Mechanistic Analysis of Anodic Oxidation of Gold in KOH (0.1 M) Solution Using the Point Defect Model

Abstract

The potentiostatic, anodic formation of gold oxide at potentials of 0.55 to 0.80 V versus SHE in aqueous KOH (0.1 M) was studied using an impedance-based Point Defect Model (PDM). The film thickness and refractive indices at each formation potential were estimated using spectroscopic ellipsometry. The thickness of the oxide increases linearly with increasing applied voltage within this range. Mott-Schottky (MS) analysis showed that gold oxide formed in KOH (0.1 M) is an n-type semiconductor, and the dominant defect (${Au}_i^{3+}$) density is calculated to be in the order of 10²¹–10²² (1/cm³). The steady-state current density of the oxide formation was independent of voltage, also in agreement with an n-type oxide. Reasonable agreement between PDM predictions and experimental observations of dominant defect density, steady-state current density, and thickness, demonstrates the value of the PDM in this system.

1. Introduction

Gold and its oxide have many applications, such as electrocatalysts for water splitting to produce H₂ and O₂, CO₂ reduction and the hydrogen evolution reaction [1,2,3]. Gold is generally considered a noble metal and its inertness with respect to aerobic oxidation reflects the positive change in Gibbs free energy ($\mathsf{\Delta }{G}_R$) for formation of the oxide under ordinary conditions (2Au + 1.5O₂ = Au₂O₃). This value is related to the standard change in Gibbs free energy of formation reaction, $\mathsf{\Delta }{G}_f^0$, by:

$$\mathsf{\Delta }{G}_R=\mathsf{\Delta }{G}_f^0+RTln\left(f_{O_2}^{-1.5}\right)=\mathsf{\Delta }{G}_f^0-1.5RTln\left(f_{O_2}\right)$$

(1)

where $f_{O_2}$ is the oxygen fugacity and $\mathsf{\Delta }{G}_f^0$~50 kJ/mol at 298 K for the most stable form of the oxide, Au₂O₃ (the “α-oxide”) [4]. By noting that at equilibrium, $\mathsf{\Delta }{G}_R$ = 0, Equation (1) can be rearranged to calculate the equilibrium oxygen fugacity as follows:

$$f_{O_2}^{eqil.}=e^{\mathsf{\Delta }{G}_f^0/1.5RT}$$

(2)

Substituting values for the parameters into Equation (2) yields $f_{O_2}^{eqil.}=$ 7 × 10⁵ b. Thus, Au₂O₃ is unstable at $f_{O_2}$ < 7 × 10⁵ b and forms spontaneously in the absence of an imposed potential only for $f_{O_2}$ > 7 × 10⁵ b. Nonetheless, in electrochemical systems, gold undergoes oxidation and subsequent formation of the oxide, Au₂O₃, at sufficiently anodic potentials in aqueous acidic and alkaline solutions [5,6,7]. The magnitude of the calculated partial pressure of oxygen ensures that Au₂O₃ is not found in nature. Noble metals, especially platinum and gold, have frequently been utilized as model systems in mechanistic investigations of anodic metal oxide formation/dissolution [8,9,10,11]. The High Field Model [12,13] and Place Exchange Model [14,15] have frequently been used to analyze the formation of gold oxide in the past, however, these models do not account for the steady-state growth that has been observed experimentally, among other issues [16]. We have therefore employed the Point Defect Model (PDM), which was developed to describe the mechanism of metal-oxide formation in aqueous solutions [17,18,19,20,21]. We recently reported the successful use of the PDM to account for the observed steady-state current density and oxide thickness in the anodic formation of gold oxide in H₂SO₄ (0.1 and 0.5 M) solutions [22].

We extend this work in the current study, using electrochemical impedance spectroscopy (EIS) within the framework of the PDM, to describe the anodic formation of gold oxide in an alkaline solution in KOH (0.1 M).

1.1. Experimental

1.1.1. Materials

The following materials were used as received: gold was used to produce the electrodes (99.999%, VEM Vacuum Engineering, Santa Clara, CA, USA), titanium was used as adhesion promoter, potassium hydroxide pellets (Merck, Rahway, NJ, USA), hydrochloric acid (37%, EMD, La Grange, IL, USA), nitric acid (EMD, 70%), and silver nitrate (99.8%, Fisher Scientific, Hampton, NH, USA). The water used to prepare the aqueous solutions was purified using a Millipore Simplicity UV system to 18.2 MΩ·cm.

1.1.2. Electrode Preparation

Polycrystalline gold electrodes (100 nm gold on 5 nm of titanium as an adhesion promoter) were vapor-deposited on glass substrates using a procedure reported previously [23]. The working electrodes were initially cleaned chemically in dilute aqua regia (0.5 M HNO₃, 1.5 M HCl) for 7 min. As a final cleaning step, the working electrodes were cleaned electrochemically by cyclic voltammetry (10 cycles) using the potential range of −0.10 to 1.0 V vs. SHE in deaerated KOH (0.1 M).

1.1.3. Anodic Formation of Gold Oxide and Characterization by in Situ Ellipsometry

All electrochemical measurements were performed using a conventional, three-electrode system, consisting of gold working electrode, a platinum-wire counter electrode, and an Ag/AgNO₃ (10 mM) (+0.8 V vs. SHE) reference electrode. Unless stated otherwise, all potentials are reported vs. SHE. Gold oxide was formed by applying a constant potential in the range of 0.55–0.80 V vs. SHE for about 1 h, until the system reached a steady state, as indicated by a constant current. After the system reached a steady-state current, while it was still under the anodic potential control, the thickness of the oxide was measured ellipsometrically using a J.A. Woollam WVASE32 spectroscopic ellipsometer (Lincoln, NE, USA) equipped with an electrochemical cell constructed of polyvinyl chloride (PVC). Measurements were collected at wavelengths of 350–800 nm and an angle of incidence of 70°. The thickness and refractive indices of the oxide layer at each potential were estimated from the experimental values of Ψ (azimuth angle) and Δ (relative phase difference) using an optical model comprising a bare gold as substrate, gold oxide overlayer, and a user-defined material file with a refractive index of 1.3345, measured independently for the contacting KOH solution [24]. The ellipsometry data was analyzed using a WvlbyWvl mat file contained in the CompleteEase software (version 4.68). All electrochemical experiments were performed using a VersaSTAT3 potentiostat (AMETEK SCIENTIFIC INSTRUMENT, Berwyn, PA, USA).

1.1.4. Electrochemical Impedance Spectroscopy (EIS) and Mott-Schottky (MS) Measurements

Electrochemical impedance spectra (EIS) were recorded over a frequency range of 20 kHz–0.01 Hz using a sinusoidal excitation of 10 mV (peak-to-peak) at the various potentials in the oxide-formation region (0.55–0.80 V vs. SHE). Mott-Schottky (MS) experiments were carried out by sweeping the potential in the negative direction from 0.90 V to 0.60 V vs. SHE at a sweep-rate of −21.43 mV/s while the capacitance was measured continuously at 1 kHz. This procedure was adopted to “freeze-in” the point-defect structure (concentration) established at the film-formation potential.

2. Point Defect Model Development

The PDM was originally developed as a mechanistic model to quantitatively describe the anodic formation of metal oxides, and the breakdown in aqueous solutions [17,18,19,20,21,22]. It treats the passive films that form on the surfaces of metals and alloys as having a two-layer structure that includes a barrier layer that grows into the metal by the formation of oxygen vacancies at the metal/barrier layer interface and a porous outer layer that develops through the precipitation of cations that pass through the barrier layer with anions from the solution [25]. The inner, barrier layer may contain point defects, such as cation vacancies, cation interstitials, oxygen vacancies, and seven interfacial reactions are used to describe the generation and annihilation of these defects at the metal/barrier (inner) layer (m/bl) and the barrier layer/solution (bl/s) interfaces (Scheme 1).

In the PDM, the seven reactions are categorized into two types based on whether they alter the boundary locations: lattice-conservative and lattice-non-conservative processes. Reactions (1), (2) and (4)–(6) are lattice-conservative, as they do not change the boundary locations. In contrast, Reactions (3) and (7), which are responsible for the formation and destruction of the oxide film, respectively, are classified as lattice-non-conservative, thus resulting in changes to the boundaries [22,26]. Analytical expressions for the rate constants of the reactions depicted in Scheme 1, as derived from the method of partial charges (vide infra), are summarized in

Table 1. Coefficients of the rate constants ($k_i=k_i^0{e}^{a_{i}V}{e}^{b_{i}L}{e}^{c_{i}pH}$) for the reactions that generate and annihilate point defects at the m/bl and bl/ol interfaces [17,18,22,27].

