Journal of The Electrochemical Society, 163 (4) H3173-H3181 (2016) H3173

JES FOCUS ISSUE HONORING ALLEN J. BARD Film Permeation by Rotating Disk Voltammetry at Electrodes Modified with Electrochemically Inert Layers and Heterogeneous Composites Krysti L. Knoche,∗ Chaminda Hettige,∗ Lois Anne Zook-Gerdau, and Johna Leddy∗∗,z Department of Chemistry, University of Iowa, Iowa City, Iowa 52242, USA

Steady state rotating disk voltammetry provides excellent measurement of permeability for films and layers on electrodes because the hydrodynamic control of rotating disks establishes well defined boundary layers normal to the electrode. For a redox probe present in solution and pre-equilibrated in the layer, voltammetry measures electrolysis current for the probe as the probe transports from solution, through the film, and to the electrode where the probe is electrolyzed. Diagnostic equations relate the steady state, mass transport limited current iss to the rotation rate ω. The incompressible layer is electroinactive about the probe formal potential. Diagnostics are available for a single layer, uniform film (Gough and Leypoldt). Uniform films are homogeneous with no structural features. Here diagnostics are provided for bilayer and multilayer uniform films, where multilayers include discretely and continuously graded films. Consideration of serial mass transport resistances normal to the electrode allows diagnostics for uniform films. Heterogeneous, micro- and nano-structured layers are structured in the plane of the electrode. Consideration of parallel mass transport resistances as part of the layer mass transport resistance yields diagnostics for heterogeneous films. Diagnostics are vetted with literature data. Theoretical and practical advantages and limitations are noted. © The Author(s) 2016. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution Non-Commercial No Derivatives 4.0 License (CC BY-NC-ND, http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial reuse, distribution, and reproduction in any medium, provided the original work is not changed in any way and is properly cited. For permission for commercial reuse, please email: [email protected]. [DOI: 10.1149/2.0241604jes] All rights reserved.

Manuscript submitted November 23, 2015; revised manuscript received February 12, 2016. Published February 24, 2016. This paper is part of the JES Focus Issue Honoring Allen J. Bard.

Rotating disk voltammetry is often used to evaluate electrodes the effective permeability of the layers. Additional information is modified with incompressible films of various materials.1–10 In steady needed to further deconvolve the thicknesses, diffusion coefficients, state rotating disk measurements, redox probe present in solution is and partition parameters that characterize permeability. Without ad- transported from solution through the inert film to the electrode where ditional information, an effective permeability for the entire layer is the probe is electrolyzed. Because probe interactions with the film found for single, bi- and multi-layers. Additional characterization in- limit flux through the film, flux maps film structure and properties; cludes independent film evaluations (e.g., microscopy, ellipsometry,16 steady state currents measure flux. An inert film is electroinactive in and electrochemical quartz crystal microbalance17,18); voltammetric the voltage range where the redox probe is electrolyzed. Inert films protocols that vary experimental parameters (e.g., layer thicknesses); include electrochemically silent layers and ion exchange polymers. and coupling steady state voltammetry with transient methods (e.g., Redox and electron conducting polymers are inert outside potential chronoamperometry and cyclic voltammetry19). ranges where the polymer itself is electrolyzed. Some micro- and Uniform films are homogeneous. Only flux normal to the electrode nano-structured, heterogeneous films are inert. Porous, electrochem- and serial resistances are needed to model uniform layers. Microstruc- ically silent layers include supports for heterogeneous catalysts and tured, heterogeneous and porous films have structures that vary in the electrodes as well as particles such as zeolites formed into a coherent plane of the electrode, which disrupt transport normal to the elec- layer. In pharmaceutical and biomembrane studies, radiotracer mea- trode. Here, equations for permeability of microstructured and porous surements are commonly employed, but rotating disk allows better layers are developed on inclusion of parallel transport paths (parallel measures of permeability because the refined hydrodynamic control mass transport resistances) within the layer. Limitations of rotating establishes well defined steady state boundary layers. disk measurements, such as statistical constraints, times to establish Rotating disk voltammetry establishes excellent hydrodynamic steady state, and failure of the methodology are discussed. Litera- control of flux normal to the electrode.11 Control of flux is exploited to ture data are used to illustrate the various methodologies outlined for determine film permeability. Voltammetric diagnostics of steady state steady state rotating disk voltammetry at electrodes modified with flux through a single, electroinactive, uniform layer are reported.12–15 electrochemically inert layers, both uniform and heterogeneous. The single layer and solution each establish a mass transport resis- tance for the probe that limits flux and measured current. Comparison of flux (current) at modified and unmodified electrodes measures film Rotating Disk Experiments mass transport resistance and permeability. Here, equations are presented for steady state permeation of re- Macroscopically, when a flat, unmodified electrode is rotated, fluid 11,20 dox probes through inert films that are composed of more than a flows from the bulk toward and normal to the electrode surface. single layer. Bilayers may be formed by successive coatings or as Immediately at the electrode surface, solvent and electrolyte species a natural consequence of a crust forming at the film solution inter- are carried with the rotating electrode. The surface layer drags the face. Multilayers, which may also be formed by successive coatings, adjacent fluid layer and so with subsequent layers for a substantial serve as models for both discretely layered and continuously graded distance such that a stagnant layer is established at the electrode films. Equations are derived for uniform films from linearized con- surface. To a reasonable approximation, transport in the stagnant layer centration profiles and from serial mass transport resistances normal occurs only by diffusion. Outside of this layer, solution is moved by to the electrode surface. Rotating disk voltammetry alone yields only convection. For convection-diffusion in an isotropic, incompressible fluid under laminar (non-turbulent) flow, the thickness of the stagnant or boundary layer, δ0 (cm), is well approximated as ∗Electrochemical Society Student Member. ∗∗ Electrochemical Society Fellow. / / − / z δ = . ν1 6 1 3ω 1 2 E-mail: [email protected] 0 1 61 Ds [1]

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of transport in solution and, independent of any other limits on the current, an intercept of zero. For an unmodified electrode, a Levich plot that flattens at higher ω indicates current limited by effects other than mass transport in solution. A linear Koutecky Levich plot embeds this limiting, ω-independent information in the intercept. Embedding of ω-independent data in the intercept of the Koutecky Levich plot forms the basis for characterization of films on electrodes. There are several practical considerations to good rotating disk measurements at unmodified electrodes. (i) To maintain laminar flow in a simple electrochemical cell, f is typically limited to less than a few thousand rotations per minute. (ii) For unmodified electrodes, f > 50 to 100 rpm establish boundary layers sufficient to avoid disruption by density and thermal gradients. (iii) Electrodes are best centered in a straight shaft. (iv) For a disk of radius r, an insulating shroud around the disk should be at least r thick to minimize radial diffusion not considered in the Levich equation derivation. (v) Disk radius r at least several fold larger than δ0 further establishes linear diffusion as the only significant transport process. (vi) A well executed rotating disk experiment typically allows three significant figures in the current. (vii) At an unmodified electrode, it is common to sweep potential slowly ( 10 mV/s) to map current from zero to the mass transport limited rate. A potential sweep for each ω is practical because steady state is rapidly established at unmodified electrodes.

