Mechanism of Anodic Dissolution of Tungsten in Sulfate–Fluoride Solutions

Abstract

Thin passive films on tungsten play an important role during the surface levelling of the metal for various applications and during the initial stages of electrochemical synthesis of thick, nanoporous layers that perform well as photo-absorbers and photo-catalysts for light-assisted water splitting. In the present work, the passivation of tungsten featuring metal dissolution and thin oxide film formation is studied by a combination of in situ electrochemical (voltammetry and impedance spectroscopy) and spectro-electrochemical methods coupled with ex situ surface oxide characterization by XPS. Voltametric and impedance data are successfully reproduced by a kinetic model featuring oxide growth and dissolution coupled with the recombination of point defects, as well as a multistep tungsten dissolution reaction at the oxide/electrolyte interface. The model is in good agreement with the spectro-electrochemical data on soluble oxidation products and the surface chemical composition of the passive oxide.

1. Introduction

In the last two decades, there has been a growing interest in the electropolishing [1,2,3,4,5], electro-etching [5,6,7], chemical–mechanical planarization [8,9,10,11,12,13,14], and electrochemical machining of tungsten [15,16,17] in order to use it as a material for nuclear fusion [15,16,17] and as a component of integrated circuits [8,9,10,11,12,13,14]. The general purpose of those studies was to ensure a reproducible method to obtain a homogeneous planar tungsten surface via the control of tungsten dissolution and growth of the thin passive film, and some of the works are discussed in more detail in relation to such a mechanism.

The evolution of surface morphology, roughness, and current density during tungsten electropolishing has been experimentally investigated and analyzed [1,2,3,4,5]. Based on changes in the surface morphology and current density, tungsten electropolishing was divided into three stages: forming a rough surface by crystallographic etching, a brightening stage yielding ultra-smooth surfaces with clear boundaries, and the pitting stage [2]. The mechanism of tungsten electropolishing in the H₂SO₄-CH₃OH electrolyte was investigated from the point of view of etching isotropy [5]. The tungsten anodic dissolution behavior indicates that anisotropic and isotropic etchings are observed when the rate-controlling stages are charge transfer or mass transport, respectively. It was found that there is a transition from anisotropic to isotropic etching as a result of the accumulation of reaction products.

In a series of works, a generalized chemical–mechanical planarization (CMP) model of tungsten has been proposed [9,13,14]. It is based on fundamental hypotheses about the polishing mechanism by considering the nature of the surface film formed, the chemical reactions at the interfacial boundary, and mechanical scraping. Based on the steady-state approximation of chemical growth kinetics, the synergistic effect of slurry components and mechanical attrition was systematically investigated to capture the smoothing mechanism. The role of the oxidant type in the CMP mechanism was investigated using single and mixed oxidants such as Fe(NO₃)₃ and H₂O₂ [12]. The high polishing rate when using mixed oxidants is probably due to the formation of a thin passive WO₃ layer on the surface via a chemical reaction with OH radicals. This hypothesis was confirmed by the low values of the corrosion current density in the presence of mixed oxidants.

Microstructures such as micro-holes and grooves are needed on the tungsten surface to improve heat-transfer efficiency in thermoelectric applications. Electrochemical machining (ECM) is a promising method for generating such microstructures. In order to improve the efficiency and accuracy of the method, a new mixed electrolyte consisting of NaClO₃ with a low concentration of NaOH has been proposed [17]. NaClO₃ is used both to increase conductivity and to promote the formation of oxide films on the tungsten surface to suppress corrosion by stray currents. NaOH is used to provide continuous treatment by removing the oxide film in the treatment site. The influence of electrolyte composition and concentration on the electrochemical dissolution of tungsten was analyzed by measuring anodic current–potential curves and open circuit–time potential curves. As a result, an optimized electrolyte with 10 wt.% NaClO₃ and 0.3 wt.% NaOH was identified, in which holes with ca. 0.5 mm diameter are formed at a processing rate four times higher than that in classical electrolytes.

The effect of mass transfer on the dissolution of tungsten has been thoroughly studied in alkaline solutions only [18,19], and the effect of different ions on mass transport has been quantitatively modeled [19]. On the other hand, the effect of mass transport is very small in weakly acidic solutions, as demonstrated by rotating disk measurements [20].

Simultaneously occurring spatially separated oxide film growth and tungsten dissolution processes imply the use of a combination of electrochemical and analytical methods to estimate their partial rates. One such attempt was made by Krebsz et al. [21]. By coupling electrochemical techniques (including scanning electrochemical microscopy with a flow-through drop-cell) and the measurement of dissolved tungsten concentration by inductively coupled plasma mass spectrometry (ICP-MS), the in situ electrochemical dissolution of tungsten was investigated. The results show that the electrochemically formed thin oxide layer dissolves chemically in an acidic environment under the influence of the constant flow of electrolytes to the surface of W, concluding that in each of the subsequent electrochemical cycles, a new oxide formation is present.

The aim of the present paper is to study the primary passivation of tungsten in sulfate–fluoride solutions at low potentials (less than 1.0 V) using a combination of steady-state and transient electrochemical methods coupled to in situ analytical methods to quantify soluble products of anodic dissolution and ex situ characterization of the formed oxides. The choice of electrolytes is motivated by their use to grow nano porous oxides on W with photocatalytic activity [22,23,24,25,26,27,28,29,30]. On the basis of the obtained experimental data, a quantitative model of the dissolution and primary passivation of the metal is proposed and quantitatively compared to experimental data. Conclusions on the role of different process parameters—applied potential, electrolyte composition, and concentration—on the mechanism of anodic oxidation are drawn. The effect of illumination on passive film growth and dissolution at much higher potentials (4–10 V) was studied in a companion paper with various photo-electrochemical methods coupled with surface analysis [31].

2. Materials and Methods

Working electrodes were cut from 0.25 mm thick tungsten plates (99.99%, Goodfellow, