Nonequilibrium Thermodynamics Formalism for Charge Transport in

J. Phys. Chem. B 2001, 105, 2465-2473

2465

Nonequilibrium Thermodynamics Formalism for Charge Transport in Redox Polymer Electrodes J. Umamaheswari and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras- 600 036, India ReceiVed: May 31, 2000; In Final Form: NoVember 5, 2000

Onsager’s formalism of nonequilibrium Thermodynamics is employed to analyze competitive diffusion processes pertaining to charge transport through redox polymer electrodes. The validity of reciprocity relations is demonstrated for electron hopping in the case of [Ru(NH3)6]3+/2+ couple on Nafion coated Glassy Carbon (GC) electrodes. The phenomenological coefficients pertaining to electron hopping and physical diffusion are estimated using the experimental data of Cp2FeTMA2+/1+ on Nafion coated GC electrodes. The redox conductivity and transport numbers are expressed in terms of Onsager’s coefficients and the calculation of the phenomenological coefficients directly from the charge propagation behavior is demonstrated. The expression for entropy production is shown to yield new insights regarding the interpretation of diffusion coefficients in redox polymeric systems.

Introduction Nonequilibrium thermodynamics finds applicability in diverse contexts such as (i) heat and mass transfer in multicomponent systems, (ii) electrokinetic phenomena, (iii) electron-transfer reactions at electrode/electrolyte interface, (iv) viscoelastic behavior in polymeric materials, and so forth.1-4 An essential merit of this methodology consists of the unification it offers among several apparently unrelated phenomena. Although various versions of nonequilibrium thermodynamics exist, the formalism propounded by Onsager5 is widely employed, especially if the linear flux-force regime is valid. Further, the statistical thermodynamic perspective of this approach provides new insights into macroscopic phenomenology. The general expression for flux Ji of a multicomponent system involving the corresponding forces Xi is represented as

Ji )

∑j Lij Xj

(1)

where Lij's are referred as Onsager’s phenomenological coefficients. Depending upon the system under consideration, Lij 's are then represented in terms of diffusion coefficients (Dij), thermal conductivities, and so forth. The estimation of diffusion coefficients (Dij) in multicomponent liquid systems constitutes a thoroughly-analyzed exercise that illustrates the power of Onsager’s formalism in rationalizing the sign and magnitude of Dij’s in terms of Onsager’s coefficients Lij. Further, the validity of Onsager’s Reciprocity Relations (ORR) has been demonstrated using the transport data pertaining to a variety of liquid mixtures.6 The tour de force of the formalism however consists in analyzing coupled effects such as thermoelectric phenomena, electro-osmosis and thermomechanical effects. The thermodynamic interpretation of kinetic data within a microscopic framework is also feasible. While one of the earliest illustrations of ORR deals with the measurement of transport numbers in electrolyte solutions, subsequent applications of nonequilibrium thermodynamics to electrochemical problems have been relatively scarce, the derivation of Bulter-Volmer

equation for electrode kinetics using Keizer’s approach being a notable exception3. This is surprising especially in view of novel phenomena and/or patterns that arise when (time-dependent) electrode potential is used as a perturbation in conjuction with other driving forces such as thermal or concentration gradients, hydrodynamic influences, and so forth. The purpose of this article is to (i) investigate diffusion migration equations for redox polymer electrodes using Onsager’s version of nonequilibrium thermodynamics; (ii) verify the validity of ORR when electron-hopping between redox species occurs; (iii) estimate the phenomenological coefficients pertaining to charge transport of [Cp2 Fe TMA]2+/1+ on Nafion coated Glassy carbon electrodes and relate these to experimental observables such as redox conductivity, transport numbers and so forth; and (iv) illustrate the use of entropy production concept in the interpretation of diffusion coefficient in restricted environments such as polymeric systems. Transport Equation. The chemically modified electrodes in which redox centers are immobilized belong to either covalently attached or ion exchange redox polymers. In the covalently attached polymers such as poly[Os(bpy)2(vpy)2]Xn where n varies from 1 to 3, the polymer itself is electroactive. Since the redox species are strongly bounded by the polymer backbone, physical displacement of the species is hindered on account of the covalent bond existing between the redox centers and the polymer backbone. Hence, electron hopping between spatially separated redox centers is the predominant mode of charge transport. However, polymer side chain motion can help in reorienting the redox species prior to electron hopping, which is normally referred to as bounded diffusion.7 On the other hand, in the case of ionexchange redox polymers, electroactive species is incorporated within the polymer by electrostatic binding, e.g., [Ru(NH3)6]3+/2+ incorporated in Nafion membranes. Hence in these cases, physical diffusion of the redox species is possible in addition to electron hopping. Presently, restricting ourselves to the (more involved) ion-exchange redox polymers,8 Onsager’s formalism

10.1021/jp001960a CCC: $20.00 © 2001 American Chemical Society Published on Web 03/01/2001

2466 J. Phys. Chem. B, Vol. 105, No. 12, 2001

Umamaheswari and Sangaranarayanan

of nonequilibrium thermodynamics (cf. Appendix A) leads to the following equation for the flux JA of the reduced species A viz.

