Methods—Design Guidelines for Tubular Flow-through Electrodes for Use in Electroanalytical Studies of Redox Reaction Kinetics

A simple graphite tubular flow-through electrode (FTE) was prepared by drilling a 1-mm-diameter hole (d = 1 mm) through a graphite rod (6.15 mm diameter, 99.9995% metals basis, Alfa Aesar). The rod was cut to 4.5 mm in length (L = 4.5 mm). A conductive silver wire was glued with conductive epoxy to the outer surface of the graphite electrode in order to make proper electrical contact. Non-conductive epoxy (thickness ≈ 0.02 cm) was applied on all outer surfaces of the electrode. Thus, only the interior surface of the FTE was exposed to the electrolyte. The electrode was glued with non-conductive epoxy to the end of a glass tube (ID ≈ 5.6 mm), which was used in a three-electrode cell described in Fig. 1. Before electrochemical experiments, the FTE was cured in the oven at 100 °C for > 12 h and then soaked in isopropyl alcohol (IPA) for 10 min. The IPA was allowed to evaporate in the oven for 5 min.

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The RDE was manufactured from a graphite rod. The graphite rod was shaped with 600-grit silicon carbide paper, polished with 5, 0.3, and 0.05-μm alumina polishing solutions on a micro-fiber polishing cloth, and washed with DI water. The diameter of the RDE was 0.5 cm.

As shown in Fig. 1a, the graphite tubular FTE was tested in a custom two-compartment cell with a three-electrode configuration consisting of the FTE as the working electrode, a coiled platinum (Pt) counter electrode, and a saturated Ag/AgCl reference electrode (Pine Research Instrumentation, Inc.). The two compartments were separated by a Nafion membrane to avoid re-oxidation and re-reduction of species electrochemically generated at the Pt counter electrode. The electrolytes consisted of varying equimolar concentrations (1, 0.5, 0.25, and 0.1 mM) of potassium hexacyanoferrate (II) trihydrate (K₄[Fe(CN)₆]·3H₂O, Sigma Aldrich) and potassium hexacyanoferrate (III) (K₃Fe(CN)₆, >99.0% purity, Sigma Aldrich) supported by 1 M KCl. The equilibrium potential (E_eq) of the working electrode was well-defined, measurable and stable in the presence of equimolar amounts of [Fe(CN)₆]^4– and [Fe(CN)₆]^3– in the electrolytes. The electrolyte flow direction was upward through the graphite tubular FTE at a flow rate (V₀) of 5 cm³ min⁻¹ controlled by a syringe pump (KDS200, KD Scientific). A SP-300 Potentiostat (Bio-Logic Science Instruments, France) was used for electrochemical measurements. Before steady-state polarization experiments were performed on the FTE and RDE, cyclic voltammetry (potential range of −0.4 to 0.2 V vs Ag/AgCl) at a scan rate of 0.02 V s⁻¹ was applied to the FTE for 5 cycles in 1 M KCl to reactivate the electrode surface. To obtain steady-state polarization curves, chronoamperometry experiments were performed as follows: the working electrode potential was changed stepwise in increments of 0.01 V and the transient current response at each potential was recorded until steady-state current was achieved. Applied potentials were within the electrochemical stability window of water. Electrochemical impedance spectroscopy (EIS) was performed to determine the ohmic resistance, which was needed for iR_Ω correction. Steady-state polarization measurements and EIS were also performed on the RDE at a rotation rate of 500 rpm.

We first perform scaling analysis for a simplified one-dimensional tubular FTE geometry shown in Fig. 1 to assess the current distribution uniformity over the FTE surface. For clarity, Table I lists definitions of all the symbols used in this work. For scaling analysis, we assume: (i) A high aspect ratio FTE with L ≫ d; (ii) Negligible potential and concentration gradients in the radial direction within the tubular FTE; and (iii) Current density on the interior surface of the FTE depends only on the axial position z.

For scaling analysis, the activation, ohmic and mass transport resistances associated with the tubular FTE were first calculated. The activation resistance (R_a) is the reaction resistance due to irreversible charge transfer at the FTE-electrolyte interface. The ohmic resistance (R_Ω) is the resistance of the electrolyte through which ionic current flows in the presence of an electric field. Finally, the mass transport resistance (R_mt) is the resistance associated with diffusional or convective transport of species to the reacting interface.