General notes about rotating disk at modified electrodes.—When rotating disk experiments are performed at an electrode modified with a layer of thickness , the stagnant layer forms in solution just as at an unmodified electrode. The stagnant layer is still of thickness δ0, but instead of being formed between x = 0andx = δ0, δ0 is formed between x =  and x =  + δ0. Transport in the stagnant layer and Figure 1. Linearized concentration profiles, c (x), for three steady state cases. the modifying layer is by diffusion. At steady state, flux in the film For serial mass transport resistances, steady state is achieved when the con- centration profiles, shown as dashed lines, are linear. (A) For an unmodified and flux in the solution must be equal. electrode; (B) for an electrode modified with a single layer, uniform film, Practical consideration for modified electrodes include those for shown for κ > 1andD < D ; and (C) for an electrode modified with unmodified electrodes and several specific to films. (i) Modifying f s  a bilayer, shown for the outer layer as a crust, where κ1  1, κ < 1, and layers must be incompressible. (ii) Thickness does not affect mea- Ds > D1 > D2. surement quality provided the uniform layer is isotropic and time to steady state is not prohibitive. (iii) Time to steady state for a uniform 2 film scales as tss ∝  /D f where D f is the diffusion coefficient in the Rotation rate, ω, is expressed in radians per second. For a rotational 2 layer; typically, steady state is achieved when tss  5 /D f ,where frequency, f , in rotations per minute, ω = 2πf/60. Probe diffusion < 2 tss can be long because D f Ds . (iv) Because tss can be much longer coefficient in solution is Ds (cm /s). Kinematic viscosity, ν is the ratio 2 for a modified electrode, rotating disk measurements are often made of the solution viscosity to solution density, (cm /s). Boundary layer by holding the potential at the mass transport limit (c(0) = 0) and ω ◦ ν = . 2 thickness decreases with . For water at 25 C, 0 00893 cm /s; waiting for steady state at some ω.Oncei is established constant, × −6 2 = ω = −1 ss for typical Ds of 8 10 cm /s and f 1000 rpm ( 105 s ), ω can be changed without changing the potential. Again, steady state δ μ = δ μ 0 is 15 m; for f 100 rpm, 0 is 46 m. must be established, but the time required is less than under potential Levich developed the current expression for steady state, mass sweep. (v) Data recorded in random ω order tends to prevent incorpo- transport limited electrolysis at a rotating disk. The Levich equation ration of film degradation as a systematic error. (vi) As for unmodified derives from a linearized approximation of diffusion through the stag- electrodes, cavitation sets the upper limit on ω. From practical expe- nant layer. Let the coordinate normal to the electrode surface be x. ∗ 3 rience, however, f as low as a few rpm can be used at modified For a redox probe in solution at bulk concentration c (mol/cm )and electrodes, especially when flux in the film sets i . (vii) To avoid = ss electrolyzed at the electrode surface (x 0) at the mass transport radial diffusion in the film, the layer should completely cover but not = = limited rate (c (x 0) 0), a steady state concentration profile c (x) extend beyond the electrode area. This is typically not difficult for hy- is well-approximated linearly as shown in Figure 1A. Steady state flux drophobic electrodes shrouds (e.g., polytetrafluoroethylene (PTFE)) = in solution, Jsoln,atx 0 sets steady state current in solution, isoln.In and films that allow probe diffusion from a higher dielectric constant / Equation 2, the discretized approximation to dc(x) dx is shown. For electrolyte. Films may swell in the electrolyte, but cannot dissolve or = electrolysis at the mass transport limit, c(0) 0, the Levich equation detach from the electrode. Because electrochemical solvents cover a is shown in Equation 3.  range of dielectric constants (e.g., 78 for water, 37 for acetonitrile, and i dc(x)  c∗ − c(0) 9 for dichloromethane), a solvent can usually be found. Some films, soln =−J x = = D  = D soln ( 0) s  s δ [2] especially polymers, swell to different extents in different solvents. nFA dx x=0 0 δ  Because of well controlled transport ( 0), rotating disk methods  ∗ characterize films with greater precision than concentration cells used isoln  c ∗ 2/3 −1/6 1/2  = Ds = 0.62c D ν ω [3] in permeation studies. Steady state voltammetry is an attractive and nFA δ s Levich 0 more accurate alternative to radiotracer studies of films and coatings n, F,andA are the number of electrons transferred (O + ne  used, for example, in pharmaceutical and coating studies. R), Faraday constant, and the geometric area of the electrode. For an unmodified electrode, the experimental, steady state current i ss Models for Uniform Films measures only flux in solution (isoln). A Levich plot of iss versus ω1/2 will be linear. A double reciprocal plot, a Koutecky Levich plot, For modified electrodes, the steady state, linear approximation −1 ω−1/2 δ of iss versus is common and will yield a slope characteristic to the boundary layer thickness in solution, 0, remains. At steady

