modelling mobile species population changes in electroactive films under thermodynamically and...

Modelling mobile species population changes in electroactive filmsunder thermodynamically and kinetically controlled conditions

Angela Jackson a, A. Robert Hillman a,*, Stanley Bruckenstein b, Irena Jureviciute c

a Department of Chemistry, University of Leicester, Leicester LE1 7RH, UKb Department of Chemistry, University at Buffalo, Buffalo, NY 14260-3000, USA

c Institute of Chemistry, Vilnius 2600, Lithuania

Received 23 November 2001; received in revised form 25 January 2002; accepted 27 January 2002

Abstract

We describe a new model for electroactive film mobile species (ion and solvent) populations under a range of thermodynamically

and kinetically controlled conditions, that allows the film state to be visualized in 3D E ,Q ,L -space, where L represents film

composition. Under thermodynamically controlled conditions, when film composition is a single-valued function of potential, the

model allows one to calculate and represent solvent content as a function of redox state under both ideal and non-ideal conditions.

Curvature in the solvent population-charge relationship can result from either thermodynamic or kinetic factors. Under kinetically

controlled conditions (with no transport limitations) the model can describe slow electrochemical and solvation processes, both in

the absence and presence of non-ideal solvation thermodynamics. The film compositional signature in E ,Q ,L -space allows visual

diagnosis of thermodynamic versus kinetic control and the identification of various possible phenomena; these include film

reconfiguration, ion and solvent trapping, relative rates of ion versus solvent transfer, and relative rates of solvent entry versus exit.

Application of the model is demonstrated for the case of polyvinylferrocene films exposed to aqueous media under permselective

conditions. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Electrochemical quartz crystal microbalance; Simulation; Kinetic model; Polymer modified electrode; Solvation; Polyvinylferrocene

1. Introduction

1.1. Overview

In this study we attempt to model the population

changes of mobile species (ions and solvent) withinelectroactive films as they undergo electrochemically

driven redox switching. It is generally recognized that

this overall process requires the exchange with the

bathing solution of ions and solvent, respectively, in

order to satisfy electroneutrality and activity constraints

[1,2]. More recently, it has been recognized that these

exchange processes may drive (or be driven by) struc-

tural changes within the film (which we choose to referto as ‘reconfiguration’ processes [3]). Quite generally,

these various processes occur on very different time

scales, which may in turn respond in quite different ways

to (electro)chemical or other physical control para-

meters. Consequently, both the acquisition and unequi-

vocal interpretation of electroactive film responses is

complex.

In kinetically controlled situations it is not always

trivial to arrive at clear mechanistic conclusions*/

indeed it may not even be clear whether the system is

under kinetic or thermodynamic control. Hence, the

purposes of this work are, firstly, to provide simple

experimentally applicable diagnostics for thermody-

namic versus kinetic control, secondly, to provide a

means of making qualitative mechanistic deductions

under kinetically controlled conditions and, thirdly, to

point the way towards extracting the relevant kinetic or

thermodynamic parameters. For illustrative purposes

we restrict discussion to the case of a redox polymer

film, but the methodology we develop is equally applic-

able to any type of electroactive film, including redox

* Corresponding author. Tel.: �/44-116-252-2144; fax: �/44-116-

252-5227.

E-mail address: [email protected] (A.R. Hillman).

Journal of Electroanalytical Chemistry 524�/525 (2002) 90�/102

www.elsevier.com/locate/jelechem

0022-0728/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 0 7 2 8 ( 0 2 ) 0 0 7 6 5 - 9

mailto:[email protected]

polymers, conducting polymers, metal oxides and inor-

ganic complexes.

Film properties, and in particular how they dictate

performance in a given application, are determined byfilm structure and composition, which can be repre-

sented in terms of redox state and solvation. Conse-

quently, we develop a methodology that allows

straightforward visualization of overall film composi-

tion (and its constituents) as a function of electroche-

mical control parameters. It is the relationship between

these control and response parameters that provides

mechanistic insight. An issue of particular importance isthe experimental time scale, since mobile species trans-

fers will likely occur on different time scales to each

other and certainly to the energetically more demanding

process(es) of polymer reconfiguration. Our earlier work

[4,5] has alluded to the interplay of these processes and

the expectation of alternation of (quasi-)equilibrium and

kinetic control as the time scale is systematically

shortened. However, we have not previously been ableto describe these phenomena quantitatively: the present

work is a step in this direction.

1.2. Previous studies

Electrochemical (‘IVt ’) techniques*/notably cyclic

voltammetry*/have been used widely to extract overall

kinetic information for electroactive films [6�/8]. How-

ever, information on the elementary steps (ion transfers,solvent transfer, structural change, etc.), derived from

the associated film population changes, requires specific

in situ non-electrochemical probes [9] such as the

electrochemical quartz crystal microbalance (EQCM)

[10,11], UV�/vis spectroscopy [12], ESR spectroscopy

[13,14], ellipsometry [15], and neutron reflectivity [16].

Taking one example, the EQCM [10,11,17�/19] has

become a routine technique in determining ion andsolvent fluxes through the associated film inertial mass

changes. Indeed early studies by Buttry and coworkers

of poly(vinylferrocene) (PVF) [20] and polyaniline [21]

films were instrumental in progressing the recognition of

solvent transfer from a qualitatively acknowledged

notion to the quantitatively monitored phenomenon

represented in more recent literature [3,10,22�/24].

Given the demonstrated capabilities of these in situnon-electrochemical techniques in observing aspects of

film composition and dynamics, surprisingly little pro-

gress has been made in terms of modelling the resultant

data. The majority of theoretical studies have focused

on the analysis of voltammetric I �/E responses, and the

underlying coupled motions of electrons and counter

ions. Examples include waveshape analysis for a rever-

sible and an irreversible surface redox reaction [25�/27]and electron transfer kinetics at the electrode j film

interface [28]. In some cases, the focus of interest was

a diffusional process, e.g. using a multilayer model [29]

or considering a film containing non-equivalent redox

sites [30] while in others the emphasis was on kinetics

[22,23]. Some mathematical approaches have been

analytical in nature, including the use of semi-integralmethods [31], while others have been numerical [30,32].

1.3. Objectives

Previous work has tended to centre on investigation

of electron/coupled counter ion kinetics; this is the

natural focus for analysis of data from IVt techniques.

We aim to couple this with analysis of associated solvent

population changes and polymer reconfiguration, toprovide a more complete picture of events during film

redox switching. The microscopic processes underlying

the macroscopic responses can then be visualised by the

previously described scheme-of-cubes [5].

We commence with consideration of the thermody-

namically controlled situation (manifested in the re-

sponse as t 0/�) that provides the boundary condition

to which a kinetically controlled system aspires, thenconsider some kinetically controlled situations.