Reaction	ai (V−1)	bi (cm−1)	Ci	Units of k0i
$\left(1\right)Au+{\mathrm{V}}_{\mathrm{A}\mathrm{u}}^{{\mathsf{\chi }}^{\prime }}\stackrel{k_1}{\to }\mathrm{A}{\mathrm{u}}_{\mathrm{A}\mathrm{u}}+{\mathsf{\nu }}_{\mathrm{A}\mathrm{u}}+\mathsf{\chi }{\mathrm{e}}^{-}$	${\alpha }_1(1-\alpha )\chi \gamma$	$-{\alpha }_1\chi \gamma \epsilon$	$-{\alpha }_1\chi \beta \gamma$	${\displaystyle \frac{\mathrm{c}\mathrm{m}}{\mathrm{s}}}$
$\left(2\right)Au\stackrel{k_2}{\to }A{u}_i^{\chi +}+{\mathsf{\nu }}_{\mathrm{A}\mathrm{u}}+\mathsf{\chi }{\mathrm{e}}^{-}$	${\alpha }_2(1-\alpha )\chi \gamma$	$-{\alpha }_2\chi \gamma \epsilon$	$-{\alpha }_2\chi \beta \gamma$	${\displaystyle \frac{\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{c}{\mathrm{m}}^2\mathrm{s}}}$
$\left(3\right)Au\stackrel{k_3}{\to }A{u}_{Au}+{\displaystyle \frac{\mathsf{\chi }}{2}}{\mathrm{V}}_{\ddot{\mathrm{O}}}+\mathsf{\chi }{\mathrm{e}}^{-}$	${\alpha }_3(1-\alpha )\chi \gamma$	$-{\alpha }_3\chi \gamma \epsilon$	$-{\alpha }_3\chi \beta \gamma$	${\displaystyle \frac{\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{c}{\mathrm{m}}^2\mathrm{s}}}$
$\left(4\right)A{u}_{Au}\stackrel{k_4}{\to }A{u}_i^{\mathsf{\Gamma }+}+{\mathrm{V}}_{Au}^{\mathsf{\chi }}+\left(\mathsf{\Gamma }-\chi \right)e^{-}$	${\alpha }_4\alpha \mathsf{\Gamma }\gamma$		${\alpha }_4\mathsf{\Gamma }\beta \gamma$	${\displaystyle \frac{\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{c}{\mathrm{m}}^2\mathrm{s}}}$
$\left(5\right)A{u}_i^{\chi +}\stackrel{k_5}{\to }A{u}^{\mathsf{\Gamma }+}+\left(\mathsf{\Gamma }-\chi \right)e^{-}$	${\alpha }_5\alpha \mathsf{\Gamma }\gamma$		${\alpha }_5\mathsf{\Gamma }\beta \gamma$	${\displaystyle \frac{\mathrm{c}\mathrm{m}}{\mathrm{s}}}$
$\left(6\right){\mathrm{V}}_{\ddot{\mathrm{O}}}+H_{2}O\stackrel{k_6}{\to }{O}_O+2{\mathrm{H}}^{+}$	${2\alpha }_6\alpha \gamma$		${\alpha }_6\mathsf{\Gamma }\beta \gamma$	${\displaystyle \frac{\mathrm{c}\mathrm{m}}{\mathrm{s}}}$
$\left(7\right)Au{O}_{\chi /2}+\chi {\mathrm{H}}^{+}\stackrel{k_7}{\to }A{u}^{\mathsf{\Gamma }+}+{\displaystyle \frac{\chi }{2}}{H}_{2}O+\left(\mathsf{\Gamma }-\chi \right)e^{-}$	${\alpha }_7\alpha \left(\mathsf{\Gamma }-\chi \right)\gamma$		${\alpha }_7\left(\mathsf{\Gamma }-\chi \right)\beta \gamma$	${\displaystyle \frac{\mathrm{m}\mathrm{o}\mathrm{l}}{\mathrm{c}{\mathrm{m}}^2\mathrm{s}}}$

and

Table 2. Definition of the standard rate constants for the interfacial defect generation and annihilation reactions employed in the point defect model [22].

Reaction	ki0
$(1)Au+{\mathrm{V}}_{\mathrm{A}\mathrm{u}}^{{\mathsf{\chi }}^{\prime }}\stackrel{k_1}{\to }\mathrm{A}{\mathrm{u}}_{\mathrm{A}\mathrm{u}}+{\mathsf{\nu }}_{\mathrm{A}\mathrm{u}}+\mathsf{\chi }{\mathrm{e}}^{-}$	$k_1^{00}\mathrm{e}\mathrm{x}\mathrm{p}(-\chi {\alpha }_1\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(2)Au\stackrel{k_2}{\to }A{u}_i^{\chi +}+{\mathsf{\nu }}_{\mathrm{A}\mathrm{u}}+\mathsf{\chi }{\mathrm{e}}^{-}$	$k_2^{00}\mathrm{e}\mathrm{x}\mathrm{p}(-\chi {\alpha }_2\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(3)Au\stackrel{k_3}{\to }A{u}_{Au}+{\displaystyle \frac{\mathsf{\chi }}{2}}{\mathrm{V}}_{\ddot{\mathrm{O}}}+\mathsf{\chi }{\mathrm{e}}^{-}$	$k_3^{00}\mathrm{e}\mathrm{x}\mathrm{p}(-\chi {\alpha }_3\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(4)A{u}_{Au}\stackrel{k_4}{\to }A{u}_i^{\mathsf{\Gamma }+}+{\mathrm{V}}_{Au}^{\mathsf{\chi }}+(\mathsf{\Gamma }-\chi )e^{-}$	$k_4^{00}\mathrm{e}\mathrm{x}\mathrm{p}(\mathsf{\Gamma }{\alpha }_4\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(5)A{u}_i^{\chi +}\stackrel{k_5}{\to }A{u}^{\mathsf{\Gamma }+}+(\mathsf{\Gamma }-\chi )e^{-}$	$k_5^{00}\mathrm{e}\mathrm{x}\mathrm{p}(\mathsf{\Gamma }{\alpha }_5\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(6){\mathrm{V}}_{\ddot{\mathrm{O}}}+H_{2}O\stackrel{k_6}{\to }{O}_O+2{\mathrm{H}}^{+}$	$k_6^{00}\mathrm{e}\mathrm{x}\mathrm{p}(2{\alpha }_6\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$
$(7)Au{O}_{\chi /2}+\chi {\mathrm{H}}^{+}\stackrel{k_7}{\to }A{u}^{\mathsf{\Gamma }+}+{\displaystyle \frac{\chi }{2}}{H}_{2}O+(\mathsf{\Gamma }-\chi )e^{-}$	$k_7^{00}\mathrm{e}\mathrm{x}\mathrm{p}((\mathsf{\Gamma }-\chi ){\alpha }_7\gamma ({\phi }_{{\displaystyle \raisebox{1ex}{$f$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}}^0+\beta pH))$

.

The method of partial charges (MPC) is a little-known technique for deriving the rate constants for chemical and electrochemical reactions. The basic postulate is that the composition of the transition state (activated complex) may be written as a weighted sum of the reactants and products. As an example, we may consider the metal dissolution reaction $m\to M^{n+}+n{e}^{-}$ in an aqueous solution. In the MPC, the reaction is postulated to pass through the transition state contained within the square brackets and designated by the superscript $‡$.

$$m\to [{\left(1-\alpha \right)m+\alpha M^{n+}+\alpha n{e}^{-}]}^{‡}\to M^{n+}+n{e}^{-}$$

(3)

The Gibbs energy of activation is now written as follows:

$$∆G^{‡}=\left(1-\alpha \right){\overline{\mu }}_m+\alpha {\overline{\mu }}_{M^{n+}}+\alpha n{\overline{\mu }}_{e^{-}}-{\overline{\mu }}_m$$

(4)

The electrochemical potentials (${\overline{\mu }}_i$) are then expressed as, ${\overline{\mu }}_i={\mu }_i^0+RTln\left(a_i\right)+zF\phi$, where ${\mu }_i^0$ is the standard chemical potential, $a_i$ is the activity of Species $i$, and $\phi$ is the local electrical potential. Assuming that the electrons are in their standard state such that the activity is one and noting that the metal is in its standard state, the Gibbs energy of activation now becomes:

$$∆G^{‡}=\alpha ∆{\mu }_R^0+\alpha RTln\left(a_{M^{n+}}\right)-\alpha nF\left({\phi }_m-{\phi }_s\right)$$

(5)

Note that the electrons reside in the metal for which the electrical potential is ${\phi }_m$ and the cation resides in the solution phase at potential ${\phi }_s$. The quantity $∆{\mu }_R^0$ is the change in standard chemical potential (Gibbs energy) for the reaction.