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state, Fick’s second law (∂c(x, t)/∂t = 0 = D∂2c (x, t) /∂x 2 = various kinetic mechanisms.) Substitution of c(−) yields −∂ J x, t /∂x  ∗    ( ) ) dictates flux in solution and flux through each layer of Ds c ∗ κD f i D δ D c the film must be equal. ss = f 0 = s  κ [7] For uniform films, each layer is homogeneous, isotropic, and in- nFA  D f + Ds δ D f + Ds  κδ 0  δ compressible. Uniform layers are without structure in the plane paral- 0 0 1/2 lel to the electrode. Films may be composed of one (single layer), two This yields a Levich plot (iss vs ω ) that approaches ω-independence (bilayer), or more (multilayer) layers. Multilayers can be discrete or at high ω. Data analysis is simplified by a Koutecky Levich plot, the continuous. Each uniform layer is characterized by thickness, trans- reciprocal of Equation 7. port, and concentration. Generally, the film is confined to the area A ∗ nFAc δ  of the electrode to eliminate steady state radial flux in the film and = 0 + κ [8] the cross sectional area of the film (A) is large compared to the film iss Ds D f thickness . For uniform films, rotating disk measurements yield at ∗ −1 ω−1/2 The plot nFAc iss vs yields a slope characteristic of transport least an effective permeability or viscosity. in solution and an ω-independent intercept characteristic of transport Uniform films are defined as electrochemically inert layers. The . 2/3ν−1/6 −1 /κ in the film. The slope is [0 62Ds ] and the intercept is D f . film itself is either electroinactive or probe electrolysis is undertaken ∗ ∗ −1 ω−1/2 For the same A and c , the slopes of nFAc iss versus should be at a potential range where the film is not electrolyzed. For example, the same for the unmodified and modified electrodes. Similar slopes Nafion is not electroactive but the probes that diffuse through Nafion serve as a check that the measurements are made in an appropriate can be electrolyzed. Redox polymers can be studied in potential ranges manner and that isoln is ω-dependent and i film is ω-independent. Flux where the monomer units are not themselves electroactive but a redox in the film J is represented as a current, i =−nFAJ .The probe is. film film film permeability of the film, Pfilm = κD f /, is found from the inverse intercept. If film and solution flux are expressed as the corresponding Single layer, uniform films.—Gough and Leypoldt12–15 first de- currents, rived the steady current measured for the electrolysis of a redox probe 1 1 1 transported from solution through an electroinactive, uniform, single = + [9] i i i layer film to the electrode surface. Film is characterized by thickness ss soln film  and probe diffusion coefficient D f . The probe present in solution A sufficiently small current, isoln or i film can dominate the measured ∗ at concentration c is equilibrated in the film prior to measurement. iss. Film concentration is reported as the equilibrium ratio of the probe between the solution and film, characterized as Specification by resistance.—Equation 9 also derives from mass transport resistance. Redox probes move through solution to the film − c( ) solution interface and through the film to the electrode surface, where κ = [4] c(+) probes undergo mass transport limited electrolysis. On motion through the film and the solution, the probe is subject to mass transport resis- − + c( )andc( ) are probe concentrations just inside and outside the tances R film and Rsoln. For transport across the film solution interface film solution interface located at x = .Note,κ also parameterizes facile, the interface introduces no resistance. Transport is normal to the probe concentration in the film relative to the solution prior to the electrode and the resistances are in series for total resistance, electrolysis. κ is not a true equilibrium constant. If the concentration RTotal = Rsoln + R film. As in Ohm’s law, resistance and current are ∗ −1 = −1 + −1 in the film is fixed independent of c , as it may be for an ion exchange inversely related, so iss isoln i film (Equation 9). This requires polymer, then the parameter κ will vary with c∗. Steady state concen- that all currents flow through the same area (A) subject to the same tration profiles for single layer, uniform films are illustrated in Figure driving force. 1B, where as drawn, κ > 1 and diffusion coefficient in the solution is higher than in the film. Data analysis.—The intercept of the Koutecky Levich plot yields permeability P = κD f /. To separate parameters κ, D f ,and, additional information is needed. Transient voltammetry, such as Specification by flux balance.—At steady state in a uniform film, κ flux through every plane parallel to the electrode surface is equal. chronoamperometry and cyclic voltammetry yield D f . If the tran- sient experiment is executed for time sufficient that the diffusion length By Fick’s second law in one dimension, if the fluxes are not equal,  2/ spatial variation in flux will lead to a time dependent variation in exceeds , D f is also found. The equation for the chronoamper- concentration inconsistent with steady state. Flux in the film and ometric response for permeation of a probe through an inert, single layer, uniform film is reported by Peerce and Bard.22 A method to solution are equal. Consider interfaces where flux is well specified: κ 2/ the electrode surface and the film solution interface. extract D f and D f from cyclic voltammetry is reported by 23 2   Knoche et al. Given P = κD f /, κ D f and perhaps  /D f ,the   κ  iss = dc(x)  = dc(x)  three parameters , D f ,and cannot be determined individually with- D f  Ds  [5] . nFA dx = dx =+ out additional information. Typically, this is x 0 x Because films swell in solvents, the values of  are frequently Discretized, linear approximation of the gradients and incorpora- poorly determined. Often, the amount of material on the electrode is tion of Equation 4 yields better specified in other ways. The mass of material on the electrode m (g) is determined either by the amount of material delivered to − + iss c( ) − c(0) c ( + δ0) − c( ) the electrode or by microbalance methods. Spectroscopy, coulometry, = D = D  2 nFA f  s δ and titration yield probe surface coverage (mol/cm ). Neither m nor 0  suffice to separate the three parameters. However, if permeability  + δ − − /κ = κ /  = c ( 0) c( ) P D f is measured for several values of m or ,thenother Ds [6] / δ0 specification of transport through the film can be found. If Pm A is invariant with m,thenPm/A has units of viscosity (g/cm s). Similarly, ∗    The concentration at  + δ0 is c . For electrolysis at the mass trans- if P is invariant with ,thenP has units mol/cm s. port limited rate, c(0) = 0. Equation 6 is solved to find c(−) = Consider an interfacial resistance to transport at the film solu-  − ∗ D 1 tion interface, R ,whereR is independent of . Then, Ds c f + Ds interface interface δ  κδ . (For potential controlled, steady state electrolysis 0 0 RTotal = Rsoln + R film() + Rinterface so that a Koutecky Levich where c(0) ≥ 0, see Reference 21 for both simple permeation and −1 = −1 +  −1 + −1 plot of iss isoln i film( ) iinterface yields an intercept of

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 −1 + −1  −1 ∝  24  / i film( ) iinterface where i film( ) . If is varied, then a plot for each emulsion. The units of Pm A are g/cm s, the units of vis- −1 −1 cosity, where 0.01 g/ cm s is 1 centipoise. Where P varies with m/A, of the measured i film() + i versus  will yield a slope pro- interface Pm/A does not vary systematically with m/A so Pm/A provides an portional to κD f and positive intercept proportional to Rinterface. Here, intensive metric to compare the polymers. Average values of Pm/A Rinterface characterizes transport across the film solution interface, but the measurement yields any -independent resistance through any are shown with standard deviations. In all cases, median Pm/A is plane parallel to the electrode. The measurement does not distinguish comparable to average Pm/A. The values of Pm/A are similar for −4 interfacial charge transfer, resistance at x = 0 from interfacial mass all emulsions, averaging to 3 × 10 cp, which is well below the vis- transport resistance at x = . cosity of water (1 cp). The intensive Pm/A is then a good figure of merit for these films but not a direct measure of the film viscosity. 36 4− Examples.—Rotating disk voltammetry is commonly used to char- Example 3: Cosnier et al., measured permeabilities of Fe(CN)6 acterize single, electroinactive, uniform films on electrodes, as for through layers of latex (5-hydroxymethyl-bicyclo[2.2.1]-hept-2-ene) 19,24–32 where the latex layers are applied in volume increments that each add examples, in studies from Allen J. Bard’s labs. Other literature 2 −4 reports for single, incompressible layers include clay and nanostruc- 0.250 mg of latex per 0.20 cm . Measured permeabilities P (10 tured clay composites, emulsion polymers and copolymers, latex, and cm/s) are reported as 16.2, 8.10, and 5.40 for 1, 2, and 3 layers. The × −3 redox polymers (where the redox potential of the probe is well sepa- product of measured P with the number of layers is fixed (1.62 10 rated from the electroactive monomer). Cases where R > 0are cm/s), so that the latex is intensively parameterized with the invariant, interface μ −1 discussed below. effective viscosity (2.06 g [cms] ), independent of the number of Example 1: Rotating disk voltammetry33 of montmorillonite clay layers. and clay + PVA (poly(vinyl alcohol)) composites 5:1 and 5:2 by Example 4: Interfacial resistance was evaluated for several redox −1  weight on basal plane pyrolytic graphite were evaluated for probe polymers by plotting measured P versus , where a positive inter- − 24 4  cept characterizes Rinterface. Where Koutecky Levich slopes were the of 1 mM Mo(CN)8 in 0.2 M sodium triflate. Film thickness was either calculated from known clay density or profilometry on wetted same as for the unmodified electrodes and positive intercepts were −1  films. Levich plots approach a limiting current at higher ω. Koutecky found from measured P versus , interfacial rates were in the range of 0.01 to 0.1 cm/s. These data were for electroactive polymers such as Levich plots have the same slope as the unmodified electrode. Plots 2+ 2+ 2+ 37 of nFAc*κD / vs −1 are linear with zero intercepts, consistent poly[(Ru(vbpy)3] , poly[VDQ] , poly[Ru(bpy)2(p-cinn)2] , and f poly(vinyl ferrocene)19 in acetonitrile with redox probes such as ben- with Rinterface = 0. Slopes yield κD f that are reported to increase with PVA content as clay ((1.1 ± 0.2) × 10−8 cm2/s) < clay + PVA zoquinone and ferrocene; VDQ is vinyl diquat, vbpy is a vinyl bipyri- (5:1) ((2.2 ± 0.7) × 10−8 cm2/s) < clay + PVA (5:2) ((25.6 ± 3.0) × dine, and p-cinn is N-(4-pyridine)cinnamamide. Yet slower rates of 10−8 cm2/s). interfacial transport may be sufficient to alter the slope of the Koutecky Levich plots; see statistical limitations in the measurements in Statis- Example 2: Various acrylate emulsion polymers were evaluated −1 for permeability and pinholes by Hall and coworkers by rotating tical limitations section. Negative intercepts of measured P versus  disk voltammetry.34 Emulsion polymers and copolymers were formed do not correspond to any known models of transport (and coupled of methyl methacrylate (MMA), glycidyl methacrylate (GMA), and transport-kinetics) for incompressible films, but can reflect statisti- butyl acrylate (BA) with poly(vinyl alcohol) stabilizer. Four sta- cally insignificant intercepts and inconsistency in the measurements −1  ble films of MMA:GMA:BA were formed in varying component and analysis. When intercepts of P versus are negative, interfacial ratios: Poly 12 (65:0:35), Poly 14 (35:0:65), Poly X1 (0:35:65), mass transport resistance likely does not dominate limitation to the and Poly X6 (0:0:100). The redox probe was 1 mM tetramethyl-o- measured current. phenylenediamine (TMPD) in KCl, phosphate buffer at pH 7.0 and the electrode was 0.1141 cm2 Pt. Permeability was reported as a Bilayers, two uniform layers.—Bilayers may be formed when two function of mass of emulsion on the electrode as shown in Table I. layers are sequentially applied to an electrode or when a film develops Where pinholes were found according to the model of Landsberg and a crust, such as might be observed as latex paint dries. Steady state Thiele,35 an x is shown in the Table. Added to the Table is P(m/A) concentration profiles for a bilayer are shown in Figure 1C. The layer nearest the electrode is of thickness 1 and diffusion coefficient D1.A ratio of concentration between the first and second layer characterized − + −3 κ =  /   Table I. Permeability (10 cm/s) of TMPD through Four yields 1 c( 1 ) c( 1 ). Similarly, for the outer layer, 2, D2,and κ =  + − /  + + Emulsions. P of TMPD is reported with mass of the emulsion 2 c( 1 2 ) c( 1 2 ) apply. per area m/A (mg/cm2).34 Specification by flux balance.—Equations for a bilayer develop Mass/Area Permeability Pm/A (Average / / ± analogously to a single layer. Equal flux through each plane parallel m A emulsion PPmA st dev) to the electrode is required. For three interfaces: at the electrode x = 0, (mg/cm2)(10−3 cm/s) (10−6 g/cm s) (10−6 g/cm s) between the first and second layer x = 1 and between the outer layer Poly 12 1.10 5.32 5.85 and the solution x = 2: ±    (65:0:35) 2.19 1.32 2.96 (4 2)    3.33 x iss dc(x)  dc(x)  dc(x)  = D1  = D2  = Ds  4.39 x nFA dx = dx =+ dx = ++ x 0 x 1 x 1 2 Poly 14 1.10 2.28 2.51 [10] (35:0:65) 2.19 1.07 2.34 . ± . Discretized, linear approximations for the gradients and introduction 4.39 0.703 3.09 (2 6 0 4) of κ and κ yield 8.78 x 1 2 Poly X1 0.53 3.43 1.82 − −  + − − + iss c( 1 ) c(0) c( 1 2 ) c( 1 ) (0:35:65) 1.10 2.21 2.43 = D1 = D2 2.19 1.51 3.31 nFA 1 2 3.33 0.95 3.16  +  + δ −  + + . ± . c ( 1 2 0) c( 1 2 ) 4.39 0.89 3.91 (2 9 0 8) = Ds [11] 6.58 x δ0 Poly X6 1.10 3.92 4.31 (0:0:100) 2.19 1.30 2.85 c( + −) − c(−)/κ c∗ − c( + −)/κ 4.39 0.833 3.66 = D 1 2 1 1 = D 1 2 2 2  s δ [12] 6.58 0.547 3.60 (3.6 ± 0.6) 2 0

