J. Phys. Chem. B 1998, 102, 2131-2137
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ARTICLES Flux-Force Formalism for Charge Transport Dynamics in Supramolecular Structures. 1. Activity Coefficient and Interaction Energy Considerations R. Aldrin Denny and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras-600 036, India ReceiVed: June 2, 1997; In Final Form: September 24, 1997
A comprehensive analysis starting from the formalism of irreversible thermodynamics, incorporating activity coefficients and interparticle interactions, is developed to derive expressions for mixed conductance, transport numbers, etc., for diffusive and electric field assisted electron hopping through polymer matrixes attached to electrode surfaces. The transport equation obtained is a combination of Dahms-Ruff diffusion and the Levich migration terms and includes short-range interparticle interactions under molecular field approximation as well as activity coefficient terms. The derived dynamical equation for electron flux follows a second-order law in species concentrations in contrast to the classical Nernst-Planck equation for ion migration. Applicable systems are thought to be organic π-conjugated electroactive polymers, electronically conducting polymers with covalently linked redox groups, i.e., metal ion redox site membranes, and ion-exchange polymers. The concept of self and tracer diffusion coefficients is brought into the formalism of charge transport in supramolecular structures. Some new insights regarding the identification of Onsager’s coefficient in diffusion through supramolecular structures are also provided.
1. Introduction The analysis of charge transport in supramolecular structures has been great interest for more than two decades.1-9 The charge propagation in these structures consists of several dynamical processes such as activation-controlled heterogenous electron transport at the metal/polymer film interface, segmental motion of the polymer films to provide proper orientation of redox active species, electric field assisted electron transport between spatially separated redox centers, and movement of counterions for maintaining electroneutrality. In addition to these, possible structural changes brought about by polymer morphologies also need to be considered. Amid such a scenario, it is therefore imperative to focus attention on different types of processes in isolation and presume their simultaneous nonoccurrence in any given experimental situation. While this strategy may be theoretically rewarding, the requisite data reduction of experimental parameters, which are invariably global and composite in nature, may invalidate methodologies that are isolated in objectives. Hence a general analysis, even if approximate, is likely to yield correct insights concerning the observed behavior. Consequently, we provide a first step in this direction by analyzing the combined effects of electron hopping between redox centers, counterion diffusion, ion pairing between redox sites and counterions, and influence of supporting electrolyte on the charge transport mechanism. Theoretical studies carried out in this laboratory10-12 and elsewhere13-18 to study electron transport in chemically modified electrodes over the past decade indicate several interesting features concerning the nature of electron hopping (nearest neighbor vs long distance), estimation of redox capacity, influence of counterion movement, etc. The ability of these
electroactive polymer films to store electrical charge or to act as electrocatalysts depends on the rate of electron hopping in the polymeric region. The formulation of the dynamics of electron hopping via electron self-exchange constitutes the primary step in this analysis. The flux-force equation for this adjacent molecule-dependent bimolecular electron hopping is substantially different from the classical Nernst-Planck equation applicable for single-particle movement. The appropriate diffusion-migration equation governing the electron-transfer dynamics was provided by Save´ant15 and Buck16 for systems without interaction and for interacting species by Chidsey and Murray.17 The formalism based on irreversible thermodynamics is particularly attractive in this context,19 because of the ease with which different coupled phenomena such as diffusion under thermal and electrical gradients, particle transport with chemical reactions/electron transfer, and hydrodynamic effects can easily be handled even for multicomponent situations. Furthermore, it is possible to analyze a hierarchy of different models which take into account additional sophisticated features encompassing interparticle interactions, distance dependence of electron transfer, system fluctuations, and the nature of stable steady states. However, a few difficulties (by no means unsurmountable) existing in such a pursuit need to be indicated, and these are (i) proper formulation of thermodynamic fluxes and forces for different diffusing species, (ii) identification of Onsager’s phenomenological coefficients in terms of conventional diffusion parameters, and (iii) proper choice of reference frames needed for multicomponent analysis, etc. There have been a few earlier attempts using an irreversible thermodynamics formalism to comprehend charge transport in
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2132 J. Phys. Chem. B, Vol. 102, No. 12, 1998
Denny and Sangaranarayanan
Figure 1. Schematic representation of charge conduction in a redox polymer film sandwiched between the electrode and the solution phase. The redox conduction occurs by electron self-exchange reactions between adjacent oxidized (acceptor) and reduced (donor) polymer sites. The figure depicts the processes at the negative (reductive) potential; however under oxidative potential, counterion diffusion occurs from the electrode to the solution phase, which is opposite the movement of electrons. d represents the thickness of the polymer film layer.
