. Chem., in press. (30) Hepel, M. Electrochim. Acta 1996, 41, 63. (31) Dusemund, C.; Schwitzgebel, G. Synth. Met. 1993, 55, 1396. (32) Bohnke, O.; Vuillemin, B.; Gabrielli, C.; Keddam, M.; Perrot, H. Electrochim. Acta 1995, 40, 2765. (33) Inaba, H.; Iwaku, M.; Tatsuma, T.; Oyama, N. J. Electroanal. Chem. 1995, 387, 71. (34) Hillman, A. R.; Hughes, N. A.; Bruckenstein, S. Analyst 1994, 119, 167. Kinetic permselecticiy is a special situation in which transient field effects cause counterion transfer to outrun all other mobile species transfers on short time scales. (35) Identification of multiple phase behavior is discussed below. We note that the addition of a phase transformation corresponds to the addition of another dimension to the problem. The creation of a species in the “wrong” phase corresponds to generation of a metastable species, which must then participate in a phase transfer. This is entirely analogous to a reconfiguration, for example, and corresponds to increasing the dimensionality of the (hyper) cube by 1 for each additional phase present. This in no way compromises the validity of the model, though we concede it complicates interpretation of experimental data. (36) Jacq, J. Electrochim. Acta 1967, 12, 1345. (37) Solvation will produce a small potential change that is much less than the one seen when there is a polymer configurational change. (38) Shin, M.; Kim, E.-Y.; Kwak, J.; Jeon, I. C. J. Electroanal. Chem. 1995, 394, 87. (39) Schiavon, G.; Zotti, G.; Comisso, N.; Berlin, A.; Pagani, G. J. Phys. Chem. 1994, 98, 4861. (40) Reynolds, J. R.; Pyo, M.; Qiu, Y.-J. Synth. Met. 1993, 55, 1388. (41) Bode, H.; Dehmelt, K.; Witte, J. Electrochim. Acta, 1966, 11, 1079. (42) Robinson, J.; Walsh, F. C. Corros. Sci. 1993, 35, 791. (43) Daum, P.; Murray, R. W. J. Phys. Chem. 1981, 85, 389. (44) Bruckenstein, S.; Wilde, C. P.; Hillman, A. R. J. Phys. Chem. 1990, 94, 6458. (45) Bruckenstein, S.; Wilde, C. P.; Hillman, A. R. J. Phys. Chem. 1993, 97, 6857. (46) Ward, M. D.; Buttry, D. A. Science 1990, 249, 1000. (47) Schumacher, R. Angew. Chem., Int. Ed. Engl. 1990, 29, 329. (48) Buttry, D. A. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1991; Vol. 17, p 1. (49) Bruckenstein, S.; Hillman, A. R. J. Phys. Chem. 1988, 92, 4837. (50) Bruckenstein, S.; Hillman, A. R. J. Phys. Chem. 1991, 95, 10748. (51) Kaufman, F. B.; Schroeder, A. H.; Engler, E. M.; Kramer, S. R.; Chambers, J. Q. J. Am. Chem. Soc. 1980, 102, 483. (52) Richardson, R. M.; Swann, M. J.; Hillman, A. R.; Roser, S. J. Faraday Discuss. 1992, 94, 295.

Hyphenated Techniques in Dynamic Electrochemistry. 1. Mechanistic

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