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For a step to the mass transport limit (c(0) = 0), algebra yields c(1 + Multilayers, all layers uniform.—Multiple layers built sequen- − − −  ) = [ Ds c∗ + D2 1 c( )]/[ D2 + 1 Ds ]andc( ) = Ds c∗/[ D1 + tially on an electrode establish different mass transport resistances. At 2 δ0 2 κ1 1 2 κ2 δ0 1 δ0 1  D1 2 Ds + 1 Ds steady state, flux through each layer and the solution must be equal.  κ δ κ κ δ ]. On substitution, 1 D2 2 0 1 2 0 Algebra can be developed as for the single layer and bilayer, but the − Ds ∗ same result is found by viewing the system as series mass transport i c( ) D δ c ss = 1 = 1 0 D1  [13] resistances. For n layers on the electrode, allow each j layer to be char- nFA   D1 + D1 2 Ds + 1 Ds 1 1   D κ δ κ κ δ acterized by  j , D j ,andκ j , analogously to above. Series resistances, 1 1 2 2 0 √ 1 2 0 expressed as currents, i j , yield As for a single film, the Levich plot (iss versus ω) is nonlinear. Data analysis is simplified by a double reciprocal (Koutecky Levich) plot. 1 1 n 1 The reciprocal of Equation 13 yields = + [16] iss isoln i j nFAc∗ δ   j=1 = 0 + 1 + 2 [14] iss Ds κ1κ2 D1 κ2 D2 Let the initial concentration in layer j of the film be defined relative n 1 1 1 1 to that in the solution as the partition coefficient κ , = κ = = + + [15] j soln k = iss isoln i1 i2 k j c j (t=0) The currents i1 and i2 represent the currents (proportional to flux) for c∗ . Then, layers 1 (inner) and 2. ∗  nFAc = j κ [17] Specification by resistance.—The three layers of solution, outer i j j,soln D j film, and inner film, each slow probe transport from the solution and to the electrode surface. Each layer presents a resistance to mass ∗ δ n  δ n transport (R , R ,andR ) and the resistances are in series. By nFAc 0 j 0 1 soln 2 1 = + = + [18] description analogous to a single uniform layer, Equation 15 is found. κ iss Ds = j,soln D j Ds = Pj For series resistance, the measured current can be limited by the j 1 j 1 highest resistance. For a sufficiently high film resistance, the measured ∗ −1 ω−1/2 A Koutecky Levich plot of nFAc iss versus has slope of i can be ω-independent. . 2/3ν−1/6 −1 ss [0 62Ds ] characteristic of solution transport and an inter- cept proportional to the sum of the mass transport resistance provided −1 ω−1/2 Data analysis.—Aplotofiss versus will yield a linear by all the layers in the film (Equation 16). The inverse of the last term plot where the slope is characteristic of diffusion in solution and the n −1 −1   of Equation 18,[ j=1 Pj ] is an effective, net permeability for the − −   1 + 1 = ∗ −1 1 + 2 κ multilayer. Absent other information, rotating disk data alone does intercept is i1 i2 (nFAc ) κ κ D κ D .Note values 1 2 1 2 2 not allow the parameters for each layer of the film to separated. also describe probe concentrations prior to electrolysis, κ2 = c(1 + − /  + + = = / ∗ κ κ = − /  + + = 2 ) c( 1 ) c2(t 0) c and 1 2 c( 1 ) c( 1 2 ) ∗ Continuously graded layers.—Equation 16 is appropriate for a c1(t = 0)/c . As for the single layer, uniform film, the slopes of the Koutecky Levich plot should be the same for the unmodified and series of discrete layers. For a continuously graded film, the integral modified electrodes. version of Equation 16 is expressed as

