8670
J. Phys. Chem. B 1998, 102, 8670-8677
Electron Self-Exchange in Redox Polymers. 1. Mechanistic Analysis and Statistical Mechanical Considerations R. Aldrin Denny and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras, 600 036 India ReceiVed: January 28, 1998; In Final Form: May 26, 1998
A comprehensive analysis of electron hopping between spatially separated redox centres pertaining to supramolecular structures is carried out by taking into account different mechanistic pathways. Spin exchange dynamics due to Kawasaki is invoked to study the transport phenomena for the more appropriate, thermodynamically favored ion pairing mechanism. The generalized master equation and its reduced form are presented and the dynamics of electron hopping is analyzed both under spin-1/2 and spin-1 Ising versions. Transition probabilities chosen for the present study satisfy the condition of detailed balancing and is a function of spin exchange jump frequencies and spin variables. Energetics of electron hopping process is obtained from the Ising Hamiltonian under molecular field approximation and incorporated into the spin exchange frequencies. The apparent diffusion coefficient that emerges along with the transport equation reflects the dynamics of electron propagation in the films. The importance of blocking factor and potential independent terms in the apparent diffusion coefficient expression overlooked in hitherto known phenomenological approaches is emphasized.
1. Introduction The Ising model originally introduced to describe phase transitions in magnetic systems1,2 has subsequently been the subject of intense research activity in diverse topics such as theoretical estimation of critical solution temperatures,3 phase diagrams of microemulsions and alloys involving multicritical points,4,5 percolation theory,6 biomembranes and lipid monolayers,7 continuous phase transitions,8 etc. However, the timedependent characteristics of Ising model (sometimes referred to as the kinetic Ising model) has received considerably less attention. The earliest analysis was carried out by Glauber,9 who allowed individual spins to interact with an external agency and studied their time dependent behavior using a master equation. The stochastic model so defined forms a class of spin flip kinetic Ising models, with extensive literature support, applicable to the study of the phenomenon of segregation dynamics,10 phase transitions and critical exponents,11,12 dielectric and magnetic systems,13 polymer chain kinetic studies,14 etc. In the original Glauber model, the allowed dynamical transitions of the system are restricted to single flips. Hence, it is inadequate to describe transport phenomena. Spin-flip processes deal with a particular site and its dynamical evolution, whereas for studying transport phenomena, models, which consider at least two spin sites simultaneously at a given instant of time, are essential. In order to realize this objective, Kawasaki15-17 proposed a dynamical model describing spin exchange processes and investigated the time dependent movement of nearest neighboring Ising spins. Kawasaki model considers simultaneous flip of two nearest neighbor spins and the total spin is a conserved quantity. Interestingly, Glauber model in one dimension can be exactly solved to obtain the structure factor for arbitrary temperature exchange with surroundings, whereas the one dimensional Kawasaki model is solvable only when the temperature of the heat bath (surroundings), wherein the system is placed, is infinite. Transcription
Figure 1. Schematic representation to illustrate macroscopic kinetic process in terms of Ising model where two spin states are equivalent. (a) Using adsorption-desorption problem as an example for spin-flip process and (b) transport process along the one-dimensional lattice to illustrate the spin exchange process.