By definition, the activation resistance (R_a) is the gradient of the activation overpotential vs current curve calculated at the average value of the current. If we assume Tafel kinetics, ²¹ then

$$\begin{eqnarray}&&{R}_{a}={\left[\displaystyle \frac{\partial {\eta }_{a}}{\partial {I}_{s}}\right]}_{{i}_{s}={i}_{{\rm{avg}}}}=\left(\displaystyle \frac{1}{{A}_{s}}\right){\left[\displaystyle \frac{\partial {\eta }_{a}}{\partial {i}_{s}}\right]}_{{i}_{s}={i}_{{\rm{avg}}}}=\left(\displaystyle \frac{1}{\pi {\rm{dL}}}\right)\left(\displaystyle \frac{{\rm{RT}}}{{\alpha }_{a}F{i}_{{\rm{avg}}}}\right)\end{eqnarray} \tag{ 1 }$$

Similarly, the ohmic resistance (R_Ω) of the tubular FTE can be calculated as:

$$\begin{eqnarray}&&{R}_{{\rm{\Omega }}}=\displaystyle \frac{\partial {\eta }_{{\rm{\Omega }}}}{\partial {I}_{c}}=\left(\displaystyle \frac{1}{{A}_{c}}\right)\displaystyle \frac{\partial {\eta }_{{\rm{\Omega }}}}{\partial {i}_{c}}=\left(\displaystyle \frac{4}{\pi {d}^{2}}\right)\left(\displaystyle \frac{L}{\kappa }\right)\end{eqnarray} \tag{ 2 }$$

In a well-supported electrolyte, the resistance associated with transport of reacting species (R_mt) may be attributed to diffusional or convective transport. Since diffusional or convective transport may occur in tandem in a tubular FTE (resistances in parallel), we can write:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{R}_{{\rm{mt}}}}=\displaystyle \frac{1}{{R}_{{\rm{diff}}}}+\displaystyle \frac{1}{{R}_{{\rm{conv}}}}\end{eqnarray} \tag{ 3 }$$

The diffusional transport resistance (R_diff) is the gradient of the concentration overpotential vs current curve:

$$\begin{eqnarray}&&\begin{array}{l}{R}_{{\rm{diff}}}=\displaystyle \frac{\partial {\eta }_{c}}{\partial {I}_{c}}=\left(\displaystyle \frac{1}{{A}_{c}}\right)\displaystyle \frac{\partial {\eta }_{c}}{\partial {i}_{c}}=\left(\displaystyle \frac{4}{\pi {d}^{2}}\right)\displaystyle \frac{\partial }{\partial {i}_{c}}\\ \,\,\times \,\left[-\displaystyle \frac{{\rm{RT}}}{{\rm{nF}}}\,\mathrm{ln}\left({\rm{1}}-\displaystyle \frac{{i}_{c}L}{{\rm{nF}}{D}_{j}{C}^{b}}\right)\right]\end{array}\end{eqnarray} \tag{ 4 }$$

For small values of i_c, R_diff can be approximated as:

$$\begin{eqnarray}&&{R}_{{\rm{diff}}}\approx \left(\displaystyle \frac{4}{\pi {d}^{2}}\right)\left(\displaystyle \frac{{\rm{RTL}}}{{n}^{2}{F}^{2}{D}_{j}{C}^{b}}\right)\end{eqnarray} \tag{ 5 }$$

Similarly, the convective mass transport resistance (R_conv) can be determined:

$$\begin{eqnarray}&&{R}_{{\rm{conv}}}=\left(\displaystyle \frac{1}{{A}_{c}}\right)\displaystyle \frac{\partial {\eta }_{c}}{\partial {i}_{c}}=\left(\displaystyle \frac{4}{\pi {d}^{2}}\right)\displaystyle \frac{\partial }{\partial {i}_{c}}\left[-\displaystyle \frac{{\rm{RT}}}{{\rm{nF}}}\,\mathrm{ln}\left({\rm{1}}-\displaystyle \frac{{i}_{c}}{{\rm{nF}}{v}_{0}{C}^{b}}\right)\right]\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{R}_{{\rm{conv}}}\approx \left(\displaystyle \frac{4}{\pi {d}^{2}}\right)\left(\displaystyle \frac{{\rm{RT}}}{{n}^{2}{F}^{2}{v}_{0}{C}^{b}}\right)\end{eqnarray} \tag{ 7 }$$