The intercept does not allow the permeabilities for the inner (P = ∗  1 nFAc δ0 1 κ D / ) and outer films (P = κ D / ) to be separated. In principle, = + dλ [19] 1 1 1 2 2 2 2 i D κ λ D λ coupling the steady state measurement with a transient measurement ss s 0 ( ) ( ) would allow terms to be separated if thicknesses are known. However,  is the total film thickness, λ is the variable of integration, and for a bilayer, equations for chronoamperometry and simulations for κ (x) and D (x) are the spatially dependent partition and diffusion cyclic voltammetry are not yet reported. Steady state measurements coefficients that vary normal to the electrode. κ (x) parameterizes the on the bilayer will yield at least a net permeability for the bilayer of initial concentration at point x in the film relative to the bulk solution, −1 + −1   ∗ P1 P2 ,andvariationof 1 and 2 may allow separation of some c . An effective, net permeability for the film is found from the inverse parameter groups. It may be possible to characterize the bilayer by of the intercept. This assumes a mechanically rigid film that does not evaluating one layer at a time. In the limit of a very thin 2 and low deform under rotation. κD2, a bilayer may be characterized as a single layer with measurable Rinterface. Data analysis.—For any multilayer system, the inverse intercept of the Koutecky Levich plot yields an effective permeability through Example of bilayers.—An excellent paper on multilayers is pre- the multilayer. The plot is indistinguishable from that found by Gough sented by Rehak and Hall38 who evaluated bilayers formed of various and Leypoldt12–15 for a uniform single layer. To discriminate multilay- polymers on a gold electrode and the polymer overlaid with a lipid ers from single layers, additional information such as film formation, 3− layer; 1 mM Fe(CN)6 was the probe. On initial formation of 3 nm imaging studies, and transient voltammetry must be employed to un- lipid layers over 100 nm sol gel films, Koutecky Levich plots exhibited tangle the various system parameters of extraction, diffusion, and two different slope domains, where slopes at low ω were the same as thicknesses. the unmodified electrode but as ω increases, slopes increased. After 18 hours, the lipid layer equilibrated to a uniform film and the slopes for Example of multilayers.—As an example of multilayers, Lu and all ω were the same as for the unmodified electrode. The permeability 39 − Hu carefully evaluated permeability of several different multilayers of sol gel cured for 10 minute was 4 ×10 10 cm/s, which was twice the that contain myoglobin (Mb) where the multilayers were assembled value for those cured for 15 minutes. Similar lag times to establish lin- electrostatically on pyrolytic graphite electrodes stepwise to set film earity of the Koutecky Levich plots were observed for lipid monolay- thickness. All films were established over an initial cationic layer of ers over 1-hexadecylamine (HDA) modified 11-mercaptoundecanoic poly(diallyldimethylammonium), PDDA. The film thickness was pa- acid (MUA) self assembled monolayers. A smooth surface is needed rameterized by n, the number of bilayers of Mb + anionic coating if the slopes for the unmodified and modified electrodes are to match. layer. Electrodes were modified with either TiO sol gel+Mb [SG- ω 2 The difference in slope at early times and high is consistent with TiO2/Mb] or TiO2 nanoparticles +Mb [NP-TiO2/Mb] or polystyrene surface roughness of the fresh lipid layer before the lipids healed to +Mb [PSS/Mb]. From quartz crystal microbalance (QCM) studies, a lower energy, flat film. For a successful measurement, a film is flat the thickness of a single bilayer  ranked as [NP-TiO /Mb] (8.4 δ bilayer 2 when 0 is large compared to any interfacial structural features. nm) > [PSS/Mb] (5.0 nm) > [SG-TiO2/Mb] (3.1 nm). Myoglobin

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Table II. Permeability of Fc(COOH) through n Bilayers Specification by resistance.—In sections Specification by resis- of Myoglobin (Mb) and TiO2 Nanoparticles [NP-TiO2/Mb], tance in single layer, uniform films, Specification by resistance in bi- 39 Polystyrene, [PSS/Mb] and Sol Gel [SG-TiO2/Mb]. layers, two uniform layers, and Multilayers, all layers uniform, steady state current measured through uniform films is modeled as mass [NP-TiO2/Mb] [PSS/Mb] [SG-TiO2/Mb] transport resistance. Probe motion through the homogeneous solution

PNP−TIO2/Mb PPSS/Mb PSG−TIO2/Mb is resisted by probe interactions with the electrolyte. Addition of a n (10−3 cm/s) (10−3 cm/s) (10−3 cm/s) film leads to additional resistances For uniform, incompressible films, mass transport is captured as serial resistances because net transport 1 9.4 6.5 20.3 is only normal to the electrode. Equation 16 for multilayer films also 2 6.3 5.05 14.0 captures single and bilayer films, i −1 = i −1 + n i −1. Current is 3 5.0 4.2 10.8 ss soln j=1 j related to flux, J as i/nF =−AJ, where any current, i , is propor- 4 3.6 3.1 7.2 k 5 3.1 2.6 5.7 tional to the flux Jk across the cross sectional area Ak where Jk is 6 3.1 2.3 5.3 established. For unmodified electrodes and uniform films, the cross 7 3.0 2.5 5.0 sectional area is the electrode area, A. Thus, for the series mass trans- 85.1port resistances for uniform films, substitution into Equation 16 yields 95.0 bilayer (nm) 8.4 5.0 3.1 n −1 −1 −1 nof fset 0.65 2.2 0.8 (Jtotal A) = (Jsoln A) + (J j A) [20] −1 P vs [n − nof fset] j=1 −3 slope (10 cm/s) 36.0 ± 1.8 21.5 ± 1.3 14.1 ± 0.6 −3 ± − . ± . ± n intercept (10 cm/s) 0.51 0.48 (0 16 0 26) 0.91 0.20 −1 −1 −1 2 Jtotal = Jsoln + J j [21] R  0.983 0.981 0.990 =  −8 2 j 1 κD f  (10 cm /s) 1.18 1.07 1.12 ef f Consider a nonuniform, heterogenous film composed of two materials a and b where each has a cross section, Aa and Ab, in the plane of the electrode. Components a and b can each be distributed in numerous is electroactive but the reduction potentials for the redox probes fer- small domains across total area A but the sum of the total cross rocene carboxylic acid is sufficiently positive of myoglobin that the section of each is such that A = Aa + Ab. An example is an array films were electrochemically inert during permeation studies. Values of colinear, cylindrical pores that traverse a support. A probe moves of P for the three types of films captured from Figure 3B in Reference through the material where components a and b provide two parallel 39 are shown in Table II, where 0.5 mM ferrocene carboxylic acid paths through the layer. Each component provides resistance to mass Fc(COOH) at pH 7 is the probe. −1 transport Ra and Rb. Because the resistances are in parallel, mass Permeability scales with  where  is approximated by n.From −1 −1 −1 transport resistance through the film is R , = R + R .For the data at the top of Table II for each class of multilayers, P values film total a b parallel currents in the film, i , = i + i . On substitution of − −1 film total a b were linearized to [n nof fset] where nof fset was optimized to max- =− 2 i j nFAj J j , imize correlation coefficient, R . The introduction of nof fset improved the regression statistics and included the thickness of the underlying AJfilm,total = Aa Ja + Ab Jb [22] PDDA layer. The value of nof fset and slope and intercept of P versus − −1 [n nof fset] are shown in the Table. In each case, the intercepts ap- Aa Ab J , = J + J [23] proach zero consistent with no significant interfacial resistance. The film total A a A b slopes yield κD f n/,where/n = bilayer . An effective κD f |ef f for the multilayers is found as the product of slope and  Listed in bilayer = fa Ja + fb Jb [24] Table II,valuesofκD f |ef f , are invariant with types of coating, and − average to (1.12 ± 0.05) × 10 8 cm2/s. The Mb layers may be the fa and fb are the fractional cross sectional areas of each component, f = A /A. For multiple components, total flux in the film is set by main limitation to flux. Because κD f |ef f is invariant, κ is likely the k k same in all three films, so that Fc(COOH) has either little electrostatic the flux of each component Jk and the fractional cross sectional area = interaction with NP-TiO , PSS, and SG-TiO or interacts only with fk through which Jk is established, where k fk 1. 2 2  Mb. If the concentration of Fc(COOH) is the same in the multilayers −8 2 J film,total = fk Jk [25] as it is in solution, then κ ≈ 1and1× 10 cm /s approximates D f . k If there is a non-permeable component on the electrode, its fractional area is included but its flux is zero. For a multicomponent, heteroge- Models for Heterogeneous Films and Composites neous film on an electrode, mass transport resistance for the solution For uniform films, the layers are isotropic and only gradients nor- and entire film remains serial, Rtotal = Rsoln + R film,total.Thecor- −1 = −1 + −1 mal to the plane of the electrode affect transport. For heterogeneous responding currents are itotal isoln i film,total. For each itotal, isoln, films, the layers are not isotropic and the structure varies in the plane of and i film,total, the total cross section is the geometric area of the the electrode. Flux of the redox probe is impacted by these structures. electrode, A. Heterogeneous films and composites are formed of multiple com- −1 −1 −1 J = J + J , [26] ponents. Spatially inhomogeneous layers include films with pores and total soln film total = composites of two or more components. Composites include adsorp- On substitution of J film,total k fk Jk for a heterogeneous film, tion of one component into a micro- or nano-structured matrix. Het- Equation 26 yields erogeneous films and composites are differentiated from uniform films −1 −1 1 by a microstructural characteristic dimension parallel to the plane of Jtotal = Jsoln + [27] the electrode, where the characteristic dimension is either compara- k fk Jk  δ ble to or smaller than either or 0. As the characteristic dimension Because transport in solution is not perturbed by the modifying approaches molecular to nanoscale and is substantially smaller than  layer and there are no reaction kinetics, the slope of the Koutecky δ and 0, films will be characterized as uniform by rotating disk voltam- Levich plot remains the same as for the unmodified electrode. The metry. To model steady state voltammetry for heterogeneous films and −1 intercept characterizes [ k fk Jk ] . If data are collected where fk composites, mass transport resistance applies. is varied in a known manner, Jk through each component can be