of the kinetic process in terms of Ising model formalism and the distinction between spin flip and spin exchange processes can be visualized using the schematic representation shown in Figure 1. Although the study of Ising models is usually centered around solving problems related to physics, a large volume of literature is now available on the use of equilibrium Ising models in electrochemistry.18-22 It is shown to be useful in analyzing double layer structures, adsorption-desorption processes, phase transitions on electrode surfaces, etc. This is largely due to the effectiveness of the model in handling systems involving interacting species in a hierarchical manner. Extension of kinetic Ising model formalism to various fields in chemistry is relatively less explored. Recently, we have demonstrated23-25
10.1021/jp9809235 CCC: $15.00 © 1998 American Chemical Society Published on Web 10/10/1998
Electron Self-Exchange in Redox Polymers. 1
J. Phys. Chem. B, Vol. 102, No. 44, 1998 8671
SCHEME 1
the use of kinetic Ising models for studying diffusion-migration equations pertaining to charge transport in redox polymers and interactions between redox molecules were effortlessly incorporated into the spatiotemporal process starting from the Ising Hamiltonian. Transport of electrons across polymeric surfaces have been studied for the past 2 decades.26-51 This study is of great importance due to its wide variety of promising applications in electrocatalysis and electrosynthesis, biosensors, energy devices, and immobilization of enzymes onto electrode surfaces. Charge transport inside polymer films is considered to occur via electron hopping process between spatially separated oxidized and reduced species. It has been recently suggested by Save´ant34 that all the initially present redox species need not be electron transfer active and introduced activity effects of molecules in transporting charge (or current). These activity effects essentially arise due to the ion pairing nature of the polymeric system. Thus, under suitable experimental conditions, redox active ions undergo aggregation leading to ion pairs and these ion pairs interact with each other to form a cluster. The objectives of studying the present microscopic model are (i) to analyze in detail, various possible mechanisms of electron hopping in supramolecular structures and to delineate the suitable pathway on the basis of thermodynamic and kinetic considerations. Earlier, Anson et al.30 have considered two different mechanisms for electron transfer and Save´ant34 has indicated briefly an alternate possibility. However, no study to date has considered all the pathways simultaneously so as to decipher the most appropriate one, which in turn depends upon the nature of redox speciesscounterions as well as polymer characteristics; (ii) to derive spatio temporal diffusion-migration equations incorporating intermolecular interaction energy effects so as to emphasize their influence on charge transport. This results in new features concerning the influence of transfer coefficients, polymer morphology, “blocking effects”, etc. and, (iii) to compare the microscopic framework provided by kinetic
Ising model formalism with other phenomenological approaches. For example, we have recently demonstrated the applicablity of the concepts of irreversible thermodynamics52,53 for electron hopping, and this provides scope for comparing nonequilibrium thermodynamics formalism with kinetic Ising model versions. The hitherto unknown connectivity between Onsager’s phenomenological coefficients and microscopic transition probabilities seems therefore possible. In this paper, we investigate different mechanisms by which electron transfer (coupled with counterion motion) can occur and subsequently decipher the most plausible mechanism of electron hopping under a given experimental situation (section 2). In section 3 we write the master equation corresponding to the thermodynamically favored stepwise mechanism and demonstrate how the derivation of transport equation can be accomplished for electron hopping between redox sites incorporating nearest neighbor interaction energies using two different statistical models viz two-state spin-1/2 Ising and three state spin-1 version. 2. Mechanism of Electron Self-Exchange In redox polymers, the concentration of ionic species inside the polymeric matrix is high and hence ion association between opposite charges due to weakly polar nature of the large moiety inside the polymer is anticipated.54,55 In addition to ion association between cations and anions leading to an ion pair, there exists a diffuse interaction between the ion paired species with all the other ions. Thus the ion paired and free ions engage in a chemical equilibrium governed by an association constant. Several constituent processes, such as segmental motion of the polymer to bring oxidized and reduced species close to each other, electron transfer, counterion motion to maintain electroneutrality, etc., comprise charge propagation in general. On the basis of the order in which these processes occur, three probable pathways of electron hopping can be envisaged, as