JA ) -Dap

{

]}

[

DhopχCA ∂CA ∂φ F + (C z + CBzB) CAzA ∂x RT Dap C° A A ∂x (2)

CA and CB are the concentrations and zA and zB are the charges of the reduced and oxidized species, respectively. C0 is the total concentration of the exchanged sites (C0 ) CA + CB) and Dap denotes the apparent diffusion coefficient which is a weighted arithmetic mean of the constituent diffusion coefficients, i.e.,

Dap ) Dhop χ + Dphys(1 - χ)

(3)

where Dhop and Dphys denote respectively, the diffusion coefficients characterizing electron hopping and physical diffusion of the redox species, χ is the fraction of exchanged sites involved in electron-transfer process and (1 - χ) is the fraction of unexchanged sites not participating in the electron hopping. Equation 3 assumes the absence of local correlations. A few limiting cases of eq 3 are worth pointing out. When the rate of physical diffusion is negligible

Dap ) Dhop χ

(4)

Thus, the diffusion coefficient observed in the experimental studies of electron hopping is Dhopχ and since Dhop itself is a function of χ, this leads to a nonlinear variation of the apparent diffusion coefficient with the fractional loading χ. Under this condition, the flux eq 2 becomes

{

JA ≈ -Dhopχ

}

∂CA nF CB∂φ - CA ∂x RT C° ∂x

(6)

The corresponding flux equation is given by

}

∂CA ZACAF∂φ + ∂x RT ∂x

(7)

which is the classical Nernst-Planck eq10 for ionic movement under a bias electric field. This eq predicts a decrease in the observed diffusion coefficient with increase in χ. Thus, eqs 4 and 6 illustrate the diverse patterns of Dap with χ which are encountered in several experimental studies depending upon the system characteristics.7,11 Verification of Onsager’s Reciprocity Relations (ORR). The reciprocity relationship derived by Onsager based on the principle of microscopic reVersibility and represented as Lij ) Lji for a multicomponent system constitutes an important relation concerning Onsager’s coefficients since it provides a prescription for reducing the number of phenomenological coefficients in describing the force-flux formalism. The experimental data of electron transfer through chemically modified electrodes can be employed for verifying the validity of ORR. In the case of ion-exchange redox polymers, this requires partitioning of the experimentally observed diffusion

RT CB ln nF CA

(8)

where E°′ is the formal potential of the redox couple. Using the above eq in conjunction with eq A-15, we obtain

LAB )

Dap ) Dphys (1 - χ)

{

E ) E°' +

(5)

which is consistent with the diffusion-migration eq of Save´ant,9a Chidsey and Murray9b and Buck9c for nearest neighbor electron hopping, χ being unity. In the alternate limit, when the physical diffusion is the pre-dominant mode of charge transport, eq 3 leads to

JA ≈ -Dphys(1 - χ)

coefficients into Dhopχ and Dphys(1 - χ) since these are needed to estimate Lij's (cf. eqns A-15 and A-16 of Appendix A). However, since the precise estimates of Dhopχ and Dphys(1 - χ) are not un-ambiguously available, we first analyze an experimental situation wherein electron hopping has been shown to be the major mode of charge transport. A particularly illuminating example is the study of [Ru(bpy)3]3+/2+ immobilized in Nafion coated GC electrodes using chronocoulometry reported by Martin and Dollard.12 With a proper choice of electrode potential, the oxidation and reduction behavior of the above Ruthenium complex has been analyzed. The electron hopping diffusion coefficients are found to be identical for oxidation and reduction reactions of the above complex, thus demonstrating the validity of ORR.13 Note that when the diffusion coefficients are estimated at a predetermined potential CB and CA are constants. Consequently, according to eq A-15, the equality of the electron hopping diffusion coefficients implies the equality of the phenomenological coefficients, i.e., LAB ) LBA. Evaluation of Onsager’s Phenomenological Coefficients. Although the illustration of the validity of ORR has been straightforward as shown above, the estimation of Lij's is not a trivial exercise because Lij's are represented in terms of the concentration of redox species which can be varied over a wide range of potentials (E). However, the relation governing CA and CB is given by the familiar Nernst eq

-C0 exp (θ) Dhop χ (1 + exp (θ))2 RT

(9)

where θ ) (E - E°′)nF/RT represents the dimensionless potential. The experimental data involving chronocoulometry pertaining to the behavior of [Ru(bpy)3]3+/2+ on Nafion coated GC provides the magnitude of various parameters appearing in eq 8. The coulometric data yields the surface concentration as 9.5 × 10-9 mol‚cm-2 from which the bulk concentration C° is determined as 1.319 × 10-4 mol‚cm-3 using the value of wet thickness (0.72 µm). The value of formal potential is reported as 1.0 V and is found to be 1.75 × 10-10 cm2 s-1. Substituting these parameters in eq 9, the dependence of LAB on potential is obtained. Figure 1 indicates the variation of LAB with the potential. The magnitude of LAB is maximum near the formal potential as expected since (i) this is the region where electron transfer is most facile and (ii) the oxidized and reduced species are equal at this potential. A subtle feature worth emphasising here is that the diffusion coefficient Dhopχ employed in eq 9 is not dependent on the applied potential E, whereas the phenomenological coefficients vary with the potential. This behavior arises due to the occurrence of other potential dependent terms in the expression of LAB, namely, the dimensionless potential term, θ in eq 9 or the potential dependent product CACB in the eq A-15 of Appendix A. The foregoing analysis is valid if electron hopping constitutes the primary mode of charge transport. It is instructive to enquire whether any new insights based on the concepts of nonequilibrium thermodynamics arise if physical diffusion and electron hopping occur simultaneously. For this purpose, we provide a