Although most of the theoretical models discussed

above and elsewhere do offer the opportunity to

consider partially redox-converted films, almost all the

interpretations of experimental data have focused on

‘end-to-end’ (i.e. complete) redox transformations. In

the present work, we consider partial redox conversion

in both the theoretical and experimental contexts; thisoffers the crucial opportunity to explore non-linear

charge-composition effects, the kinetic or thermody-

namic origins of which are commonly confused. For

reasons discussed below, our coverage of non-equili-

brium cases is restricted to those in which transport

processes are not rate limiting.

2. The model

Quite generally, we represent the compositional and

redox state of the film at any time (t ) by a vector in 3D

E ,Q ,L -space [33], where E (V) is the applied potential,

Q (C cm�2) is the charge density and, in the most

general case, L encapsulates all aspects of the film

composition. L can be considered to contain theindividual mobile species population(s), Gi (mol

cm�2). In this paper, we consider an electroactive

(polymer) film under permselective conditions, so that

film compositional (L ) changes are associated only with

counter ion and solvent transfers and there is no

contribution from salt movement [34]. In this particular

circumstance, the value of Q defines the ion population

(say, counter anion population GA) through Faraday’slaw, so L immediately represents film solvent popula-

tion, GS. We choose to define L and Q to be zero for the

reduced state of the film; for the illustrative ‘�/1/0’-type

A. Jackson et al. / Journal of Electroanalytical Chemistry 524�/525 (2002) 90�/102 91

redox system we consider later, this corresponds to a

film with zero ion content. Consequently, the state of

the system at any instant can be described in E ,Q ,L -

space by a vector V(t):

V(t)�Q(t)i�E(t)j�L(t)k (1)

i , j and k are the unit vectors for the Q -, E - and L -axes.

3. Thermodynamically controlled responses

3.1. Film ion population

Since the relationship between film redox state (under

permselective conditions, equivalent to ion population)

has been discussed by many authors, we simply sum-marize what is required to develop the (new) aspects of

solvation. For the surface-bound redox process:

RedUOx�e� (2)

the populations (G /mol cm�2) of redox species Ox and

Red respond to potential (E ) in a Nernstian manner:

E � E��RT

nFln

GOx

GT � GOx

(3)

where the total population of redox sites, GT�/GOx�/

GRed and the remaining symbols have their usualmeanings. Here we will assume that there are no, or

equal, interactions between redox sites; such differing

interactions could readily be included (as has been well

illustrated by several authors [35,36]), but in the present

analysis this would cloud the algebra without adding

new insight. Based on Faraday’s law, the equilibrium

film charge density also follows a Nernstian response:

E�E��RT

nFln

Q

QT � Q(4)

where QT is the charge density required for total film

redox conversion.

Based upon Eqs. (3) and (4), the 3D locus of film

composition, V (E ,Q ,GOx) for Eq. (2) is illustrated inFig. 1 for the case of a one-electron couple in a polymer

film of typical coverage GT �/10�8 mol cm�2. In this

simple case L reduces to GA, which is (through the

electroneutrality constraint) equivalent to GOx. 2D

projections on the E ,Q -, E ,GOx- and Q ,GOx-planes are

also shown. As required of thermodynamic control, the

locus is single valued, independent of time scale and

direction of redox switching. The E ,Q -projection corre-sponds to the Nernst equation and the linearity of the

Q ,GOx-projection is simply a statement of Faraday’s

law, GA�/Q /F .

3.2. Film solvent population

For convenience (and representative of the frequency

change data one would obtain from the EQCM) we

consider the change in solvation consequent upon redox

switching. For the purpose of illustration, we consider

the case in which the oxidised film is more solvated; thiscorresponds to the experimental example we will con-

sider later, but transposition from the case of a more

lyophilic oxidised film to a more lyophilic reduced film

is trivial. Thus, we write the equilibrium for redox-

induced film solvation state change:

Ox�xSUOxS (5)

where Ox represents an unsolvated oxidised site, xS (in

which x may be non-integer) represents the change in

solvent (S) content per redox site and the subscript Sdenotes a solvated species. In reaction (5), we denote the

equilibrium constant by KS and make no distinction

between bound and ‘free’ solvent within the film. In the

simplest, ‘ideal’ case in which each site and its local

environment (including solvent) behaves independently,

there is a linear relationship between film solvent

content and charge. We can then express the solvent

population change, GS, as:

GS��

KS

KS � 1

�xQ

F(6)

More generally, the solvent population change may not

be linearly related to the charge density, i.e. the activity

coefficient for solvent within the film may be a function

of charge state within the film. Based on the behaviourof solutions, in which activity coefficients vary non-

linearly with ionic strength, this is more likely to be the

case than the ideal situation represented by Eq. (6).

Fig. 1. Redox state V vector for the equilibrium path taken upon film

redox conversion. Axes x , y and z , respectively, represent potential

(E ), charge density (Q ) and population of oxidised species (GOx).

A. Jackson et al. / Journal of Electroanalytical Chemistry 524�/525 (2002) 90�/10292

Now, the solvent population will be related to some

function of the extent of redox state switching. We

represent this by the general expression:

GS��

KS

KS � 1

�xQT

Ff (Q=QT) (7)

where the non-idealities are contained within the func-

tion f(Q /QT), that varies from zero to unity as redox

conversion takes place.The complexities of solution phase chemistry are such

that there are no generally applicable equations for

activity coefficients in anything other than dilute solu-

tions. At the effective concentrations associated with

solvated polymer films*/and with the other complex-

ities they bring*/there is no immediate prospect of an

analytical expression for f(Q /QT). We are therefore

restricted to exploring the type of thermodynamicresponse one might see for some empirical expressions,

although we note that if a general analytical solution

emerges it could immediately be inserted into Eq. (7).

For the purposes of illustration, we explore expressions

of the form:

f

�Q

QT

��

�Q

QT

�m

(8)

where m is a ‘non-ideality parameter’ that defines the

relationship between solvent population change and

redox state and Eq. (6) is the ideal case that m is unity.

Since activity coefficients can vary continuously withcomposition, m can in principle take any value. Figs. 2

and 3, respectively, show the results of calculating the

equilibrium solvent content (using Eqs. (7) and (8) for

reaction (5)) as a function of redox state for values of

m �/1 and m B/1. The key result is that the solvent

content is (except for the special, ideal case of m�/1)

non-linear with charge density. This is most obviously

seen in the Q ,GS-projection (effectively, a DM �/Q or F �/

Q plot for an EQCM experiment), but also has

significant effects on the 3D equilibrium compositional

locus traced out during redox conversion. Thus we

arrive at the important conclusion that curvature per se

is not a diagnostic for kinetic effects; the test is whether

there is hysteresis for the oxidation and reduction half-

cycles. An additional feature is that, regardless of the

value of m , there is no change in the Q ,E -projection (the

Nernst equation for the system, which is insensitive to

solvent transfer).