8672 J. Phys. Chem. B, Vol. 102, No. 44, 1998 shown in Scheme 1. Firstly, the mechanism can proceed in a stepwise manner (mechanism I) wherein dissociation of oxidized species, electron transfer, and association takes place in succession. Secondly, the initial step may be the association of reduced species, and then electron transfer followed by dissociation (mechanism II). Finally, all the three steps take place simultaneously in a concerted fashion (mechanism III). The precise knowledge about the mechanism of charge transport is essential in order to model the system for further fundamental studies. Thus, if mechanism III is operative, then both the oxidized and reduced species present in the system will be electron transfer active and contribute to electron hopping. However, if mechanism I or II is the actual pathway, then a reduced rate of charge transport will be observed compared with the anticipated rate when total number of redox species is considered as in mechanism III. In the case of mechanism II, association of reduced species which exists in chemical equilibrium with electron transfer active species is the first step. Thus an increase in the concentration of electroinactive counterions which is involved in ion pairing should enhance charge transport and hence the current. On the other hand, in the case of mechanism I, because the dissociation of initially present oxidized species leads to electron transfer active molecules, charge transport rate becomes dependent on the dissociation constant of oxidized species and any increase in supporting counterion concentration would result in a decrease in the charge transport rate. Hence, it is imperative to comprehend the actual mechanism of electron transport for deriving (oxidized and reduced species dependent) diffusion migration equations and thereby predict current-potential characteristics, under different experimental situations. The presence of ions in polymers either as a part of the backbone or as a pendant group imparts a wide range of properties to the resulting materials. Ionogenic nature of the polymer makes the system rheologically complex and inhomogeneous and produces profound changes in system behavior. Otherwise, a homogenous system bifurcates the polymeric region into organic and aqueous (or more polar) phases. The initially present redox species are uncharged and hence persist in organic part of the polymer where it is stabilized. However, when they undergo association or dissociation, their ions are localized in the polar region due to better solvation. For mechanism II to be operative, association has to take place between neutral electroactive reduced species in polymeric phase and mobile electroinactive counteranion in polar phase. The tendency of electroinactive species such as ClO4- to form an ion pair with a reduced species, viz. [M], involves transfer of ClO4- from polar to organic phase or movement of [M] from organic to polar phase and these are considered unfavorable due to the strong repulsive forces of van der Waals type. Furthermore, metal complexes considered in redox polymers have an effective atomic number of 36 and have a saturated outer electronic configuration. Hence, the probability for ClO4to undergo association with this undissociated reduced species is lower owing to the excess negative charge already present around the complex. Thus the association barrier for mechanism II becomes high so as to eliminate its participation in electron transport. For the concerted process (mechanism III), we again encounter a similar situation where electron transfer takes place between [MClO4] and [M] in the organic phase and the simultaneous transfer of ClO4- has to occur in a concerted manner in the organic medium itself which is energetically unfavorable. In contrast to these, mechanism I involves dissociation of [MClO4] in organic medium (in plane j) and
Denny and Sangaranarayanan results in a passage of ClO4- and [M]+ from organic to the polar part of system which is energetically favorable. After electron transfer, [M]+ becomes [M] and diffuses back to organic region, and in the adjacent plane i, a reverse process takes place, viz. passage of ClO4- from organic to polar medium and subsequently involving association with [M]+ in polar part of the system which results in a low activation barrier. On the basis of the above energetic considerations, the order in which electron transport occurs is