According to activated complex theory, the rate of reaction ($R$) is written as follows:

$$R=\varsigma {\displaystyle \frac{kT}{h}}{e}^{-∆G^{‡}/RT}$$

(6)

in which it is assumed that a quasi-equilibrium exists between the reactants (m) and the activated complex. In Equation (6), $k$ is Boltzmann’s constant, $h$ is Planck’s constant, and $\varsigma$ is the “crossing frequency”, which is the probability that the system, having sufficient energy to surmount the barrier, actually passes over per unit time and does not revert to the reactant. Substitution of Equation (5) into Equation (6) therefore yields the following:

$$R=\varsigma {\displaystyle \frac{kT}{h}}{e}^{\left[-\alpha ∆{\mu }_R^0-\alpha RTln\left(a_{M^{n+}}\right)+\alpha nF\left({\phi }_m-{\phi }_s\right)\right]/RT}=\varsigma {\displaystyle \frac{kT}{h}}\left[e^{-\alpha ∆{\mu }_R^0/RT}{e}^{\alpha nF\left({\phi }_m-{\phi }_s\right)/RT}\right]e^{ln{\left(a_{M^{n+}}\right)}^{-\alpha }}$$

(7)

Using the identity, $e^{ln\left(X\right)}=X$, we obtain the rate of reaction as:

$$R=\varsigma {\displaystyle \frac{kT}{h}}\left[e^{-\alpha ∆{\mu }_R^0/RT}\right]e^{\alpha nF\left({\phi }_m-{\phi }_s\right)/RT}{\left(a_{M^{n+}}\right)}^{\alpha }=k{\left(a_{M^{n+}}\right)}^{-\alpha }$$

(8)

where $a_{M^{n+}}$ is the activity of $M^{n+}$ in the solution. The rate constant for the reaction ($k$) is therefore:

$$k=\varsigma {\displaystyle \frac{kT}{h}}\left[e^{-\alpha ∆{\mu }_R^0/RT}\right]e^{\alpha nF\left({\phi }_m-{\phi }_s\right)/RT}$$

(9)

In an electrochemical reaction, the potential of a metal electrode is measured with respect to a reference electrode ($E$). Thus, designating the reference electrode potential as ${\phi }_R$, we write

$${\phi }_m-{\phi }_s={\phi }_m-{\phi }_R+{\phi }_R-{\phi }_s=E+{\phi }_R-{\phi }_s$$

(10)

Accordingly, the rate constant becomes:

$$k=\varsigma {\displaystyle \frac{kT}{h}}\left[e^{-\alpha ∆{\mu }_R^0/RT}\right]e^{\alpha nF\left({\phi }_R-{\phi }_s\right)/RT}{e}^{\alpha nFE/RT}$$

(11)

or

$$k=k^0{e}^{\alpha nFE/RT}$$

(12)

where the standard rate constant is

$$k^0=\varsigma {\displaystyle \frac{kT}{h}}\left[e^{-\alpha ∆{\mu }_R^0/RT}\right]e^{\alpha nF\left({\phi }_R-{\phi }_s\right)/RT}$$

(13)

The reader will recognize Equation (12), which is ubiquitous in electrochemical kinetics. However, the present derivation via the MPC highlights several important factors. First, Equation (13) demonstrates a strong connection between the kinetics and the thermodynamics of a reaction, a relationship whose existence is often overlooked despite being a central concept of Marcus theory for electron transfer [28]. Indeed, Equation (13) would seem to provide a path towards estimating the standard rate constant from $∆{\mu }_R^0$, ${\phi }_R-{\phi }_s$, and $\varsigma$. The most difficult quantity to estimate is $\varsigma$, though recent advances in quantum mechanics/density functional theory provide some encouragement in this area [29]. Second, the parameter $\alpha$, which is synonymous with the symmetry factor $\beta$ in classical electrochemical kinetic theory, is identified in MPC with the extent of mixing of the initial and final states in formulating the activated complex. Thus, for $\alpha <$ 0.5, the activated complex is said to be “reactant-like” but if $\alpha >$ 0.5 it is “product-like”. We use this concept later in interpreting the electrochemical kinetic data for gold in KOH solutions.

The current density, I, as observed in an external circuit and denoted by Equation (12), results from the processes that generate and consume electrons.

$$I=F\{\chi k_1{C}_v^L+\chi k_2+\chi k_3+\left(\mathsf{\Gamma }-\chi \right)k_4+\left(\mathsf{\Gamma }-\chi \right)k_5{C}_i^0+\left(\mathsf{\Gamma }-\chi \right)k_7{\left({\displaystyle \frac{C_{H^{+}}}{C_{H^{+}}^0}}\right)}^n\}$$

(14)

Here, F is Faraday’s constant, and n is the reaction order of the dissolution of the bl (Reaction (7), Scheme 1) with respect to the proton concentration. $C_v^L$ and $C_i^0$ are the concentrations of the cation vacancy and metal interstitial at the m/bl and bl/s interfaces, respectively. Note that, provided $\mathsf{\Gamma }\ne \chi$, all reactions except Reaction (6) contribute to the electron current; if $\mathsf{\Gamma }=\chi$ (the case for gold), only Reactions (1)–(3) contribute to the current. This is so for both the transient and steady-state regimes. In this case, the current density is expressed as follows:

$$I=F(\chi k_1{C}_v^L+\chi k_2+\chi k_3)$$

(15)

Note that this expression is not a function of the concentrations of the metal interstitial or the oxygen vacancy at either interface, which has major consequences for the derivation of the relaxations in these quantities in response to the modulated voltage, as explained below. Note further, that in the steady state (and only in the steady state), $k_1{C}_v^L=k_4$ so if a method is available for determining $k_4$ independently of the current, then the steady-state current density simplifies to

$$I_{ss}=\mathsf{\Gamma }F(k_2+k_3+k_4)$$

(16)

and the need to know $C_v^L$ is avoided. Optimization of the PDM on experimental EIS data provides such a means, as is described later in this paper.

The rate constants for the reactions are obtained based on the MPC as follows (see Table 1):

$$k_i=k_i^0\exp \left[a_i\left(V-R_{ol}I\right)+b_{i}L+c_{i}pH\right]\hspace{1em}i=1,2,3$$

(17)

$$k_j=k_j^0\exp \left[a_j\left(V-R_{ol}I\right)+c_{j}pH\right]\hspace{1em}j=4,5,6,7$$

(18)

In the above expressions, we treat the oxide layer as a bi-layer with a resistive outer layer (R_ol) formed on the barrier layer. Therefore, the potential that exists at the bl/s interface differs from the value applied at the outer layer/solution (ol/s) interface due to the potential drop across the outer layer, if the current flowed to a remote cathode, which is the case in these experiments.

