Electrochemical Estimation of Diffusion Anisotropy of N, N, N′, N

ARTICLE pubs.acs.org/JPCB

Electrochemical Estimation of Diffusion Anisotropy of N,N,N0,N0-Tetramethyl-para-phenylenediamine within the Normal Hexagonal Lyotropic Mesophase of Triton X 100/Light Water: When Can the Effects of Cross-Pseudophase Electron Transfer be Neglected for Partitioned Reagents? Jonathan E. Halls,†,‡,§ Nathan S. Lawrence,‡ and Jay D. Wadhawan*,†,§ †

Department of Physical Sciences (CHEMISTRY), The University of Hull, Cottingham Road, Kingston-upon-Hull HU6 7RX, United Kingdom ‡ Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom ABSTRACT: The H1 lyotropic liquid crystalline phase of Triton X 100 with aqueous 0.1 M potassium chloride is examined as a medium in which to determine the axiosymmetric anisotropy in the diffusion flux of N,N,N0 ,N0 -tetramethyl-para-phenylenediamine using electrochemical methods (voltammetry and potential step chronoamperometry) at both planar electrodes and two-dimensional flux microdisk electrodes. Comparison of experiment with theory suggests the ratio of anisotropic diffusion coefficients in the directions tangential and perpendicular to the electrode surface varies over two orders of magnitude (from 0.04 to 3.3) with increasing concentration of the redox analyte. This is understood through the occurrence of a long-range charge transfer across the pseudophase | pseudophase boundary interface, occurring as a result of differential diffusivities of the redox probe within the surfactant and aqueous subphases. These data and their dependence on the analyte concentration empower, in a proof-of-concept, the estimation of the partition equilibrium constant (KP); the value estimated for the small electroactive-drug mimetic considered is log KP = 2.01 ( 0.05 (at 294 ( 2 K) and is in agreement with that envisaged for its partition between n-octanol and water. It is suggested that only measurements at low analyte loadings allow for interphase electron transfers to be neglected, since then percolation effects appear to dominate the Faradaic current.

’ INTRODUCTION Anisotropic mass transport is a feature of many chemical systems and includes atomic diffusion in minerals,1 material transfer within devices that employ membranes such as gas sensors2 and fuel cells,3 polymeric materials4 and liquid crystals.5 The latter systems, with their long-range orientational order, and which may also exhibit a positional order, are almost ideal systems in which to study the effects of anisotropy, as has been investigated for lyotropic systems using ionic conductivity6 and NMR measurements.7 Albeit that there is a paucity of literature, electrochemical methods for the quantification of diffusion anisotropy have been na and co-workers9 for the investigated by Herino et al.8 and Abru~ case of thermotropic liquid crystals, by Murray et al.10 for lyotropic liquid crystals, and White and co-workers for the voltammetric determination of anisotropy in opal nanostructures.11 These works, though seminal, require the use of macroscopically aligned monodomain materials and the use of millimetric electodes, for which planar diffusion restricts each experiment in providing information in merely one dimension. In a previous work,12 we r 2011 American Chemical Society

developed a unifying global theory for the facile quantification of axiosymmetric diffusion anisotropy, exploiting the transition between planar unidimensional transport and convergent, twodimensional diffusion at microdisk electrodes, using potential-step chronoamperometry. We have subsequently initiated13 a systematic investigation of electron transport through three-dimensional, autoassembled molecular electrochemical nanosystems based on lyotropic liquid crystals, where the medium into which the electrode is immersed has some form of long-range redox structure, with a view to exploit these lightweight, self-annealing and flexible soft-matter systems for technological application, for example, as conduits for information transfer or energy storage and conversion. In this work, we report the estimation of the anisotropic diffusion of N,N,N0 ,N0 -tetramethyl-para-phenylenediamine (TMPD) from voltammetric measurements made in the Received: February 23, 2011 Revised: April 1, 2011 Published: May 02, 2011 6509

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hexagonal (H1) phase of the lyotropic liquid crystal formed via the addition of aqueous 0.1 M KCl electrolyte to Triton X 100 in a 58:42 wt % ratio; the self-assembly of the lyotropic mesophase of ordered ensembles of micelles only occurs at high concentrations of surfactant in water, when the micelles pack together into geometric arrangements, parametrized by the Israelachvili packing ratio,14 depending on the micellar shape, giving rise to a single thermodynamic phase of high viscosity (ca. 22 kPa s for the systems considered herein).15 TMPD is a well-known redox mediator used extensively in the oxidized form (W€urster’s Blue) as an electron acceptor from ascorbic acid during the study of biological electron transfer processes16 and is known to partition between aqueous and organic media; it is estimated17 to have an n-octanol | water partition coefficient of log Kp = 2.08 ( 0.47 at 298 K (Kp = [i]org/[i]aq) for species i partitioning between organic, org, and aqueous, aq, phases), in agreement with literature data on the partition coefficients of ortho- and paraphenylenediamines.17 Triton X 100 was chosen as the surfactant as it is a well-known detergent used in biology to disrupt biomembranes so as to elicit cellular apoptosis,18 while being nonionic so that there should be no electrical migration contribution of the cylindrical micelles observed in the voltammetry. Moreover, the surfactant pseudophase (palisade layer or hydrophobic interior core) will have a different affinity for TMPD compared with the aqueous pseudophase, the static dielectric constant (εs) for the palisade layer of Triton X 100 micelles has been estimated as being19 25.7, which compares favorably with that for ortho-nitrophenyl octyl ether (24.2)  the solvent preferred by Girault and co-workers over n-octanol (or the more volatile 1,2-dichloroethane),20 as water | n-octanol partition coefficients sometimes do not give a good estimate of drug absorption or permeability.20e There is an additional complication to the system, since TMPD can be protonated in water21a,b (pKBHþ = 2.2 and 6.35 at 298 K), so that in aqueous 0.1 M KCl, the TMPD will buffer the solution, depending on the amount of dissolved atmospheric CO2 present. Since TMPD is a respiratory electron transfer chain uncoupler, as a result of proton translocation across lipid membranes,21c we may assume that deprotonation of this Eigen-“normal” acid will be fast, and under equilibrium control.21d Then, analysis through the thermodynamic cycle constructed around the following relationships, TMPDðaqÞ a TMPDðsurf Þ

KP 1

TMPDHþ ðaqÞ a TMPDðaqÞ þ Hþ ðaqÞ þ

TMPDHþ ðsurf Þ a TMPDðsurf Þ þ Hþ ðsurf Þ TMPDHþ ðaqÞ a TMPDHþ ðsurf Þ

ðKBHþ Þaq ðKBHþ Þsurf KP 2

indicates that the partition coefficient for the monoprotonated form (KP2) is equivalent to that of the free base form (KP1) weighted by the ratio of the acid dissociation constants within the two subphases: Kp2 = [(KBHþ)aq/(KBHþ)surf]KP1. The implicit notion that the surfactant pseudophase may contain charged species is not without experimental justification, as elegantly evidenced through the radiolytic generation of solvated electrons within the palisade layer of Triton X 100 micelles when in isotropic solution (at concentrations above the cmc), and also when present in the H1 and LR phases.22a However, what is more difficult to quantify is the pKBHþ of the