Charge Transport in Redox Polymer Electrodes

J. Phys. Chem. B, Vol. 105, No. 12, 2001 2467

Figure 1. Schematic variation of LAB with applied potential for [Ru(bpy)3]3+/2+on Nafion coated GC electrodes calculated from eq 9.The parameters employed are C° ) 1.319 × 10-4 mol‚cm-3, E ) 1.0V and Dhopχ ) 1.75 × 10-10 cm2 s-1and are taken from the data of Martin and Dollard.12

plausible framework of analysis, by considering the study of Cp2FeTMA2+/1+- Nafion/GC system carried out by Bard et al. (Cp2FeTMA+ refers to [(trimethylammonio)methyl]ferrocene).14 Let us recall the expression for the apparent diffusion coefficient, eq 3. If two diffusion coefficients are known in that eq viz Dap using potential step or sweep techniques and Dphys(1 - χ) by permeation experiments, then it becomes possible to calculate Dhopχ and hence LAB with the help of eq 9. In the experimental study by Bard et al.14 (on Cp2FeTMA2+/1+-Nafion/GC) the following values are reported: Dap ) 1.7 × 10-10 cm2 s-1 ; Dphys(1- χ) ) 1.6 × 10-10 cm2 s-1 and E°′ ) 0.32 V. Using these estimates and assuming a value of 1 × 10-4 mol‚cm-3 for C° in eq 9, the variation of LAB with potential is obtained which is shown in Figure 2a. Analogously, the phenomenological coefficient LAV for physical diffusion is obtained from eq A-16 as follows

LAB )

- C0 Dphys(1 - χ) (1 + exp (θ))RT

(10)

The dependence of LAV on potential is shown in Figure 2b. We may note that the absolute value of LAV decreases with potential for physical diffusion. In the case of electron hopping, the variation of LAB with potential is symmetric on either side of formal potential. This behavior is not un-anticipated since electron hopping will occur at the maximum rate when the product CACB becomes maximum, i.e., at the formal potential. Hence, according to eq A-15 of Appendix A, LAB should be minimum at that potential. However, in the case of physical displacement, the variation of LAV (or LBV) with potential depends on the changes in the concentration of reduced (or oxidized) species, since only one concentration term is present in the expression for the phenomenological coefficient pertaining to physical diffusion (cf. eq A-16 of Appendix A). In other words, the phenomenological coefficient for physical diffusion of a species depends on its own concentration (which in turn varies with the applied potential) and not on the concentration of any other species. Redox Conductivity and Transport Numbers. The specific conductivity of a system is in general represented as the proportionality constant between steady-state current density and potential gradient viz.

Figure 2. (a) Schematic variation of LAB with applied potential for Cp2FeTMA2+/1+-Nafion coated GC electrodes calculated using eq 9. The parameters employed are C0 ) 1 × 10-4 molcm-3, Dhopχ ) 1 × 10-11 cm-2s-1 and E ) 0.32V and are from the data of Bard et al.14 (b) Schematic variation of LAV with applied potential calculated from eq 10 for Cp2FeTMA2+/1+system on Nafion coated GC electrode considered in (a) with Dphys(1 - χ) ) 1.6 × 10-10cm2 s-1.

∂φ ∂x

jdc ) - κ

(11)

According to eq 11, to estimate the conductivity of a system, the current density arising from potential gradients is needed. Further, the current density is related to flux as

jdc ) F

∑i ziJsi

(12)

where zi is the charge of the migrating species ′i′ and Jsi is the flux of the species with respect to solvent fixed reference frame.15 In the case of redox polymer electrodes, the conductivities are generally measured using impedance spectroscopy or steady state potential gradient experiments. In impedance measurements, at the high frequency region, the measured conductivity includes the conductivities of all the mobile species, which is customarily referred to as high-frequency conductivity9c κHF,

κHF ) κhop + κphys + κc

(13)

where κhop and κphys refer to the conductivities of the redox polymer due to electron hopping and physical displacement of the redox species, and κc refers to the conductivities of other species like counterions, co-ions, and so forth.



For the sake of simplicity, let us neglect the third term in eq 13, i.e., conductivities due to counterions and co-ions. Hence from eqs 11 and 12

jdc ) F[zAJA + zBJB]

(14)

since zV ) 0. Rewriting the fluxes in terms of the phenomenological coefficients, the current density becomes

jdc ) F{zA[LAAXA + LABXB + LAVXV] + zB[LBAXA + LBBXB + LBVXV]} (15) Since16 LAA+LAB+LAV ) 0 and LBA+LBB+ LBV ) 0, eq 15 is rewritten as

jdc ) F{zA[LAB(XB - XA) + LAV(XV - XA)] + zB[LBA(XA - XB) + LVB(XV - XB)]} (16) j ) is the j ) where (µ The driving force Xi is defined as -grad (µ i i electrochemical potential of ′i′ and after substituting for the same in eq 16, we obtain redox conductivity as

κredox ) -F2{LAB(zB - zA)2 + LAVzA2 + LBVzB2} (17) where the first term in the bracket corresponds to electron hopping between the redox species and the last two terms correspond to the physical diffusion of reduced and oxidized species through the polymer matrix. Equation 17 becomes

{

2 F2 n CACB κredox ) Dhopχ + zA2CADphys(1 - χ) + RT C°

zB2CBDphys(1 - χ)

form of eq 15 is then obtained as

j ) F{zA[LAAXA + LABXB] + zB[LBAXA + LBBXB]} (20) After making use of the conditions LAB + LAA ) 0 and LBA + LBB ) 0 and, substituting for the driving forces in terms of the electrochemical potentials and using Onsager’s reciprocity relationship LAB ) LBA, we have

κhop ) -(zB - zA)2F2LAB ) -n2F2LAB

}

(18)

after substituting the appropriate expressions for LAB, LAV, and LBV and so forth and assuming κc ) 0. The above eq can be rewritten after making use of the fact that Dap ) Dhop χ + Dphys(1 - χ) as

κredox )

Figure 3. Schematic variation of κredox with applied potential, from eq 18 for Cp2FeTMA2+/1+ sNafion coated GC electrodes with the same parameters as in figures (2a) and (2b).

