J. Phys. Chem. B 1997, 101, 9593-9599
9593
Charge Transport in Ferrocene-Containing Polyacrylamide-Based Redox Gels Hai-zhi Bu, Ann M. English,* and Susan R. Mikkelsen* Department of Chemistry and Biochemistry, Concordia UniVersity, 1455 de MaisonneuVe BouleVard West, Montreal, Quebec, Canada H3G 1M8 ReceiVed: July 9, 1997X
The charge transport properties of the redox gels formed from the copolymerization of vinylferrocene (VF) with acrylamide and N,N′-methylenebisacrylamide (PVAB gels) are described. The apparent electron diffusion coefficients (Dap) measured by cyclic voltammetry and chronocoulometry exhibit an upward bowlike curvature vs immobilized VF concentration ([VF]imm) within the range 0.1-1.60 mM in 0.10 M supporting electrolyte (NaH2PO4/Na2HPO4, NaClO4, or NaNO3). The range of measured Dap values [(0.6-6.0) × 10-7 cm2 s-1] can be interpreted in terms of the mean-field model developed by Blauch and Save´ant (Blauch, D. N.; Save´ant, J. M. J. Am. Chem. Soc. 1992, 114, 3323). The large Dap values are attributed to the large λ (range of molecular motion permitted to the ferrocene residues) and large kex (bimolecular electron self-exchange rate constant of the ferrocenes in the PVAB gels). It was further observed that Dap decreases on increasing the electrolyte (NaClO4 or NaNO3) concentration from 0.10 to 0.50 M, especially at low (0.1-0.6 mM) and high (1.1-1.6 mM) [VF]imm, resulting in almost flat Dap vs [VF]imm profiles at 0.50 M electrolyte. This observation is attributed to increased nonbonded interactions between the polymer matrix, the redox sites, and the electrolyte upon increasing the electrolyte concentration.
Introduction Investigations of polymers containing covalently attached redox sites have considered both their fundamental properties, such as charge propagation mechanisms and dependence of charge transport rates on redox site concentrations,1-8 and their applications in amperometric enzyme electrodes,9-14 electrocatalysis,15,16 electrosynthesis,16 and energy conversion.17 There is no doubt that an understanding of charge transport mechanisms and rates in redox polymers is essential for optimizing their performance in various applications. Electron transport through polymeric systems containing redox centers occurs via physical displacement of the redox centers and electron hopping between nearby centers. For free redox molecules in solution, the apparent electron diffusion coefficient, Dap, representing a combination of physical motion and electron hopping, is given by the Dahms-Ruff equation
Dap ) Dphys + kexδ2C*/6
(1)
where Dphys is the diffusion coefficient for physical displacement of the redox molecules, kex the bimolecular rate constant for electron hopping, C* the total concentration of redox species, and δ the center-to-center distance between redox centers at the time of electron transfer.18 When the redox centers are attached to a polymer matrix by way of covalent, coordinate, or strong electrostatic binding such that the physical diffusion of the redox centers does not contribute to charge transport, Dap is then expressed by the Laviron-Andrieux-Save´ant equation
Dap ) kexδ2C*/6
(2)
which is regarded as an extreme case (Dphys ) 0) of eq 1.18 Equations 1 and 2, predicting a linear increase of Dap with increasing C*, have been widely used to qualitatively elucidate * Authors to whom correspondence should be addressed. Fax: 514-8482868. Tel: 514-848-3355. X Abstract published in AdVance ACS Abstracts, October 15, 1997.