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than solution,42,43 i κD − J = film = f ε = Pε [28] film nFA  If a film contains pores of uniform radius r separated by 2R center to center, but the pores are not straight, the above equations apply when modified for the tortuosity, τ = pore length/ as i κD ε ε film = f = P [29] nFA  τ τ where τ is the path length for diffusion through the film. If permeabilities are determined as a function of the composition and dimensions of the microstructural components, additional infor- mation about mass transport associated with the microstructures can be extracted.43,44 When δ0 is comparable to the scale of the microstructural charac- δ Figure 2. The boundary layer of thickness 0 is shown as dash at an un- teristic lengths, these analyses will likely fail. Under these conditions, modified electrode and at electrodes modified with layers containing pinholes. scanning electrochemical microscopy (SECM), first introduced by Measurements by rotating disk voltammetry map the transit of a probe from the Bard and coworkers,45 may be appropriate as SECM exploits inter- outer edge of the boundary layer to the electrode surface. For the unmodified rogation of the boundary layers established about micro- to nano- electrode, the path is normal to the electrode, as shown by the vector. When the electrode is modified with a layer containing pinholes, the probe moves from structures. Quantitative aspects of SECM measurements have been 46 outside the boundary layer to the electrode surface but the path of the probe described by Cornut and Lefrou. For rotating disk measurement, δ may be blocked by material on the electrode. If δ0 is large compared to the if 0 can be increased to several fold the microstructural character- features in the film, then the deviation in the path of the probe to the electrode istic lengths, then effective porosity of the layer can be measured. is negligible. For the case of small pinholes well separated (middle panel), the These methods may be useful in characterizing films with character- δ radial diffusion layer is small compared to 0 and rotating disk data yield an istic dimensions that approach molecular such as well formed films of effective permeability. In the case of large pinholes, the radial diffusion profiles zeolites and covalent (COF) and metal organic (MOF) frameworks. merge to a linear profile a short distance from the layer as shown in the right Transport around and through the structures must be differentiated δ , the path of the panel. Where the merged diffusion layer is thin compared to 0 as has been undertaken for clay films on electrodes.6 At the other probe is lengthened neglibly as the probe reaches the electrode and effective δ permeability is again measured. extreme, 0 may not be large compared to structural features and measurement constraints are not satisfied; for example, non-linear Koutecky Levich plots are sometimes observed for large pinholes and rough surfaces. determined by evaluating total flux as a function of the characteristic dimensions that set fk . Limitations of the Measurements

Pores and microstructured composites.—Rotating disk voltam- When compared to an unmodified electrode, the presence of a film metry measures effective permeability through layers that contain or membrane on the electrode surface alters the time to achieve steady pores and other regular micro- and nano-structural components. Effec- state and the allowed rotation rates. It also introduces some statistical limitations on the data analysis. tive permeability is measured once the boundary layer, δ0, is approxi- mately five times the largest characteristic dimension of the structures in the plane of the electrode. Like concentration profiles at microelec- Time to steady state.—The presence of a film on the electrode in- trodes, steady state, boundary layers that perturb concentration from troduces a lag time to establish steady state because physical diffusion bulk drop across a distance less than five electrode radii. coefficients within films are often many orders of magnitude lower For straight, colinear pores of radius r at a density of N pores per than diffusion coefficients in solution. The time needed to achieve cm2 and pores separated center to center by 2R at steady state, a radial steady state is again set by approximately five times the diffusion  2/  diffusion profile of length ∼5r is established about each pore at the length, tss 5 D f .As increases, the time to steady state in- film solution interface, if the pores are well separated. See the middle creases dramatically. For example, in Nafion films of a micrometer panel of Figure 2. If pores are not well separated, the radial diffusion thickness, the time to achieve steady state is on the order of a few 47 profiles merge into a linear diffusion profile, as shown in the right panel hours. Experimentally, the time to achieve steady state is found by of Figure 2. As long as either 5r or the linear diffusion profile formed as applying a fixed potential and monitoring the current until it is in- the radial profiles merge have characteristic lengths well less than δ , variant. Measurements are most efficiently made by maintaining the 0 ω rotating disk will yield an effective permeability P. The diffusion path electrode at this potential when changing . Scanning the potential ω for probes diffusing across δ will be negligibly lengthened compared for each complicates determining whether the system is at steady 0  to the normal by the film structures on the electrode. A rough estimate state. Once films are characterized, comparison of the values and D to the time to steady state serves as a crude check of measurement of a sufficiently thick boundary layer is that δ0 exceeds five times the f dimension of the largest structural feature, so that the probe diffusion quality. path differs negligibly from the normal and the effective permeability is measured. For colinear, cylindrical pores, 2R > r,so2R sets one Rotation rates.—At unmodified electrodes, the upper limit on ω diffusion length. As long as δ0  10R, probes diffusing from δ0+ to  is set by the onset of turbulence. The lower limit is set when the will establish diffusion paths near normal to the electrode, so effective hydrodynamic boundary layer thickness exceeds the radius of the 20 ω > ν/ 2 permeability is measured. For cases where 2R and r are comparable to disk electrode as 10 rdisk. For modified electrodes, the up- 35,40,41 ω δ0, Landsberg and coworkers derived the equations for pinholes. per limit on is similarly set. Practical experience has shown that For straight pores through an inert substrate, the porosity ε is the the lower limit of ω is lower than for an unmodified electrode. For open cross section of the pores normalized by A, ε = πNr2/A; ε is example, Nafion modified electrodes can be studied at a few rota- thesameas fopen. Expressed in terms of surface coverage, θ, ε = 1−θ. tions per minute. Slower rotation rates may be achieved when the If the pores are solution filled, a linear concentration gradient within mass transport resistance in the film is high compared to that in the the pores yields −J film = εDs /. For pores filled with material other solution.