F2 {D (C z 2 + CBzB2) + RT ap A A Dphys(1 - χ)(CAzA + CBzB)2} (19)

which upon rearrangement becomes identical with the earlier derivation for conductivity.5 Although eq 19 is quite general, the measurement of redox conductivity per se is difficult, since as mentioned earlier, the high frequency conductivity includes the contribution from the mobility of counterions and co-ions in the polymer as well as in the solution phase in addition to the redox conductivity. Hence, isolation of individual conductivities is precluded. However, it is possible to observe the conductivity arising exclusively from electron hopping using the polymer modified electrode in sandwich or inter digitated array configurations and conducting steady-state potential gradient experiments.17 This is achieved by employing either of the following strategies: (i) high scan rate in potential sweep techniques (Linear and Cyclic Voltammetry) and (ii) low temperatures. Both of the conditions lead to the reduction in counterion diffusion within the polymer, which thereby halts the development of concentration gradient inside the redox polymeric system. When high scan rates are employed the counterions remain essentially immobile. Similarly, at lower temperatures,physical diffusion18 of the redox species and the counterions is precluded18b since they are “frozen”. A limiting

(21)

where n equals zB - zA. Thus, it is possible to obtain Onsager’s cross coefficient for electron hopping directly from experimental data using eq 21 instead of evaluating via eq 9. When LBA is expressed in terms of diffusion coefficient, we obtain the electronic conductivity derived earlier by Chidsey and Murray,9b

κhop )

n2F2CACBDhopχ C° RT

(22)

Note that in eq 19 κredox denotes conductivity due to electron hopping as well as physical diffusion, whereas κhop in eq 22 is that arising from electron hopping alone. κhop is implicitly dependent upon potential via CA and CB. To infer the dependence of κredox on E, the experimental data of charge transport for Cp2FeTMA2+/1+in Nafion coated GC electrode (reported by Bard et al.,14 cf. section 2.2) is employed. For various potentials, CA and CB are estimated with the help of Nernst eq (eq 8). After substituting for the diffusion coefficients, CA, CB and C° in eq 18, the variation of redox conductivity with electrode potential E is obtained, as shown in Figure 3. Similarly in order to interpret the variation of κhop with E, a system in which electron hopping is the predominant mode of charge transport is needed. For this purpose the experimental data for Ru(bpy)33+/2+-Nafion/GC system as reported by Martin and Dollard12 was chosen (cf. section 2.1). The concentrations CA and CB are calculated as described above. After substituting for the appropriate terms in eq 22, the variation of κhop with potential is obtained as shown in Figure 4. Note that in electron hopping transport, the conductivity is maximum only at the formal potential E°′ where both oxidized and reduced species are present at equal proportions (cf. Figure 4). The conductivity is negligibly small at other values of the electrode potential. In the case of redox conductivity, the conductivity increases even after the formal potential due to the enhanced contribution of


J. Phys. Chem. B, Vol. 105, No. 12, 2001 2469 transport within the polymer. Thus, the transport number of electrons arising solely from hopping is written as

thop )

n2LAB n2LAB + zA2LAV + zB2LBV

(27)

Similarly, the ionic transport number describing the physical displacement of the redox species can be represented as

tphys )

Figure 4. Schematic variation of κhop with potential for [Ru(bpy)3]3+/2+Nafion coated GC electrodes calculated using eq 22 for the parameters indicated in Figure 1.

physical diffusion to the conductivity in this potential region. Infact, as per eq 19, the physical diffusion of redox ions should contribute to conductivity at all potentials, due to the higher fraction of reduced species at E < E°′ and oxidized species at E > E°′. But since this ionic conductivity varies as the square of the ionic charges and since always zB > zA, its contribution to the total redox conductivity is more when the fraction of oxidized speices is higher, i.e., when E > E°′. Analogously, in the study of ionic transport through polymers and biological membranes19 the measured conductivity corresponds to the physical diffusion of the electroactive species through the disordered matrix. Thus, for polymers consisting of reduced (A) and oxidized (B) species, the steady-state current due to the ionic diffusion can be represented as

jdc ) F{zA[LAAXA + LAVXV] + zB[LBBXB + LBVXV]} (23) Since electron hopping is absent, the electronic flux and, consequently LAB, is zero. By incorporating Onsager’s relationships, LBB ) -LBV and LAA ) -LAV and substituting for the driving forces in eq 23, we obtain the ionic conductivity of the redox species as follows

κphys ) -F2{zA2LAV + zB2LBV}

(24)

After substituting for the phenomenological coefficients, we have

F2Dphys(1 - χ) 2 {zA CA + zB2CB} κphys ) RT

(25)