S1089-5647(97)02226-8 CCC: $14.00
the charge transport phenomena in redox polymer systems.1 However, decreases in Dap with increasing C*, or constant Dap values regardless of C*, have been observed and appear to be common, especially in cases where multiply charged redox species are incorporated into polymer films by electrostatic interactions. Such interactions between the redox species and the polymer matrix reduce the diffusion rate of the redox species, the segmental motion of the polymer backbone, the rates of charge-compensating counterion motion, and/or the motion of solvent.1.19 In addition, the diffusing redox molecules, which must move between more or less fixed sites within a polymeric matrix, are considered to have their rate of motion limited by the decreasing availability of sites as C* increases (single-file diffusion effect).19 Both electrostatic interaction and singlefile diffusion effects lead to decreased Dap values with increasing C*. Depending on whether (a) electron hopping or (b) electrostatic interaction and/or single-file diffusion predominates, three typical patterns of charge propagation may occur as follows: (1) when the contribution from (a) is dominant, Dap increases with increasing C*; (2) when the effects in (b) are dominant, Dap decreases with increasing C*; and (3) when (a) and (b) counterbalance each other, no variation of Dap with C* is observed.19 Furthermore, examples of systems where the electron transport rates increase sharply at high site concentrations and less steeply at lower concentrations have also been reported.1,19 Several theories have been proposed to explain the nonlinear increase of Dap with C*, such as the He-Chen model,20 FritschFaules-Faulkner model,21 percolation theory,22 and AndrieuxSave´ant theory.23 Although extensive experimental and theoretical work has been carried out, detailed characterization of charge propagation mechanisms in redox polymers often encounters difficulties due to the limited information available on the microscopic distribution of redox sites and on the dynamic behavior of the polymer matrix. The situation is further complicated by the fact that different experimental measurements of the same property can yield different and even contradictory results, and sometimes the same observations are interpreted in different ways.1,24 © 1997 American Chemical Society
9594 J. Phys. Chem. B, Vol. 101, No. 46, 1997 Of all the redox polymers used to modified electrodes, ferrocene (Fc)-containing ones, notably poly(vinylferrocene) (PVF), have been most widely used because of the well-behaved electrochemical properties of the Fc0/Fc+ couple.1,25 PVFmodified electrodes are generally prepared by either solvent evaporation,26,27 electrochemical deposition,28,29 or plasma polymerization30 procedures in organic solvents. Charge transport mechanisms through these relatively hydrophobic redox polymeric films have been extensively explored with respect to various factors such as redox states, the nature and concentration of the supporting electrolyte and counterions, temperature, solvent swelling, solvent incorporation, electron hopping, polymer chain motion, and polymer morphology.1,25 In addition, the dependence of charge transport rate on Fc concentration has been studied in a copolymer of vinylferrocene (VF) and (δ-methacrylylpropyl)trimethoxysilane31 and in blends of PVF and poly(tetracyanoquinodimethane).32 In the copolymer system, the Fc content was varied from 38 to 88%, but unfortunately, the interpretation of the data was complicated by the incomplete electroactivity of the Fc sites and consequent ambiguity in redox site concentration. The Dap values determined by chronoamperometry increased with concentration when nominal Fc concentrations were used, but Dap was independent of concentration when the actual electroactive site concentrations as determined by cyclic voltammetry were substituted for the nominal concentrations.31 Similar problems emerged for the blended polymer system. We have recently developed a novel Fc-containing polyacrylamide-based redox gel (designated PVAB) which can be formed in a one-step procedure on VF, acrylamide (AA), and N,N′methylenebisacrylamide (BIS) copolymerization.9 Since hydroxypropyl-β-cyclodextrin (HPCD) was used to convert waterinsoluble VF into a water-soluble VF-HPCD inclusion complex, the polymerization was performed in aqueous solution and initiated by a ternary catalyst system consisting of flavin mononucleotide, hydrogen peroxide, and N,N,N′,N′-tetramethylethylenediamine. PVAB gels containing glucose oxidase (GOx-PVAB) were prepared by polymerization from GOxcontaining solutions, which resulted in GOx entrapment in the polymer matrix. A GOx-containing redox gel electrode (GOxRGE) was constructed by covering the GOx-PVAB gel on the surface of a carbon paste electrode with a dialysis membrane. The PVAB gels contain up to ∼94.5% (w/w) water, and only 0.19-1.60 mM immobilized Fc which corresponds to 0.0630.54% and 0.0035-0.030% (w/w) Fc residues in the dried polymer matrices and hydrated