Figs. 2 and 3 were calculated for the case that KS0/�,

i.e. complete solvation of all oxidized sites. In this case,

the first bracket on the right hand side of Eq. (7)

simplifies to unity. Physically, this corresponds to the

simpler case of interconversion of ‘single start and end

states’, which we have discussed elsewhere [4]. In the

event that there is incomplete solvation at equilibrium

(‘multiple start and end states’ [4]), then the solvent

population simply scales with the factor KS/(KS�/1);

visually, this corresponds to a vertical ‘compression’ of

the vertical coordinates in Figs. 2 and 3. For example, in

the case that KS�/1, GS is decreased by a factor of two,

and in the case that KS0/0, the figures collapse to a 2D

E ,Q -plot representing only coupled electron/ion trans-

fer.

Fig. 2. Equilibrium compositional path describing redox conversion

and solvation of an electroactive film for solvent non-ideality para-

meter values m �/1: m�/5 (diamonds), m�/4 (circles), m�/3 (trian-

gles) and m�/2 (squares). Solid lines represent 3D V -vector; points

represent corresponding 2D projections on each plane. Data calculated

using Eqs. (7) and (8) with KS�/� for reaction (5).

Fig. 3. Equilibrium compositional path describing redox conversion

and solvation of an electroactive film for solvent non-ideality para-

meter values m B/1: m�/1/5 (diamonds), m�/1/4 (circles), m�/1/3

(triangles) and m�/1/2 (squares). Solid lines represent 3D V -vector;

points represent corresponding 2D projections on each plane. Data

calculated using Eqs. (7) and (8) with KS�/� for reaction (5).


3.3. Polymer reconfiguration

We do not debate the physical origins of polymer

reconfiguration (or other material structural change)*/

be it driven by electrostatics, solvation, free volume

phenomena, etc.*/but rather focus on the effects it has

on the film compositional response in E ,Q ,L -space.

Analogous to our considerations of redox and solva-

tion states, we consider a model in which there are two

‘configurations’ [5]. The equilibrium between these

configurations is characterized by an equilibrium con-

stant Kconfig, that is defined by the difference in theirstandard Gibbs energies. A consequence of this is a

difference in standard electrode potentials DE8 for

differently ‘configured’ Ox/Red couples (of the same

solvation state). Kconfig and DE8 can be related through

the standard expressions:

�RT ln(Kconfig)�DG��nFDE� (9)

We note that the notion of different polymer config-

urations influencing energetics has been considered by

Evans and coworkers [37�/39]. Their considerations of

this phenomenon suggested one might encounter stan-

dard potential shifts, DE8, on the order of 100 mV,corresponding to DG8 on the order of 10 kJ mol�1 or

Kconfig:/50.

On the basis of this, E ,Q ,L -space responses were

calculated for ‘configurations’ spaced at 100 mV (10 kJ

mol�1) intervals. Representative results (for the simple

case of m�/1 and KS�/�) are shown in Fig. 4;

extension to other values of m and KS would be

straightforward, and would simply involve lateral shiftsof loci shown in Figs. 2 and 3.

At this point, we address a major practical issue in the

interpretation of experimental data. It is anecdotally

acknowledged that polymer reconfiguration processes

can be very slow, commonly occurring on time scales

much longer than typical electrochemical measurements.

It is easy to imagine situations in which film redox

switching and solvation (as described in previoussections) occur for a film in a given configuration,

following which there is rapid (e.g. electrostatically-

driven) reconfiguration to the new equilibrium config-

uration, which is maintained during the reverse redox

half-cycle. The result, in E ,Q ,L -space, would be a non-

single valued locus, but which was apparently indepen-

dent of time scale, at least over a certain range, i.e.

hysteresis (the common signature of kinetic control) butindependence of response on time scale (commonly used

as a diagnostic for thermodynamic control). This subtle

interplay of kinetic and thermodynamic effects prompts

us to consider now the kinetics of the processes whose

equilibria we have so far discussed.

4. Kinetically controlled responses

4.1. Overview

The large number of possible elementary steps, the

wide choice of electrochemical control functions, and

the diversity of mechanistic possibilities make it im-

practical to offer encyclopaedic coverage of all possibi-

lities. We therefore restrict our attention here to somerepresentative cases that we perceive might be com-

monly encountered; in support of this notion, we are

(later) able to apply them to experimental data.

Specifically, we will consider coupled electron/counter

ion and solvent transfers as possible rate limiting

processes. This corresponds to operating (i) under

permselective conditions (low or moderate electrolyte

concentrations) and (ii) on moderate time scales, i.e. notso extended that (typically slow) polymer reconfigura-

tional processes have a chance to occur. In the scheme-

of-cubes type visualization, this corresponds to restrict-

ing events to a 2D plane within a single cube [5]. In

common with the work of Gabrielli et al. [22,23], and as

discussed in the light of experimental data below, we do

not consider diffusional processes within the film.

Furthermore, we will only consider single ‘start’ and‘end’ states. As discussed in Section 3.2, this corresponds

to considering Ox and Red species with very different

solvation characteristics (large KS); as discussed above,

we have the machinery to normalize the GS responses

with a KS/(KS�/1) factor, but inclusion of this here will

complicate the algebra without adding insight. We then

choose to drive the redox conversion using a potentio-

dynamic (linear potential scan) control function, sincethis is the most commonly used electrochemical techni-

que. Arbitrarily, we start with an equilibrated reduced

film, then oxidise and subsequently reduce it; transposi-

Fig. 4. V vector for the equilibrium path following film reconfigura-

tion. KS�/�; z -axis shows the oxidised species population (GOx) for

E 8/V�/0.6 (squares), 0.5 (triangles), 0.4 (squares).


tion to the case of an initially oxidized film is straight-

forward.

4.2. Electron/ion transfer response to potentiodynamic

control function

The problem of electron (and, via the electroneutrality

constraint, counter ion) population change has been

considered by a number of workers (see above). Without

undue detail, we summarize the approach in the mini-

mum detail required to develop the solvent transfer

problem, which has received little or no quantitative

attention.