If we assume that the applied potential at the ol/s interface changes sinusoidally around an average value of $\overline{V}$, the applied potential can be written as follows:

$$V=\overline{V}+\delta V=\overline{V}+∆V{e}^{j\omega t}$$

(19)

in which $\omega$, $∆V$, and $\overline{V}$ are angular frequency, the voltage amplitude at zero frequency, and an average voltage under steady-state conditions. In this case, the Faradic admittance is calculated as follows:

$$Y_F={\displaystyle \frac{1}{Z_F}}={\displaystyle \frac{\delta I}{\delta V}}={\displaystyle \frac{∆I}{∆V}}$$

(20)

Here, Z_F is the Faradic impedance. Also, the potential at the bl/s interface (U) and the modulated potential at the tip of the Luggin probe of the reference electrode (V) are related to the current density, I as follows:

$$U=V-I{R}_{ol}$$

(21)

Then,

$$Y_F={\left({\displaystyle \frac{dI}{dV}}\right)}_{V=\overline{V}}={\left({\displaystyle \frac{dI}{dU}}\right)}_{U=\overline{U}}{\left({\displaystyle \frac{dU}{dV}}\right)}_{V=\overline{V}}={\left({\displaystyle \frac{dI}{dU}}\right)}_{U=\overline{U}}\left[1-R_{ol}{\left({\displaystyle \frac{dI}{dV}}\right)}_{V=\overline{V}}\right]$$

(22)

Rearrangement gives Equation (23), as follows:

$${\displaystyle \frac{1}{Y_F}}={\displaystyle \frac{1}{Y_F^0}}+R_{ol}or{Y}_F={\displaystyle \frac{Y_F^0}{1+R_{ol}{Y}_F^0}}$$

(23)

Here, $Y_F^0={\left({\displaystyle \frac{dI}{dU}}\right)}_{U=\overline{U}}$ represents the admittance in the absence of the outer layer, assuming the applied voltage at the bl/s interface is $\overline{U}$ under steady-state conditions. If $R_{ol}$ goes to zero, then $Y_F=Y_F^0$, implies that the barrier layer controls the interfacial impedance. On the other hand, if $R_{ol}{Y}_F^0\gg 1$, $Y_F$ = 1/$R_{ol}$, indicating that the outer layer determines the interfacial impedance. However, as the outer layer does not form in the case of the current experiments, $Y_F=Y_F^0$, a single-layer model is used to interpret the EIS data.

The rate of change of the barrier layer thickness with time can be expressed by Equation (24),

$${\displaystyle \frac{dL}{dt}}={\displaystyle \frac{d{L}^{+}}{dt}}-{\displaystyle \frac{d{L}^{-}}{dt}}=\mathsf{\Omega }{k}_3^0{e}^{a_{3}V}{e}^{b_{3}L}{e}^{c_{3}pH}-\mathsf{\Omega }{k}_7^0{e}^{a_{7}V}{e}^{c_{7}pH}{\left({\displaystyle \frac{c_{H^{+}}}{c_{H^{+}}^0}}\right)}^n$$

(24)

where ${\displaystyle \frac{d{L}^{+}}{dt}}$ and ${\displaystyle \frac{d{L}^{-}}{dt}}$ are the rate of film growth at the m/bl interface and the rate of dissolution of the bl at the bl/s interface, respectively. Under steady-state conditions, the rate of change of the barrier layer thickness is zero; therefore, the right side of Equation (24) is set to zero (${\displaystyle \frac{dL}{dt}}=0$) and the steady-state thickness, L_ss, is derived as Equation (25), (assume χ = Γ, which is the case for Au).

$$L_{ss}=\left[{\displaystyle \frac{1-\alpha }{\epsilon }}\right]V+\left[{\displaystyle \frac{2.303n}{{\alpha }_3\epsilon \chi \gamma }}-{\displaystyle \frac{\beta }{\epsilon }}\right]pH+{\displaystyle \frac{1}{{\alpha }_3\epsilon \chi \gamma }}ln\left({\displaystyle \frac{k_3^0}{k_7^0}}\right)$$

(25)

The response of the current and barrier layer thickness to the imposed sinusoidal variation of the applied voltage are used to calculate the Faradic admittance from the current density expression (Equation (26)). Here, to simplify the nomenclature, we set $\left[{\mathrm{V}}_{\mathrm{A}\mathrm{u}}^{{\mathsf{\chi }}^{\prime }}\right]=C_v^L$, and ${\left[A{u}_i^{\chi +}\right]=C}_i^0$, so that the mean current becomes:

$$\overline{I}=\mathsf{\Gamma }\mathrm{F}(\overline{k_2}+\overline{k_3}+\overline{k_4})$$

(26)

The superscript bar means the system is under steady-state conditions. To simplify the nomenclature further, we define $k_7^{\prime }$ = ${k_7\left(C_{H^{+}}/C_{H^{+}}^0\right)}^n$ because no outer layer forms under these experimental conditions. Accordingly, $Y_F^0$ is expressed as follows:

$$Y_F^0={\displaystyle \frac{\delta I}{\delta U}}={\displaystyle \frac{∆I}{∆U}}=I_U+I_L{\displaystyle \frac{∆L}{∆U}}+I_v^L{\displaystyle \frac{∆C_v^L}{∆U}}+I_i^0{\displaystyle \frac{∆c_i^0}{∆U}}$$

(27)

in which

$$I_U=\mathrm{F}\left\{\chi a_1\overline{k_1}\right.\overline{C_v^L}+\chi a_2\overline{k_2}+\chi a_3\overline{k_3}+\left(\mathsf{\Gamma }-\chi \right)a_4\overline{k_4}+\left(\mathsf{\Gamma }-\chi \right)a_5\overline{k_5}\overline{C_i^0}+\left(\mathsf{\Gamma }-\chi \right)a_7\overline{k_7^{\prime }}\}$$

(28)

$$I_L=-\mathrm{F}\left\{\chi b_1\overline{k_1}\right.\overline{C_v^L}+\chi b_2\overline{k_2}+\left.\chi b_3\overline{k_3}\right\}$$

(29)

$$I_v^L=F\chi \overline{k_1}$$

(30)

$$I_i^0=F\left(\mathsf{\Gamma }-\chi \right)\overline{k_5}$$

(31)

Here, we stipulate that the applied potential changes sinusoidally at an average value, ($U=\overline{U}+\delta U=\overline{U}+∆U{e}^{j\omega t})$, leading to the relaxation of the barrier layer’s thickness and the concentrations of cation interstitials ($C_i^0$, and cation vacancies ($C_v^L\left)\right)$. We assume that Reaction (3) is irreversible, therefore the concentration of anion vacancies does not appear in the current density expression. This assumption results in the impedance response not including the relaxation of the oxygen vacancy concentration and Reaction (6) rate constant (k₆), which is an artifact of considering Reaction (3) to be irreversible [22,30]. The case of the reversible formation of oxygen vacancies has been published elsewhere [29] but the mathematics are inhibitive. The calculation of the terms, ${\displaystyle \frac{∆L}{∆U}},{\displaystyle \frac{∆C_i^0}{∆U}}\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{∆c_v^L}{∆U}}$ is reported elsewhere [22,30], however, we repeat the derivations here for clarity.

The first term can be derived by taking the total differential of Equation (10).

$${\displaystyle \frac{d\delta L}{dt}}=j\omega \delta L{e}^{j\omega t}=\mathsf{\Omega }\delta k_3-\mathsf{\Omega }{\left({\displaystyle \frac{c_{H^{+}}}{c_{H^{+}}^0}}\right)}^n\delta k_7=\mathsf{\Omega }\left(\overline{k_3}{a}_3\delta U-\overline{k_3}{b}_3\delta L\right)-\mathsf{\Omega }\overline{k_7^{\prime }}{a}_7{\left({\displaystyle \frac{c_{H^{+}}}{c_{H^{+}}^0}}\right)}^n\delta U$$

(32)

or

$$L_U\equiv {\displaystyle \frac{∆L}{∆U}}={\displaystyle \frac{\mathsf{\Omega }\left(\overline{k_3}{a}_3-\overline{k_7^{\prime }}{a}_7{\left(\frac{c_{H^{+}}}{c_{H^{+}}^0}\right)}^n\right)}{j\omega +\mathsf{\Omega }\overline{k_3}{b}_3}}$$

(33)

and ${\displaystyle \frac{∆C_i^0}{∆U}}$ can be calculated similarly, as follows. The flux density of the cation interstitials is given by the first Nernst-Planck equation:

$$J_i=-D_i{\displaystyle \frac{\partial C_i}{\partial x}}-\chi D_{i}K{C}_i$$

(34)

in which D_i is the diffusion coefficient of the interstitials (cm²/s), $K={\displaystyle \frac{F\epsilon }{RT}}$, ε is the electric field strength in the barrier layer (V/cm), R is the universal gas constant (8.314 J/mol⋅K), and T is the temperature in Kelvin (K).