mechanism I > mechanism III > mechanism II Thus, mechanism I is thermodynamically suitable for electron transfer and increased anion-cation attraction is possible because of reduced availability of water (compared to their large concentration of ionic aqueous solution) and low dielectric constant of the organic polymer. The charge transport dynamics is thus dependent on the nature of ion pairing, concentrations of free oxidized and reduced electroactive ions, and mobile electroinactive counterions. Furthermore, when the concentration of ionic species is large and ion pairing between electroactive components and counterions is strong, the rate of electron hopping via mechanism I becomes slow, and hence, mechanism I is likely to be replaced by a kinetically favoured mechanism III, because of its low activation barrier (Eact) as compared with mechanism II. The preferential pathway for electron hopping (EH) process under the strongly ion paired situation is
mechanism III > mechanism II g mechanism I Thus, under any given condition, electron hopping in redox polymers proceeds via mechanism I or III depending on the nature and extent of ion pairing between electroactive oxidized species and counterions. 3. Kinetic Ising Model Formalism and Transport Equations Here we employ a kinetic Ising model version to study the thermodynamically favoured mechanism I with the notations A (≡[M]+), B (≡[M]), C (≡ClO4-), and D (≡[MClO4]). We consider a lattice consisting of particles A, B, and D, in which A possesses a single positive charge, whereas B and D are neutral species. A and D are oxidized species, while B is the reduced species. The only mode of transport that exists in this lattice is via electron hopping which occurs by electron selfexchange between A and B. D is considered to be electroinactive and hence does not participate in electron hopping process. Due to electron transfer, there exists an “apparent” movement of species A and B (i.e., when electron self exchange takes place, A and B interchange their respective positions). (Note that particle A differs from B only by an electron, which is involved in the hopping process. Because A is charged, when electron hopping takes place, charge propagates, and in order to maintain the system electrically neutral, an equivalent amount of electroinactive counterion C moves in the lattice provided by A, B, and D (counterions do not form a part of the regular lattice). Thus, electron hopping is inherently coupled with the counterion movement. When A ion pairs with C, it is converted into D. Thus particles A, B, and D can be interconverted. In our present analysis, we assume that electron hopping and electroinactive counterion movement take place between nearest neighboring sites. Furthermore, the coupling between electron and electroinactive counterion movements is viewed as resulting solely from electroneutrality. In brief, electron gets transported
Electron Self-Exchange in Redox Polymers. 1
J. Phys. Chem. B, Vol. 102, No. 44, 1998 8673
when two adjacent molecules are A and B, but A is also involved in ion pairing with C to give D and because D is electron transfer inactive, the presence of D next to B prohibits electron transfer. 3.1. Generalized Master Equation Approach. The classical n-dimensional Ising model (where n ) 1, 2, 3, etc.) postulates an ordered array of N lattice sites, and the state of each lattice site is completely assigned a specific particle to it, customarily denoted as S. For example, the state variable at site j is denoted by Sj. In addition, each site j is given a spin variable σj, which can take different values which depend upon states (Sj) of site j, where σj means σ(Sj). Hence the choice of states represented in the system determines the particular version of Ising model. The derivation of diffusion-migration equations can be accomplished using either a two-state (or equivalently spin-1/2 Ising) version or a three-state (spin-1 Ising) model. The hopping process under consideration is taking place under the influence of an external electric field, and this applied field is assumed to be felt over a discrete region of multilayers. The statistical description of electron hopping is provided by the following generalized master equation:
∂
PN({σ}N;t) )
∂t
∑ 〈ij〉
∑Wij(σiσj{σ}ij)PN({σ}ijN;t) 〈ij〉
Wji(σjσi{σ}ij)PN({σ}N;t) +
∑Wkj(σkσj{σ}jk)PN({σ}jkN;t) 〈jk〉
∑Wjk(σjσk{σ}jk)PN({σ}N;t)
(1)
〈jk〉