We now define two dimensionless parameters. Here, we refer to these as Wa_I, i.e., the classical Wagner number widely used to analyze electrochemical systems, ^22–24 and Wa_II, i.e., a modified version of the Wagner number incorporating the effect of mass transport. Wa_I represents the ratio of the activation resistance to the electrolyte ohmic resistance, and Wa_II represents the ratio of the activation resistance to the mass transport resistance: ^22,25,26

$$\begin{eqnarray}&&{{\rm{Wa}}}_{I}=\displaystyle \frac{{R}_{a}}{{R}_{{\rm{\Omega }}}}=\left(\displaystyle \frac{{\rm{RT}}\kappa }{{\alpha }_{a}F{i}_{{\rm{avg}}}}\right)\left(\displaystyle \frac{d}{4{L}^{2}}\right)\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{{\rm{Wa}}}_{{\rm{II}}}=\displaystyle \frac{{R}_{a}}{{R}_{{\rm{mt}}}}=\left(\displaystyle \frac{n}{{\alpha }_{a}}\right)\left[\displaystyle \frac{{\rm{nF}}\left({D}_{j}\,+\,L{v}_{0}\right){C}^{b}}{{i}_{{\rm{avg}}}}\right]\left(\displaystyle \frac{d}{4{L}^{2}}\right)\end{eqnarray} \tag{ 9 }$$

For any practical tubular FTE operating at typical flow rates, Lv₀ is of the order of 1 cm² s⁻¹ and thus Lv₀ ≫ D_j implying that convection is significantly more effective at transporting reactant species compared to diffusion. Thus, for a typical tubular FTE with convection, Eq. 9 simplifies to:

$$\begin{eqnarray}&&{{\rm{Wa}}}_{{\rm{II}}}\approx \left(\displaystyle \frac{n}{{\alpha }_{a}}\right)\left[\displaystyle \frac{{\rm{nFL}}{v}_{0}{C}^{b}}{{i}_{{\rm{avg}}}}\right]\left(\displaystyle \frac{d}{4{L}^{2}}\right)\end{eqnarray} \tag{ 10 }$$

Note that Wa_II is inversely correlated to the Thiele modulus (Th) and Damköhler number (Da) commonly found in literature on chemical kinetics. ^27,28 From Eqs. 8 and 9, it can be concluded that for uniform current distribution in a tubular FTE whereby the overall process is activation controlled (ohmic and transport resistances are small in comparison to the activation resistance), Wa_I and Wa_II must be large quantities.

Figure 2 shows Wa_I and Wa_II values plotted as a function of the average current density (i_avg). For this plot, parameter values, e.g., C^b, n, v₀, α_a, and κ, are reported in Table II. These values represent closely the ferri/ferrocyanide reaction on graphite, as discussed below. To achieve large values of Wa_I and Wa_II, i.e., uniform current distribution, Fig. 2 shows that the tubular FTE must be operated at a low average current density because Wa_I and Wa_II both depend inversely on i_avg. At typical flow rates through the FTE, e.g., v₀ = 10.6 cm s⁻¹ used in Fig. 2, we observe that Wa_II is very large (Wa_II > 100 over the entire range of current densities shown) and that Wa_II ≫ Wa_I. This is a manifestation of the fact that convective transport through the tubular FTE considerably lowers its transport resistance compared to its typical ohmic resistance.

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In this section, we perform analytical modeling to quantify the extent of current distribution non-uniformity as a function of Wa_I and Wa_II. The current distribution over the tubular FTE surface was modeled for the simplified axisymmetric geometry shown in Fig. 3a. An analytical expression for the secondary current distribution (small Wa_I, large Wa_II) for this geometry, assuming linearization of the kinetics near i_avg, ²² was provided by Akolkar: ³⁰

$$\begin{eqnarray}&&\displaystyle \frac{{i}_{s}}{{i}_{{\rm{avg}}}}=m\left[\displaystyle \frac{{e}^{m\left({\rm{1}}-\tfrac{z}{L}\right)}+{e}^{-m\left({\rm{1}}-\tfrac{z}{L}\right)}}{{e}^{m}-{e}^{-m}}\right]\end{eqnarray} \tag{ 11 }$$