The continuity equation is as follows:

$${\displaystyle \frac{\partial C_i}{\partial t}}=D_i{\displaystyle \frac{{\partial }^2{c}_i}{\partial x^2}}+\chi D_{i}K{\displaystyle \frac{\partial c_i}{\partial x}}$$

(35)

which must be solved for the respective boundary conditions:

$$-k_5{C}_i=-D_i{\displaystyle \frac{\partial c_i}{\partial x}}-\chi D_{i}K{C}_{i}atx=0$$

(36)

$$-k_2=-D_i{\displaystyle \frac{\partial c_i}{\partial x}}-\chi D_{i}K{C}_{i}atx=L$$

(37)

By substituting $C_i=\overline{C_i}+∆C_i{e}^{j\omega t}$ into Equations (34) to (37) and linearizing the boundary conditions relative to ΔU and ΔL gives:

$$j\omega {∆C}_i={\displaystyle \frac{{\partial }^2{∆c}_i}{\partial x^2}}+\chi D_{i}K{\displaystyle \frac{\partial ∆c_i}{\partial x}}$$

(38)

or

$$-{\overline{k}}_5\left({\overline{C}}_i^0{a}_5∆U+∆{\overline{C}}_i^0\right)=-D_i{\left({\displaystyle \frac{\partial {∆c}_i}{\partial x}}\right)}_{x=0}-\chi D_{i}K∆C_i^{0}atx=0$$

(39)

$$-{\overline{k}}_2\left(a_2∆U-b_2∆L\right)=-D_i{\left({\displaystyle \frac{\partial ∆C_i}{\partial x}}\right)}_{x=L}-{\chi D}_{i}K∆C_i^{L}atx=L$$

(40)

Analytical solution of the Equations (38)–(40) gives ${\displaystyle \frac{∆C_i^0}{∆U}}$ as follows:

$${\displaystyle \frac{∆C_i^0}{∆U}}={\displaystyle \frac{A+B}{∆U}}=∆C_{iU}^0+∆C_{iL}^0{\displaystyle \frac{∆L}{∆U}}$$

(41)

where

$$∆C_{iU}^0={\displaystyle \frac{b_{1U}\left(a_{22}-a_{21}\right)+b_{2U}(a_{11}-a_{12})}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$$

(42)

$$∆C_{iL}^0={\displaystyle \frac{b_{2L}(a_{11}-a_{12})}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$$

(43)

$$a_{11}=\left(r_1+\chi K\right)D_i-{\overline{k}}_5$$

(44)

$$a_{12}=\left(r_2+\chi K\right)D_i-{\overline{k}}_5$$

(45)

$$a_{21}=\left(r_1+\chi K\right)D_i{e}^{r_{1}L}$$

(46)

$$a_{22}=\left(r_2+\chi K\right)D_i{e}^{r_{2}L}$$

(47)

$$r_{\mathrm{1,2}}={\displaystyle \frac{-\chi K\pm \sqrt{{\chi }^2{K}^2+\raisebox{1ex}{$4j\omega $}\!\left/ \!\raisebox{-1ex}{$D_i$}\right.}}{2}}$$

(48)

$$b_{1U}={\overline{k}}_5{a}_5{\overline{C}}_i^0$$

(49)

$$b_{2U}={\overline{k}}_2{a}_2$$

(50)

$$b_{2L}={-\overline{k}}_2{b}_2$$

(51)

Using the same method for estimating the cation interstitials, it is possible to determine the change in the concentration of anion (oxygen) vacancies, except that subscripts 2 and 5 are replaced with 3 and 6 associated with the Reactions (3) and (6), and the charge on an oxygen vacancy is +2 in place of +$\chi$ for a metal interstitial. Also, the diffusivity of the oxygen vacancy is uniquely different from that for the metal interstitial. However, because the formation of the interstitial (Reaction (2), Scheme 1) is also assumed to be irreversible, the same problem exists in deriving a comparable expression for the relaxation of that point defect (metal interstitial)

The term ${\displaystyle \frac{∆C_v^L}{∆U}}$ is derived as follows:

$${\displaystyle \frac{∆C_v^L}{∆U}}={\displaystyle \frac{A{e}^{r_{2}L}+B{e}^{r_{2}L}}{∆U}}=∆C_{vU}^L+∆C_{vL}^L{\displaystyle \frac{∆L}{∆U}}∆L$$

(52)

in which

$$∆C_{vU}^L={\displaystyle \frac{\left(b_{1U}{a}_{22}-b_{2U}{a}_{12}\right)e^{r_{1}L}+\left(b_{2U}{a}_{11}-b_{1U}{a}_{21}\right)e^{r_{2}L}}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$$

(53)

$$∆C_{vL}^L={\displaystyle \frac{b_{2L}{a}_{11}{e}^{r_{2}L}-b_{2L}{a}_{12}{e}^{r_{1}L}}{a_{11}{a}_{22}-a_{12}{a}_{21}}}$$

(54)

$$r_{\mathrm{1,2}}={\displaystyle \frac{\chi K\pm \sqrt{{\chi }^2{K}^2+\raisebox{1ex}{$4j\omega $}\!\left/ \!\raisebox{-1ex}{$D_v$}\right.}}{2}}$$

(55)

$$a_{11}=\left(r_1-\chi K\right)D_v$$

(56)

$$a_{12}=\left(r_2-\chi K\right)D_v$$

(57)

$$a_{21}=[\left(r_1-\chi K\right)D_v{+{\overline{k}}_1]e}^{r_{1}L}$$

(58)

$$a_{22}=[\left(r_2-\chi K\right)D_v+{\overline{k}}_1]e^{r_{2}L}$$

(59)

$$b_{1U}=-{\overline{k}}_4{a}_4$$

(60)

$$b_{2U}=-{\overline{k}}_1{a}_1{\overline{C}}_v^L$$

(61)

$$b_{2L}={\overline{k}}_1{\overline{C}}_v^L{b}_1$$

(62)

Substitution of Equations (33), (41) and (52) into Equation (27) gives the final expression as follows:

$$Y_F^0=I_U+I_L{L}_U+I_v^L\left(∆C_{vU}^L+∆C_{vL}^L{L}_U\right)+I_i^0(∆C_{iU}^0+∆C_{iL}^0{L}_U)$$

(63)

Here, $Y_F=Y_F^0$ and hence the impedance is written as $Z_F={\displaystyle \frac{1}{Y_F}}$.

3. Impedance Model

The electrical equivalent circuit (EEC) presented in Figure 1 was used to analyze the impedance of the anodic oxide film formed on a gold electrode.

The total impedance of the system can be expressed as:

$$Z_t={\left[{\displaystyle \frac{1}{{(Z}_F+Z_w)}}+{\displaystyle \frac{1}{Z_{CPE-g}}}+{\displaystyle \frac{1}{Z_{Re,h}}}\right]}^{-1}+R_s$$

(64)

$$Z_w={\displaystyle \frac{\sigma }{\sqrt{w}}}-{\displaystyle \frac{j\sigma }{\sqrt{w}}}$$

(65)

$$Z_{CPE-g}={\displaystyle \frac{1}{C{\left(j\omega \right)}^m}}$$

(66)

in which ${\mathrm{Z}}_{\mathrm{t}}$ represent the total impedance, which is the summation of several impedance components such as Z_F, the Faradaic impedance associated with the PDM interfacial reactions taking place at the m/bl and bl/s interfaces; Z_W, the Warburg impedance attributed to the transport of the dominant defect through the barrier layer; and σ, the corresponding Warburg coefficient. Due to the distribution of the oxide film thickness on the various facets of a polycrystalline substrate, the geometric capacitance of the barrier layer is denoted by the constant phase element ZCPE-g with m < 1. The term Z_Re,h is the impedance of the movement of the free electrons and holes through the barrier layer, and R_s is the resistance of the solution between the oxide barrier layer and the tip of the Luggin probe. If oxygen evolution had occurred, then a Randles sub-circuit would be added to the solution side of the bl/s interface to represent that reaction in Figure 1. No oxygen evolution was observed.