protonated TMPD within the subphases (both surfactant and aqueous), since there the confined and restricted environment of both pseudophases will affect the acid/base properties of water by virtue of disrupting its hydrogen bonded structure,22b,c so that Brønsted acidity is better described in terms of “proton transfer efficiency”, parametrized by the rate constants for protonation and deprotonation rather than the classical notion of pH. Can then KBHþ in the surfactant subphase be equal to that in the aqueous subphase? This then depends on the structure of the subphases, and the location of the TMPD within the surfactant subphase (the palisade layer or the hydrophobic interior). For isotropic micellar solutions of Triton X 100, TMPD (estimated radius23 of 5 Å), like most aromatic amines, resides within the palisade layer of the nonionic surfactant24 (of typical thickness25 25 Å). This latter site is hydrated owing to “thermodynamically-bound” water hydrogen-bonded to the oxoethylene groups of the surfactant side chain. Electrolytes such as KCl when present within the aqueous pseudophase are thought to enter into the palisade layer,24 causing several effects to occur, such as providing suitable counterions for any protonated form of TMPD, coupled with an increase in micellar size and aggregation number,26 and also an increase in the hydration of the palisade layer due to the enhancement of “mechanicallytrapped” water.27 It has been proposed that the latter effect has dramatic consequences, specifically, (i) the migration of palisade layer probes toward the micellar core28 and (ii) a partial dehydration of the surfactant chains (tantamount to a reduction in the amount of “thermodynamically-bound” water),29 which may lead to an enhanced “microviscosity” of the palisade layer,30 and slower solvation dynamics,24 due to the collapse of the surfactant chains. Nevertheless, these effects do suggest that we may be able to consider the rates of protonation and deprotonation of the TMPD within each subphase as being equivalent. This notion is further justified by the reported second pKBHþ for ortho-phenylenediamine and meta-phenylenediamine being 4.61 and 5.01, respectively, at 298 K in aqueous 0.1 M KCl, with these values reducing only slightly to 4.41 and 4.31, respectively, at 298 K for ortho-phenylenediamine (in a 70:30 vol/vol dioxane:water mixed solvent containing 0.1 M KCl) and meta-phenylenediamine (in a 80:20 wt/wt dimethyl sulfoxide:water solvent containing 0.1 M NaClO4).31 Hence, defining a global speciation partition coefficient, KP, as Kp = (cTMPDsurf þ cTMPDHþsurf)/(cTMPDaq þ cTMPDHþaq), we observe that, under the assumption of unity activity coefficients, this parameter is equivalent to the single partition coefficient if K P 1 = K P 2 , or reduces to the following, if not, K P = (KP1 þ KP2{cHþ/(KBHþ)aq})/(1 þ cHþ/(KBHþ)aq). Thus, treating TMPD/TMPDHþ partitioning as though it is tantamount to TMPD partitioning only, we accordingly invite the question as to whether it is possible to use dynamic electrochemical measurements to estimate the partition coefficient of TMPD within this system? To begin to answer this problem, we consider the illustration in Figure 1a, in which the cylindrical micelles are depicted as bananas oriented within an aqueous system. Introducing an electrode into this medium such that the close-packed cylindrical micelles are perpendicular to the electrode surface (homeotropic alignment) and the H1 phase is axiosymmetric with respect to rotation around the direction normal to the electrode surface, and under the assumption that the electrode does not distort the lyotropic phase,10 oxidation of TMPD both within the aqueous and the surfactant pseudophases can occur at 6510

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factor of π/(2 3 31/2) to account for the fraction of the electrode surface exposed to the surfactant pseudophase. The alternate possibility that partition of TMPD from the surfactant micelles into the aqueous subphase precedes electron transfer that occurs exclusively within the aqueous pseudophase, in an overall CE process, is unlikely to take place, given the considerations outlined above on the palisade layer. Thus, since the mutual diffusion coefficient34 of TMPD is likely to be smaller in the micellar (surf, DP), cf. aqueous (aq, DA), pseudophase, there is a possibility that TMPDþ•(aq) may actually mediate the oxidation of TMPD(surf) via long-range electron transfer at the pseudophase | pseudophase interface (in a scheme of squares),35 which complicates any analysis; see Figure 1b. How may we assess the occurrence of this mediated self-exchange and thence estimate the value of KP? First, we may use voltammetric measurements at planar electrodes to obtain a single apparent (one-dimensional) diffusion coefficient (Dap) of TMPD under the assumption of isotropic diffusion of TMPD within the H1 phase,36 and with separate estimation of the TMPD diffusion coefficients in the aqueous and micellar pseudophases, we may exploit a MillerMajda formulation33a,b of the relationships between these in the presence, eq 1, or absence, eq 2, of self-exchange (see Appendix 1 for the relevant derivations) to determine the partition coefficient. It is important to stress that the equilibrium constant for the mediated reaction may not necessarily be unity (corresponding to self-exchange); this actually will vary with the formal electrode potentials of the TMPD forms in the aqueous subphase relative to those in the surfactant pseudophase, ΔE00 : K = exp[(F/RT)ΔE00 ]. For typical solvents,23c the variation of the formal potential for TMPD oxidation from solvent to solvent is not very much (typically approximately 50 mV across a wide range of aprotic and hydrogen-bonded solvents, with an order of magnitude smaller variation in alcohol solvents). Given that the surfactant pseudophase comprises an alkylaromatic inner core and a palisade layer comprising ethylene oxide moieties, we suggest that there may be a small difference in the formal potentials (vide infra), so that K is within an order of magnitude from unity. KP ¼

DA  Dap Dap  DP

pffiffiffiffiffiffi pffiffiffiffiffiffiffi DA  Dap KP ¼ pffiffiffiffiffiffiffi pffiffiffiffiffiffi Dap  DP Figure 1. (a) Orientation of part of one domain of the H1 lyotropic liquid crystal system, and the coordinate system employed herein for electrochemical measurement. Cylindrical Triton X 100 micelles are represented as orange-yellow bananas embedded within the aqueous pseudophase (blue). (b) Partitioning of TMPD between the aqueous (aq) and surfactant (surf) pseudophases allows for electron transfer within each pseudophase, in addition to electron transfer across the pseudophase | pseudophase interface (illustrated by the arrows drawn in magenta). (c) Variation of KP with apparent diffusion coefficient (Dap) following the MillerMajda formulation (q.v. Appendix 1) for DA = 105 cm2 s1 and DP = 107 cm2 s1. Note that Dap is kept so that it does not equal DA or DP. The red line indicates eq 1, corresponding to mediated self-exchange; the blue line is the response via eq 2, no selfexchange.

the electrode, in a manner reminiscent of earlier work on redox polymers and microporous alumina films,32,33 with a geometric

ð1Þ

ð2Þ

It is seen from Figure 1c that, as expected, the presence of the self-exchange process results in a larger value of the partition coefficient, except when Dap f DA or Dap f DP, thereby constraining the ranges over which the partition coefficient may be measured to log KP < 1 and log KP > 2; measurement in the range likely relevant for TMPD  1 g log KP g 2 is ambiguous (and with large error in KP) unless it can be ascertained whether mediated self-exchange occurs or not. Recognizing that the mediated electron transfer process occurs in a transport regime perpendicular to the physical translation diffusion of the redox molecules normal to the electrode surface (for the case of Figure 1a), the anisotropy in the diffusion coefficients in each of the orthogonal directions axial (z) and radial (r) to the electrode surface will be a function of the loading of the micellar subphase, and thus, of the concentration of TMPD within the 6511

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medium, under the assumption that the viscosity of the H1 phase is not a function of TMPD concentration. Note that if the transfer of electroactive species at the pseudophase | pseudophase interface occurs rapidly compared with any electron transfer, then the distribution equilibrium is always maintained and the diffusion coefficients observed should not be a function of the TMPD concentration. This has been the case considered by Murray and co-workers,10 for which they suggested, on the basis of an approximate model extended from the work by Peerce and Bard,37 that, for one-dimensional diffusion, the apparent diffusion coefficient may be expressed as

Dap

l2 DA DP KP  ¼ d ðl  dÞDA þld þ dDP KP KP

te0 a¼1 tf¥ a¼0 Da ¼0 DZ a¼1

"R, Z R e 1, Z ¼ 0

ðbulk solutionÞ ðtransport-limited electrolysisÞ

R > 1, Z ¼ 0

ðinsulation boundaryÞ

R, Z f ¥

ðbulk solutionÞ

and demonstrated that the following expression for the normalized current, ψ, is suitable for experimental comparisons:41 i pffiffiffiffiffiffiffiffiffiffi 4F Dr Dz r0 c0 rffiffiffi   1 π 0:7824 þ 0:2146exp  pffiffiffi ¼ 0:7854 þ 2 τ τ

ψ¼ ð3Þ

where d is the length of the aqueous subphase, and (l  d) is the length of the micellar pseudophase. We will report the rigorous examination of this expression in a future work,38 which will additionally consider the effects due to slow partition kinetics. We begin by outlining the theoretical principles for voltammetry and chronoamperometry within axiosymmetric anisotropic media, keeping our analyses within a pragmatic vein by focusing on how experimental data may be compared with theory via working curves and expressions so as to extract anisotropic diffusion coefficients.