where PN({σ}N;t) represents the normalized N spin distribution function at time t and {σ}Nij and {σ}Njk are the configurations of spin ij and jk that have interchanged their spin configuration with that of the corresponding {σ}N. In principle, the master equation can be solved to obtain the time-dependent probability distribution PN({σ}N;t).56 However, this function is too detailed to be useful, because the observable parameters are not PN({σ}N;t) but certain averages over PN({σ}N;t) such as
〈σj〉 )
∑ σjPN({σ}N;t)
(2)
∑ σjσiPN({σ}N;t)
(3)
{σ}N
and
〈σjσi〉 )
{σ}N
where the sum ∑{σ}N runs over all possible spin configurations of the system:
∑ )σ∑ ∑ ‚‚‚σ ∑ ) (1σ ) (1 ) (1
{σ}N
1
2
∂t
) -2
∑ 〈σjWji(σjσi{σ}ij)〉 - 2 ∑ 〈σjWjk(σjσk{σ}jk)〉
〈ij〉j
〈jk〉j
(4)
state of occupany (S′)
σ′
χT
χD
occupied by A or B occupied by D
+1 -1
1 0
0 1
where 〈σj〉 as defined earlier is the expectation value of σj. The summation index 〈ij〉j means summing over all pairs of ij which involve j as the terminal site. Equation 4 is also known as the first moment equation or reduced master equation. 3.2. Spin-1/2 Ising Model. In spin-1/2 Ising model the allowed state variables are two, but because three particles (A, B, and D) are present in the lattice, a careful accounting of particles is required. Hence, these particles are first separated as similar and dissimilar species using a first set of state variable.57 Subsequently, a further distinction is made with another set of state variable to demarcate the two similar entities. In this analysis, the regular lattice is populated by particles A, B, and D. For describing the system, electron transfer active sites (A and B) are distinguished from the electron transfer silent (inactive) sites D using a state variable S′. The microscopic spin variable corresponding to this state variable is σ′, and it can take a value of +1 or -1 depending upon the site being occupied by either redox active species (A or B) or redox inactive species (D). The ensemble average of the spin variables leads to the macroscopic observables. Here the macroscopic state variables namely occupancy of particles A, B, D, and total electroactive species are respectively designated as χA, χB, χD, and χT ()χA + χB). The connectivity between the state variable S′, spin variable σ′, and their ensemble averages are represented in Table 1. Consequently, we can write
χT )
(
χD )
(
)
(5)
)
(6)
1 + 〈σ′〉 2
and
1 - 〈σ′〉 2
The specification of S′ enables us to distinguish whether a particular site is occupied by an electroactive A (or B) or electroinactive species, whereas by using the second state variable S′′ distinction between A and B is taken into account. The spin variable corresponding to the state variable S′′ is σ′′ and represents two possible values +ξ and -ξ (which are microscopic) corresponding to that site being occupied by A and B, respectively. The relation between the state variable (S′′) and spin variable (σ′′) is given in Table 2. In general, we have
N
and the angular bracket 〈‚‚‚〉 signifies the ensemble average. The first-order averages such as 〈σj〉 are useful because they relate to the local concentration (or density of species transported), and one may also study as a next step the dynamics of second-order averages 〈σiσj〉 giving site-to-site correlation and are important for the study of phenomena such as clustering. The above generalized master eq 1 can be written in a more compact form as
∂〈σj〉
TABLE 1: State Variable S′ and Its Interpretation for Spin-1/2 Ising Model
χA )
(
χB )
(
)
(7)
)
(8)
〈ξ〉 + 〈σ′′〉 2
and
〈ξ〉 - 〈σ′′〉 2
χT ) 〈ξ〉
(9)
χD + χT ) 1
(10)
Thus, a complete description of a given site requires specification of both the state variables, namely, S′ and S′′. These state
8674 J. Phys. Chem. B, Vol. 102, No. 44, 1998
Denny and Sangaranarayanan and
TABLE 2: State Variable S′′ and Its Relation with Spin Variable σ′′ state of occupancy (S′′)
σ′′
χA
χB
occupied by A occupied by B
+ξ -ξ
ξ 0
0 ξ
variables are connected to each other through ξ, and this connectivity is brought about through eqs 5 and 9. With the above relations between the state variables and spin variables, the transition probability for the spin exchange between sites j and i can be written as
(
Wji(σjσi{σ}ij) ) kji
)(
) (
ξ - σ′′j ξ + σ′′i + 2 2 ξ - σ′′i ξ + σ′′j kij 2 2
)(
)
(
)(
)
(
(11)
)(
ξ + σ′′j 2
)
(12)
(18)
]
(19)
The individual jump frequencies may be written using potential independent rate k0 as
kji ) k0 exp(-τ2/kBT)
(20)
kij ) k0 exp(-τ1/kBT)
(21)
For kji and kij given by eqs 20 and 21 to satisfy eq 18, we further demand that
τ1 - τ2 ) e(Φi - Φj) + J(χAj - χAi)
(22)
〈ij〉
Wjk(σjσk{σ}jk)PeN({σ}N;t) ) Wkj(σkσj{σ}jk)PeN({σ}jkN ;t)
(15)
where PNe represents the equilibrium probability and is related to the Hamiltonian as