gels, respectively. To the best of our knowledge, this is the lowest concentration range of redox sites reported for Fc-containing redox polymers or gels used in polymer-modified electrodes and biosensors. Nonetheless, the catalytic current densities of the GOx-RGEs were surprisingly high considering the extremely low Fc loading of the gels. For instance, a catalytic current density of ∼13.5 µA cm-2 was obtained for a GOx-RGE containing 0.91 mM immobilized VF and 1.0 mg mL-1 entrapped GOx in N2-saturated phosphate buffer with 10 mM glucose,9 while under the same conditions glucose sensors constructed from pure polymerized ferrocenes33-35 exhibited catalytic current densities (24-38 µA cm-2) that were only 2-3-fold higher. These results indicate that the immobilized Fc mediators in the PVAB gel are highly efficient. The PVAB gel represents an ideal redox polymeric material in view of its unique features, which include very high water content, low polymer loading (∼5.5%, w/w), extremely low redox site concentration, and excellent mediation performance. To probe the mechanism and dynamics of charge transport through the PVAB redox gel system, Dap measurements by
Bu et al. cyclic voltammetry and chronocoulometry were carried out under different experimental conditions. As reported here, Dap was found to be strongly dependent on redox site concentration, and an upward bowlike curvature of Dap vs immobilized VF concentration within the range 0.19-1.60 mM was unexpectedly observed. The distribution and separation of the redox centers within the PVAB gels were examined, as was the dependence of Dap on the ratio of BIS to AA. The effects of supporting electrolyte on charge transport rates were also investigated. The results of these studies are discussed mainly in terms of the mean-field model developed by Blauch and Save´ant.18 Experimental Section Gel and Electrode Preparation. The PVAB gels and electrodes (RGEs) were prepared as described previously.9 Briefly, stock solutions of 400 mg mL-I AA and 25 mg mL-1 BIS (Aldrich) in 0.1 M sodium phosphate buffer, pH 7.0 (PB), and 10 mM VF (Aldrich) in 50 mM HPCD9 were prepared, and stored in the dark at 4 °C. AA (48 mg) and BIS (7.5 mg) were mixed with VF (0.2-2.0 µmol), and to catalyze the polymerization, FMN (30 nmol), H2O2 (1.76 µmol), and TEMED (3.30 µmol) were added. The volume was adjusted to 1.0 mL. Deoxygenation was performed by flowing watersaturated N2 over the surface of the solution for 10 min prior to irradiation with a 9-W UV lamp (Ultra-Violet Products). The polymerization was considered complete when a semirigid PVAB gel block was observed (30-250 min). A RGE was fabricated by placing a small piece (∼2 × 2 × 1 mm) of the PVAB gel on the sensing surface of a carbon paste electrode (CPE). The gel layer was covered and flatten to a thickness of ∼100-200 µm with a dialysis membrane (6000-8000 MW cutoff, Spectrum Medical Industries), which was anchored by a rubber O-ring. The resulting RGE was exhaustively washed with PB and subjected to cyclic voltammetry (0.0-0.6 V vs Ag/AgCl) until a constant profile was reached before recording the electrochemical measurements reported here. Electrochemical Measurements. Cyclic voltammetry and chronocoulometry were performed using a Bioanalytical Systems BAS 100A potentiostat. A three-electrode single-compartment cell configuration was adopted with a RGE, an Ag/AgCl reference electrode (BAS), and a Pt wire (Fisher) auxiliary electrode. All measurements were carried out at ambient temperatures (23 ( 1 °C) in solutions deaerated by purging with water-saturated N2. Cyclic voltammograms were recorded from 0.0 to 0.5 V (vs Ag/AgCl) at scan rates of 0.01-0.5 V s-1. Chronocoulometric graphs were recorded by stepping the potential from -0.1 to 0.6 V (vs Ag/AgCl) with a quiet time of 10 s and a pulse width of 250 ms. Results and Discussion Redox Site Concentration and Distribution. The immobilized VF concentration ([VF]imm), which includes only copolymerized VF residues that cannot diffuse out of the gel phase, was determined spectrophotometrically as described previously.9 The procedure involves demetalation of the Fc groups by 5% trichloroacetic acid, followed by complexation of the free Fe2+ with ferrozine, a colorimetric reagent for iron.36 No measurable volume change occurred during polymerization of the redox hydrogels which are >94% water with ∼5.5% w/w polymer matrix and 0.0035-0.030% w/w redox residues. Solvation or hydration of the polymer is similar to that of the monomers, so no distinct solvent swelling or shrinking occurs on polymerization. Thus, ignoring possible minute volume changes on polymerization, it was found that ∼94% VF in the polymerization solutions was immobilized in the concentration range 0.1-0.8 mM and ∼80% at 2.0 mM.9
Charge Transport in Ferrocene-Containing Redox Gels
J. Phys. Chem. B, Vol. 101, No. 46, 1997 9595
Figure 1. Plot of the average edge-to-edge distance (dNN) between ferrocene residues vs the immobilized VF concentration ([VF]imm) in the PVAB gels. Values were calculated using eq 3 in the text.