4. Optimization

A genetic curve fitting methodology was used to optimize the PDM on the EIS data to retrieve the parameter values that represent the properties of the gold oxide layer formed on gold [30,31]. In this study, the optimization was performed using the Igor Pro software (Version 8.04, WaveMetrics, Inc., Portland, OR, USA) with a custom software interface powered by Andrew Nelson’s “gencurvefit” package [32]. The methodology is illustrated by the flowchart in Figure 2. The success of the optimization was determined by whether the parameter values obtained through the PDM optimization satisfied specified convergence criteria. If adequate convergence was not attained, the previous step was repeated and the process started over using new initial estimates for the model parameters. The convergence conditions evaluated included the following criteria [22,30]:

(1)

all parameter values were physically reasonable and within the defined limits;

(2)

both the computed Z’(w) and Z’(w) in the Nyquist and Bode planes adequately matched the corresponding experimental data;

(3)

the standard rate constants ${(k}_i^0)$ the polarizability of the bl/s interface (α), and the transfer coefficients for the point-defect generation and annihilation reactions at the barrier layer interfaces (α_i), were potential-independent within experimental precision; and

(4)

the steady-state current density (I_ss) and oxide film thickness (L_ss), predicted using the parameter values obtained from the model optimization, agreed with the measured experimental values. To be clear, the parameters gained through model optimization on the experimental EIS data were used to predict L_ss and I_ss values instead of using their values in the optimization process. As such, I_ss and L_ss are both valuable independent evaluations of the physico-electrochemical validity of the optimization process as well as of the PDM itself.

5. Results and Discussion

5.1. Cyclic Voltammetry

Figure 3 shows the cyclic voltammogram of a polycrystalline gold electrode in aqueous KOH (0.1 M). The potential region of −0.10–0.30 V is associated with the electrochemical adsorption of hydroxide (OH⁻) [33]. In the forward scan direction, oxide formation commences at 0.42 V, and in the reverse scan, oxide reduction starts at 0.60 V with the peak maximum at 0.41 V.

5.2. Potentiostatic Gold Oxide Formation

Figure 4 shows the current density versus time for oxide formation on gold in KOH (0.10 M) solution at various potentials under potentiostatic conditions. The anodic current density reached a steady state after around 1 h of constant applied potential at the gold working electrode. The cause(s) of the brief periods of instability in the current density, which appear as intense oscillations, remains undetermined because there was no evidence that the impedance becomes negative from load line analysis.

5.3. Mott-Schottky Analysis

Figure 5 displays the Mott-Schottky (MS) plot obtained for a polycrystalline gold working electrode in aqueous 0.1 M KOH. Mott-Schottky analysis reveals the electronic properties of the oxide layer, such as its semiconductor character (p-type vs. n-type), as well as provides a semi-quantitative estimate of the defect density. A semiconductor’s space charge capacitance for an n-type semiconductor is given by the Mott-Schottky equation [34]:

$${\displaystyle \frac{1}{C_{sc}^2}}={\displaystyle \frac{2}{\widehat{\epsilon }{\epsilon }_{0}e{N}_D}}\left(V-V_{FB}+{\displaystyle \frac{kT}{e}}\right)$$

(67)

Here, $\widehat{\epsilon }$(9.57) is the average dielectric constant for gold oxide, which we determined ellipsometrically, as detailed later in this work. The constant ε₀ represents the vacuum permittivity (8.854 × 10⁻¹⁴ F/cm), e is the electron charge (1.602 × 10⁻¹⁹ C), k is the Boltzmann constant (1.38 × 10⁻²³ J/K), V is the applied potential, V_FB is the flat-band potential, T is the temperature (K), and N_D is the donor density (cm⁻³). The 1/C² vs. V plot has a positive slope at potentials less than 0.86 V vs. SHE, indicating that gold oxide is an n-type semiconductor with metal interstitials and/or oxygen vacancies as its main defects. The calculated donor densities at a temperature of 295.15 K using Equation (67) and best-fit lines from the MS plot are included in Figure 5. The donor densities are in orders of 10²¹ and 10²² (cm⁻³), in good agreement with the values obtained for gold oxide formed in acidic and near-neutral solutions [22,35].

The two linear regions of the Mott-Schottky plot (Figure 5) presumably reflect a change in the dominant donor within the two different voltage regions, though it could reflect a phase change related to the oxidation/reduction of the gold cations from one oxidation state to another in this range. However, there is no spectroscopic evidence supporting the existence of lower oxidation states of gold such as Au(+1) and Au(+2), and the layer is assumed to be Au(+3), i.e., Au₂₊xO_3−y. Hence, this hypothesis cannot be accepted.

Furthermore, calculated defect densities using PDM parameters are consistent with cation interstitials as the dominant defects throughout the entire potential range in the n-type region. The change in slope of the Mott-Schottky plot might be due to the ionization of defects occurring in two steps. e.g., $V{O}^{--}\to V{O}^{.}+e^{-}$ (CB) and, $V{O}^{.}\to V{O}^{..}+e^{-}$ (CB), where “CB” denotes the conduction band.

5.4. Ellipsometric Oxide Thickness and Refractive Index

A three-phase model (metal/film/electrolyte) was used to analyze the data, with bare gold serving as the substrate, gold oxide (Au₂O₃) acting as a thin film on top of the substrate, and aqueous KOH (0.1 M) serving as the ambient environment. Figure 6 shows the experimental spectroscopic ellipsometric values of Ψ and ∆ recorded for the gold oxide formed at different anodic potentials in KOH (0.1 M) and fits to the data. Using the previously reported indices [22], the thickness of the oxide formed at each potential was initially calculated. This estimated thickness was then entered into the ellipsometric model as a known parameter and refractive indices were generated using a WvlbyWvl mat file and Levenberg-Marquardt regression analysis [22].

Table 3. Refractive index (n), extinction coefficient (k) and dielectric constant (ε) of the gold oxide formed at different potentials obtained through ellipsometry at 500 nm.

Potential (V) vs. SHE	n	k	$\mathsf{\epsilon }={\mathit{n}}^2-{\mathit{k}}^2$ [36]
0.55	3.58	1.67	10.0
0.65	3.40	1.52	9.26
0.70	3.37	1.33	9.55
0.80	3.37	1.38	9.45

lists the refractive indices generated for each potential at a wavelength of 500 nm.

5.5. Validity of the Impedance Data

The Kramers-Kronig (K-K) transformation was used to verify the casualty, linearity, and stability of the impedance data [37,38,39]. Figure 7 demonstrates that the system complies with these constraints of linear system theory by showing excellent agreement between the predicted imaginary and real components from K-K transforms and the experimental data.

5.6. Optimization Results and Discussion

5.6.1. PDM Optimization Results and Extraction of the Model Parameters

The experimental and PDM predictions of the EIS data are shown in Figure 8 as Nyquist (a) and Bode magnitude and phase (b) plots at the anodic potentials studied. The points represent experimental data and the red solid lines represent PDM predictions using the optimized parameters. The strong agreement between the experimental data and calculated values demonstrates the success of the model in describing the behavior of this system.

Table 4. The kinetic parameters generated from the optimized PDM on the experimental impedance data in aqueous 0.1 M KOH as a function of applied potentials.