Here, m = Wa_I ^−0.5. When ohmic resistance is negligibly small (large Wa_I) but diffusion resistance is large (v₀ = 0 providing small Wa_II), the current distribution obeys: ³¹

$$\begin{eqnarray}&&\displaystyle \frac{{i}_{s}}{{i}_{{\rm{avg}}}}=\mu \left[\displaystyle \frac{{e}^{\mu \left({\rm{1}}-\tfrac{z}{L}\right)}+{e}^{-\mu \left({\rm{1}}-\tfrac{z}{L}\right)}}{{e}^{\mu }-{e}^{-\mu }}\right]\end{eqnarray} \tag{ 12 }$$

where μ is related to Wa_II as: μ·tanh(μ) = (α_a·Wa_II)⁻¹ for a one-electron transfer reaction (n = 1). The analytically modeled current profiles over the FTE surface at various Wa_I and Wa_II values are shown in Fig. 3b. In the absence of flow (v₀ = 0 cm s⁻¹) and at very low average current density, Wa_I is large (negligible ohmic resistance compared to activation resistance) but Wa_II is small (substantial diffusional mass transport resistance compared to activation resistance). This leads to significant non-uniformity in the current distribution (Fig. 3b) with less current able to penetrate deep into the FTE due to diffusion limitations in the absence of flow. The presence of flow through an FTE is thus of paramount importance in eliminating diffusion limitations and achieving large Wa_II as seen in Eq. 9. In the presence of flow through the tubular FTE (v₀ = 10.6 cm s⁻¹), Wa_I and Wa_II values decrease as i_avg increases. Increase in i_avg makes the current increasingly non-uniform (Fig. 3b). In all cases, Wa_II ≫ Wa_I due to the fact that mass transport resistance is nearly eliminated by the flow attributes of the FTE. Notably, the ratio i_s/i_avg (which is maximum at the FTE entrance z/L = 0) is equal to or less than 1.05 for Wa_I > 6. We take i_s/i_avg ≤ 1.05 to represent relatively uniform current distribution and thus the criterion Wa_II ≫ Wa_I > 6 becomes the key design criterion to ensure reasonably uniform current distribution across the tubular FTE.

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The above evaluation of current distribution using scaling analysis and analytical modeling provided key design criterion for tubular FTEs: Wa_II ≫ Wa_I > 6. Plots of Wa_I and Wa_II at various i_avg, d, and L values are shown in Fig. 4. Analysis presented in Fig. 4 assumes a constant flow velocity (v₀ = 10.6 cm s⁻¹). To achieve large values of Wa_I and Wa_II, FTE must be operated at a small average current density (i_avg) and must have a large diameter (d) or a small length (L). Figure 4 shows that, to achieve uniform current distribution, i.e., Wa_II ≫ Wa_I > 6, the FTE must be designed to have a diameter d ≥ 0.2 cm and be operated at an average current density i_avg ≤ 0.23 mA cm⁻² when L is fixed at 0.45 cm. Alternatively, the tubular FTE must have a length L ≤ 1 cm and be operated at i_avg ≤ 0.02 mA cm⁻² when d is fixed at 0.1 cm. In Fig. 4, conditions below the horizontal solid line (Wa_II = Wa_I = 6) should be avoided as these would lead to non-uniform current distribution.

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Guided by Fig. 4, we now investigate the ferri/ferrocyanide reaction kinetics on a graphite tubular FTE. We constructed a tubular FTE setup as described under "methods" with L = 0.45 cm and d = 0.1 cm. Figure 5a shows steady-state polarization curves measured on a graphite FTE at various equimolar concentrations of K₃Fe(CN)₆ and K₂Fe(CN)₆. A stable equilibrium potential (E_eq) was established at each [Fe(CN)₆]^4–/[Fe(CN)₆]^3– concentration (e.g., E_eq = 0.33 V vs Ag/AgCl at C_II = C_III = 1 mM). At large overpotentials (V_app—IR_Ω—E_eq > 0.2 V), mass transport limitations are observed in Fig. 5a. Below the mass transport limit, the anodic voltammogram in the Tafel range can be described by:

$$\begin{eqnarray}&&{i}_{{\rm{avg}}}={i}_{0}\left({\rm{1}}-\displaystyle \frac{{i}_{{\rm{avg}}}}{{i}_{L}}\right)\exp \left[\displaystyle \frac{{\alpha }_{a}F}{{\rm{RT}}}\left({V}_{{\rm{app}}}-I{R}_{{\rm{\Omega }}}-{E}_{{\rm{eq}}}\right)\right]\end{eqnarray} \tag{ 13 }$$