The distance between neighboring redox centers, an important rate-determining parameter in charge propagation, is controlled not only by the total concentration of redox centers but also by their microscopic distribution in fixed-site redox polymers.1,3,21 Various theoretical models of charge transport in redox polymers assume that the redox centers are randomly distributed in the polymeric phases. In reality, this assumption is not always satisfied, and examples of markedly nonrandom distributions of redox sites in polymeric films have been reported.37,38 It is not easy to examine redox center distributions in polymers since no direct methods are available. In the present study, the random distribution of the Fc residues in the PVAB redox gels was accomplished by sparging water-saturated N2 into the viscous pregel solution to ensure efficient mixing and eliminate nonuniform redox site and polymer matrix distributions. Uniform redox-center distribution in the gel can be indirectly confirmed by the high reproducibility of RGE fabrication. A number of RGEs were fabricated using the same CPE and small pieces of gel that were cut from different regions of a gel block containing 0.93 mM immobilized VF. The mean ( SD of the anodic peak currents obtained at 10 such RGEs was determined to be 808 ( 24 nA (by cyclic voltammetry at 0.01 V s-1 in PB) with a coefficient of variation (CV) of 3.1%. Such a small CV value indicates that the uniformity of the PVAB gel is excellent, and it is reasonable to assume that the Fc residues within the PVAB gel or the RGE gel layer are randomly distributed. For a gel system with randomly distributed redox sites, the mean nearest-neighbor center-to-center separation (rNN) between redox groups can be estimated using a microscopic hard-sphere model developed by Fritsch-Faules and Faulkner.21 By considering an excluded volume occupied by each redox group within which no other groups can intrude, rNN is given by
rNN )
( ) [() 3
4πc
1/3
γ
e Γ
∞
4
3
-
∑ n!(n + 4/3)
n)0
]
(-1)nγ(n+4/3)
(3)
where Γ(4/3) ) 0.892 98, c is the number concentration of redox centers (1 M corresponds to 6.02 × 1020 cm-3), and γ ) (4/ 3)πr03c where r0 is the molecular diameter. With r0 ) 0.6 nm for Fc,21 average nearest-neighbor edge-to-edge distances (dNN ) rNN - r0) between redox sites were calculated from eq 3 for the PVAB redox gels containing different concentrations of immobilized VF residues. The curve of dNN vs [VF]imm (Figure 1) indicates that dNN greatly decreases with increasing [VF]imm
Figure 2. (a) Steady-state cyclic voltammograms recorded for a RGE containing 1.09 mM immobilized VF in 0.10 M sodium phosphate buffer, pH 7.0, at different scan rates. (b) Chronocoulometric plot of Q vs t1/2 observed for the same RGE as in (a).
up to ∼0.8 mM, while only a small reduction in dNN occurs between 0.8 and 1.60 mM. These data provide a starting point for elucidating the electron propagation mechanisms within the PVAB redox gel. Dependence of Dap on [VF]imm. Dap is usually determined by methods such as cyclic voltammetry,5,3 chronoamperometry,4,5 chronocoulometry,3,8 impedance spectroscopy,3,7 and pulse techniques,40,41 as well as steady-state voltammetry.2,6 Cyclic voltammetry and chronocoulometry were used in this study to determine the Dap values for the PVAB gels. Cyclic voltammograms of redox-polymer-modified electrodes exhibit diffusional responses with varying scan rate (ν) under semi-infinite linear diffusion conditions.25,42 The voltammetric peak current (ip) is directly proportional to ν1/2 and is given by the Randles-Sevcˇik equation, which also describes ip for species dissolved in solution and diffusing to the electrode25
ip ) 2.69 × 105n3/2A(Dap1/2C*)ν1/2
(4)
where A is the electrode surface area and C* ) [VF]imm in the present work. Hence, the slope of the ip vs ν1/2 plot provides the product Dap1/2C*, from which Dap can be estimated when C* is known. Typical steady-state cyclic voltammograms at different sweep rates from a RGE containing 1.09 mM immobilized VF are shown in Figure 2a. The peak potential separation is ∼90 mV at a scan rate of 0.01 V s-1, but greater peak separations are seen at faster scan rates, indicating quasireversible electrochemical behavior.42 The plot of ip vs ν1/2 (not shown) is linear for the RGE, which reveals diffusion-controlled charge transport through the PVAB gel to the CPE surface.25 It is somewhat dangerous to draw too many conclusions from cyclic voltammetric information alone due to the continuous variation of the boundary condition (potential) and difficulties associated with uncompensated iR drops and background currents.43 In addition, eq 4 is strictly applicable only to electrochemically reversible redox couples, while the RGE voltammograms indicate quasi-reversibility. Chronocoulometry is a good alternative for determining Dap since resistance effects can be overcome by using larger potential steps.25 In this case,
9596 J. Phys. Chem. B, Vol. 101, No. 46, 1997
Bu et al.
Figure 3. Variation of the apparent electron diffusion coefficient (Dap) with the immobilized VF concentration ([VF]imm) for the PVAB gel electrodes as measured by cyclic voltammetry (b) and chronocoulometry (O) in 0.10 M sodium phosphate buffer, pH 7.0. The Dap values are the average of two to six measurements, and the coefficients of variation (where available) are ∼5%.
film charge transport (rather than interfacial kinetics) is ratelimiting, and a diffusion-limited response is observed. For chronocoulometry, the passage of charge (Q) is given by the integrated Cottrell equation which predicts a linear Q vs t1/2 relationship3,8
Q ) 2nFA(Dap1/2C*)π-1/2t1/2
(5)
where t is time within the 250-ms pulse. Thus, Dap values can be calculated from the slopes of plots of Q against t1/2 at a given redox site concentration. A Cottrell plot of Q vs t1/2 (Figure 2b) was obtained for the RGE in Figure 2a, and the good linearity corroborates diffusion-limited charge propagation in the PVAB gel to the CPE surface. Dap values were measured by both cyclic voltammetry and chronocoulometry in 0.1 M sodium phosphate buffer, pH 7.0 (PB) for RGEs containing varying concentrations of immobilized VF residues. The profiles of Dap vs [VF]imm from both methods (Figure 3) are similar: in the [VF]imm range 0.81.6 mM, Dap increases with increasing [VF]imm, while an inverse dependence of Dap on [VF]imm was obtained in the 0.1-0.8 mM range. However, the voltammetric method yields a Dap value equal to about half that obtained by chronocoulometry at each [VF]imm. A large difference in Dap values measured by the two methods was also observed by Forster and co-workers, who explained that the difference arose since ion migration is “long range” in cyclic voltammetry, but “short range” in chronocoulometry, due to the relative extent of the redox reaction in the two techniques.44 In addition, the Dap values [(0.6-6.0) × 10-7 cm2 s-1] calculated for the PVAB redox gels are surprisingly high compared to those (10-11-10-9 cm2 s-1) reported for other Fc-containing polymer systems,1,25,43 and only ∼5-50-fold lower than Dap for free Fc species in homogeneous solution (3 × 10-6 cm2 s-1).45 Whenever charge-transport rates are interpreted using diffusional models, it is necessary to determine whether migration makes a distinct contribution to charge propagation. Although diffusional behavior of electron transport through the PVAB gel was observed as discussed above, possible ionic migration effects must also be considered. Andrieux and Save´ant23 pointed out that migrational enhancement of diffusional charge transport rates is possible only when the mobility of electroinactive counterions, which maintain electroneutrality throughout the
redox polymer film, is low compared to the electron mobility. Considering that the diffusion coefficients for Fe(EDTA)-, a relatively large anion, are approximately equal in both methylated cross-linked poly(4-vinylpyridine) film and homogeneous solution at ∼5 × 10-6 cm2 s-1,3,46 it is anticipated that the diffusion coefficients of the supporting counterions (H2PO4-, HPO42-) used in our experiments with the PVAB gels should be of the order of 10-5 cm2 s-1, similar to those of the free species in solution.47 This estimate is over 10-fold higher than the largest measured Dap value for the PVAB gels (6 × 10-7 cm2 s-1) and suggests a negligible role for ionic migration in charge propagation through the PVAB gels. In addition, the measured Dap values were found to be insensitive to the thickness of the gel layer from approximately 0.1 to 1 mm (data not shown), again suggesting that movement of the chargecompensating counterions through the gel is very facile and does not limit the overall charge transport rate.5,48 Charge Transport Mechanisms. Blauch and Save´ant18 performed a systematic study of the interdependence of electron hopping and physical displacement in electron transport through supramolecular redox systems. Influences arising from the solvent, spectator ions, and the supramolecular structure were factored out. They concluded that when physical motion is either nonexistent or much slower than electron hopping, charge propagation is fundamentally a percolation process22 in which electron transfer occurs by nearest-neighbor reactions within a cluster, but not between clusters isolated by nonredox matrices. In the opposite limit, rapid molecular motion thoroughly rearranges the molecular distribution between successive electron hops, thus leading to mean-field behavior. Monte Carlo simulations were employed to study the transition (dynamic percolation) between static percolation and mean-field behavior as a function of the relative rates of electron hopping and physical motion.18 In the simulations, electron hopping was characterized by a time constant (te) representing the average time between attempted electron hops and by the electron hopping distance (δ). Likewise, the rate of physical displacement was governed by a time constant (tp) representing the average time between attempted molecular hops in the absence of a potential energy gradient and by a displacement constant (λ) describing the range of motion of a molecule from its equilibrium position. Thus, the variation of Dap with C* is determined by two ratios, te/tp and λ/δ. Static percolation is expected when the redox centers are either strictly immobile (λ f 0) or have large tp (te/tp f 0), while mean-field behavior is anticipated to prevail when the physical motions are both extensive and rapid (λ > δ and te > tp).18 For systems where redox centers are irreversibly attached to a surrounding supramolecular structure, Blauch and Save´ant18 introduced the concept of bounded diffusion to describe the physical displacement of the redox centers around their anchoring points. Provided the rate of bounded diffusion exceeds that of electron hopping, and the range of bounded diffusion is sufficiently extensive to allow interactions between neighboring redox centers, the mean-field approximation leads to a modified form of the Laviron-Andrieux-Save´ant expression (eq 2)18
Dap ) kact(δ2 + 3λ2)C*/6
(6)
where kact is the bimolecular activation-limited rate constant for electron hopping (l/kact ) 1/kex - 1/kdiff where kdiff is the diffusion-limited bimolecular rate constant) and the 3λ2 term accounts for contributions from physical motion. Equation 6 predicts a linear increase of Dap with increasing C* if λ remains essentially constant over the C* range examined, while different or more complicated profiles of Dap vs C* may be observed if
Charge Transport in Ferrocene-Containing Redox Gels
J. Phys. Chem. B, Vol. 101, No. 46, 1997 9597
Figure 4. Effect of the N,N′-methylenebisacrylamide (BIS) cross-linker content used in copolymerization on the apparent charge diffusion coefficient (Dap) obtained by cyclic voltammetry (b) and chronocoulometry (O) in 0.10 M sodium phosphate buffer, pH 7.0. The total content of acrylamide and BIS was kept constant at 55.5 mg mL-1 for all the gel preparations, and 1 mM VF was present in each polymerization solution, yielding an immobilized VF concentration of ∼0.93 mM.
λ varies significantly with C*. Note that although no contribution to charge transport arises from bounded physical diffusion in the absence of electron exchange (kact ) 0), the presence of exchange allows physical motion to contribute significantly to charge propagation. Hence, a synergy exists between bounded diffusion and electron hopping in that physical displacement continuously redistributes the redox centers and permits them to encounter each other, while electron exchange circumvents the diffusive barriers imposed by the polymer host, so that physical motion can contribute to charge propagation.18 Considering the high content of water, small percentage of copolymer matrix, extremely low loading of redox centers (hence very large dNN; Figure 1), and the surprisingly high Dap values for the measured PVAB gels, it is not hard to imagine that the physical movement of the Fc groups around their anchoring points must be very rapid (very small tp or te/tp ) Dphys/De > 1 where De is the electron hopping diffusion coefficient) and extensive (very large λ) such that physical motion makes a large contribution to charge transport. The dominant role of physical displacement on the overall charge transport rates is indirectly demonstrated by the results shown in Figure 4, where it is observed that Dap decreases significantly on increasing the extent of cross-linking by increasing the BIS content of the PVAB gels. Obviously, greater cross-linking lowers the fluidity of the polymer structure and makes the gel more rigid, leading to a decrease in λ in eq 6. Thus, the sensitivity of Dap to gel rigidity implies that charge propagation through the PVAB gel is dominated by physical motion rather than electron hopping, i.e. Dphys > De, and the mean-field model (eq 6) of bounded diffusion of the redox centers can be ideally applied. The optimal content of BIS was found to be 7.5 mg mL-1 for both the preparation and performance of the PVAB gels for use in sensors. Since λ is undoubtedly much greater than δ (0.6 nm) due to the large range of dNN values (5-15 nm; Figure 1), the term δ2 + 3λ2 approaches 3λ2, and eq 6 can be simplified to
Dap ) kactλ2C*/2
(7)
which shows that Dap is more sensitive to variation in λ than in C*.