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Surface roughness.—In vetting measurements on modified elec- trodes, good measurements require δ0 large compared to characteristic lengths for structures in the layer, as in Figure 2. When the coating on the electrode is not flat and has features of lengths comparable to δ0, then the rotating disk measurement may not be useful. The boundary layer is established as solvent molecules are carried by the flat rotat- ing surface. If the layer is not flat, then solvent will be trapped in the valleys at the film solvent interface and the boundary layer will be superimposed above the entrapped solvent. Because the zone of en- trapped solvent is poorly defined, the Koutecky Levich analysis does not serve to characterize the film. If linear, the slopes may deviate from those in solution. Large pinholes where δ0 is not sufficiently large cause non-linear Koutecky Levich plots especially at higher ω. Examples are noted in References 34, 38.

Statistical limitations.—Where the Koutecky Levich equation Figure 3. Koutecky Levich plots are represented for the unmodified electrode −1 −1 −1 characterizes current, i = i + i , i film embeds the permeabil- of slope m KL,un mod and intercept zero (gray open circles, solid gray regression ss soln film = . × −3 ity, whether that is the permeability or the effective net permeability line); a film of P 3 2 10 cm/s with slope m KL,un mod and intercept −1 for the layer. When mass transport resistances for the film and solu- P ; and a series of examples where disruption of the boundary layer by −1 −1 thermal motion and density gradients leads to a decrease in the experimental tion are comparable, such that both isoln and i film are significant, the δ δ ω boundary layer thickness 0 as compared to the intended 0 associated with the Koutecky Levich plot yields an -dependent slope that is the same ω δ /δ experimental .As 0 0 (the labels for the lines) decreases, slope decreases as that for the unmodified electrode and an ω-independent intercept δ /δ δ /δ as ( 0 0)m KL,un mod. (For illustration, the simplistic model is that 0 0 is characteristic of film flux. In some cases, flux in the film is so low that ω − − the same for all .) The disruption in the boundary decreases the slope. The 1  1 −1 −1 i film isoln and within the experimental uncertainties, iss is limited common intercept P is an artifact of the simple model; P is not anticipated solely by film flux. The Koutecky Levich plot will be ω-independent well measured from experimental data. The plot is based on water at room × −6 2 and iss = i film. It is sometimes observed that the Koutecky Levich plot temperature, a common P,andDs of 8 10 cm /s. is linear, but the slope falls between that for the unmodified electrode and zero. This may occur where film flux is substantially less than ω1/2 solution flux but not yet small enough that the measured current is ω- solved Koutecky Levich plots, a Levich plot of iss versus may independent. For small currents, measurement precision can obscure be more appropriate, where i film is extrapolated to the limiting iss at infinite ω. the ω dependence. When i film isoln but iss is not yet ω-independent, In cases where the Koutecky Levich plot is ω-independent and several factors can adversely impact measurement quality. When i film slope zero, J J and the measured i = i . Thus, if data is low, long tss is common; if insufficient time is allowed to achieve film soln ss film steady state, then the steady state model is not applicable because are ω-independent, then the film is still parameterized by rotating disk. Time to steady state may be long. J film = Jsoln and iss will be overestimate at given ω. If steady state is achieved, the model constrains J film = Jsoln,wheresmallJ film re- quires extremely long δ0 be established at very low ω. If for example, Non-linear Koutecky Levich plots.—Koutecky Levich plots may density gradients and thermal motion disrupt the boundary layer, the be nonlinear, even when rotation rates are appropriate. Non-linear δ δ measurement boundary layer 0 is thinner than the intended 0 based plots arise when the model for series mass transport resistances fails. ω on applied and the measured iss will be higher than the iss that Non-linear effects include kinetic effects for reactions within the films, δ δ 48 ´ 24,49–54 correlates with 0.Wheniss is analyzed with 0, both the Levich and such as those described by Albery and Saveant and coworkers. Koutecky Levich plots can be misinterpreted. Effects other than transport in the solution that are rotation rate de- A simplistic representation of boundary layer disruption on pendent can yield plots that are either non-linear or have a Koutecky Koutecky Levich plots is shown in Figure 3, where data are cal- Levich slope other than that observed for the unmodified electrode. A culated for common experimental conditions in water. The plot film that is fluid and distorts on rotation because of shear would not ∗ 55 of nFAc Ds /iss versus δ0 for the unmodified electrode is shown be well described by the Koutecky Levich analyses. Pinholes and by the open gray circles and the solid gray regression line, with surface roughness can yield non-linear Koutecky Levich plots. slope m KL,un mod and intercept zero. When a uniform film with P = 3.2 × 10−3 cm2/s is added, the plot shown with solid gray Conclusions circles and a thin solid line results with slope m KL,un mod and intercept P−1, as anticipated for films evaluated well within model constraints. The well-controlled mass transport established at steady state ro- Koutecky Levich plots are shown for a simple model where the bound- tating disk electrodes yields excellent conditions for determination of ω δ /δ δ /δ = . ary layers at all are decreased with fixed 0 0.For 0 0 0 99, flux through films and membranes. Because of the highly effective the long dashed line (slope 0.99m KL,un mod) is nearly coincident with stirring, measurements are more reliable and less hazardous than ra- δ /δ 12–15 the film with slope m KL,un mod.As 0 0 decreases, plots are linear of diotracer methods. Gough and Leypoldt first presented the equa- δ /δ δ /δ slope ( 0 0)m KL,un mod. For the smallest 0 0, the slope approaches tion characteristic of steady state transport of a redox probe from zero. Here, all lines converge to intercept P−1, but this is an artifact solution through a homogeneous film to the electrode where the probe of the simple model as posed. In experiments, the extent of boundary undergoes diffusion limited electrolysis. Measurements are made on layer disruption will likely increase with thickness (ω−1/2). A range electrochemically inert films that are either not electroactive or eval- of slopes between m KL,un mod and zero maps behavior between those uated in a voltage range where the film is not electrolyzed. Here, anticipated based on the model and a film flux so low that the ω equations are presented to characterize flux through bi- and multi- dependence is lost to the precision of the measurements. layers that include discretely and continuously graded films. When Koutecky Levich plots that have slopes less than the solution slope only rotating disk voltammetric data are used and other information but greater than zero do not allow ready measurement of film per- about the layers is absent (e.g., thicknesses), only an effective perme- meability. Experimentally, changing the range of ω or  by an order ability through the film is found. Discussion of the conditions where of magnitude may shift the behavior into either ω independence or rotating disk voltammetry can be used to characterize microstructured ω dependence of the unmodified electrode. When lower ω values films and films with pores is presented. experimentally accessible for modified electrodes are used, verify Electrodes modified with multilayers can be characterized by ro- −1 −1/2 the Koutecky Levich slope is the same as m KL,un mod. For poorly re- tating disk voltammetry and Koutecky Levich (iss (ω) versus ω )