	E/V vs. SHE	0.55	0.65	0.70	0.8	Average	Stdv
Parameter
$\alpha$		0.305	0.305	0.305	0.305	0.305	0.000
${\alpha }_1$		0.133	0.142	0.142	0.142	0.140	0.00390
${\alpha }_2$		0.0557	0.0524	0.0524	0.0524	0.053	0.00143
${\alpha }_3$		0.0160	0.016	0.016	0.016	0.016	0.000
${\alpha }_4$		0.0517	0.0455	0.0455	0.0455	0.047	0.00268
${\alpha }_5$		0.154	0.154	0.154	0.154	0.154	0.000
${\alpha }_6$		0.125	0.125	0.125	0.125	0.125	0.000
${\alpha }_7$		0.157	0.157	0.157	0.157	0.157	0.000
n		0.5	0.5	0.5	0.5	Assumed
β/V		−0.0592	−0.0592	−0.0592	−0.0592	Assumed [22]
${\phi }^0$		−0.05	−0.05	−0.05	−0.05	Assumed [22]
$\epsilon$/V·cm−1		6.75 × 106	6.75 × 106	6.75 × 106	6.75 × 106	1st Opt
pH		13.3	13.3	13.3	13.3
$\chi /\mathsf{\Gamma }$		3/3	3/3	3/3	3/3	Assumed
T/⁰K		295	295	295	295
$k_1^{00}$/cm·s−1		$9.15\times {10}^{-23}$	$4.81\times {10}^{-23}$	$3.17\times {10}^{-23}$	$2.87\times {10}^{-23}$	$5.00\times {10}^{-23}$	$2.51\times {10}^{-23}$
$k_2^{00}$/mol·cm−2·s−1		$6.05\times {10}^{-16}$	$2.62\times {10}^{-16}$	$3.14\times {10}^{-16}$	$1.36\times {10}^{-16}$	$3.29\times {10}^{-16}$	$1.72\times {10}^{-16}$
$k_3^{00}$/mol·cm−2·s−1		$5.98\times {10}^{-21}$	$8.66\times {10}^{-21}$	$3.41\times {10}^{-21}$	$1.01\times {10}^{-20}$	$7.03\times {10}^{-21}$	$2.56\times {10}^{-21}$
$k_4^{00}$/mol·cm−2·s−1		$6.40\times {10}^{-19}$	$1.90\times {10}^{-20}$	$8.98\times {10}^{-21}$	$1.34\times {10}^{-19}$	$2.00\times {10}^{-19}$	$2.58\times {10}^{-19}$
$k_5^{00}$/cm·s−1		$6.20\times {10}^{-6}$	$6.80\times {10}^{-6}$	$3.68\times {10}^{-6}$	$3.40\times {10}^{-6}$	$5.02\times {10}^{-06}$	$1.50\times {10}^{-06}$
$k_6^{00}$/cm·s−1		$4.83\times {10}^{-6}$	$5.80\times {10}^{-6}$	$3.64\times {10}^{-6}$	$7.36\times {10}^{-6}$	$5.16\times {10}^{-06}$	$1.71\times {10}^{-06}$
$k_7^{00}$/mol·cm−2·s−1		$2.18\times {10}^{-13}$	$3.13\times {10}^{-13}$	$1.27\times {10}^{-13}$	$3.67\times {10}^{-13}$	$2.61\times {10}^{-13}$	$9.48\times {10}^{-14}$
$k_1^0$/cm·s−1		$1.71\times {10}^{-22}$	$9.42\times {10}^{-22}$	$6.20\times {10}^{-23}$	$5.62\times {10}^{-23}$	$9.60\times {10}^{-23}$	$4.59\times {10}^{-23}$
$k_2^0$/mol·cm−2·s−1		$7.87\times {10}^{-16}$	$3.36\times {10}^{-16}$	$4.02\times {10}^{-16}$	$1.74\times {10}^{-16}$	$4.24\times {10}^{-16}$	$2.25\times {10}^{-16}$
$k_3^0$/mol·cm−2·s−1		$6.44\times {10}^{-21}$	$9.34\times {10}^{-19}$	$3.68\times {10}^{-21}$	$1.09\times {10}^{-20}$	$7.59\times {10}^{-21}$	$2.76\times {10}^{-21}$
$k_4^0$/mol·cm−2·s−1		$5.01\times {10}^{-19}$	$1.53\times {10}^{-20}$	$7.25\times {10}^{-21}$	$1.08\times {10}^{-19}$	$1.58\times {10}^{-19}$	$2.02\times {10}^{-19}$
$k_5^0$/cm·s−1		$3.00\times {10}^{-6}$	$3.29\times {10}^{-6}$	$1.78\times {10}^{-6}$	$1.65\times {10}^{-6}$	$2.43\times {10}^{-06}$	$2.75\times {10}^{-07}$
$k_6^0$/cm·s−1		$3.26\times {10}^{-6}$	$3.92\times {10}^{-6}$	$1.78\times {10}^{-6}$	$4.97\times {10}^{-6}$	$3.48\times {10}^{-6}$	$1.16\times {10}^{-6}$
$k_7^0$/mol·cm−2·s−1		$2.18\times {10}^{-13}$	$3.31\times {10}^{-13}$	$1.27\times {10}^{-13}$	$3.67\times {10}^{-13}$	$2.61\times {10}^{-13}$	$9.48\times {10}^{-14}$
$k_1$/cm·s−1		$1.61\times {10}^{-14}$	$1.02\times {10}^{-13}$	$1.20\times {10}^{-13}$	$3.49\times {10}^{-13}$	$1.47\times {10}^{-13}$	$1.23\times {10}^{-13}$
$k_2$/mol·cm−2·s−1		$1.71\times {10}^{-12}$	$7.15\times {10}^{-13}$	$1.06\times {10}^{-12}$	$7.04\times {10}^{-13}$	$1.05\times {10}^{-12}$	$4.09\times {10}^{-13}$
$k_3$/mol·cm−2·s−1		$5.85\times {10}^{-20}$	$9.67\times {10}^{-20}$	$4.06\times {10}^{-20}$	$1.37\times {10}^{-19}$	$8.33\times {10}^{-20}$	$3.71\times {10}^{-20}$
$k_4$/mol·cm−2·s−1		$1.14\times {10}^{-20}$	$6.51\times {10}^{-22}$	$3.34\times {10}^{-22}$	$5.86\times {10}^{-21}$	$4.57\times {10}^{-21}$	$4.53\times {10}^{-21}$
$k_5$/cm·s−1		$3.93\times {10}^{-11}$	$7.50\times {10}^{-11}$	$5.35\times {10}^{-11}$	$1.65\times {10}^{-11}$	$6.35\times {10}^{-11}$	$1.82\times {10}^{-11}$
$k_6$/cm·s−1		$3.89\times {10}^{-14}$	$1.15\times {10}^{-13}$	$8.17\times {10}^{-14}$	$5.61\times {10}^{-13}$	$1.99\times {10}^{-13}$	$2.11\times {10}^{-13}$
$k_7$/mol·cm−2·s−1		$2.18\times {10}^{-13}$	$3.31\times {10}^{-13}$	$1.27\times {10}^{-13}$	$3.67\times {10}^{-13}$	$2.61\times {10}^{-13}$	$9.48\times {10}^{-14}$
Re,h Ω·cm2		2.56 × 106	1.53 × 107	2.22 × 106	1.37 × 107	8.45 × 106	9.92 × 106
$CPE-g$/F·cm−2		$5.89\times {10}^{-5}$	$5.52\times {10}^{-5}$	$5.39\times {10}^{-5}$	$5.48\times {10}^{-5}$
$CPE-\alpha$/F·cm−2		0.90	0.92	0.95	0.92
$R_s$Ω·cm2		24	24	24	24
σ (Warburg coefficient)		1.10 × 105	5.00 × 103	2.13 × 105	5.00 × 103	8.33 × 104	7.66 × 104
$D_i$/cm2·s−1		$6.30\times {10}^{-17}$	$2.27\times {10}^{-20}$	$9.06\times {10}^{-17}$	$1.09\times {10}^{-20}$	$3.84\times {10}^{-17}$	$3.23\times {10}^{-17}$
Iss (A·cm−2)/EXP		$3.82\times {10}^{-8}$	$3.49\times {10}^{-8}$	$3.13\times {10}^{-8}$	$3.25\times {10}^{-8}$
Iss (A·cm−2)/PDM		$4.95\times {10}^{-7}$	$2.07\times {10}^{-7}$	$3.07\times {10}^{-7}$	$2.04\times {10}^{-7}$
$L_{ss}$ nm/Ellipsometry		0.148	0.210	0.272	0.443
$L_{ss}$ nm/PDM		0.143	0.211	0.286	0.410

provides a summary of the model parameters, including reaction rate constants, transfer coefficients, the dominant defect’s diffusivity, steady-state thickness, and current density.

Plots the kinetic parameters generated from the optimized PDM as a function of applied potential (Figure 9) revealed that these parameters are nearly independent of applied potential in all cases, as expected from electrochemical theory [40]. A further significant result is that the rate constants for Reaction (2) (k₂⁰⁰, k₂⁰ and k₂) are larger than those for Reaction (3) (k₃⁰⁰, k₃⁰ and k₃), which indicates that gold interstitials are the primary defects in the defective barrier oxide layer throughout the potential range, and that the passive film is an n-type semiconductor. Furthermore, the average values of α₂ (0.053), and α₃ (0.016), and the order of magnitude of k₂⁰⁰ (10⁻¹⁶ mol/cm²⋅s), k₂⁰ (10⁻¹⁶ mol/cm²·s), k₃⁰⁰ (10⁻²¹ mol/cm²·s), and k₃⁰ (10⁻²¹ mol/cm²·s) are similar to those reported previously in the n-type region for the case of gold oxide formed in H₂SO₄ (0.1 M) solution, α₂ (0.096), α₃ (0.09), k₂⁰⁰ (10⁻¹⁷ mol/cm²·s), k₂⁰ (10⁻¹⁷ mol/cm²·s), k₃⁰⁰ (10⁻¹⁹ mol/cm²·s), and k₃⁰ (10⁻¹⁹ mol/cm²·s) [22].