Equation 24 is consistent with the equations describing ionic conductivities in dilute electrolytic solutions20 except that the Onsager’s coefficients LAV and LBV, now take into account the blocking effect offered by the polymer matrices for the diffusion of ionic species through them. Transport numbers can be expressed as the ratio of conductivity of the appropriate species to the total conductivity,21 i.e.,

ti )

κi

∑i

(26) κi

In the case of redox polymer electrodes ′i′ refers to electrons and electroactive species if we exclude counterion and co-ion

zA2LAV + zB2LBV n2LAB + zA2LAV + zB2LBV

(28)

It follows from the above equation that if Onsager’s coefficients can be estimated heuristicallly or otherwise, all the experimental observables pertaining to redox polymer electrodes become available per se. Entropy Production. For systems not far away from thermodynamic equilibrium, it is possible to express the rate of entropy production as16

dSint dt

)

1 T

∑i JiXi

(29)

and the fluxes are given by eq 1. At first, let us confine ourselves to systems wherein electron hopping is the only mode of charge transport (i.e., neglecting ionic diffusion through vacancies). In this case, the subscript ′i′ in eq 1 represents the species A or B. The flux of the reduced species A is represented as

JA ) LAAXA + LABXB

(30)

After substituting for the phenomenological coefficients and the driving forces in eq 1, we obtain the rate of entropy production as

dSint 1 ) [LAAXA2 + 2LABXAXB + LBBXB2] dt T

(31)

The entropy production should be positive or zero depending upon whether the process is irreversible or reversible, and hence, the following inequalities must hold true: LAA g 0, LBB g 0 and LAALBB-LABLBA g 0. The coefficient LAB corresponds to the diffusion of species A in the field generated by B, which can be expressed in terms of the interdiffusion coefficient using eq A-15. Similarly, the coefficients LAA and LBB should represent the diffusion of a species in its own field. In the case of systems with electron hopping as the predominant mode of transport, the physical diffusion of the redox species is almost negligible. Hence, the only manner in which A or B can “apparently” move, is by electron hopping. Thus, the direct coefficients LAA and LBB have no direct significance but can be represented in terms of LAB (or LBA) as

LAA ) -LAB )

C AC B D χ C°RT hop

(32)

LBB ) -LBA )

C AC B D χ C°RT hop

(33)

which leads to the condition that LAALBB-LABLBA ) 0. On physical grounds, too, this equality is expected as the redox couple A/B is reversible. Similarly, in the case of polymer systems in which the physical diffusion of oxidized and reduced species alone takes



place and electron self-exchange is absent, the rate of entropy production is given as

dSint dt

)

1 T

∑i JiXi

i ) A, B, V

(34)

Upon substitution of appropriate terms in eq 34 and making use of the condition LAB ) LBA ) 0 (since electron hopping is absent), we have

dSint 1 ) [LAAX2A + LBBXB2 + LVVXV2 + dt T 2LAVXAXV + 2LBVXBXV] (35)

dSint 1 ) [LAAXA2 + LABXAXB + LAVXAXV + LBAXBXA + dt T LBBXB2 + LBVXBXV + LVAXVXA + LVBXVXB + LVVXV2] (40) After making use of the reciprocity relationships LAB ) LBA, LAV ) LVA and LBV ) LVB, eq 40 becomes

dSint 1 ) [LAAXA2 + LBBXB2 + LVVXV2 + 2LABXAXB + dt T 2LAVXAXV + 2LBVXBXV] (41) Accordingly, we obtain the following limits for Onsager’s direct and cross phenomenological coefficients for the case of competing diffusion process

which leads to the condition

LAA g 0 LBB g 0 Lvv g 0

LAA g 0 LBB g 0 LVV g 0

(36)

and

LAALBBLvv g LAAL2BV + LBBL2AV +

and

LAALBBLVV g

LAAL2BV

+

LBBL2AV

(37)

For the above inequality to hold true, the magnitude of the cross coefficients should be smaller than the self-phenomenological coefficients. Thus, we can write LAA g LAV and LBB g LBV so that the entropy production due to internal fluxes will be g 0. LAA is given by CADA/RT where DA refers to the intradiffusion coefficient. Similarly, LAV is represented22 as CADA(1 - χ)/RT. Comparing the expressions for LAA and LAV, we obtain

DA g DA(1 - χ)

(38)

Therefore, if χV ) 1, the observed diffusion coefficient in a polymer matrix can become equal to that observed in dilute solutions. Since χV will be always less than unity in polymer systems, the physical diffusion coefficient of an ionic species in a polymeric matrix (DA(1 - χ)) is, in general, lower than that observed in electrolytic solutions (DA). Similarly, the above argument when applied to the physical diffusion of the oxidized species leads to the analogous result, i.e.,

DB g DB(1 - χ)

LVVL2AB - 2LABLAVLBV (42)

(39)

Equations 38 and 39 provide a method of estimating the blocking factor if the physical diffusion coefficient of ions within a polymer matrix, Di(1 - χ) and in dilute electrolytic solutions, Di is known. The analysis of blocking factor χ in eq 39 when competing diffusion processes occur, is a profound exercise in random walk models of transport phenomena wherein its influence on dimensionality of the lattice, nature of interactions (hard core vs Coulombic), applied electric field and so forth are investigated using a variety of theoretical approaches. An elementary account of the blocking factor in this context is given elsewhere23. Nernst-Planck equation corresponding to the physical diffusion of A or B arising from Onsager’s formalism is provided in Appendix B. In the case of competitive diffusion processes where both electron hopping and physical diffusion can take place simultaneously, the rate of entropy production is obtained by making use of eq 1 as follows