’ THEORY In this section, we examine the main results of axiosymmetric diffusion anisotropy observed at microdisk electrodes under potential step chronoamperometry and voltammetric conditions, so as to elicit strategies for the simultaneous determination of diffusion coefficients in the directions normal and tangential to the electrode surface. The model developed is adequate to represent transport of small molecules within and in the solvent between cylinders of large and nonflexible molecules that are assumed to not move much during either mutual diffusion of the solute, or self-diffusion of the solvent. Note that these channels are assumed to be at least twice the size of the solute, so that Fickean diffusion of the latter occurs.39 Formulation of the Two-Dimensional Diffusion Anisotropic Problem at Disk Electrodes. We consider the general one-

electron oxidation reaction, Az  e a Bzþ1, where z refers to the charge on species A. In the following we assume that species A and B have similar solvation structures and do not undergo electrical migration. In an axiosymmetric, cylindrical, diffusionally anisotropic environment, if species A and B have identical anisotropic diffusion coefficients for transport in the directions perpendicular (z) and tangential (r) to the electrode, we may write down Fick’s second law for species A:40 1 Da D2 a 1 Da Dz D2 a ¼ 2þ þ Dr Dt DR R DR Dr DZ2

current (i) during transport-limited chronoamperometry, viz. subject to the following boundary conditions,

where τ = 4(Drt/r02) is the dimensionless time variable. Compared with the case of isotropic diffusion, the above suggest that the effective diffusion coefficient is merely the geometric mean of the diffusion coefficients in the two orthogonal directions, viz. D = (DrDz)1/2. Equation 5 is useful for comparison with experimental data for the quantitative extraction of the diffusion coefficients in both radial (Dr) and normal (Dz) directions, as the dimensionless experimental current depends on both diffusion coefficients, while the reduced time variable merely depends on the radial diffusion coefficient; indeed, iterative optimization of Dr and Dz via comparison of adimensional experimental currents, ψexpt = iexpt/4F(DrDz)1/2r0c0 with ψtheory given by the expression in eq 5 over the whole normalized temporal domain, τ (dummy variable s), subject to a difference-minimized parameter, ℵ = (1/Σs)Σs(|ψexpt  ψtheory|)/ψexpt readily enables determination of Dr and Dz from experimental data. Formulation of the Voltammetric Scenario for Axiosymmetric Diffusion Anisotropy at Disk Electrodes. Accommodation of the Electrode | Insulator Singularity with Heterogeneous Electrode Kinetics. We are interested in the voltammetric response in the presence of axiosymmetric cylindrical diffusion anisotropy, and begin solution of eq 4 via transformation from semi-infinite, cylindrical (R, Z) space into closed (Θ, Γ) quasiconformal space using,12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi π  ð6Þ R ¼ 1  Θ2 sec Γ 2 Z¼Θ

rffiffiffiffiffi   Dz π tan Γ 2 Dr

Note that the above expressions slightly differ from those reported in our previous work.12 Defining the reduced potential, in terms of the linear potential sweep used in voltammetry: ξ¼

ð4Þ

where we have introduced the adimensional parameters for concentration, a = cA/c0 (c0 is the effective concentration of species A in the bulk medium), and cylindrical space, R = r/r0; Z = z/r0 (r0 is the radius of the disk electrode). Note that Dr cannot equate to zero. An expression similar to eq 4 can be written to describe the transport of species B. In a previous work,12 we solved eq 4 to obtain the temporal (t) variation of the Faradaic

ð5Þ

F F F 0 0 ðE  E0 Þ ¼ ðEi  E0 Þ þ vt RT RT RT

ð7Þ

in which F is the Faraday constant, R is the molar gas constant, T is the absolute temperature, E is the applied potential difference between working and reference electrodes (we do not consider Ohmic losses), Ei is the initial potential of the voltammetric sweep, E00 is the formal potential of the A | B redox couple, and v = dE/dt, enables the re-expression of eq 4 in a manner that readily allows the mathematical singularity at the electrode | insulation 6512

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boundary to be addressed such that numerical solution does not require sophisticated finite difference grids,12 Da ¼ Dξ

1

 Θ

p2

2

þ tan2



π Γ 2



(

)   2 2 4 Da Da 2 D a 2 π  2 cos Γ þ ð1  Θ Þ 2  2Θ π 2 DΓ2 DΘ DΘ

ð8Þ

In the above, p = and solely depends on the radial diffusion coefficient. The effect of these transformations is that the semi-infinite [0, ¥) cylindrical polar coordinate diffusion field in both R and Z directions maps onto closed [0, 1] quasiconformal space in each, as well as allowing direction-dependent diffusion coefficient ratio to be absorbed within the expression, so that eq 8 is identical to that of an isotropic system of diffusion coefficient Dr.12 Defining the initial potential by the parameter, u = (F/RT)(Ei  E00 ), the pertinent boundary conditions under the coordinate change become

case considered herein, thermal variation will significantly affect the voltammetry owing to changes in micellar aggregation number and, at worst, will result in phase change. Current Evaluation. The current flowing through the disk electrode is described by the expression Z 1  Da R dR ð9Þ i ¼ 2πFDz c0 r0 DZ 0 0 which, in reduced form becomes ψ¼

(r0/Dr1/2)(Fv/RT)1/2

i pffiffiffiffiffiffiffiffiffiffi ¼ 4F Dr Dz r0 c0

Z 0

1



Da DΓ

 dΘ

ð10Þ

0

0 e Θ e 1; Γ ¼ 1

Thus, for numerical solution, eq 8 was converted into finite difference form for a uniformly spaced (dΘ = dΓ = h) Θ, Γ grid (dummy variables k and j), and uniform temporal grid and solved using the alternating direction implicit (ADI) method using a tridiagonal matrix alogorithmic solver, with the electrode surface condition (a0,k) expressed in the following finite difference form. 8 9 2 > > > > < = Rξ πΛΘ he k  a1, k a0, k ¼  > > 2 > > : 1 þ eξ þ ; Rξ πΛΘk he 1  þ ð11Þ 2 1 þ eξ þ πΛΘk heRξ

in which a0 is the dimensionless surface concentration of species A and ButlerVolmer kinetics (variables ks and R) are parametrized by the dimensionless rate constant, Λ = ks[r0/(DrDz)1/2]. Note that the surface boundary condition at Γ = 0 is considered only for the case where the diffusion coefficients of species A and B are identical, viz. DA = (DArDAz)1/2 = DB = (DBrDBz)1/2. A corollary of this is equality of the diffusion coefficients in each of the two directors, viz., DAr = DBr and DAz = DBz. If this is not the case, the result outlined in Appendix 2 suggests that the surface boundary condition becomes (2/π)(1/Θ)(∂a/∂Γ)Γ=0 = ΛeRξ{a0[(DArDAz/DBrDBz)1/4 þ eξ]  1}, where Λ is redefined in terms of species A. This case will not be considered herein. Significance of Parameter p . The dimensionless parameter p is a convenient measure of the timescale of the voltammetric scan, since it characterizes any voltammogram with respect to the electrode size, the voltage sweep rate, and the relative importance of “edge effects” via the radial diffusion coefficient. For the anisotropic case here, as for the isotropic case, p ,1 corresponds to steady-state voltammetric behavior, and is observed whenever any of the following criteria are met, r0 f 0, v f 0, Dr f ¥. Likewise, p . 1 affords the linear sweep response generally observed at large electrodes in isotropic media and occurs whenever any of the following limits occur, r0 f ¥, v f ¥, Dr f 0. Thus, this parameter elegantly captures the essence of the diffusion anisotropy inherent within the system within the transformed space. Similarly, for the effects of shorter voltammetric time scale and smaller electrode size. Note that although the experimental temperature features in p , its effects are not readily discernible due to the thermal variation of solution viscosity (and therefore diffusion coefficients). For the experimental