exp(-βH ) Z
(16)
Z being the partition function. Using the Hamiltonian in eq 13, eq 14 for detailed balancing can be simplified to yield
PeN({σ}ijN;t) PeN({σ}N;t)
) exp[2β(Ei - Ej)] (17)
Equation 22 implies the following form for τ1 and τ2
τ1 ) τ0 + R1e(Φi - Φj) + R1J(χAj - χAi)
(23)
τ2 ) τ0 - R2e(Φi - Φj) - R2J(χAj - χAi)
(24)
R1 + R2 ) 1
(25)
and
where R1 and R2 can be considered analogous to the transfer coefficients in electrode kinetics. At this stage it is important to emphasize that the constant R1 (or R2) appearing in the potential and interaction energy expression in eq 23 (or eq 24) need not necessarily be equal. In an earlier study on adsorption-desorption kinetics using Ising model versions, two different R’s were employed.59 However, here R1 (or R2) is adequate because eq 25 is always valid. τ0 in eqs 23 and 24 is introduced for the sake of generality and its physical interpretation will be discussed later. The spin-exchange jump frequencies hence follow as
[
]
kij ) kE exp -
R1e R1J (Φi - Φj) (χ - χAi) kBT kBT Aj
kji ) kE exp
where
Ej )
]
Similarly using eqs 12 and 15,
(13)
(14)
)
∑ σ′′j + const
[
Wji(σjσi{σ}ij)PeN({σ}N;t) ) Wij(σiσj{σ}ij)PeN({σ}ijN;t)
Wij(σiσj{σ}ij)
eΦi J + 2 4
[
where hj ) eΦj + const and Φj represents the potential at site j. Under equilibrium conditions, transition probabilities represented in terms of system parameters, satisfy the condition of detailed balancing. Hence the following equality results:
Wji(σjσi{σ}ij)
Ei )
kjk J e ) exp (Φ - Φj) + (χ - χAk) kkj kBT k kBT Aj
∑j hjσj - J∑σjσi
PeN )
∑ σ′′i + const
kji J e ) exp (Φ - Φj) + (χ - χAi) kij kBT i kBT Aj
where k’s represent the jump frequencies between the appropriate sites. 3.3. Ising Hamiltonian and Spin Exchange Frequencies. Energetics of electron hopping can be obtained from the Ising Hamiltonian incorporating the composite interaction energy term J and single-particle energy hj as23
Hj ) -
eΦj J + 2 4
Substituting expressions for Ei and Ej into eq 17 and using the transition probability equations,58 we obtain
and for spin exchange between sites j and k
ξ - σ′′j ξ + σ′′k + Wjk(σjσk{σ}jk) ) kjk 2 2 ξ - σ′′k kkj 2
Ej )
eΦj J + 2 4
∑ σ′′i + const
[
R2J R2e (Φ - Φj) + (χ - χAi) kBT i kBT Aj
(26)
]
(27)
Electron Self-Exchange in Redox Polymers. 1
J. Phys. Chem. B, Vol. 102, No. 44, 1998 8675
TABLE 3: State Variables and Their Interpretation in Spin-1 Model state of occupancy (S) occupied by A occupied by B occupied by D
σ
χA
χB
χD
+1 -1 0
1 0 0
0 1 0
0 0 1
Similarly from eqs 18 and 19 we obtain
[
]
kkj ) kE exp -
R1e R1J (Φk - Φj) (χ - χAk) kBT kBT Aj
R2J R2e kjk ) kE exp (Φk - Φj) + (χ - χAi) kBT kBT Aj
[
(28)
]
( )
〈σ2〉 + 〈σ〉 χA ) 2
(31)
〈σ2〉 - 〈σ〉 2
(32)
χB )
χD ) 1 - 〈σ 〉 2
(33)
We have emphasized earlier that the transition probability is dependent on the nature of the neighboring site under consideration. Transition probability corresponding to the spin exchange between sites i and j can be represented for the three state Ising model as
( )( ) ( )( )
Wji(σjσi{σ}ij) ) σiσjkji
σj + 1 σi - 1 + 2 2 σi + 1 σj - 1 (34) σiσjkij 2 2
Similarly the transition probability for spin exchange between sites j and k is
(
Wjk(σjσk{σ}jk) ) σjσkkkj
)( ) ( )(
dχD ) kasC0EχAχC - kdisχD dt
(30)
) )
}
where the rate of change of D (electron transfer silent) can be written as
3.4. Spin-1 Ising Model. As mentioned earlier the presence or absence of a particle at a particular site is indicated in terms of a spin variable. The allowed spin values for a three state Ising model are +1, 0, and -1. Table 3 represents the relationship between the state variable S and the spin variable, σ for the spin-1 case. Table 3 indicates that when σj is +1 the corresponding occupancy χAi is 1 (if σj is -1 or 0 then χAj is zero). Thus microscopic spin variables can be converted into macroscopic occupancies using eqs 31-33,
( (
{
∂χA ∂χA ∂χB ∂ ∂Φ e ) D0E χB - χA + χ χ + ∂t ∂x ∂x ∂x kBT A B ∂x ∂χD ∂χA J χ χ (36) kBT A B ∂x ∂t
(29)
where
τ0 kE ) k0 exp kBT
3.5. Phenomenological Transport Equations. The transition probabilities for both spin-1/2 (eqs 11 and 12) and spin-1 (eqs 34 and 35) are substituted in the reduced master eq 4 using the expressions for χA and χB (eqs 7 and 8 for spin-1/2 and eqs 31 and 32 for spin-1) to derive a difference equation. Converting the difference equation into the corresponding differential equation, we obtain
σk + 1 σj - 1 + 2 2 σj + 1 σk - 1 σjσkkjk (35) 2 2
)
Thus W’s represented in eqs 34 and 35 exist only for those configurations where adjacent neighbors are A and B and vanishes for all other configurations. The spin-exchange frequencies represented in eqs 34 and 35, however, are assumed to be given by eqs 26-29 as before.