When the system is disturbed from an equilibratedfully reduced state by a linear potential sweep (voltam-

metric experiment, scan rate v in V s�1), the variation of

the surface population (GOx) of the oxidised species can

be expressed in terms of a time or potential variable,

respectively, according to:

dGOx

dt�ke[GRedh

(1�a)�GOxh(�a)] (10)

or

dGOx

dE�ke

�1

haGT�

�1

ha�1�

1

ha

�GOx

�nF

RTh (11)

where ke (s�1) is the rate constant for coupled electron/

ion transfer into/out of the film, ke is a dimensionless

rate constant defined as ke�/RTke/nFv , h�/exp((nF /

RT (E�/E8)), and all the remaining symbols have their

usual significance. Obviously, ke0/� corresponds to

reversible electrochemical behaviour (considered above);

as will be seen later, ke:/10 gives essentially reversible

electron/ion transfer characteristics, and the onset ofelectron/ion transfer kinetic effects becomes visible for

keB/3. These aspects have been discussed previously by

Laviron and Roullier [26], so we do not discuss them

further, but proceed to the coupled solvent transfer

process.

4.3. Solvent transfer response to potentiodynamic control

function

We consider the following solvation model

Ox�xSUk0f

kb

OxS (12)

where Ox is an oxidised site in the polymer, S is solvent

in the solution and OxS is a solvated oxidised site in the

polymer. The rate equation for the film solvent popula-

tion is:

dGS

dt�k

0

f [GOx][S]x�kb[GOx;S] (13)

where k ?f and kb are the heterogeneous rate constants for

solvent transfer into and out of the film defined by

reaction (12). Since a relatively small absolute amount

of solvent transfers across the film j solution interface

(as compared to the vast excess in bulk solution), thechange in solvent concentration in the bulk solution is

essentially zero, i.e. [S]:/constant. Therefore, reaction

(12) is pseudo-first order and the rate Eq. (13) simplifies

to:

dGS

dt�kf [GOx]�kb[GOx;S] (14)

where kf�/k ?f [S]x .

The population (GOx) of unsolvated oxidised sites can

be calculated from the total number (solvated and non

solvated) of oxidised sites (GOx,T):

GOx�GOx;T�GOx;S (15)

Therefore, the rate equation can now be written:

dGS

dt�kf [GOx;T�GOx;S]�kb[GOx;S] (16)

The total solvent population is the mean number of

solvent molecules associated with all oxidised sites and

their local environments (without distinction as to

whether the solvent is ‘bound’ or ‘free’). Thus, GS�/

xGOx,S where x is the number of solvent molecules

(whether ‘bound’ or ‘free’) per oxidised site. Hence:

dGS

dt�kf

�GOx;T�

GS

x

��kb

�GS

x

�(17)

Using our previous definition for the equilibrium

constant for solvation of the oxidised film, KS�kf=kb :

dGS

dt�kfGOx;T�

�kf �

kf

KS

�GS

x(18)

In the context of a linear potential scan (scan rate, dE /

dt�/v V s�1):

dGS

dE�

kf

v

�GOx;T�

�1�

1

KS

�GS

x

�(19)

Although the focus of this present section is on kinetics,

we recognize that the thermodynamic issues must all be

borne in mind, including activity effects. To allow

discussion of the latter, we use the same empirical

approach as discussed above, in which non-idealitiesare symbolized by the parameter m in a power law

representation:

dGS

dE�kS

�GOx;T

�GOX

GT

�m

�GS

x

�1�

1

KS

��(20)

In Eq. (20), we have partly normalised the solvation rate

constant by writing kS�/kf/v . Simultaneous solution of

Eqs. (12) and (20) should then yield the electron (charge)

and solvent populations as functions of potential for the


specified scan rate, and the counter ion population via

Faraday’s law:

GA�Q

nF(21)

On the basis of the above equations, construction of

3D plots in E ,Q ,GS-space*/specifying instantaneous

film composition*/for selected values of ke and kS was

accomplished as follows. First, the ‘electron/ion’ pro-

blem was solved. Eq. (12) for the coupled electron/ionflux was solved by the input of ke (with fixed ‘typical’

values of a�/0.5, n�/1, T�/298 K, GT�/10�8 mol

cm�2) to define ke. Integration of the flux with respect

to time (defined by the instantaneous potential and

potential scan rate) then provided the film ion popula-

tion (i.e. charge density). This information was then

used to solve for the solvent flux defined by Eq. (20). In

the latter case, the additional input parameters were kf

(to define kS) and (if desired) m ; for reasons discussed

above, we considered only a large value of solvation

equilibrium constant, KS�/1000. Integration of the

solvent flux with respect to time then gave the film

solvent population. The mechanics of these calculations

were carried out using commercially available software

(Maple).

In the above exposition, we discussed the ‘forward-going’ half of a cyclic voltammetric experiment. For

comparison with experimental data, it is necessary to

obtain the response (in film populations) over the

complete cycle. In order to achieve this, we simply set

the ‘initial’ composition for the reverse sweep (in terms

of redox state, i.e. electron/counter ion population, and

solvent population) as that achieved at the end of the

forward half cycle. Intuitively, for long time scaleexperiments (effectively, kS�/1), the reverse half cycle

response will be independent of the forward half cycle,

since equilibrium is established at the end of each half

cycle; sample simulations with large values of kS bear

this out. Conversely, for short time scale experiments

(effectively, kS5/1), equilibrium will not be established

at the end of the forward scan*/to an extent dependent

upon time scale*/and the reverse scan response willcommence from a non-equilibrated state. For multiple

voltammetric cycles, the final film ion and solvent

populations for the n th redox cycle were set as initial

boundary conditions for the (n�/1)th redox cycle and

the procedure repeated.

4.4. Results for kinetically controlled situations

We present calculations for an initially equilibrated

reduced film under permselective conditions (see above),taken through a complete redox cycle. For the purposes

of generalization, we present the solvent population

changes normalised with respect to the overall solvent

population change; in the terminology of reaction (12)

and Eq. (20), this corresponds to plotting GS/x as the

compositional parameter L . In total, there are four

possible cases*/the combinations of fast and slow

electron/ion and solvent transfers. The ‘fast electron/

ion and fast solvent’ transfer case corresponds to the

thermodynamically controlled case discussed in Section

3 and illustrated in Figs. 1�/3. Hence, we now consider

the kinetically controlled cases of slow (rate limiting)

electron/ion transfer with rapid solvent transfer and

slow (rate limiting) solvent transfer with rapid electron/

ion transfer; we also consider the case that both mobile

species transfers are slow, in which case the slower will

be rate limiting.

Figs. 5 and 6, respectively, show film ion population

(GA, represented as Q , via Eq. (21)) and film solvent

population (GS, normalized with respect to the total

solvent transfer, represented by the stoichiometric

coefficient x ) responses for rate limiting solvent transfer

and rate limiting electron/ion transfer. In each case the

non-rate limiting process is rapid (cf. the case of both

processes slow, in Fig. 7). We show both the 3D E ,Q ,L -

vector locus (where L in this case corresponds to GS)

and the three projections on the 2D planes. At this

point, we show representative calculations for ‘sym-

metric’ mobile species entry/exit dynamics; the case of

asymmetric entry/exit dynamics*/which we simplisti-

cally anticipate to assume most commonly a ‘slow entry/

rapid exit’ form*/is considered later. The apparently

unusual units of the partially normalised solvent transfer

rate constant kS are to allow representation of temporal

effects for any potential scan rate (in this linear potential

scan scenario) on a single potential axis.