Equation 42 when expressed in terms of the appropriate diffusion coefficients leads to interesting inequalities among them. Thus, from the concept of entropy production it is possible to infer the limits to the magnitude of phenomenological coefficients vis a vis diffusion coefficients.24c Perspectives and Summary It is of interest here to comment upon the relevance of nonequilibrium thermodynamics formalism to experimental studies on charge transport through redox polymers. We note that an important objective in such studies is to extract “diffusion coefficients” corresponding to various processes associated with charge compensation process, using chronocoulometry and related experimental techniques. These in turn provide qualitative information regarding the influence of polymer morphology, interaction between redox centers, solvent dynamics, and so forth. Our present approach provides a first step in comprehending the observed magnitude of diffusion coefficients via Onsager’s coefficients. In particular, eqs 38 and 39 indicate the extent to which physical diffusion of a species is hindered in a polymeric environment. Further, since the magnitude of Dphys and Dhop can vary over a wide range, the availability of upper and lower bounds on them arising out of the present version, is of importance in the design of electrocatalysts. The foregoing analysis has demonstrated the applicability of Onsager’s formalism of NonequilibriumThermodynamics to the study of charge transport through redox polymer electrodes. The evaluation of Onsager’s coefficients Lij is carried out for electron hopping between [Ru(NH3)6]3+ and [Ru(NH3)6]2+ incorporated onto Nafion covered GC electrodes and the reciprocity relation is shown to be valid. While the formulation of transport equations in itself can be accomplished by different methods, the perspective offered by nonequilibrium thermodynamics is especially attractive since it gives a thermodynamic basis for the processes taking place within the polymer modified electrodes. A hierarchy of diffusion migration equations can be derived by employing improved representations for electrochemical potentials. It is interesting to note that the composition of Lij in terms of Dij given by eqn (A-9) is valid even when diverse interaction schemes are incorporated.23 Further, several hitherto


J. Phys. Chem. B, Vol. 105, No. 12, 2001 2471

unknown issues such as (i) choice of different reference frames and their implications, (ii) onset and stability of nonequilibrium stationary states using entropy production formulas, and (iii) dichotomy between thermodynamic and kinetic versions regarding their equivalence can be brought into focus in the context of electrocatalysis. Recent trends in the realm of nonequilibrium thermodynamics is concerned with the equivalence between different formalisms of nonequilibrium thermodynamics (Onsager, Lagrange, Keizer) as well as extension to nonlinear regimes (cf. Extended Irreversible Thermodynamics19), wherein processes far from equilibrium can also be treated.24 Consequently, electrochemical data pertaining to charge transport through redox polymer electrodes is expected to predict novel phenomena or provide new insights into well-known results. Hence, a systematic study using the approach outlined here will be worth pursuing. A limitation of the present approach is the noninclusion of percolation and correlation effects. These factors, despite their importance can be incorporated at the present level of development, only in a phenomenological manner in the case of competitive diffusion processes. We may also emphasize that the transport equation analyzed above have not considered other perturbations such as the effect of temperature gradient on the current response; nonequilibrium thermodynamics formalism is especially suitable to incorporate these miscellaneous effects solely via additional inclusion of the driving force term in the flux-force relation. We may also recall here that one of the earliest methods of verifying ORR has been on the basis of triangular cyclic reaction sequence given by

Hence, the analysis expounded here can alternatively be carried out via rate constants kij's pertaining to physical diffusion and electron transfer by keeping the mean square displacement as a constant of approximately equal magnitude which may lead to interesting inequalities among the rate constants of the various processes. Acknowledgment. We thank the reviewers for valuable comments and the Department of Science and Technology of the Government of India for financial support. Appendix A In this Appendix, we illustrate the methodology of deriving the flux equation using Onsager’s formalism of nonequilibrium thermodynamics. In the case of ion-exchange polymers (cf. Verification of Onsager’s Reciprocity Relations section), electron hopping and physical displacement occur simultaneously and the flux-force relation is given by

JA ) LAAXA + LABXB + LAVXV

(A-1)

JB ) LBAXA + LBBXB + LBVXV

(A-2)

JV ) LVAXA + LVBXB + LVVXV

(A-3)

where JA, JB, and JV denote the fluxes due to reduced species, oxidized species and vacancies respectively and XA, XB, and XV denote the corresponding “forces”. For clarity, we focus attention on the flux JA. It is well-known that in a ternary system such as the one considered here, Onsager’s formalism leads to

LAA + LAB + LAV ) 0. Consequently, we can write eq A-1 as

JA ) LAB(XB - XA) + LAV(XV - XA)

(A-4)

XA, XB, and XV are the negative gradients of appropriate electrochemical potentials. Assuming that vacancies are uncharged, we have

XB - XA ) -grad[Fφ(zB - zA) + RTlnCB - RTlnCA] (A-5) and

XV - XA ) -grad[- zAFφ + RTlnCV - RTlnCA]

(A-6)

wherein activity coefficient effects and interactions between species have been neglected. If we consider the transport to be one-dimensional, eq A-4 becomes

[

]

∂lnCB ∂lnCA ∂φ + RT - RT ∂x ∂x ∂x ∂lnCV ∂lnCA ∂φ LAV -zAF + RT - RT (A-7) ∂x ∂x ∂x