In addition, in terms of the method of partial charges (MPC, vide supra) for the kinetics of electrochemical processes, the charge-transfer coefficient can be viewed as a measure of the location of the activated complex (transition state) between the starting and final states, with 0 < α < 0.2 indicating a “reactant-like” transition state, 0.8 < α < 1 indicates a “product-like” transition state, and 0.2 < α < 0.8 indicating a transition state intermediate between reactants and products. The average value of charge transfer for Reaction (7), α₇ (0.157) is much smaller than the values reported previously for gold oxide formed in H₂SO₄ (0.1 and 0.5 M) solutions (0.850 and 0.860), [22] which suggests a difference in the position of the transition state on the reaction coordinate [41].

5.6.2. Anodic Current Density

Figure 10 shows the experimental (black triangles) and predicted (red circles) values of steady-state current density for the formation of gold oxide in KOH (0.1 M) at different applied voltages. Equation (14) and the optimized model parameters listed in Table 4 were used to calculate the predicted steady-state current densities. The logarithm of these values is almost independent of applied voltage, in agreement with the PDM prediction for an n-type oxide in which (Γ = χ).

5.6.3. PDM Calculation of the Oxide Thickness

Figure 11 provides a comparison of the steady-state oxide thickness values calculated using Equation (25) and the PDM-optimized parameters listed in Table 4 and those measured experimentally by ellipsometry. The experimental and PDM values both increase linearly with applied voltage and are in excellent agreement, indicating that the model has accurately described the properties of the oxide.

5.6.4. Defects Concentration and Defect Diffusivity

The steady-state defect concentrations were calculated at each applied potential using Equations (68)–(70):

$$C_i^L=\left({\displaystyle \frac{k_2}{k_5}}\right)·N_A$$

(68)

$$C_{OV}^L=\left({\displaystyle \frac{k_3}{k_6}}\right)·N_A$$

(69)

$$C_{CV}^0=\left({\displaystyle \frac{k_4}{k_1}}\right)·N_A$$

(70)

where N_A is Avogadro’s number (6.023 × 10²³), $C_i^L$, and $C_{OV}^L$ are the concentrations of the metal interstitial and the oxygen vacancies in the bl at the m/bl interface, and $C_{CV}^0$ is the concentration of the cation vacancies in the bl at the bl/s interface. Furthermore, the diffusivity of the dominant defects can be calculated based on the parameters listed in Table 4 and Equation (71) [17].

$$D_i=2{\sigma }^2{\left(1-\alpha \right)}^2{I}_{ss}^2/{\epsilon }^2$$

(71)

Here, σ is the Warburg coefficient, α is the polarizability of the bl/s interface, I_ss is the anodic current density, ε is the electric field strength, and $D_i$ is the diffusivity of the dominant defects, which is the cation interstitials in this study. Furthermore, $D_i$ the diffusivity of the cation interstitial as the dominant defect in this study is estimated through the PDM optimization.

Table 5. Steady-state defect densities and diffusivity of the dominant defect in gold oxide formed in aqueous KOH (0.1 M) at different potentials.

	E/V vs. SHE	0.55	0.65	0.70	0.80
Parameter
$C_i^L/\mathrm{c}{\mathrm{m}}^{-3}$		$2.17\times {10}^{22}$	$8.63\times {10}^{21}$	$1.19\times {10}^{22}$	$4.92\times {10}^{21}$
$C_{OV}^L/\mathrm{c}{\mathrm{m}}^{-3}$		$9.06\times {10}^{17}$	$5.08\times {10}^{17}$	$3.00\times {10}^{17}$	$1.47\times {10}^{17}$
$C_{CV}^0/\mathrm{c}{\mathrm{m}}^{-3}$		$5.53\times {10}^{17}$	$3.86\times {10}^{15}$	$1.68\times {10}^{15}$	$1.01\times {10}^{16}$
$D_i/\mathrm{c}{\mathrm{m}}^2·{\mathrm{s}}^{-1}$ (Calculated)		$6.30\times {10}^{-17}$	2.27 × ${10}^{-20}$	9.06 × ${10}^{-17}$	1.09 × ${10}^{-20}$
$D_i/\mathrm{c}{\mathrm{m}}^2·{\mathrm{s}}^{-1}$ (From PDM Optimization)		$6.37\times {10}^{-16}$	$6.18\times {10}^{-16}$	$2.31\times {10}^{-16}$	$2.76\times {10}^{-16}$

includes the dominant defect concentrations and diffusivities, showing that the concentration of cation interstitials is higher than oxygen vacancies and cation vacancies, as expected. In addition, the diffusivity is relatively independent of voltage, as predicted by the PDM.

5.6.5. Point Defects Distribution in the Barrier Layer

The distribution of dominant defects—metal interstitials, oxygen vacancies, and cation vacancies—in the barrier layer, [42] were calculated using Equations (72)–(74) and the results are shown in Figure 12, Figure 13 and Figure 14.

$$C_{{\mathrm{M}}_{\mathrm{i}}}=C_{{\mathrm{M}}_{\mathrm{i}}}^0\left(1-{\displaystyle \frac{k_5}{\chi K{D}_{{\mathrm{M}}_{\mathrm{i}}}}}\right)\exp \left(-\chi Kx\right)+{\displaystyle \frac{k_2}{\chi K{D}_{{\mathrm{M}}_{\mathrm{i}}}}}$$

(72)

$$C_{{\mathrm{V}}_{\mathrm{o}}}=C_{{\mathrm{V}}_{\mathrm{o}}}^0\left(1-{\displaystyle \frac{k_6}{2K{D}_{{\mathrm{V}}_{\mathrm{o}}}}}\right)\exp (-2Kx)+{\displaystyle \frac{\chi k_3}{4K{D}_{{\mathrm{V}}_{\mathrm{o}}}}}$$

(73)

$$C_{{\mathrm{V}}_{\mathrm{M}}}=C_{{\mathrm{V}}_{\mathrm{M}}}^L\left(1-{\displaystyle \frac{k_1}{\chi K{D}_{{\mathrm{V}}_{\mathrm{M}}}}}\right)\exp \left[\chi K(x-L)\right]+{\displaystyle \frac{k_4}{\chi K{D}_{{\mathrm{V}}_{\mathrm{M}}}}}$$

(74)

Here $C_{{\mathrm{M}}_{\mathrm{i}}}^0$, $C_{{\mathrm{V}}_{\mathrm{o}}}^0$ and $C_{CV}^L$ are steady-state concentrations of metal interstitials, oxygen vacancies and cation vacancies. The rest of the parameters are chosen from Table 4. For both the metal interstitials and oxygen vacancies, the defect concentration is constant with x/L except for the region closest to the m/bl interface. For the cation vacancies, the concentration is constant throughout the layer.

Because the PDM is a continuum model, the maximum thickness of these layers is estimated to be x = 10⁻⁹ L_bl, where L_bl is the thickness of the barrier layer. Since the maximum thickness of the barrier layer is about 0.4 nm, the thickness of the barrier layer is estimated to be 0.4 × 10⁻⁹ m or 0.4 × 10⁻⁷ cm, a physically small number as the minimum dimension is a single crystallographic plane (about 0.14 nm for Au₂O₃). Thus, the result is best regarded as being a “model parameter” with little actual physical meaning because the PDM is a continuum model and does not recognize effects due to the discrete sizes of atoms and ions.

6. Conclusions

This study’s results can be summarized as follows:

• The properties of anodic gold oxide in KOH (0.1 M) can be adequately described using parameters determined through optimization of the PDM on the experimental EIS data.
• The thickness of the anodic oxide resulting from the PDM optimization on the EIS data is in excellent agreement with the experimental values measured using spectroscopic ellipsometry.
• Values of steady-state current density during the formation of gold oxide at different potentials, derived from optimized PDM parameters, are in reasonable agreement with the experimentally measured values.
• Consistent with the PDM for the case where no change in cation oxidation state occurs upon barrier layer dissolution or cation ejection, the thickness of the barrier layer was observed to increase linearly with applied voltage, and the passive current density was found to be voltage-independent.
• Defect densities estimated through MS analysis and from the PDM are in good agreement. The dominant defect was found to be the cation interstitial, with a density in the order of (10²¹–10²²) cm⁻³.