Fig. 5. Film ion population (GA, represented as Q , via Eq. (21)) and

film solvent population (GS) responses for a representative rate limiting

solvent transfer case. Arrows indicate potential scan direction.

Simulation parameters: ke�/10 (fast electron/ion transfer); KS�/

1000; kS�/10 V�1 (see main text for significance of units); equal

forward and reverse solvent transfer rates.


In the case of rate limiting solvent transfer (Fig. 5),

the E ,Q -plane projection corresponds to the Nernst

equation: it is single valued, as equilibrium is maintained

over the complete cycle. By contrast, both the E ,GS- and

Q ,GS-planes show substantial hysteresis. For presenta-

tional clarity we show only a representative ‘slow

solvent’ case, but additional calculations (not shown)

demonstrate that the extent of this hysteresis increases

with decreasing kS. The case illustrated (large equili-

brium constant for solvation) effectively encapsulates

other solvation thermodynamics cases, since (as indi-

cated earlier in the discussion of Fig. 3) one can

normalize the equilibrium solvent population change

with the factor KS/(KS�/1). In both the forward and

reverse half cycles, film (de)solvation is only partial, i.e.

the equilibrium solvation state is not achieved at either

end of the scan. Repetitive cycling accentuates this effect

and leads to solvent ‘trapping’ (see below).In the case of rate limiting electron/counter ion

transfer (Fig. 6), the E ,Q - and E ,GS-planes show

hysteresis, but the Q ,GS-plane does not. This is a

consequence of the solvent population effectively ‘track-

ing’ the film redox state, the latter now not being in

equilibrium with the applied potential. Repetitive cy-

cling accentuates this effect and leads to ion ‘trapping’

(see below).We now move from the somewhat simplistic pictures

of Figs. 5 and 6 to incorporate two experimentally more

plausible situations. First, we consider the case that both

electron/ion and solvent transfers are slow. A represen-

tative case is shown in Fig. 7. Here we see that there is

hysteresis in all three projections. Second, we remove the

restriction that the rates of mobile species entry and exit

are the same. A representative example is shown in Fig.

8. We have selected what we anticipate to be the more

likely case (also, see experimental data below) that

solvent entry is slower than solvent expulsion; the

reverse situation could be straightforwardly described

by our methodology.As indicated above, the spectrum of possible ion and

solvent entry and exit rate parameters, both in absolute

and relative terms, prompt us to consider the possibility

of ‘trapping’ processes. By this, we imply the situation in

which there is entry of a mobile species (ion or solvent)


film solvent population (GS) responses for a representative rate limiting

electron/ion transfer case. Arrows indicate potential scan direction.

Simulation parameters: ke�/1 (slow electron/ion transfer); KS�/1000;

kS�/1000 V�1 (see main text for significance of units); equal forward

and reverse solvent transfer rates.


film solvent population (GS) responses for a representative case of slow

electron/ion and slow solvent transfers. Arrows indicate potential scan

direction. Simulation parameters: ke�/ 0.05; KS�/1000; kS�/30 V�1

(see main text for significance of units); equal forward and reverse

solvent transfer rates.


film solvent population (GS) responses for a representative case of slow

and asymmetric solvent transfers. Arrows indicate potential scan

direction. Simulation parameters: ke�/0.05; KS�/1000; kS�/20 V�1

for forward (oxidation) half cycle and kS�/100 V�1 for reverse

(reduction) half cycle, i.e. slower solvent entry than exit.


into the film during redox switching in one direction, but

incomplete expulsion of that species in the reverse half

cycle. Figs. 9 and 10, respectively, show representative

examples of scenarios in which counter ion and solvent

are trapped within the film. Full 3D E ,Q ,L -space plots

were presentationally crowded, so we illustrate the effect

through the 2D E ,L -plane projections. In these simula-

tions, we start with an equilibrated reduced film, which

takes in counter ions and solvent upon oxidation, but

which incompletely expels them upon re-reduction.

Hence, the reduced film at the end of the first complete

potential cycle has a composition that differs (substan-

tially) from that at the start of the potential cycle. To

illustrate the effect further, we also show the result of asecond potential cycle, immediately following the first

(i.e. with the ‘end’ state of the first cycle constituting the

‘start’ state of the second cycle). The result is a

progressive ratchetting up of the film mobile species

populations. In the mirror image case that the mobile

species were initially present, then expelled but incom-

pletely replenished within the film, the result is a

progressive ratchetting down of the film mobile speciespopulations. We have observed*/but not hitherto been

able to describe quantitatively*/such effects for poly-

pyrrole films upon multiple cycling under kinetically

controlled conditions [40]. This underscores the point we

have repeatedly made concerning the general impor-

tance of film history [4,5]. In particular, the common

practice of cycling to a ‘reproducible’ response (after an

arbitrary number of redox cycles) will lead to inter-change of an unknown set of redox/solvation states, to

the extent that extraction of meaningful kinetic informa-

tion will be practically impossible. The recommendation

of recording the mobile species population responses for

the first cycle after film equilibration can be considered

analogous to the corresponding classical solution vol-

tammetry protocol of recording the I�/E curve for the

first cycle.

5. Application to experimental data

The previous two sections have attempted to predict

film population changes for a variety of thermodynamic

and kinetic scenarios. Correspondingly, in EQCM

experiments on a range of electroactive films we have

observed previously a range of phenomena (e.g. massand charge trapping [41]) that we have interpreted

qualitatively in terms of redox-driven mobile species

population changes within the film. A test of the new

theoretical approach*/which we now undertake*/is to

apply it to representative experimental data and explore

the extent to which it can enhance our knowledge of the

system.

For the purposes of illustration, we will considerEQCM data acquired for a system that we have studied

in some detail [3,7,24,41�/44]: poly(vinylferrocene) ex-

posed to aqueous 0.1 mol dm�3 NaClO4 solution.

Justification is given in the cited literature, but here we

simply assert (i) that the films are rigid (acoustically

thin), so that QCM frequency responses can (via the

Sauerbrey equation) be simply interpreted in terms of

film population changes; (ii) that the films are permse-lective [45], so that ‘salt’ transfer need not be considered;

(iii) that we are able to assay the total redox population

coulometrically on the basis of ferrocene/ferricinium

Fig. 9. Ion population (represented through GOx) as a function of

applied potential for two consecutive voltammetric redox cycles.