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plots. Intercept of linear plots yields an effective permeability through 13. D. Gough and J. Leypoldt, A.I.Ch.E.J., 26, 1013 (1980). the layers but, in the absence of other information or transient voltam- 14. D. Gough and J. Leypoldt, Anal. Chem., 52, 1126 (1980). metric results, neither the permeabilities that characterize the indi- 15. D. A. Gough and J. K. Leypoldt, J. Electrochem. Soc., 127, 1278 (1980). 16. C. M. Carlin, L. J. Kepley, and A. J. Bard, J. Electrochem. Soc., 132, 353 (1985). vidual layers nor the three characteristics of partition parameter, dif- 17. S. Bruckenstein and A. R. Hillman, Electrochemical quartz crystal microbalance fusion coefficient, and film thickness can be determined separately. studies of electroactive surface films, in Handbook of Surface Imaging and Visual- Heterogeneous films can also be characterized by Koutecky Levich ization, A. T. Hubbard, Editor, CRC Press, Boca Raton, FL, 101–13 (1995). analyses provided the characteristic dimensions of the film structures 18. D. A. Buttry and M. D. Ward, Chem. Rev., 92, 1355 (1992). δ 19. J. Leddy and A. J. Bard, J. Electroanal. Chem., 153, 223 (1983). are well below the thickness of the boundary layer, 0. Modified 20. A. Bard and L. Faulkner, Electrochemical Methods, John Wiley & Sons, Inc., New electrodes require longer times to establish steady state but measure- York, (2001) Second ed. ments can be made at lower rotation rates. Two conditions allow ready 21. A. Amarasinghe, T. Chen, H. Paul, P. Moberg, F. Tinoco, L. Zook, and J. Leddy, determination of permeability, where either the mass transport resis- Anal. Chim. ACTA, 307, 227 (1995). 22. P. Peerce and A. Bard, J. Electroanal. Chem., 112, 97 (1980). tance in the film and solution are comparable or where the mass trans- 23. K. L. Knoche, C. Hettige, P. D. Moberg, S. Amarasinghe, and J. Leddy, J. Elec- port resistance in the film is dominant and the steady state current is trochem. Soc., 16 H285 (2013). ω-independent. Intermediate cases are not well interpreted. Change 24. J. Leddy, A. Bard, J. Maloy, and J. Saveant, J. Electroanal. Chem., 187, 205 (1985). of  or ω may shift results into either i film comparable to isoln or 25. M. Krishnan, X. Zhang, and A. J. Bard, J. Am. Chem. Soc, 106, 7371 (1984). i film isoln. Limitations to the method include nonlinear Koutecky 26. D. Ege, P. K. Ghosh, J. R. White, J.-F. Equey, and A. J. Bard, J. Am. Chem. Soc, 107, Levich plots that can arise from several factors that include reactions 5644 (1985). within the films, films sufficiently fluid that the film distorts on ro- 27. P. Henning, H. S. White, and A. J. Bard, J. Am. Chem. Soc, 104, 5862 (1982). tation, and large pinholes. Consideration of serial and parallel mass 28. R. A. Bull, F.-R. Fan, and A. J. Bard, J. Electrochem. Soc., 131, 687 (1984). 29. R. Bull, F.-R. Fan, and A. Bard, J. Electrochem. Soc., 130, 1636 (1983). transport resistances in the layer generates diagnostics appropriate to 30. J. Leddy and A. Bard, J. Electroanal. Chem., 153, 223 (1983). the film microstructure. 31. J. Leddy and A. Bard, J. Electroanal. Chem., 189, 203 (1985). 32. X. Zhang, J. Leddy, and A. Bard, J. Am. Chem. Soc, 107, 3719 (1985). 33. T. Okajima, T. Ohsaka, O. Hatozaki, and N. Oyama, Electrochimica Acta, 37, 1865 (1992). Acknowledgments 34. J. J. Gooding, C. E. Hall, and E. A. H. Hall, Analytica Chimica Acta, 349, 131 (1997). 35. R. Landsberg and R. Thiele, Electrochimica Acta, 11, 1243 (1966). The financial assistance of the Army Research Office (ARO 43808- 36. S. Cosnier, S. Szuneritus, R. S. Marks, A. Novoa, L. Pueh, E. Peres, and I. Rico-Lattes, CH / DAAD19-02-1-0443) and the National Science Foundation (NSF Talanta, 55, 889 (2001). 1309366) are most gratefully acknowledged. A Fellowship in Resi- 37. T. Ikeda, R. Schmehl, P. Denisevich, K. Willman, and R. W. Murray, J. Am. Chem. dence at the Obermann Center for Advanced Studies facilitated com- Soc, 104, 2683 (1982). 38. M. Rehak and E. A. H. Hall, Analyst, 129, 1014 (2004). pletion of this paper. Dedicated to Allen J. Bard on the occassion of 39. H. Lu and H. Naifei, Electroanalysis, 15, 1511 (2006). awarding the first Allen J. Bard Award of the Electrochemical Society, 40. F. Scheller, S. Muller, R. Landsberg, and H.-J. Spitzer, J. Electroanal. Chem., 19, 2015. 187 (1968). 41. F. Scheller, R. Landsberg, and S. Muller, J. Electroanal. Chem., 20, 375 (1969). 42. J. Leddy and N. E. Vanderborgh, J. Electroanal. Chem., 235, 299 (1987). 43. Y. Fang and J. Leddy, J. Phys. Chem., 99, 6064 (1995). References 44. L. A. Zook and J. Leddy, J. Phys. Chem. B, 102, 10013 (1998). 45. A.J.Bard,F.R.F.Fan,J.Kwak,andO.Lev,Anal. Chem., 61, 132 (1989). 1. R. Murray, Acc. Chem. Res, 13, 135 (1980). 46. C. Lefrou and R. Cornut, ChemPhysChem, 11, 547 (2010). 2. R. Murray, Molecular Design of Electrode Surfaces Vol. 22 of Techniques of Chem- 47. L. A. Zook, Morphological Modification of Nafion For Improved Electrochemical istry, Interscience, New York, (1992). Flux, Ph.D. thesis, University of Iowa, 1996 (1996). 3. M. Kaneko and D. Worhle, Adv. Polym. Sci, 84, 141 (1988). 48. W. Albery and A. Hillman, Annu. Rep. Prog. Chem., Sect. C, 78, 377 (1981). 4. H. Abruna, Coord. Chem. Rev., 86, 135 (1986). 49. C. P. Andrieux and J. M. Saveant, Catalysis at Redox Polymer Coated Electrodes, in 5. K. Snell and A. Keenan, Chem. Soc. Rev., 8, 259 (1979). Molecular Design of Electrode Surfaces, R. W. Murray, Editor, Vol. XXII John Wiley 6. A. Fitch, Clays and Clay Minerals, 38, 391 (1990). & Sons, Inc., New York, 207–270 (1992). 7. R. Durst, A. Baumner, R. Murray, R. Buck, and C. Andrieux, Pure and Appl. Chem, 50. C. Andrieux, J. Dumas-Bouchiat, and J. M. Saveant, J. Electroanal. Chem., 131,1 69, 1317 (1997). (1982). 8. M. Opallo and A. A. Lesniewski, J. Electroanal. Chem., 656, 2 (2011). 51. C. Andrieux and J. Saveant, J. Electroanal. Chem., 142, 1 (1982). 9. P. Cervini and E. Cavalheiro, Analytical Letters, 45, 297 (2012). 52. C. Andrieux, J. Dumas-Bouchiat, and J. Saveant, J. Electroanal. Chem., 169, 9 (1984). 10. A. Walcarius, Analytica Chimica Acta, 384, 1 (1999). 53. C. Andrieux and J. Saveant, J. Electroanal. Chem., 17165 (1984). 11. S. Bruckenstein and B. Miller, Acc. Chem. Res, 10, 54 (1977). 54. C. P. Andrieux and J. M. Saveant, J. Electroanal. Chem., 134, 163 (1982). 12. D. Gough and J. Leypoldt, Anal. Chem., 51, 439 (1979). 55. A. Oron, S. H. Davis, and S. G. Bankoff, Reviews of Modern Physics, 69, 931 (1997).

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