polymer films. Among them, mention may be made of Buck16 in the study of electron hopping in mixed conductors, so as to derive a second-order flux equation incorporating the correct Fermi level of electrons. A form of the Ricatti equation is shown to result from this analysis. Further, Chidsey and Murray17 have also indicated the usefulness of the approach based on irreversible thermodynamics by deriving appropriate conductance equations. The purpose of this and the subsequent article (part 2) is (i) to provide a formalism based on nonequilibrium thermodynamics to describe the electron exchange between spatially separated redox centers, aided by an external electric field (the spatiotemporal diffusion-migration equation derived therefore has several novel features such as influence of activity coefficient of the redox species and interparticle interaction energy), (ii) to demonstrate the validity of identifying Onsager’s phenomenological coeffficients Lij in terms of diffusion coefficients, well known in the transport of solutes in multicomponent liquid systems, (iii) to incorporate the influence of ion pairing and diffusivity of the redox active and electroinactive counterions on the conductivity and transport numbers, and (iv) to compare the predictions of the earlier simplified versions with the present unified model. In this report, we consider a redox polymer that can exist in either an oxidized or reduced state involving an n electron transfer process,
O + n e- h R Since these oxidation-reduction couples are connected to each other through the polymer backbone, their movement inside the matrix is restricted, and hence charge propagation is assumed to occur via site-to-site electron hopping. During the electronhopping process, the oxidized and reduced electroactive ions get interconverted, and hence there exists an apparent motion of these molecules. When an electron from a reduced species hops to an adjacent oxidized species, an electroinactive counterion moves accordingly inside the polymer matrix in order to sustain electroneutrality. For example, during reduction of a film of poly[Os(bpy)2(vby)2](ClO4)3, when electrons propagate to the other end of the polymer chain via donor/acceptor electron self-exchange, counterions (ClO4-) move opposite the direction of electron hopping and thereby the film maintains electroneutrality, thus additional ions will migrate out to the solution from the film, as shown in Figure 1. The movement of counterions is usually described by the classical Nernst-Planck equation20
- Jc ) Dc
{
}
∂Cc zce ∂φ + C ∂x kBT c ∂x
It must be recognized that maintenance of electroneutrality implies inherently that electron hopping is coupled to the physical motion of electroinactive counterions. The dynamics of electron hopping thus depends on the nature of the electroactive ion and various association and dissociation processes that take place due to interaction of ionic species that affect electron transport. The incorporation of activity coefficients in modeling ion transport across membranes for symmetric cells is not new.21 For example, Eisenman and co-workers22 have analyzed different types such as fixed site, ionized mobile site, and liquid mobile site membranes with ion-pairing effects. However theoretical models in the context of electron transport through chemically modified electrodes have ignored activity coefficient variations under concentration gradient and electric field gradient conditions on the charge propagation dynamics assuming that interaction between particles and others remain the same when the ionic composition of the redox polymer is changed. As stated by Saveant,3g such an assumption is unlikely to hold when ion concentrations involved are not negligible and ion aggregation occurs under these conditions. The role of activity on the equilibrium properties of electroactive polymer films has been first analyzed,2c,3c and later its role in electrochemical responses has been considered.3e-g It is therefore necessary to estimate the flux due to the apparent motion of oxidized and reduced species by means of electron self-exchange movement, which dictates the current during the electron transport process in terms of activity coefficient and interaction energy parameters. 2. Irreversible Thermodynamics Formalism 2.1. Flux Equations. The instantaneous state of reductant and oxidant may be characterized in terms of local concentrations (number density) FR and FO. In the conventional description, the transport of two components relative to the solvent is given by the molar flux densities Ji,23
Ji ) Fi(Vi - V0)
i ) R, O
(1)
Here, V0 and Vi are the velocities of the solvent and the ith solute constituent. The fluxes are driven by generalized forces Xi,