Simulation parameter ke�/0.01 (equal forward and reverse transfer

rates). Simulation starts with an equilibrated reduced film, defined as

zero ion population; cycle 1 (squares), cycle 2 (circles), arrows indicate

scan direction. Failure of ion population to achieve equilibrium at

either extreme of potential results in ‘ion trapping’.

Fig. 10. Solvent population change as a function of applied potential

for two consecutive voltammetric redox cycles. Simulation parameters:

ke�/0.01; KS�/1000; ks�/30 V�1 (equal forward and reverse solvent

transfer rates); m�/1. Simulation starts with an equilibrated reduced

film, defined as zero population change; cycle 1 (squares), cycle 2

(circles), arrows indicate scan direction. Failure of solvent population

to achieve equilibrium at either extreme of potential results in ‘solvent

trapping’.


(one electron) redox chemistry; (iv) that the reduced film

is less solvated than the oxidised film. These conditions

are encapsulated within the range of scenarios we have

considered theoretically in this work.

The experimental details are given elsewhere [24,41]

but measurements were made for films deposited on Au

electrodes supported on 10 MHz AT-cut crystals (ICM,

Oklahoma), and potential scan rates in the range 45/v /

mV s�15/80 were used. In terms of the generic

nomenclature represented by reactions (2), (5) and (12)

above, Red�/VF, Ox�/VF�ClO4� and S�/H2O, where

VF represents a vinylferrocene unit on a polymer chain.

On the basis of Faraday’s law (Eq. (21)), values of

charge (Q ) were converted to the mass of perchlorate

counter ion transferred; these values were subtracted

from the observed mass change to yield the solvent

population change, GS.

Representative data are shown in Fig. 11 for three

complete redox cycles of an initially fully equilibrated

reduced PVF film. At the outset, we make three

qualitative observations on the data which we shall set

out to rationalize and parameterize on the basis of our

new theoretical treatment. First, the anodic half cycle

response for the first scan is systematically shifted to

positive potentials as compared to that for all subse-

quent scans (which are identical); this is consistent with

the reconfiguration characteristic seen in Fig. 4. Second,

Fig. 11 shows mass trapping*/failure of the solvent

population to return at the end of the first full cycle to

its initial value. Third, the data show hysteresis in all

three 2D planes of E ,Q ,L -space, so there is no question

that the system is under kinetic control*/although we

emphasise (see below) that this does not preclude the

presence of thermodynamic non-idealities. All of the

above phenomena have been discussed in our model and

thus we now proceed to explore each of them in more

detail.

The shift on the potential axis in the charge responsefor the first anodic half cycle is on the order of 25 mV.

Based on Eq. (9), this corresponds to a ‘drop’ on a

‘scheme-of-cubes’ visualization into a second oxidized

state configuration that is ca. 2.4 kJ mol�1 more stable.

The corresponding value of Kconfig:/3. Although the

thermodynamic energy change is only on the order of

RT , there is no further ‘reconfiguration’ signature, so it

must be kinetically very slow. Thus, we do not dwellfurther on this issue, but explore the movement of the

system within the electron/ion- and solvation-based

‘square scheme’ at fixed configuration.

The extent of mass trapping is relatively small, but

nonetheless clearly visible. Although there is no direct

structural evidence from the EQCM data, we speculate

that it may be associated with the change in film

configuration. Specifically, a change in polymer struc-ture is synonymous with a change in packing of the

chains, and thus of the volume available to solvent.

We now come to the main feature of the data, which

takes predominant place in the application of our

model: the thermodynamic and/or kinetic origins of

the curvature in the solvent/ion population relationship

and the (necessarily) kinetic origins of the hysteresis in

all responses. To the best of our knowledge, simulta-neous consideration of thermodynamic and kinetic

solvation effects has not been considered previously

for electroactive polymer films. In order to apply our

analysis, we need to identify input and variable para-

meters. In the first category, the standard electrode

potential, E8, was calculated from experimental data at

moderate scan rates (where redox equilibrium is most

closely approached); under the conditions employed, wefind E8�/0.44 V for the ‘reconfigured’ state. Based on

the overall ion and solvent compositional changes, x�/

4.8. In the second category, we need to consider: m (for

possible thermodynamic effects); ke (or its normalized

counterpart, ke) and a for electron/ion kinetic effects;

and kf and kb (whose ratio defines KS) for solvent

transfer kinetic effects. Throughout, we make the

‘typical’ assumption that a�/0.5. Based on the verydifferent charge and solvation characteristics of the two

redox states, we also assume that one solvation state

predominates in each redox form, i.e. KS is large; we

take KS�/1000, but once KS�/10 its precise value makes

little difference to the observed response.

As a first step, we attempted to fit the data using a

fixed value of m�/1 and only varying the electron/ion

and solvent transfer rate constants. This corresponds toa model in which the solvent behaves ideally in a

thermodynamic sense and in which all the ‘curvature’

in the Q ,GS-responses can be attributed to kinetic

effects. In a separate experimentally-based study [41]

Fig. 11. 3D E ,Q ,DM -plot (equivalent to a E ,Q ,L -plot) of EQCM

data for a voltammetric experiment (v�/0.08 V s�1) involving a

polyvinylferrocene film exposed to 0.1 M sodium perchlorate. Data for

three complete cycles, commencing from a fully equilibrated reduced

state, are shown; the first anodic half-cycle response is that at more

anodic potentials.


we have shown how one can use the ratio of solvent and

ion fluxes (and the potential dependence thereof) as a

diagnostic of whether ion or solvent transfer is the

slower process. For this system, the qualitative resultwas that solvent entry (during film oxidation) was

slower than counter anion entry; this experimental

evidence must necessarily constrain our selection of the

relative values of ke and kS. Surprisingly, even allowing

wide variation of the electron/ion and solvent rate

constants (spanning a range from effectively reversible

to extremely slow), we were not able to obtain accep-

table fits. (The ‘best’ fits we were able to obtain involvedslow solvent transfer rate constants, kS:/10 V�1 (i.e.

kf:/0.8 s�1 at v�/80 mV s�1) but even these were

unacceptably poor.) We are thus driven to conclude

that, although kinetic effects unquestionably are the

cause of the hysteresis (with respect to applied potential)

of the ion and solvent populations, thermodynamic

effects also contribute to the curvature in the solvent-ion

population relationship (DM �/Q plot in the context ofan EQCM experiment).