JA ) -LAB nF

[

]

where ∂φ/∂x denotes the potential gradient, and n denotes the number of electrons transferred during electron self-exchange (n ) zB - zA). Assuming uniform distribution of vacant sites (i.e., dCV/dx ) 0), we obtain

[

JA ) -LAB

]

RT∂CB RT ∂CA ∂φ + nF CB ∂x CA ∂x ∂x RT∂CA ∂φ -zAF (A-8) LAV CA ∂x ∂x

[

]

To convert the above equation into the conventional Fick’s law description, we require the compositions of the Onsager’s coeficients LAB and LAV. The general expression for Lij, where i and j refer to the concerned species is given by25

( )

Lij ) δij

( )

χi χiχj DiCT + DC RT RT ij T

(A-9)

where δij is the Kronecker delta, χi and χj are the fractional concentrations, CT denotes the total number of sites while Di and Dij denote the intra (or tracer) and inter diffusion coefficients, respectively. Equation A-9 expresses the dependence of Lij in a general manner on Di (customarily called as the tracer diffusion coefficient) and Dij (inter diffusion coefficient). While the use of tracer diffusion coefficient is common in solid-state physics, the analysis of inter diffusion coefficients Dij is crucial in multicomponent liquid systems. In the present context, we employ A-9 such that terms involving Di do not occur. Thus,

LAB )

χAχB D C RT AB T

(A-10)

LAV )

χAχV D C RT AV T

(A-11)

where χA + χB ) χ is the fraction of sites occupied by the redox species and χV deontes fraction of un-exchanged sites. (χ + χV ) 1). Since Onsager’s coefficients are related as LAA + LAV + LAB ) 0 we obtain


LAA ) -


χAχB χAχV DABCT D C RT RT AV T

(A-12)

The direct coefficients are always positive (cf. the Entropy Production section), hence the diffusion coefficients DAB and DAV in eq A-12 per force are negative whereby the phemonenological coefficients can be rewritten as

χAχB |D |C LAB ) RT AB T

(A-13)

χAχV |D |C RT AV T

(A-14)

LAV ) -

To map with experimental results, the above coefficients can be expressed as

CACB LAB ) D χ C°RT hop LAV ) -

(A-15)

CA D (1 - χ) RT phys

(A-16)

where Dhopχ ) |DAB|C°/CT and Dphys(1 - χ) ) |DAV|CV/CT. Substituting for the phenomenological coefficients in eqn A-7 and coupling the apparent diffusion coefficient as Dap ) Dhop χ + Dphys(1 - χ) we have the flux JA as

JA ) -Dap

{

[

Dhopχ CA ∂CA F + (C z + CAzA ∂x RT Dap C° A A CBzB)

] } ∂φ ∂x

(A-17)

The fluxes JB and JV can be obtained in a similar manner as follows

JB ) -Dap

{

]}

[

∂CB DhopχCB ∂φ F + (C z + CBzB) C z ∂x RT B B Dap C° A A ∂x (A-18)

{

}

∂CA zACAF∂φ + + JV ) |DVA|(1 - χ) ∂x RT ∂x ∂CB zBCBF∂φ |DVB|(1 - χ) + ∂x RT ∂x

{

}

species respectively and XA and XB denote the corresponding “forces”. As in Appendix A, we focus attention on JA. Since electron hopping does not occur, the terms LAB and LBA of eqns A-1 and A-2 are absent in eqns B-1 and B-2. Onsager’s formalism leads to LAA + LAV ) 0. Consequently, we can write eqn B-1 as

JA ) LAV(XV - XA)

(B-3)

Since we consider only the physical diffusion of redox ions through the polymer matrix both LAA and LAV describe the ionic movements. Assuming uniform distribution of uncharged vacancies (i.e., dCV/dx ) 0 and zV ) 0) and considering the transport to be one-dimensional, eqn B-4 becomes

[

]

∂lnCA ∂φ + RT ∂x ∂x

JA ) LAV zAF

(B-4)

where ∂φ/∂x denotes the potential gradient. From eq A-9, we obtain the expression for the phenomenological coefficient LAV as

LAV ) -

χAχV |D |C RT AV T

(B-5)

The negative sign arises since LAA ) - LAV and the Onsager’s self-coefficient, LAA has to be positive. Substituting for LAA in terms of the concentration units in eqn B-5, we obtain the classical Nernst-Planck eq for the diffusion of A.

JA ) -|DAV|(1 - χ)

{

}

∂CA zACAF∂φ + ∂x RT ∂x

(B-6)

The flux eq of the oxidized species B due to uni-molecular physical diffusion can be derived in a similar manner as

JB ) -|DBV|(1 - χ)

{

}

∂CB zBCBF∂φ + ∂x RT ∂x

(B-7)

The above analysis illustrates the elegance of Onsager’s formalism of nonequilibrium thermodynamics to derive diffusion migration equations for ionic transport and is amenable for generalization to multicomponent diffusion as well.