Numerical Results and Discussion. We consider the natural limits to the electrochemical reversibility of the heterogeneous electron transfer individually. Note that since the reduced variables and quasi-conformal transformation have been chosen so that eq 8 is identical for the isotropic case,42 the discussion below is brief and focused primarily on the extraction of anisotropic diffusion information from the voltammograms. Λ f ¥: Reversible Electrode Kinetics. In this limit, ks . (DrDz)1/2/r0, so that the Nernst law is upheld at the electrode surface: a0 = b0eξ. Figure 2a illustrates representative dimensionless voltammograms in this heterogeneous electron transfer limit corresponding to 1 e log p e 1. Under these conditions, there is little variation in the observed reduced voltammograms induced via the occurrence of axiosymmetric diffusion anisotropy, as anticipated; rather the dominance of p on the waveshape is apparent, as expected, so that as p f 0, ψlim f 1. Likewise, as p f ¥, ψp f (0.4463π/4)(r0/Dr1/2)(Fv/RT)1/2 = 0.351 p as expected. Note that this limit is not illustrated here. Thus, in this limit, voltammetric methods may be directly applicable for the quantification of Dr and Dz, provided E00 is known, since this allows transformation of the experimental abscissa into reduced form, with transformation of the current axis under assumed values of Dr and Dz enabling the fit as described earlier between experimental and predicted voltammograms taken over a variety of scan rates, as the latter enables the change in p which is the only variable which is a function of a single diffusion coefficient. Alternatively, the following of the peak current as a function of the voltage sweep rate allows for a more convenient form of determining Dr and Dz, rather than employing complicated waveshape-fitting procedures, since, as given by Aoki,42c the

ξeu a¼1

"Θ, Γ

ξ>u

0 e Θ e 1;Γ ¼ 0

2 1 Da ¼ ΛeRξ fa0 ð1 þ eξ Þ  1g π Θ DΓ

Da ¼0 DΘ Da ¼0 DΘ a¼1

Θ ¼ 0;0 e Γ e 1 Θ ¼ 1; 0 e Γ e 1

6513

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dimensionless peak current follows the expression, ψp ¼

ip pffiffiffiffiffiffiffiffiffiffi 4F Dr Dz r0 c0

¼ 0:34e0:66p þ 0:66  0:13e11=p þ 0:351p

ð12Þ

so that eq 12 can be used in an analogous manner as seen for eq 5 to obtain Dr and Dz. Nevertheless, the drawback with this type of method is that if microelectrodes need to be employed, and experiments must be undertaken with full compensation of any Ohmic loss term; the use of low scan rate voltammetry using electrodes of millimetric size give rise to p ∼100 for typical scan rates, leading to poor experimental resolution of Dr and Dz. Λ f 0: Nonreversible Electrode Kinetics. In the electrochemically irreversible limit, ks , (DrDz)1/2/r0. However, voltammetric waveshapes now depend on p and Λ. Moreover, since Λ varies dramatically with the presence of diffusion anisotropy, at constant ks and p , waveshape shifts on the adimensional abscissa of the voltammograms presented in Figure 2b (for the case ks = 103 cm s1 and p = 1.0) are wild. Accordingly, voltammetry within this regime requires the simultaneous knowledge of multiple parameters (ks, E00 , R), with full waveshape-fitting for variable scan rates the only method to quantify Dr and Dz. We will not consider this case further. Thus, it appears that microelectrode voltammetry alone is unlikely to be sufficiently versatile to deconvolute axiosymmetric anisotropic diffusion coefficients, unless the redox species engages in fast heterogeneous electron transfer processes, and multiple experiments are undertaken at a variety of voltage scan rates. In contrast, however, the use of potential step chronoamperometric methods at microelectrodes provides an attractive and facile, single-experiment approach by which to determine the extent of diffusion anisotropy.

’ EXPERIMENTAL METHODS

Figure 2. (a) Linear sweep voltammograms corresponding to fast heterogeneous electron transfer. Voltammograms are illustrated for 11 e log(Dz/m2 s1) e 7, with 9 e log(Dr/m2 s1) e 7 for (i) p = 0.1 and (ii) p = 10. The arrow indicates the direction of the potential sweep. (b) Linear sweep voltammograms corresponding to slow heterogeneous electron transfer. Data are for p = 1.0 (for ks = 105 m2 s1, R = 0.5), with Dr = Dz = 109 m2 s1 (Λ = 0.1, black); Dr = 100Dz = 109 m2 s1 and, equivalently, Dz = 100Dr = 109 m2 s1 (Λ = 1.0, blue); Dr = 100Dz = 107 m2 s1 and, equivalently, Dz = 100Dr = 107 m2 s1 (Λ = 0.01, red). The arrow indicates the direction of the potential sweep.

Chemical Reagents. All chemical reagents were purchased from Sigma-Aldrich or Fisher Scientific in the purest commercially available grade, and used as received. Water, with a resistivity of not less than 18 MΩ cm, was taken from an Elgastat system (Vivendi, Bucks., UK). Oxygen-free nitrogen was obtained from BOC Gases, U.K. The H1 phase43 was prepared by mixing an aqueous 0.1 M KCl solution containing a known amount of TMPD to Triton X 100 ((tert-octylphenoxy)polyethoxyethanol) in a 58.0:42.0 wt % ratio. The mixture was heated, under a stream of nitrogen, to approximately 320 K, with stirring, so as to form the micellar isotropic phase, for about 1 h, so as to achieve homogenization. The sample was then allowed to cool slowly overnight to 294 ( 2 K before further experimentation. The density of the resulting system was estimated to be ca. 1.11 g cm3. As the H1 phase is a single thermodynamic phase, the concentration of the paraphenylenediamine derivative is reported in moles per unit volume of the H1 phase. Experiments using magnetic alignment of the H1 phase (using fields on the order of 13 Gauss), to be reported in a future work,38a indicate that the resulting viscous gel formed are likely with the cylindrical micelles in a homeotropictype form, viz., with the cylindrical micelles being perpendicular to the electrode surface, consistent with existing literature.38b Instrumentation. Optical microscopy was undertaken using an Olympus BH-2 polarizing microscope. Electrochemical measurements were undertaken using a commercially available potentiostat 6514

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(Autolab, PGSTAT30, Eco Chemie, The Netherlands) controlled by a Pentium IV computer, with the media thermostated at 294 ( 2 K. The working electrode was either a 3.0 mm diameter glassy carbon disk electrode or an 11.0 μm (diameter) carbon fiber microdisk electrode (both purchased from BASi, UK). These electrodes were cleaned and polished using increasingly finer grades of carborundum paper (Presi, France), followed by polishing on a napped polishing pad using an aqueous alumina (0.3 μm, Presi, France) slurry. For the experiments with surfactant media, the working electrode was polished and carefully placed in different positions for every change in experimental parameter, so as to encompass any effects due to electrode location. A nickel wire spiral formed the counter electrode, and a saturated calomel electrode was employed as the reference electrode. Note that owing to the viscosity of the H1 phase, experiments were not conducted with inert-gas purging prior to experimentation. Computing. Adimensional voltammogram simulation, undertaken via program encoding in GNU FORTRAN using the freely available G77 compiler, was executed on an Intel Pentium processor (of speed, 2.4 GHz, with 1.98 GB of random access memory) and employed a uniform spatial and temporal finite difference grid, of size 100  100 (NΓ  NΘ), resulting in h = 0.01 with Δξ = 103, to furnish stable solutions of sufficiently adequate convergence, using a downwind difference scheme for evaluation of the current. These computations, run using r0 = 10.0 μm, c0 = 2.0 mM, 11 e log(Dr/m2 s1) e 7, 11 e log(Dz/m2 s1) e 7, and with u = 10.0 for a typical reduced potential amplitude of ζu (2 e ζ e 3), were set to run at arbitrarily defined p , using Nernstian surface conditions for reversible behavior, viz., a0,k = 1/(1 þ eξ); values of Butler Volmer heterogeneous rate constant, ks, for the nonreversible behaviors reported herein were kept at 105 m s1 with symmetry factor (R) set at 1/2. Simulation of a single dimensionless voltammogram typically took less than 5 min of CPU time. Data were imported and manipulated using either Microsoft Excel XP or Microcal Origin 6.0, and scaled-up, as appropriate, to dimensioned variables.

Figure 3. (a) Cyclic voltammograms recorded at a 3.0 mm diameter glassy carbon disk electrode immersed into the H1 phase containing 10.3 mM TMPD, recorded at a scan rate of 0.1 V s1. Three consecutive scans are shown with the first scan in black, the second in blue, and the third in red. The arrow indicates the direction of the initial voltage sweep. (b) Variation of the peak potentials of the first wave with experimental time scale. (c) Variation of the peak currents of the first wave with square root of the scan rate. (c) Note that (b) and (c) have the same key: 2.1 mM TMPD (blue squares), 5.1 mM TMPD (red circles), 10.3 mM TMPD (green triangles) for oxidation (closed symbols) and reduction (open symbols).

’ RESULTS AND DISCUSSION We consider the voltammetry of TMPD in the H1 phase first at a macroelectrode and then examine voltammetry and potential step chronoamperometry at a microelectrode. Voltammetry at a Millimetric-Sized Electrode. Figure 3a illustrates the voltammetry observed at a 3.0 mm diameter glassy carbon electrode when immersed into the H1 phase of Triton X 100 made using 42.0 wt % surfactant and 58.0 wt % aqueous 0.1 M KCl containing 10.3 mM TMPD. Although the phase was not aligned to form monodomains of oriented cylindrical micelles, the prepared medium exhibited radiance when examined between crossed-polarizers, and with a texture indicative of the anisotropic H1 phase at this surfactant/water mixture ratio. This suggests that, although the presence of the supporting electrolyte may cause phase boundary compositions to change,43 it is not manifested at the surfactant/water ratios employed herein. It is clear from the voltammograms presented that two principal redox waves occur, as anticipated; the small wave between the two primary waves is due to the quinone-imine product formed as a result of ipso attack of water on the electrogenerated dication. We shall concentrate primarily on the first wave, which corresponds to a single one-electron oxidation of TMPD to furnish the cation radical, TMPDþ•, with, 6515

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Table 1. Variation of the Diffusion Coefficients for TMPD in the H1 Phase of Triton X 100/Light Water c0/mM

107Dap /cm2 s1 a

107D/cm2 s1 b,c

107Dr/cm2 s1 d,e

107Dz/cm2 s1 d,e

107(DrDz) /cm2 s1 d,f

107Diso/cm2 s1 d,g

2.1

2.0

1.2

0.1

2.4

0.5

0.70

5.1

1.1

1.6

1.0

1.4

1.2

1.2

10.3

0.99

1.7

2.3

0.70

1.3

1.2

1/2

a Diffusion coefficients extracted using RandlesSevick eq 13 using data obtained at a 3.0 mm diameter glassy carbon disk electrode. b Data obtained at a 11.0 μm (diameter) carbon microdisk electrode from steady-state voltammetry, under the assumption of diffusive isotropy. c Obtained using the expression ilim = 4FDr0c0 where r0 is the disk radius; error bar estimated as being (0.5  108 cm2 s1. d Data obtained at a 11.0 μm (diameter) carbon microdisk electrode from chronoamperometry. e Since the iterative optimization was undertaken in logarithmic space (for computational ease); there is a variable error bar on the diffusion coefficients reported, of ca. 0.25%. f This value is the true isotropic diffusion coefficient reported from the Dr and Dz estimations. g This is the estimated isotropic diffusion coefficient, obtained using Dr = Dz in the iterative process.

in the low scan rate limit, peak potential, EpOx = 0.13 V vs SCE, and corresponding reduction wave at EpRed = 0.04 V vs SCE, affording a peak-to-peak potential separation of 93 mV. Although this latter value is larger than anticipated for a one-electron electrochemically reversible process, the cation radical species appears to be stable within this medium, as evidenced by the reproducibility of the voltammograms on continuous redox cycling. The peak oxidation and reduction potentials are generally invariant with the voltammetric time scale and TMPD concentration (see Figure 3b), as anticipated for an electrochemically reversible scenario, with the exception of the data recorded at the smallest time scales employed. It is important to note that the data have not been corrected (either through postexperiment analysis, or online electronic positive feedback compensation) for Ohmic drop, nor for double-layer effects, so that data reported for experimental time scales smaller than ca. 0.13 s may not be solely due to quasi-reversible electrode kinetics. The slow scan limit apparent resulting paradox, viz. larger than expected peak-to-peak separation, yet characteristic invariance of peak potentials with time scale may be a manifestation of the facilitated cross-pseudophase electron transfer reaction, as the forward and reverse processes do not necessarily need to follow the same pathway.44 Note that these observations are also consistent with there being a slight difference in the formal potentials for TMPD oxidation in both phases (even if this is a concerted proton electron loss from TMPDHþ). This is reasonable, since although electrolyte ions may be present within the palisade layer, so that their proximity to TMPD causes little kinetic impediment to the oxidation process, the solvation environment is altered compared with bulk water—even without added electrolyte, the H1 phase of Triton X 100 is thought to have at least one water molecule tightly bound to the detergent per oxyethylene group (of which there are between nine and ten per surfactant molecule).43 Moreover, the stability of the voltammograms on redox cycling does rule out the possibility of a CE reaction (exit of TMPD from the surfactant micelles into the aqueous channels followed by heterogeneous reaction). Examination of RandlesSevcik plots of concentration-normalized peak current with square-root of the scan rate (see Figure 3c) exhibit direct proportionality, as anticipated for diffusive mass transport. Table 1 reports the extracted diffusion coefficients, Dap, from these plots, extracted using the equation, rffiffiffiffiffiffi F pffiffiffiffiffiffiffipffiffi Dap v ip ¼ 0:4463FSc0 ð13Þ RT Note that this single-phase analysis does not require the consideration of a geometric limitation (a factor of π/(2 3 31/2)) to

the electrode area (S), to account for the fraction of the electrode surface exposed to the surfactant pseudophase, in line with other work that assumes isotropic diffusion within the H1 phase.36 It is noticeable that the data recorded at the lowest concentration give rise to the largest diffusion coefficient, with the data gathered at the higher concentrations of TMPD being apparently independent of the concentration of TMPD within the lyotropic system, within experimental error. Assuming the electrical migration toward the electrode can be neglected for TMPD (in both oxidized and reduced forms) as well as for the nonionic surfactant micelles, transport of palisade layer probes is a mixture of diffusion of the probe along the length of the micelle, lateral diffusion of the probe within the palisade layer, and rotational motion of the whole micelle. Within the framework of the Debye model of rotational Brownian motion,45 and under the assumption that the spherically capped cylindrical micelles can be approximated as cylinders of diameter 60 Å (2rm, a reasonable value given that measured for Triton X 100 micelles in the H1 phase,46 with 25 Å palisade layer thickness), of length (lm) ca. 600 Å (as estimated by Paulaitis et al.47 for C12E5 alkyl polyoxyethylene ether nonionic surfactant in the H1 phase), the time scale for micellar rotation (τrot), given by τrot ¼

πrm 2 lm η kB T

ð14Þ

in which η is the viscosity of the bathing medium (water), kB is the Boltzmann constant, and T the absolute temperature, is estimated as being ca. 4 μs, a value several orders of magnitude smaller than the time scales employed RT/Fv g 25 ms). Insofar as the H1 phase is likely non-Newtonian, this model may not be relevant, but we draw encouragement from the fact that the rotational tumbling of the near-spherical micelles within the more viscous and isotropic I1 phase of dodecyltrimethylammonium chloride is thought to occur within a few nanoseconds.48 Nevertheless, we assume that the micelles may be considered to be cylindrically isotropic for the electrochemical experiments reported herein, so that Fickean diffusion of the TMPD species is the most important transport within the H1 phase. Inasmuch as diffusion coefficients within the micellar pseudophase are strongly dependent on the palisade layer “microviscosity”, for all experiments reported herein (at constant aqueous electrolyte concentration of 0.1 M), this parameter is essentially unchanged. Thus, the mutual diffusion coefficients of TMPD reported in Table 1 correspond to the mixture of diffusion of TMPD within the aqueous and the micellar pseudophases. The slight concentration dependence of these does hint that a mediated electron transfer occurs especially in the light of the order of magnitude difference in the viscosities of water compared with that of the 6516