Tafel plots (Fig. 5b) were constructed where the kinetic current density i_k = ∣i_avg [i_L/(i_L − i_avg)]∣ was plotted vs V_app—IR_Ω—E_eq. For Tafel kinetics, semi-log plots of i_k vs V_app—IR_Ω—E_eq show a region with linearity because:

$$\begin{eqnarray}&&{\mathrm{log}}_{{\rm{10}}}{i}_{k}{=\mathrm{log}}_{{\rm{10}}}{i}_{0}+\left[\displaystyle \frac{{\alpha }_{a}F}{\left({\rm{2.303}}\right){\rm{RT}}}\right]\left({V}_{{\rm{app}}}-I{R}_{{\rm{\Omega }}}-{E}_{{\rm{eq}}}\right)\end{eqnarray} \tag{ 14 }$$

In Fig. 5b, the data points chosen from the raw polarization plot (Fig. 5a) meet the design criterion: Wa_II >> Wa_I > 6. This ensures that non-uniform current distribution does not convolute Tafel analysis. As seen in Eq. 14 and its application to Fig. 5b, the Tafel slope provided the anodic charge transfer coefficient: α_a = 0.58, and the Y-intercept at V_app = E_eq provided the exchange current density: i₀ = 0.046 mA cm⁻² (at equimolar concentration of 0.25 mM). Similar analyses were performed on a graphite RDE: the raw polarization data (Fig. 5c) was screened for uniform current distribution, followed by Tafel analysis using Eq. 14. Analysis yielded α_a = 0.56 and i₀ = 0.052 mA cm⁻² for equimolar concentration of 0.25 mM. Good agreement between the two methods is observed in Table III which compares the kinetics parameters α_a and i₀ determined using FTE and RDE over a range of concentrations. Note that, for the RDE (Fig. 5c), the conventional Wagner number takes the form: ²⁵

$$\begin{eqnarray}&&{{\rm{Wa}}}_{I\_\mathrm{RDE}}=\left(\displaystyle \frac{\mathrm{RT}{\rm{\kappa }}}{{\alpha }_{a}F{i}_{{\rm{avg}}}}\right)\left(\displaystyle \frac{4}{\pi {r}_{{\rm{RDE}}}}\right)\end{eqnarray} \tag{ 15 }$$

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From the dependence of the exchange current density on C^b (Table III), the standard rate constant can be determined: ²¹

$$\begin{eqnarray}&&{i}_{0}=\mathrm{nF}{k}_{0}{C}^{b}\end{eqnarray} \tag{ 16 }$$

Figure 6 shows a plot of i₀ vs C^b and contains kinetics data acquired on the tubular FTE as well as the RDE following the design criterion (Wa_I > 6). From the slopes in Fig. 6, the standard rate constants (k₀) were extracted: for FTE, k₀ = 1.95 × 10^–3 cm s⁻¹, and for RDE, k₀ = 2.14 × 10^–3 cm s⁻¹. While these kinetics constants are already in very good agreement (within 10%), the slightly higher apparent rate constant measured on the RDE may be attributed to its elevated roughness and thus higher electroactive surface area confirmed by Atomic Force Microscopy (discussed in detail in the Supporting Information available online at stacks.iop.org/JES/168/043505/mmedia). Our rate constant value is in the same general range as values reported previously. ^32,33 The differing electrocatalytic properties of graphite electrodes due to oxygenation ³⁴ and edge effects ³⁵ result in a wide range of rate constant values. Therefore, consistent preparation and treatment of graphite electrodes is crucial in obtaining reproducible kinetics parameters.

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To demonstrate the importance of strict adherence to the design criterion (Wa_I > 6) when conducting kinetics analysis, we evaluated the error in i₀ introduced when deviating from the design criterion. In Fig. 5, if Wa_I > 2 is followed as the selection criterion for Tafel analysis, we get i₀ = 0.167 mA cm⁻² at C^b = 1 mM and 0.067 mA cm⁻² at C^b = 0.5 mM. As seen in Fig. 6, these values are 14%–24% lower than those for the correct criterion (Wa_I > 6, which ensures highly uniform current distribution). Analogous to well-established analysis of the disk electrode, ³⁶ a mathematical treatment of the magnitude of error in kinetics measurement using FTE and its dependence on the FTE current distribution non-uniformity will be reported in a subsequent contribution.