(A-19) References and Notes

Equation A-17 is identical with that reported earlier using Kinetic Ising Model versions26 and jump frequency model5 and is a generalization of the nearest neighbor electron hopping flux eq of Save´ant9a and the classical Nernst Planck eq for ionic movement.10 Appendix B We derive here the flux equation for uni-molecular ionic transport (Nernst Planck eqn) within a polymeric matrix, using Onsager’s formalism of nonequilibrium thermodynamics. Thus, for systems having A and B, the fluxes due to physical diffusion can be represented as

JA ) LAAXA + LAVXV

(B-1)

JB ) LBBXB + LBVXV

(B-2)

and

where JA and JB denote the fluxes due to reduced and oxidized

(1) de Groot, S. R. Thermodynamics of IrreVersible Processes; NorthHolland: Amsterdam, 1952. (2) Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard University Press: Cambridge, Massachusetts, 1965. (3) Keizer, J. Statistical Thermodynamics of IrreVersible Processes; Springer-Verlag: New York, 1992, Chapter 5, pp 208-211. (4) Lebon, G. In AdVances in Thermodynamics, Vol. 7; Sieniutyz, S., Salamon, P., Eds.; Taylor and Francis: New York, 1992 p 58. (5) (a) Onsager, L. Phys. ReV. 1931, 37, 405 (b) Onsager, L. Phys. ReV. 1931, 38, 2265. (6) Cussler, E. L. DiffusionsMass Transfer in Fluid Systems; Cambridge University Press: New York, 2nd ed, 1998, Chapter 5. (7) (a) Inzelt, G. in Electroanalytical Chemistry, Vol. 18; Bard, A. J. Ed.; Marcell Dekker: New York, 1994. (b) Umamaheshwari, J.; Sangaranarayanan, M. V. J. Phys. Chem. B 1999, 103, 5687. (c) The bounded diffusion of the polymer side chains is also characterised by a diffusion coefficient, see 7d. (d) Jones, R. A. L.; Richards, R. W. in Polymers at Surfaces and Interfaces; Cambridge University Press: New York, 1999. (8) We have precluded the analysis of charge transport in polymers with bounded diffusion, since for these polymers, the estimate for physical diffusion coefficient is not known. (9) (a) Saveant, J. M. J. Electroanal. Chem. 1986, 201, 11. (b) Chidsey, C. E. D.; Murray, R. W. J. Phys. Chem. 1986, 90, 1479. (c) Buck, R. P. J. Phys. Chem. 1988, 92, 2. 4196.

Charge Transport in Redox Polymer Electrodes (10) See for example Buck, R.P. J. Membr.Sci. 1984, 17, 1. (11) Blauch, D. N.; Saveant, J. M. J. Am. Chem. Soc. 1992, 114, 3323. (12) Martin, C. R.; Dollard, K. A. J. Electroanal. Chem. 1983, 159, 127. (13) The electron hopping diffusion coefficient for oxidation and reduction can be denoted as DBA and DAB respectively where A and B are the reduced and oxidized species, respectively. (14) White, H. S.; Leddy, J.; Bard, A.J. J. Am. Chem. Soc. 1982, 104, 4811. (15) (a) We have considered the solvent fixed reference frame in order to be consistent with existing treatments for transport in electrolytic solutions, see 15b. (b) Tyrrel, H. J. V.; Harris, K. R. Diffusion in Liquids:A Theoretical and Experimental Study; Butterworth & Co: Markham, ON, Canada, 1984. (16) De Groot, S. R.; Mazur, P. In Nonequilibrium Thermodynamics; North-Holland Publishing Company: Amsterdam, 1962. (17) Dalton, E. F.; Surridge, N. A.; Jerrigan, J. C.; Wilbourn, K. O.; Facci, J. S.; Murray, R. W. Chem. Phys. 1990, 141, 143,. (18) (a) Jernigam, J. C.; Murray, R. W. J. Phys. Chem. 1987, 91, 2030. (b) Although the Arrhenius form of temperature dependence is still valid for electron hopping as well as physical diffusion, the exponential term makes a small contribution, at lower temperatures. It is presumed here that physical diffusion is more hindered when low temperatures are employed.

J. Phys. Chem. B, Vol. 105, No. 12, 2001 2473 (19) Forland, K. S.; Forland, T.; Ratkje, S. K. In AdVances in Thermodynamics; Sieniuttez, S., Salamon, P., Eds.; Taylor and Francis: New York: 1992, Vol. 6, p 340. (20) Forland, K. S.; Forland, T.; Ratkje, S. K. In IrreVersible Thermodynamics: Theory and Applications; John Wiley & Sons: 1998 eqn (4.6). Note that LAV and LBV are negative thereby making kphys positive. (21) Bard, A.J.; Faulkner, L.R. In Electrochemical Methods: Fundamentals and Applications; John Wiley & Sons: New York, 1980. (22) (a) The coefficient LBV in polymer matrix differs from LBB in dilute solutions by the blocking factor χ; (b) Zhang, L.; Macdonald, D. Electrochim. Acta 1998, 43, 679. (23) Denny, R. A.; Sangaranarayanan, M. V. J. Phys. A. Math. Gen. 1998, 31, 7671. (24) (a) Zhong, E. C.; Friedman, H. L. J. Phys. Chem. 1988, 92, 1685. (b) Miller, D. G. J. Phys. Chem. 1994, 98, 5565, and references therein (c) Since the quadratic form of eq 41 should be positive definite, the matrix associated with the Onsager’s coefficients can be subjected to further analysis, regarding the magnitude of individual coefficients and their upper and lower bounds. This aspect requires further investigation. (25) Denny, R. A.; Sangaranarayanan, M. V. J. Phys. Chem. B 1998, 102, 2131. (26) Fort, J.; Casas-Vizques, J.; Mindez, V. J. Phys. Chem. B 1999, 103, 860.

Nonequilibrium Thermodynamics Formalism for Charge Transport in

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