In order to test this deduction, we turned to data on

longer time scales, for which kinetic complications are

absent and the response is governed by thermodynamic

parameters. For this, we utilised data from an analogous

experiment to that illustrated in Fig. 11, but at a

potential scan rate of 4 mV s�1. Although simulations

of the Q ,E -plane projection were essentially insensitiveto the solvent non-ideality parameter (as one would

expect), the projections involving GS were sensitive to m

(again, as one would expect): the ideal case of m�/1

provided a very poor description of the solvent popula-

tion as a function of potential (or charge), particularly in

the first half of the oxidation process. With the same

values as before for E8, x , a and KS, and using large

values of the rate constants (to which the system isinsensitive in this non-kinetic regime), we therefore

explored the effect of varying the thermodynamic

solvent non-ideality parameter m . We found that values

of m :/2 provided a good description of the solvent

population for the first part of the oxidation process.

The non-ideality decreased with increasing conversion

to the oxidised state; although there is no means of

attributing a molecular explanation for this, we spec-ulate that water will behave less ideally in the very

hydrophobic reduced polymer than in the oxidised

polymer.

With the confidence that solvent non-ideality effects

are present, we then used this to guide the fitting of the

kinetically controlled (high scan rate) data. The deduc-

tion is that the ‘concave’ film solvent population

responses seen in the data of Fig. 11 would be in partthe result of values of m "/1. Whether we chose either to

impose a value of m�/2 and fit the data to the kinetic

parameters or to allow m to vary, we came to the same

conclusion. Specifically, we found (see Figs. 12 and 13)

that the closest visual fits resulted from values of m :/2,

consistent with the chemistry of a polar solvent inter-

acting with a film that is less polar in its reduced state

and more polar in its oxidized state. Interestingly, in this

kinetically controlled case, there was no evidence of an

approach to ideality (m 0/1) at high redox conversion.

We speculate that this is associated with the failure,

under kinetic control, to establish the equilibrium

solvation state.

Although the incorporation of solvent non-ideality

improved the fits, they were still far from the optimum.

Qualitatively, one sees solvent fluxes that are relatively

Fig. 12. Comparison of experimental and simulated 3D E ,Q ,GS -

vector representations (and the three 2D projections) for PVF

oxidation. Experimental data (full line) correspond to the first anodic

half cycle response in Fig. 11. Simulation parameters: E 8�/0.44 V;

x�/4.8; v�/0.080 V s�1; m�/2; ke�/10; KS�/1000, ks/V�1�/50

(circles), 100 (triangles), 1000 (diamonds).

Fig. 13. 2D Q ,GS-projections comparing the experimental data of Fig.

11 with simulations for kinetically controlled solvation with thermo-

dynamic non-ideality. Inset highlights low conversion region. Slightly

‘noisy’ trace corresponds to experimental data. Simulations for x�/

4.8, m�/2, ke�/10, KS�/1000 and kS/V�1�/50, 100 and 1000

(increasing solvation rate from right to left).


low during the early stages of oxidation and relatively

high during the latter stages of oxidation, i.e. the solvent

transfer rate constant appears to increase with charge.

The results of incorporating both non-ideality andkinetic acceleration are illustrated for the full 3D vector

in Fig. 12 and more clearly for the solvation coordinate

by the 2D Q ,GS-projection in Fig. 13. The experimental

curve can be modelled in the early stages of oxidation by

slow solvent transfer (kS�/50 V�1, i.e. kf�/4.0 s�1 at

v�/80 mV s�1) and in the later stages by fast solvent

transfer (kS�/1000 V�1, i.e. kf�/80 s�1 at v�/ 80 mV

s�1), with m�/2 throughout.An important test is that the deduction of a switch

between thermodynamic control at low scan rate and

kinetic control at high scan rate must be consistent with

the values of the fitted kinetic parameters. In the case of

the high scan rate experiment*/deliberately chosen to

highlight kinetic effects*/we deduced rate constants for

solvent transfer of kf�/ 4.0 s�1 in the early stages and

kf�/80 s�1 in the later stages. At a scan rate v�/80 mVs�1, these correspond to dimensionless rate constants,

(RTkf/Fv ), of 1.3 and 26, i.e. shifting from ‘slow’ to ‘fast’

kinetics. In the context of an experiment at a scan rate

v�/4 mV s�1, these correspond to dimensionless rate

constants, (RTkf/Fv ), of 26 and ca. 500, i.e. ‘fast’

throughout. Consequently, the analysis is internally

self-consistent.

Turning to the reduction half cycle, the situation issomewhat different. In terms of the raw solvent popula-

tion response, the extent of curvature is much smaller.

At this point, it is important to note that the change in

film redox state (the ‘Red’, rather than ‘Ox’, side of the

square scheme) means that the various parameters may

assume different values from those of the oxidation half

cycle. The solvent:ion flux analysis [24,41] shows that

solvent transfer during film reduction keeps pace withthe changing redox state, i.e. kf (for desolvation)�/ke.

The best fits to the cathodic half cycle data (which were

identical for all scans; see Fig. 11 above) were found

using a model with rate limiting*/but fairly rapid (i.e.

just below the ‘reversible’ value of ke:/10)*/electron/

ion transfer and rapid desolvation (kS�/1000 V�1, i.e.

kf�/80 s�1 at v�/80 mV s�1). Interestingly, in this

reduced and reconfigured state, activity effects appearedless significant as compared to the first anodic half cycle.

Thus, m�/1 (no inclusion of solvent non-ideality effects)

gave acceptable fits. This may be a manifestation of the

‘phase behaviour’ described long ago by Daum and

Murray [46] for polyvinylferrocene.

6. Conclusions

We are able to model electroactive film mobile species

(ion and solvent) populations under a range of thermo-

dynamically and kinetically controlled conditions. In

general terms, the resultant film state can be visualized

in 3D E ,Q ,L -space, where L represents film composi-

tion. In the present study, we have restricted our

attention to permselective conditions, in which case L

represents the film solvent population (with ion popula-

tion being represented by charge, Q , through the

electroneutrality condition).

Under thermodynamically controlled conditions, film

composition is (necessarily) a single-valued function of

potential, E . We are able to represent solvent content as

a function of redox state under both ideal and non-ideal

conditions. In the latter case, an empirical non-idealityfunction was selected, but the model could incorporate

any appropriate function. The results show that curva-

ture in the solvent-charge relationship will result from

non-ideality, i.e. can result from thermodynamic factors

as well as the generally appreciated kinetic factors. We

have also been able to show how irreversible structural

changes (‘reconfigurations’, in the polymer context) can

lead to changing quasi-equilibrium responses.Under kinetically controlled conditions, a wide spec-

trum of behaviour can arise. This can be attributable to

slow electrochemical and/or solvation processes. We

have restricted our attention to cases in which film

transport processes are not rate limiting, but have

considered asymmetric ion and solvent entry/exit rates

and have shown how thermodynamic effects (including

solvent non-ideality) and film reconfiguration canmodulate even dominant kinetic phenomena.