Figure 5. Variation in the range of physical motion (λ) of the ferrocene residues vs immobilized VF concentration ([VF]imm) in the PVAB gels. The λ values were calculated using eq 7 from the cyclic voltammetry (b) and chronocoulometry (O) data in Figure 3.
Murray’s group carried out a study of diffusion dynamics of Fc and its derivatives dissolved in an amorphous cross-linked poly(ethylene oxide) polymer electrolyte by voltammetry.49 It was found that the diffusion coefficients of the Fc species decreased with increasing concentration, which was attributed to reduced segmental mobility of the polymer chain as a result of dipole and/or nonpolar interactions between the polymer chain and the ferrocenes. We further interpret this as a reduction in λ (eq 7) on increasing the redox center concentration. Similarly, it is possible for the PVAB redox gel that the segmental fluidity of the polyacrylamide-based chain, or the range of physical displacement of the Fc groups (λ), decreases with increasing [VF]imm. This would likely result in a λ vs [VF]imm profile similar to the dNN vs [VF]imm profile shown in Figure 1, where λ decreases steeply over the lower [VF]imm range (0.1-0.8 mM) but much less sharply at higher [VF]imm (0.8-1.6 mM). In fact, this prediction has been qualitatively confirmed by profiles of λ vs [VF]imm (Figure 5) where λ values are calculated from the data in Figure 3 using eq 7 with kact ) 107 M-1 s-1 for ferrocenes in organic solvents.3,50 However, the calculated λ values are extraordinarily high (40-300 nm), which may be a result of underestimating kact. For a bimolecular electron transfer reaction, kex is given by51
kex ) KAκνkN
(8)
where KA is the donor-acceptor precursor complex formation constant, κ the electronic factor or transmission coefficient, ν the frequency factor, and kN the nuclear factor which contains the free energy of activation. During the preparation of the PVAB gels,9,10 the hydrophobic Fc residues were forced to disperse in the highly hydrated gel medium. The hydrophobic affinity between Fc residues together with the repulsion of water molecules should result in highly favorable work terms for Fc0Fc+ precursor complex formation. Thus, a higher KA is expected for the Fc0-Fc+ precursor complex in the PVAB gels compared to that in organic solvents, where hydrophobic interaction between the Fc residues and the solvent molecules will have adverse effects on precursor complex formation. Consequently, kact > 107 M-1 s-1 is anticipated for Fc0/Fc+ electron self-exchange in the PVAB gel. For example, if kact ) kex ) 109 M-1 s-1, the calculated λ values will fall within the more reasonable range 4-30 nm, comparable to the range of dNN values in Figure 1. Use of redox couples with known kex values in an aqueous environment, such as polypyridine
9598 J. Phys. Chem. B, Vol. 101, No. 46, 1997
Figure 6. 6. Effect of supporting electrolyte (a, NaClO4; b, NaNO3) concentration on the apparent electron diffusion coefficients (Dap) vs immobilized VF concentration ([VF]jmm) for the PVAB gel electrodes as determined by cyclic voltammetry in 0.10 (b) and 0.50 (2) M electrolyte solution, pH 6.0 ( 0.1, as well as by chronocoulometry in 0.10 (O) and 0.50 (4) M electrolyte solution.