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Figure 4. (a) Cyclic voltammograms (scan rate 0.01 V s1) obtained at an 11.0 μm diameter carbon microdisc electrode immersed in the H1 phase containing (i) 2.1 mM and (ii) 10.3 mM TMPD. In both cases, four redox cycles are depicted (black, blue, red, and green correspond to cycles 14, respectively), and the arrow indicates the direction of the initial voltage sweep. The magenta line in each case corresponds to the theoretical response for reversible heterogeneous electron transfer with E00 = E1/2 = þ0.033 V vs SCE, based on the diffusion coefficient data given in Table 1 (see text). Note that to compare theoretical data with experimental (due to reductive baseline present in the latter, see text), the former were translated on the ordinate axis by (i) 0.025 nA and (ii) 0.101 nA. (b) Chronoamperometric transients (recorded over the same temporal domain of 10 s, at a potential of þ0.30 V vs SCE) in logarithmic dimensionless variables (see text) showing the match between experiment and theory for the case when diffusion are considered anisotropic. The black solid line corresponds to eq 4, and experimental transients are plotted as a series of points with the following TMPD concentrations: 2.1 mM (blue squares), 5.1 mM (red circles), and 10.3 mM (green triangles). Note that the experimental transients reported are the average observed over three repeats; the deviations for the first few points in each case are due to capacitative charging effects. (c) Plot of concentrationnormalized anisotropic diffusion coefficient (i = r or z), as determined from Figure 4b with TMPD concentration within the H1 phase. Blue circles represent the radial diffusion coefficient (Dr); red triangles represent the diffusion coefficient perpendicular to the electrode surface (Dz).

palisade layer (∼1 cP cf. ∼27 cP). Note that the viscosity of neat Triton X 100 is 240 cP at 298 K.49 Nevertheless, these data, under the assumption that the viscosity of the phase is unaffected by the TMPD concentration, may be employed in the framework of eqs 1 and 2 to estimate the partition coefficient of TMPD, estimating the diffusion coefficient of aqueous TMPD as (4.38 ( 0.13)  106 cm2 s1 (based on a RandlesSevcik plot obtained using an aqueous 1.33 mM solution of TMPD in 0.1 M KCl, and for which EpOx = 0.04 V vs SCE and EpRed = 0.03 V vs SCE), which is ca. 1 order of magnitude larger than that observed in the H1 phase. Although the aqueous TMPD voltammograms appear to be shifted by about 80 mV relative to those within the H1 phase, this does suggest cross-phase electron transfer occurs, though probably with an interphase electron exchange that is within an order of magnitude from unity.

Under the assumption that the palisade layer can be described in terms of Newtonian fluids, we may estimate the diffusion coefficient of TMPD within the micellar palisade layer through the inverse proportionality of diffusion coefficient and viscosity, viz. (1.63 ( 0.05)  107 cm2 s1. Since this value is larger than the observed value at TMPD concentrations greater than 2.1 mM, partition coefficient estimation can only be undertaken for the smallest TMPD concentration employed, for which log KP = 2.01 ( 0.07 (if it is assumed that the mediated electron transfer occurs with K = 1), or log KP = 1.53 ( 0.06, using eq 2. These values, although in essence, similar to that calculated for the n-octanol | water system, nevertheless lie within the “region of ambiguity” of eqs 1 and 2 (see earlier). Accordingly, we next examine measurements made using microelectrodes. 6517

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Electrochemistry of TMPD in the H1 Phase at Microelectrodes. Voltammetry at a Micrometric-Sized Electrode. Figure 4a

illustrates voltammograms (scan rate 0.01 V s1) for the oxidation of TMPD within the normal hexagonal lyotropic phase of Triton X 100. These exhibit signamoidal waveforms characteristic of fast, two-dimensional transport to the electrode surface,50 with E1/2 = 0.033 ( 0.002 V vs SCE for the one-electron oxidation of TMPD, a process that is fully electrochemically reversible within the Tomes criterion (E1/4  E3/4 = 57.8 ( 12.3 mV). Moreover, in contrast to the macroeelctrode voltammograms, there is little loss of signal when the second oxidation wave is scanned, especially at the higher TMPD loadings employed, indicating that the loss of the dication from the medium | electrode interface is faster than the nucleophilic attack by water. In turning to the second voltammetric wave observed at the microelectrode, we note that the observed currents are up to twice as large as the one-electron case. Note that there is a peak shaped signal occurring at the lower TMPD concentrations for the second oxidation, which produces a larger signal than the first process. Considering the first wave, the values of the limiting current enable the estimation of the apparent diffusion coefficient of TMPD within this lyotropic phase, under the assumption of isotropic diffusion. These data are reported in Table 1, where it can be seen there is a weak dependence of the observed diffusion coefficient with the concentration of TMPD in the system. Composite systems with concentration-dependent diffusion coefficients are not unknown.40b However, for the present system, we may suggest that this is either due to a long-range interphase electron transfer, or simply, long-range mediation of the electron transfer process through the exit of the generated TMPDþ• species and reaction from there, allowing for the propagation of charge transfer over micelles that are not close to the electrode surface. Formally, these two pathways are different, and the possibility of the exit of the cation radial from the micelles occurring in concert with the electron transfer (or even through a stepwise manner) is an additional complexity. Nevertheless, we may simplify such processes by recognizing they are both tantamount to a long-range charge transfer, so that under the assumption that radial diffusion dominates, a graph of diffusion coefficient against concentration enables the determination of the diffusion coefficient for physical displacement (Dtrans), and that due to hopping to be determined, employing the RuffBotar view,51 1 D ¼ Dtrans þ Dhop ¼ Dtrans þ kδ2 c0 6

ð15Þ

In the above expression, k refers to the bimolecular electron exchange kinetics between TMPD(surf) and TMPDþ•(aq), with a center-to-center distance of closest approach (during the in phase vibrational bascules) denoted by δ. Note that the factor of 1 /6 in the above expression refers to a three-dimensional transport. The data in Table 1 and Figure 4a suggest that Dtrans = 1.2  107 cm2 s1, and Dhop = 6.0  109c0 cm2 s1 (Pearson’s product moment correlation coefficient of 0.92). The former is within experimental error of that estimated for TMPD from the macroelectrode data (vide supra). Since the TMPD/TMPDþ• species can exist either in the aqueous pseudophase or in the surfactant pseudophase, with a hydrodynamic radius of52 6.39 Å, or radius23 5 Å, respectively, we may employ the geometric average of these to afford δ = 11.3 Å as a minimum value, using δ = 25 Å (the palisade layer thickness) as a maximum value, and thus determine that the bimolecular electron transfer rate

constant is on the order of (0.63.0)  109 M1 s1. These values appear of a reasonable order of magnitude, compared with those estimated within other similar systems,32 although it is important to note that this type of analysis is crude since we additionally cannot deconvolute between physical displacement and hopping diffusion in the direction normal to the electrode surface. The reason that such long-range charge transfer propagation is not readily apparent from the data at the larger electrode is simply because different transport regimes are considered at the large and small electrodes (see later), but taken together, the data provide good evidence that a square scheme is operational, especially since the half-wave potential for the TMPD oxidation at the microelectrode is close to that observed in aqueous solution, confirming that different processes are being interrogated at large and small electrodes within the H1 phase (see later). Thus, recognizing that the isotropic assumption cannot enable accurate estimation of partition equilibria (log KP is deduced as 2.75 ( 0.49 or 2.26 ( 0.47 using eqs 1 and 2, respectively), we next examine the extraction of Dr and Dz using chronoamperometry. Chronoamperometry at a Microelectrode. Figure 4b depicts experimental transients (given in dimensionless form) at a carbon microdisk electrode immersed into the H1 phase. The protocols outlined earlier were employed to extract anisotropic diffusion coefficients from the experimental data, by varying the diffusion coefficients in the range 9 e log(Di/cm2 s1) e 6 (i = r or z) in unit logarithmic steps of 103, to afford the data presented in Table 1. Note that the experimental transients were averaged over three repeats, so as to limit any effect due to the position of a cylindrical micelle at the electrode | insulation singularity, where the electrode is most sensitive as the flux (and Ohmic drop) is greatest. Also given in Table 1 are the effective isotropic diffusion coefficient (reported as DrDz1/2), in addition to the diffusion coefficient extracted from the chronoamperometric transients under the a priori assumption of isotropic diffusion within the H1 phase (Diso). Clearly, the data under these two cases are consistent over the TMPD concentration range used. Furthermore, in noting the discrepancy in the “isotropic” diffusion coefficients estimated from macro- and microelectrodes, it is important to realize that the former values have been calculated under the assumption of isotropic flux. The anisotropic data in Table 1 have been used to simulate linear sweep voltammograms that are given in comparison with the experimental data in Figure 4a. It is seen that there is a reasonable fit between experiment and theory, with a small discrepancy at the lowest TMPD concentrations, likely stemming from the nonideal baseline currents (the H1 phase was not degassed prior to experimentation, so the baseline currents are slightly reductive in nature). The occurrence of the voltammetric peaks at the smaller TMPD loadings is apparent. It is noteworthy that the anisotropy ratio Dz/Dr decreases by 2 orders of magnitude as the TMPD loading increases, so that the measurement at the lowest TMPD concentration behaves as though transport is to a one-dimensional planar electrode, while that at the highest TMPD loading, the diffusive transport regime is similar to that anticipated at a cylinder electrode. Indeed, examination of the transient data suggests that only the 5.1 mM concentration of TMPD exhibits a good fit between the experiment and theory for the isotropic diffusion to a microdisk electrode, since it is only at this concentration when Dr and Dz have close values. Under the assumption that the orientation 6518

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(and viscosity) of the phase is independent of the TMPD concentration, and that the averaging of the experimental data enables the full removal of any dependence on electrode spatial location within the phase, noting that micellar occupancy via partitioning is a random process with Poisson statistics,53 viz. Px ¼

xx x e x!