(37)
kas and kdis denote the rate constant for dissociation and association, respectively, and CE0 is the total concentration of the redox species. A similar procedure for the movement of B results in the transport equation as
{
}
∂χB ∂χA ∂χB ∂χB ∂ ∂Φ e J ) D0E χA - χB χ χ + χ χ ∂t ∂x ∂x ∂x kBT A B ∂x kBT A B ∂x
(38) where
D0E ) k0 exp(-τ0/kBT)(∆x2)(1 - χD)
(39)
D0E ) DE/C0E
(40)
and
An identical approach can be adopted to account for the movement of electroinactive counterions, C, inside the polymer, and the transport equation can be formulated as
{
}
∂χD ∂χC ∂ ∂χC e ∂Φ ) DC χC + ∂t ∂x ∂x kBT ∂x ∂t
(41)
where DC ) k0Cδ2, k0C represents the frequency with which the counterion migrates to the adjacent site and δ denotes the average hopping distance. A few subtle features on the use of nonequilibrium kinetic Ising model may be emphasized here. The description of a system using equilibrium Ising Hamiltonian is, in general, not sufficient because it inherently has no dynamical properties. This necessitates incorporating time dependence into the model, which can be attempted in either of the two ways: (i) by adding temporal characteristics to the Hamiltonian itself or (ii) by writing a dynamical equation to account for spontaneous changes in spin configuration with time. Usually the latter approach is followed via the generalized master equation.9 The structure of eq 39 is particularly illuminating in view of the presence of blocking factor (1 - χD) and solvent dynamics vis a` vis polymer morphology. The presence of blocking factor in the diffusion coefficient expression is itself not new;60 however, the time dependent transport equation along with the appropriate diffusion coefficient is derived here simulataneously. The presence of blocking factor in the diffusion coefficient accounts for the reduction in frequency of spin exchange
8676 J. Phys. Chem. B, Vol. 102, No. 44, 1998 between particles A and B which is attributed to obstruction by particles D. Thus the effective diffusion coefficient is k0exp(-τ0/kBT)(∆x2)CE0(1 - χD) instead of k0(∆x2)CE0. When ion pairing is absent (i.e., when mechanism III in Scheme 1 is operative) the lattice is occupied by particles that are only involved in spin exchange process and hence the blocking factor vanishes. In the present formalism, only particles A, B, and D span the entire lattice, and even if other molecules which do not participate directly in spin exchange are present, the final transport equation remains unaltered. However, the structure of the diffusion coefficient formula (eq 39) gets modified in the blocking factor denoting obstruction due to additional particles as well. Another term that arises naturally in the derivation of the macroscopic transport equation is τ0. As stated earlier it is potential independent and can be attributed to solvent effects and polymer morphology.38 Several reports have demonstrated51 solvent and polymer morphological effects to contribute significantly to electron hopping rate and thereby to the electron diffusion coefficient. The effects of equilibrium61,62 and nonequilibrium63-67 aspects of the solvent medium on barrier free energy for electron transfer reactions were recognized sometime ago and recently these effects are studied for redox polymeric systems also.68 Transport eqs 36 and 38 governing electron hopping between spatially separated redox centres incorporating intermolecular interaction are obtained under mean field condition. Note that the mean field approximation is employed only to decouple spin configurations pertaining to the interaction term in the Ising Hamiltonian. We have shown recently25 that molecular field approximation need not be inVoked in the derivation of diffusion-migration equation if the interaction energy is excluded from consideration because the Hamiltonian can be described by the single-particle energy term itself. Further improvements in the structure of transport equations are possible when higher order approximations are carried out. Another point worth noting is the explicit assumption of the symmetry factor as being 0.5 in earlier phenomenological approaches.33,35 In comparison, the present kinetic Ising model formalism does not require this improvization because the spatiotemporal equation is independent of the value of the symmetry coefficient. An identical conclusion emerged from our earlier analysis52,53 of transport equations using nonequilibrium thermodynamics as well. Although the treatment presented herein has been restricted to one-dimensional spin exchange model, the qualitative insights obtained above are valid even in higher dimensions, because as demonstrated elsewhere,69 the dimensionality is largely a secondary issue, in spin-conserving systems. Such an assertion however is not true in the case of equilibrium Ising models where dimensionality plays a crucial role. 4. Perspectives and Conclusions Owing to the ionogenic nature of redox polymers, ion pairing is an unavoidable phenomenon. Ion association between electroactive and electroinactive ions are facilitated by low dielectric constant of the polymer and reduced availability of water for ion solvation. Thus it is possible for both electroactive oxidized and reduced species to have ion coupling with electroinactive counterions. On the basis of this, electron hopping between spatially separated redox species and concurrent counterion movement is postulated to occur via three different electron transfer pathways. And the most probable ones that mimic electron hopping in redox polymers are selected on the basis of kinetic and thermodynamic considerations.