complexes, would yield better estimates of λ in the PVAB gels, and such experiments are planned. In summary, Dap, representing the overall rate of electron transport, decreases with increasing [VF]imm up to ∼0.8 mM (Figure 3) since the changes in λ dominate. However, Dap increases with increasing [VF]imm above 0.8 mM (Figure 3), and the gels exhibit the charge transport behavior predicted by eq 7 with essentially constant λ. Dependence of Dap on Electrolyte Concentration. The nature and concentration of supporting electrolyte usually exert a strong influence upon charge transport rates in redox polymers. Contradictory results have been reported for a variety of redox polymeric systems since Dap has been observed to both decrease6,49,52,53 and increase5,44,54 with increasing electrolyte concentration at a fixed redox site concentration. In the present study, it was found that the Dap vs [VF]imm profiles for the PVAB gel electrodes in 0.10 M NaClO4 and NaNO3 (pH 6.0 ( 0.1) are similar to those in 0.10 M sodium phosphate buffer, pH 7.0 (Figures 3 and 6). It can be further observed from Figure 6 that when the electrolyte concentration is raised from 0.10 to 0.50 M, the Dap values exhibit a much more significant decrease at the low (0.1-0.6 mM) and high (1.1-1.6 mM) [VF]imm ranges than those measured at 0.6-1.1 mM [VF]imm. It is well-known that supporting electrolytes have nonbonded interactions with polymer matrices, resulting in a reduction in polymer chain fluidity by a magnitude related to
Bu et al. electrolyte concentration.6,49,52,53 As described above, charge transport in the PVAB gels is controlled by different factors at low and high [VF]imm; thus, it is necessary to consider the possible effects of supporting electrolyte concentration on Dap in both [VF]imm regions. At low [VF]imm (0.1-0.6 mM) Dap is dominated by changes in λ; consequently, depression of chain segmental mobility in 0.50 M electrolyte results in a greater reduction in Dap at the lowest [VF]imm, as seen in Figure 6. In the [VF]imm range 1.1-1.6 mM, the average edge-to-edge separation between redox sites is small (∼5 nm), and relatively strong nonbonded interactions exist between the polymer matrix and the Fc residues. As the electrolyte concentration is raised, interactions between the electrolyte and both the polymer chain and the Fc residues undoubtedly increase, and it is conceivable that the three components form ternary cluster-like complexes. Such cooperative interactions would significantly reduce the contribution not only of physical displacement but also of electron hopping to the overall charge transport rates, resulting in large decreases in Dap in 0.50 M electrolyte at high [VF]imm (Figure 6). Within the middle range of [VF]imm (0.6-1.1 mM), the influence of supporting electrolyte on λ and kact is probably weaker, and thus only a small reduction in Dap in 0.50 M electrolyte is observed (Figure 6). It is also seen that the Dap values obtained in NaNO3 solution are smaller than those measured in NaClO4 solution (Figure 6a vs 6b). This may arise from stronger hydrophobic interaction of NO3- with the polymer matrix and the redox sites since NO3- possesses a smaller molar volume and hydration number (larger structure-breaking hydrophobic effect) than ClO4-.55 In our previous study,10 it was found that the Dap values increased by 20-50% when negatively charged monomers [2-acrylamidoglycolic acid (AGA) or acrylic acid (ACA)] were incorporated into the neutral PVAB redox gels. The higher Dap values may result from increased polymer-chain fluidity upon introduction of hydrophilic groups (COO- and OH in AGA, and COO- in ACA), which would enhance the contribution of physical displacement to the overall charge propagation. Furthermore, it was observed that the Dap vs [VFlimm profiles for the negatively charged redox gels (data not shown) are similar to those for the neutral PVAB gels (Figure 3), suggesting that charge propagation occurs by the same mechanism in the neutral and charged redox gels. Acknowledgment. The authors thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support. H.-Z.B. gratefully acknowledges receipt of a Doctoral Fellowship from Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche (FCAR) and a Concordia University Graduate Fellowship. References and Notes (1) Inzelt, G. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1994; Vol. 18, p 89. (2) Aoki, A.; Rajagopalan, R.; Heller, A. J. Phys. Chem. 1995, 99, 5102. (3) Larsson, H.; Sharp, M. J. Electroanal. Chem. 1995, 381, 133. (4) Surridge, N. A.; Sosnoff, C. S.; Schmehl, R.; Facci, J. S.; Murray, R. W. J. Phys. Chem. 1994, 98, 917. (5) Forster, R. J.; Vos, J. G. Langmuir 1994, 10, 4330. (6) Aoki, A.; Heller, A. J. Phys. Chem. 1993, 97, 11014. (7) Mathias, M. F.; Haas, O. J. Phys. Chem. 1993, 97, 9217. (8) Elmgren, M.; Lindquist, S.-E.; Sharp, M. J. Electroanal. Chem. 1993, 362, 227. (9) Bu, H.-Z.; Mikkelsen, S. R.; English, A. M. Anal. Chem. 1995, 67, 4071. (10) Bu, H.-Z.; English, A. M.; Mikkelsen, S. R. Anal. Chem. 1996, 68, 3951. (11) Lumley-Woodyear, T. D.; Rocca, P.; Lindsay, J.; Dror, Y.; Freeman, A.; Heller, A. Anal. Chem. 1995, 67, 1332.
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