ð16Þ

where Px is the probability that a micelle contains x TMPD molecules, where the average TMPD occupancy within the micelles is x, the likelihood of long-range charge propagation is enhanced as the TMPD loading increases, above a critical percolation threshold,54 so that radial propagation then becomes a more significant pathway than axial transport. Based on the view that Dr represents the (one-dimensional) charge transport between individual species located in either the micellar or aqueous pseduophases, Dz is a mixture of physical translation and electron hopping perpendicular to the electrode surface, the negative slope of Dz with c0 is likely a manifestation of relative importance of the faster facilitated charge propagation across the pseudophase | pseudophase interface, as the concentration of analyte increases, and highlights the fact that transport within the lyotropic liquid crystal can be “tuned” via the square scheme, while the infinitely dilute limit (c0 f 0) of Dz (2.7  107 cm2 s1, a value 16.4 times smaller than that observed in aqueous 0.1 M KCl solution) represents a mixture of micellar and aqueous pseudophase translation diffusion. These views are consolidated by the self-consistency in the data for the magnitude of 1/2kδ2 (using δ = 11.3 Å) for the hopping parameter, affording bimolecular kinetics of 3.2  109 M1 s1, which is close to the result obtained earlier under the three-dimensional isotropic assumption. This large value is not a surprising result since the primary factor that affects the bimolecular electron transfer kinetics is the reorganization energy, and the two values may be of similar orders of magnitude by virtue of the oligoether side chains of the nonionic surfactant. Strikingly, the data in Table 1, when replotted in Figure 4c demonstrate that the radial diffusion coefficient, Dr, changes its variation with TMPD loading as the TMPD concentration increases, moving from a quadratic relationship at low loadings to one of direct proportionality at higher loadings. This unexpected variation can be rationalized through stochastic effects within an interphase charge transfer process, at low TMPD loadings, the probability of a micelle being unoccupied is given by eq 16 for the case of x = 0, viz., P0 = eλc0, where we recognize that the average number of TMPD molecules occupying each Triton X 100 micelle will be proportional to the TMPD concentration within the H1 phase. Thus, if λ is small, the exponent may be linearized so that P0 ≈ 1  λc0. Then, since Dr is, to a first approximation, proportional to the product of finding a TMPD molecule in the surfactant subphase and that for finding a TMPDþ• ion within the aqueous subphase, as a result of the bimolecular charge transfer process, it is then proportional to (1  P0)2, provided the partition coefficient of TMPD strongly favors the surfactant pseudophase. Thus, Dr is anticipated to vary with the square of the TMPD concentration, as observed. Why then should this variation tail off at higher TMPD loadings? In this case, again provided the TMPD strongly favors the surfactant subphase, the probability of finding a micelle that is empty is approximately zero. Under these conditions, then, interphase electron transfer is tantamount to a charge carrier hopping

process, under the assumption that vertical axial diffusion is negligible, so that Dr is expected to be directly proportional to the TMPD concentration, as observed.32,55 Thus, the results presented in Figure 4b,c suggest that only the data gathered at the smallest TMPD loading should be used in conjunction with eq 1, assuming a value of unity for the interphase electron exchange to extract partition equilibrium information, since it is only for this concentration that the experimental transient exhibits essentially planar, one-dimensional diffusive characteristics, with nonzero Dr. Under these conditions, we may formally view the TMPD diffusion as equivalent to a membrane diffusion that is related to the diffusion coefficient inside the micellar and aqueous pseudophases, Dm,56 Dz ¼

Dm ς2

ð17Þ

where ς is the tortuosity factor, defined as the ratio of the average length of the micelles and aqueous channels to the vertical distance between the electrode and the end of the micelles/ aqueous channels (essentially a ratio to account for column undulations in the nonlinear visoelastic Triton X 100 micelles).46 Given the domain size of hexagonal lyotropic liquid crystals is typically ca. 50 μm,46 a distance essentially pragmatically larger than that probed voltammetrically for these systems based on an EinsteinSmoluchowski view (as this would require potential scan rates of less than 1.0 mV s1, a sweep rate that is smaller than that considered in this work), this factor really accounts for the orientation of the H1 phase. Thus, assuming the formal potential for TMPD oxidation is independent of its psedophase location, and unity tortuosity, the partition coefficient for TMPD between aqueous and micellar pseudophases is, unequivocally, log KP = 2.01 ( 0.05 at the experimental temperature, pleasingly in agreement with that suggested at the macroelectrode (under the assumption of cross-pseudophase electron transfer) and estimated for partition between ortho-nitrophenyl octyl ether and water (log KP = 2.33 ( 0.51),20e based on the calculated value at 298 K for the partition across the n-octanol | water interface (log KP = 2.08 ( 0.47),17 suggesting that the micellar TMPD concentration is two orders of magnitude larger than in the aqueous pseudophase.

’ CONCLUSIONS We have illustrated that voltammetry is not very useful in determining the extent of anisotropic diffusion in structured nanosystems but, rather, we should rely on chronoamperometric measurements. For the case of TMPD within the normal hexagonal lyotropic liquid crystal (H1) of Triton X 100, which partitions between the subphases, we suggest that the electron transfer across the pseudophase | pseudophase interface is important, since the operation of a “square scheme” essentially acts as electron transfer “pump”, thereby allowing the anisotropic system to behave as though it is macroscopically isotropic. In returning to the question posed within the title, we note that measurements must be made at low analyte concentrations when percolation processes dominate. Insofar as we have not considered the effects due to the partitioning of the electrogenerated product herein, we note that these effects will likely be most significant for time-reversal experiments and will report our studies in this in a future work.38a 6519

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’ APPENDIX 1 Derivation of Expressions for KP Given in eqs 1 and 2. Consider the following partition equilibrium for an electroactive chemical species initially present in concentration c0 in an aqueous phase, mixed with high concentration of surfactant in an isotropic micellar phase, such that the redox molecules may partition between the aqueous (A) and surfactant (P) pseudophases, so that, after cooling, the concentrations (per unit volume of the resulting lyotropic liquid crystal, assuming that the presence of solute occurs with no change in volume of the former) at thermodynamic equilibrium are c0A and c0P, respectively, A a P. Defining the partition coefficient, Kp = c0P/c0A = xP/xA, where xi is the equilibrium mole fraction of the redox species in pseudophase i, such that xA þ xP = 1, it follows that the expression for the equilibrium concentrations within each pseudophase are c0P = [KP/(1 þ KP)]c0 and c0A = [1/(1 þ KP)]c0. For a one-dimensional system in which the aqueous and surfactant pseudophases are aligned so that they are parallel to each other and perpendicular to the electrode surface, for the case of transport-limited chronoamperometry, and ignoring any terms due to electrical migration or convection, the electrode will see diffusive transport in both pseudophases described by a Fickean diffusion coefficient Di, in addition to any mediated electron transfer across the pseudophase | pseudophase interface. Since the identity of the chemical species is the same in each pseudophase, this interfacial electron transfer is a self-exchange process of essentially unity equilibrium constant for the forward and reverse electron transfer processes. Note that we consider that the oxidized and reduced species do not partition differently. Considering the case where there is no self-exchange, we may write the Cottrell equation for the current (i)time (t) response in terms of an apparent diffusion coefficient, Dap: pffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi i πt ¼ c0A DA þ c0P DP ¼ c0 Dap FS

where F is the Faraday constant and S is the total electrode area, thereby yielding pffiffiffiffiffi pffiffiffiffiffiffiffi i πt 1 pffiffiffiffiffiffi KP pffiffiffiffiffiffi ¼ Dap ¼ DA þ DP ðA1.1Þ FSc0 1 þ KP 1 þ KP Expressing the above in terms of mole fractions affords the identical result derived by Miller and Majda,33a,b viz., pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi Dap ¼ xA DA þ xP DP

Again, expressing KP in terms of mole fractions affords the MillerMajda result as required,33a,b Dap ¼ xA DA þ xP DP It follows from these results that the partition coefficient can be determined if the mutual diffusion coefficient of the redox species is known in each of the two pseudophases, and if the measured diffusion coefficient is Dap, eqs 1 and 2 may be formulated.