Film compositional responses in E ,Q ,L -space pro-

vide rapid visual diagnostics for thermodynamic versus

kinetic control and the participation of various possible

processes. This has been illustrated for the case of

polyvinylferrocene films exposed to aqueous media

under permselective conditions, in which we have been

able to diagnose kinetic control (though with thermo-dynamic non-idealities in terms of film solvent popula-

tion) and to identify and characterize film

reconfiguration, ion and solvent trapping, and relative

rates of ion versus solvent transfer and of solvent entry

versus exit.

Acknowledgements

We thank the National Science Foundation (grant

number CHE 9616641) for financial support of this

work. A.J. thanks the EPSRC and the British Council

for financial support.

References

[1] S. Bruckenstein, A.R. Hillman, J. Phys. Chem. 92 (1988) 4837.

[2] S. Bruckenstein, A.R. Hillman, J. Phys. Chem. 95 (1991) 10748.


[3] S. Bruckenstein, P. Krtil, A.R. Hillman, J. Phys. Chem. B 102

(1998) 4994.

[4] A.R. Hillman, S. Bruckenstein, Faraday Trans. 89 (1993) 3779.

[5] A.R. Hillman, S. Bruckenstein, Faraday Trans. 89 (1993) 339.

[6] R.W. Murray, Molecular Design of Electrode Surfaces, Wiley,

New York, 1992.

[7] A.R. Hillman, N.A. Hughes, S. Bruckenstein, J. Electrochem.

Soc. 139 (1992) 74.

[8] A.R. Hillman, N.A. Hughes, S. Bruckenstein, Analyst 119 (1994)

167.

[9] A.R. Hillman, H.L. Bandey, P.M. Saville, R.W. Wilson, S.

Bruckenstein, J.G. Vos, in: R.W. Richards, S.K. Peace (Eds.),

Polymer Surfaces and Interfaces III, Wiley, New York, 1999.

[10] S. Bruckenstein, A.R. Hillman, in: A.T. Hubbard (Ed.), Hand-

book of Surface Imaging and Visualisation, CRC Press, Boca

Raton, FL, 1995.

[11] D.A. Buttry, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol.

17 (Ch. 1), Marcel Dekker, New York, 1991.

[12] R.J. Gale, Spectroelectrochemistry, Plenum Press, New York,

1988.

[13] H.L. Bandey, M. Gonsalves, A.R. Hillman, A. Glidle, S.

Bruckenstein, J. Electroanal. Chem. 410 (1996) 219.

[14] J.G. Gaudiello, P.K. Ghosh, A.J. Bard, J. Am. Chem. Soc. 107

(1985) 3027.

[15] S. Gottesfeld, in: A.J. Bard (Ed.), Electroanalytical Chemistry,

vol. 15, Marcel Dekker, New York, 1989, p. 143.

[16] R.W. Wilson, R. Cubitt, A. Glidle, A.R. Hillman, P.M. Saville,

J.G. Vos, Electrochim. Acta 44 (1999) 3533.

[17] R. Schumacher, Angew. Chem. 29 (1990) 329.

[18] D.A. Buttry, M.D. Ward, Chem. Rev. 92 (1992) 1355.

[19] N. Oyama, T. Ohsaka, Prog. Polym. Sci. 20 (1995) 761.

[20] P.T. Varineau, D.A. Buttry, J. Phys. Chem. 91 (1987) 1292.

[21] D. Orata, D. Buttry, J. Am. Chem. Soc. 109 (1987) 3574.

[22] C. Gabrielli, M. Keddam, N. Nadi, H. Perrot, J. Electroanal.

Chem. 485 (2000) 101.

[23] C. Gabrielli, M. Keddam, H. Perrot, M.C. Pham, R. Torresi,

Electrochim. Acta 44 (1999) 4217.

[24] I. Jureviciute, S. Bruckenstein, A.R. Hillman, in: M.P. Butler, P.

Vanysek, N. Yamazoe (Eds.), Chemical and Biological Sensors

and Analytical Methods, The Electrochemical Society, Penning-

ton, 2001, p. 82.

[25] E. Laviron, J. Electroanal. Chem. 112 (1980) 1.

[26] E. Laviron, L. Roullier, J. Electroanal. Chem. 115 (1980) 65.

[27] E. Laviron, J. Electroanal. Chem. 122 (1981) 37.

[28] M. Rudolph, D.P. Reddy, S.W. Feldberg, Anal. Chem. 66 (1994)

589.

[29] C.P. Andrieux, P. Hapiot, J.M. Saveant, J. Electroanal. Chem.

172 (1984) 49.

[30] P.J. Peerce, A.J. Bard, J. Electroanal. Chem. 114 (1980) 89.

[31] P.J.O. Mahon, K.B. Oldham, J. Electroanal. Chem. 445 (1997)

179.

[32] K. Aoki, J. Electroanal. Chem. 160 (1984) 33.

[33] S. Bruckenstein, A.R. Hillman, J. Phys. Chem. B 102 (1998)

10826.

[34] E. Pater, S. Bruckenstein, A.R. Hillman, Faraday Trans. 92

(1996) 4087.

[35] W.J. Albery, M.G. Boutelle, P.J. Colby, A.R. Hillman, J.

Electroanal. Chem. 133 (1992) 135.

[36] A.B. Brown, F.C. Anson, Anal. Chem. 49 (1977) 1589.

[37] E.F. Bowden, M.F. Dautartas, J.F. Evans, J. Electroanal. Chem.

219 (1987) 49.

[38] E.F. Bowden, M.F. Dautartas, J.F. Evans, J. Electroanal. Chem.

219 (1987) 91.

[39] M.F. Dautartas, E.F. Bowden, J.F. Evans, J. Electroanal. Chem.

219 (1987) 71.

[40] S. Bruckenstein, K. Brzezinska, A.R. Hillman, Phys. Chem.

Chem. Phys. 45 (2000) 3801.

[41] I. Jureviciute, S. Bruckenstein, A.R. Hillman, A. Jackson, Phys.

Chem. Chem. Phys. 2 (2000) 4193.

[42] A.R. Hillman, D.C. Loveday, S. Bruckenstein, J. Electroanal.

Chem. 300 (1991) 67.

[43] S. Bruckenstein, A. Fensore, A.R. Hillman, Faraday Disc. 107

(1997) 323.

[44] X. Zeng, S. Moon, S. Bruckenstein, A.R. Hillman, Anal. Chem.

70 (1998) 2613.

[45] A.R. Hillman, D.C. Loveday, M.J. Swann, S. Bruckenstein, C.P.

Wilde, Analyst 117 (1992) 1251.

[46] P. Daum, R.W. Murray, J. Phys. Chem. 85 (1981) 389.


modelling mobile species population changes in electroactive films under thermodynamically and...

Documents