Xi ) -grad(µi)
(2)
Flux-Force Formalism. 1
J. Phys. Chem. B, Vol. 102, No. 12, 1998 2133
where grad indicates ∂/∂x + ∂/∂y + ∂/∂z and µi is the electrochemical potential of the ith constituent. The electrochemical potential term can be split into chemical and electrical parts, viz.,
µi ) µeli + µch i
(3)
To derive an expression for chemical potential, we consider an array of F equivalent sites in which FO molecules of O and FR molecules of R are distributed randomly. The partition function for an isolated molecule R on a site is qR, and when two nearest-neighbor sites are both occupied by molecules, there is a potential energy of interaction ω between the molecules. Each molecule has Z closest neighbors, and there are in all l/2ZF pairs of these. Such pairs are of three kinds, namely, OO, RR, and OR, with the interaction energy FOOωRR, FRRωRR, and FORωOR. This is a realistic model applicable for supramolecular structures if the forces between the sites are of sufficiently short range that only nearest-neighbor interactions need to be taken into account. In the derivation of chemical potentials, we shall use a lattice model; that is, we assume that the motion of the molecules reduces merely to oscillations about some equilibrium positions. This assumption is valid for the present analysis due to the rigidity provided by the solid polymer support for electrochemically active redox molecules, thus giving the molecules some short-range order in the lattice. In this context, earlier impedance models of Mathis and Buck24 and Armstrong et al.8,25 that deal with the nature of plasticizer and dielectric constant variation may be mentioned. Summing all the neighbors of these FR molecules, we have
ZFR ) 2FRR + FOR
(4)
ZFO ) 2FOO + FOR
(5)
and similarly
The canonical ensemble partition function for two particles in a solution can be written as26 FO Q(FR, FO, T) ) qFR R qO
q(FR, FO, FOR)e-W/k T ∑ F B
(6)
OR
where W in terms of number concentrations and intermolecular dispersion forces (ω) is
W ) FRRωRR + FOOωOO + FORωOR
(7)
and g(FR, FO, FOR) is the number of arrangements of FR molecules of reduced species R and FO molecules of oxidized species O in F sites so that there are altogether FOR pairs of nearest neighbors of oxidized and reduced molecules. Converting eq 7 using27
Z ) (ωRR + ωOO - 2ωOR) 2
(8)
we obtain
Q(FR, FO, T) ) (qR e-ZωRR/2kBT)FR (qOe-ZωOO/2kBT)FO
g(FR, FO, FOR) eF ∑ F
OR/ZkBT
(9)
OR
At present, no rigorous mathematical treatment is available to evaluate g(FR, FO, FOR) that is complete and exact. However
approximations that are fairly satisfactory for our purposes exist. According to the Bragg-Williams approximation, the simplest version in the hierarchy of interacting lattice gases, the configurational degeneracy and average nearest-neighbor interaction energy are both handled on the basis of random distribution of molecules among sites. Thus eq 9 becomes
Q(FR, FO, T) ) (qR e-ZωRR/2kBT)FR (FO + FR)!eF*OR/ZkBT ) (10) FR! FO!
-ZωOO/2kBT FO
(qO e where
g(FR, FO, FOR) ) ∑ F
(FR + FO)!
OR
FR! FO!
and FOR* is the average value of FOR.
FORg(FR, FO, FOR) ∑ F F* OR )
OR
g(FR, FO, FOR) ∑ F
)Z
FRFO (11)
FR + FO
OR
Thus, the chemical potential of the reduced species can be written as
( )
(
)
2 µR° FR µch R ∂ ln Q ) )+ ln FR + ln γR kBT ∂FR kBT kBT FO + FR (12)
where
µR° + ln γR ) -ln(qR(FO + FR)e-ZωRR/2kBT) kBT and FR° denotes the standard chemical potential of reduced species, γR is the activity coefficient that arises due to the ionpairing nature of the redox species, represents the “composite interaction energy” parameter consisting of the occupied siteoccupied site interactions, and ωij is the absolute value of interaction energy between two nearest neighbors i and j. Using eqs 3 and 12, the electrochemical potential in general can be written as
( )
µi ) µi° + zie∆φ + kBT ln Fi kBT In γi -
Fj Fi + Fj
2
(13)
The above equation is valid only when the nearest-neighboring interactions are contributing to the total energy and needs modification if long-range forces such as Coulombic effects are also taken into consideration (note that zie∆φ represents the electrostatic interaction of ions with the external electric field and terms such as zi2e2, which reflect ion-ion interaction, are absent in eq 13). If we impose the condition ωOO >> ωRO and/or ωRR >> ωRO, it follows from eq 8 that is positive so that any molecule O prefers R as its nearest-neighboring partner than its own species. This leads to a perfectly alternate chain of redox active species when the oxidized and reduced species concentrations are equal and increases the electron-transfer rate. Alternatively, in the other limit, viz. ωOO