’ APPENDIX 2 Case for Nonequal Diffusion Coefficients of Reactant and Product. We demonstrate herein the relationship between the

surfacial concentrations of reactant and product when both exhibit different diffusion coefficients, via consideration of potential step chronoamperometry for which the potential difference applied between the working and reference electrodes is sufficiently negative compared with the formal electrode potential (E00 ) so that mass transport-limited reductive electrolysis occurs, A þ ne a B. In one-dimensional motion in isotropic space, Fick’s second law for each species, DcA D2 cA ¼ DA 2 Dt Dx DcB D2 cB ¼ DB 2 Dt Dx is solved subject to the following boundary conditions.

1 KP DA þ DP 1 þ KP 1 þ KP

cA ¼ c0

tg0

x¼0

cA ¼ 0

tg0

x f ¥ cA f c0

cB ¼ 0   DcA DB DcB ¼  Dx 0 DA Dx 0 cB f 0

x cA y ¼ pffiffiffiffiffiffiffiffiffi a¼ c0 D θ pffiffiffi  A   i θ Da DB Db pffiffiffiffiffiffi ¼ ψ¼ ¼  Dy 0 DA Dy 0 nFSc0 DA τ¼

t θ

b¼

cB c0

where θ is the measurement time, all other symbols taking their usual definitions, enables simplification of Fick’s second law and the corresponding boundary conditions: Da D2 a ¼ 2 Dτ Dy τ¼0 yg0 τg0 yf¥ τg0

y¼0

Db DB D 2 b ¼ Dτ DA Dy2 a¼1 b¼0 a¼1 b¼0 ! ! Da DB Db a0 ¼ 0 ¼  Dy DA Dy 0

0

Proceeding via Laplace transformation (variable s) with respect to τ, and using a bar beneath a variable to denote Laplace space, we may write D2 a ¼ s a  aτ ¼ 0 ¼ s a  1 Dy2 DB D 2 b ¼ s b  bτ ¼ 0 ¼ s b DA Dy2

and so, Dap ¼

"x

Introducing the following adimensional variables,58

For the case where self-exchange is important, following Miller and Majda,33a,b we use the pertinent result derived by Saveant et al. for self-exchange reactions at redox polymer electrodes:57 sffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi i πt c0 c0 DP 0 ¼ cA DA 1 þ 0P 1 þ 0P ¼ c0 Dap FS cA c A DA Now, insertion of the appropriate terms and rearrangement furnishes pffiffiffiffiffi pffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i πt DP DA ¼ Dap ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ KP FSc0 DA 1 þ KP

te0

ðA1.2Þ 6520

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where A and B are integration constants. It follows from the fact that b must hold a finite value as y1/2f ¥ so that B = 0. Also, when y = 1/2 0, b0 = A. Thus, b = b0ey(DB/DA) s . The dimensionless current in Laplace space relates to the concentration gradient of species B: rffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffipffi pffi DB D b DB f b0 s ey DA =DB s g0 ψ ¼  ¼ DA Dy 0 DA rffiffiffiffiffiffi rffiffiffiffiffiffi ψ DA DB p ffi ¼0 ¼ b0 s w b0  pffi s DB DA Likewise, the differential equation in species A is inhomogeneous but can be converted into homogeneous form via the substitution u = sa  1, to afford pffi pffi 1 a ¼ A0 ey s þ B0 ey s þ s where A0 and B0 are integration constants. It can be seen that B0 = 0 so that a is finite when y f ¥, and that A0 = a0  1/s. Noting the relationship in Laplace space between current and the concentration of species A:     Da 1 pffi ypffis ψ ¼ ¼  a0  g0 sfe Dy 0 s ψ pffi 1 1 ¼ pffi  a0 s w a0 þ pffi ¼ s s s Hence, combining the variables for the surface concentrations of species A and B in Laplace space reveals rffiffiffiffiffiffi DB 1 ¼ a0 þ b0 s DA so that inversion into the real space affords the following relationship between surface concentrations of species A and B in normalized form, as required. rffiffiffiffiffiffi DB a0 þ b0 ¼1 ðA2.1Þ DA Note that although this expression has been derived for the case of one-dimensional diffusion, it is valid for all geometries within isotropic media. It follows from eq A2.1 that the following relationships hold. DA ¼ DB DA > D B DA < DB

a0 þ b0 ¼ 1 a0 þ b0 > 1 a0 þ b0 < 1

’ AUTHOR INFORMATION Corresponding Author

*E.mail: [email protected]. Tel: þ44 (0) 14 82 46 63 54. Fax:þ44 (0) 14 82 46 64 10. Web: http://www2.hull.ac.uk/ science/chemistry/staff/academic_staff/dr_jay_wadhawan.aspx. Present Addresses §

Department of Chemistry, The University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom.

)

Now, the above second-order homogeneous differential equation with constant coefficients in species B is solved to furnish, pffiffiffiffiffiffiffiffiffiffipffi pffiffiffiffiffiffiffiffiffiffipffi b ¼ Aey DB =DA s þ Bey DB =DA s

ARTICLE

Laboratoire d’Electrochimie Moleculaire, UMR CNRS  P7 7591, Universite Paris Diderot  Paris 7, B^atiment Lavoisier, 15 rue Jean-Antoine de Baïf, 75205 Paris Cedex 13, France.

’ ACKNOWLEDGMENT We express gratitude to the Engineering and Physical Sciences Research Council for financing this work (grant number EP/ G020833/1). ’ GLOSSARY OF TERMS AND SYMBOLS a normalized concentration of species A concentration of species A cA effective concentration c0 d length of aqueous subphase D geometric mean of diffusion coefficients in two dimensions apparent diffusion coefficient Dap diffusion coefficient in aqueous subphase DA isotropic diffusion coefficient Diso diffusion coefficient due to electron hopping Dhop one-dimensional membrane diffusion coefficient Dm diffusion coefficient in surfactant subphase DP diffusion coefficient in direction tangential to electrode Dr surface diffusion coefficient due to physical translation Dtrans diffusion coefficient in direction perpendicular to elecDz trode surface E applied potential difference between working and reference electrodes formal potential for a redox couple E 00 F Faraday’s constant k bimolecular electron transfer rate constant Boltzmann constant kB partition coefficient KP l length of aqueous and micellar subphase micelle length lm Poisson probability function Px r direction tangential to electrode surface micelle radius rm electrode radius r0 R normalized radial director R molar gas constant S electrode area t time T absolute temperature TMPD N,N,N0 ,N0 -tetramethyl-para-phenylenediamine u normalized initial potential in linear sweep voltammetry v voltage sweep rate mole fraction of partitioned species within the aqueous xA subphase mole fraction of partitioned species within the surfacxP tant subphase z direction normal to electrode surface Z normalized axial director δ center-to-center distance for hopping electron transfer ξ parameter to describe reduced potential in voltammetry Γ transformed coordinate η viscosity Λ dimensionless ButlerVolmer standard heterogeneous 6521

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The Journal of Physical Chemistry B Θ τ τrot ς ℵ p

rate constant transformed coordinate dimensionless time in chronoamperometry rotational correlation time tortuosity factor mean-scaled absolute deviation voltammetric waveshape parameter

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