Denny and Sangaranarayanan The “apparent” movement of electroactive oxidized and reduced species arising due to electron self exchange is conveniently analyzed using Kawasaki’s spin exchange model. The master equation and its first moment are written for the spin exchange between nearest neighbors. Transition probabilities which form a part of reduced master equation can assume different forms; however, those obeying the condition of detailed balancing are chosen for the present study. The transition probabilities selected are represented in terms of their spin variables and jump frequencies. Energy required for the spin exchange between two nearest neighboring sites is obtained from the Ising Hamiltonian considering both single-particle and interaction energy terms. Decoupling between Ising spins is done using the molecular or mean field approximation. Higher approximations like Bethe ansatz, which considers the interaction of chosen spins with their neighbors as being different, may also be employed.70 We note that the mean field condition is introduced only at the Hamiltonian level and scope for studying the diffusion process from generalized master equation itself is expected to yield even newer insights concerning fluctuations, etc. Further, simulations of generalized master equation71-74 have become a routine exercise in studying transport phenomena, albeit in nonelectrochemical contexts. The description of three particles in a lattice using the two state (spin1/ ) Ising model is more involved than the three-state spin-1 2 Ising model version, but the final result concerning the the structure of the diffusion-migration equation remains unaltered for both the models. The use of spin-1 model for deriving electrochemical transport equations of redox particles is attempted here for the first time. In part 2, we investigate the steady state analysis of the transport equations derived here and discuss the dependence of diffusion coefficient on concentration of redox centers, ion pairing association constant, interaction energetics, etc. Acknowledgment. This work was supported by the Department of Science and Technology, Government of India. Supporting Information Available: Conversion of the reduced master equation into transport eq 36 and the derivation of phenomenological transport equation using a two-term GME (6 pages). Ordering information is given on any current masthead page References and Notes (1) Ising, E. Z. Physik. 1925, 31, 253. (2) Onsager, L. Phys. ReV. 1944, 65, 117. (3) Wheeler, J. C. J. Chem. Phys. 1975, 62, 433. (4) Mukamel, D.; Blume, M. Phys. ReV. A 1974, 10, 610. (5) Degiorgio, V.; Piazza, R.; Corti, M.; Minero, C. J. Chem. Phys. 1985, 82, 1025. (6) Coniglio, A.; Klein, W. J. Phys. A: Math. Gen. 1980, 13, 2775. (7) Nagle, J. F. J. Chem. Phys. 1975, 63, 1255. (8) Zimmer, M. F. Phys. ReV. E 1993, 47, 3950. (9) Glauber, R. J. J. Math. Phys. 1963, 4, 294. (10) Puri, S.; Frisch, H. L. J. Phys. A: Math. Gen. 1994, 27, 6027. (11) Menyha´rd, N. J. Phys. A: Math. Gen. 1994, 27, 6139. (12) Kamphorst Leal de Silva, J.; Moreira, A. G.; Soares, M. S.; Sa´ Barreto, F. C. Phys. ReV. E 1995, 52, 4527. (13) Gonzalo, J. A. Phys. ReV. B 1970, 1, 3125. (14) Orwoll, R. A.; Stockmayer, W. H. In Stochastic Processes in Chemical Physics; Shuler, K. E. , Ed.; Interscience: New York, 1969. (15) Kawasaki, K. Phys. ReV. 1966, 142, 164. (16) Kawasaki, K. Phys. ReV. 1966, 145, 224. (17) Kawasaki, K. Phys. ReV. . 1966, 150, 285. (18) Mott, N. F.; Watts-Tobin, R. J. Electrochim. Acta 1961, 4, 79. (19) See for example, Rangarajan, S. K. J. Electroanal. Chem. 1977, 82, 93. (20) Sangaranarayanan, M. V.; Rangarajan, S. K. J. Electroanal. Chem. 1984, 176, 119.
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J. Phys. Chem. B, Vol. 102, No. 44, 1998 8677 (54) Eisenberg, A. Macromolecules 1970, 3, 147. (55) Eisenberg, A.; King, M. In Ion-Containing Polymers; Academic: New York, 1977. (56) Kikuchi, K.; Yoshida, M.; Maekawa, T.; Watanabe, H. Chem. Phys. Lett. 1991, 185, 335. (57) The model proposed here is identical to the one introduced by Kawasaki [ref 11] for studying self-diffusion in a binary alloy AB with particle A having an isotope A*. One spin variable takes -1 and +1 to distinguish between A (including A*) and B and another spin variable which takes the value of 1 and 0 is used to distinguish the isotope A* from the other two species (namely A and B), thus describing the system completely. (58) Instead of substituting χA, one can substitute χB expression (eq 8), in which case we obtain spin-exchange frequency ratios as
[
]
[
]
kji ne J (φ - φj) + (χ - χBj) ) exp kij kBT i kBT Bi and
kjk ne J (φ - φj) + (χ - χBj) ) exp kkj kBT k kBT Bk
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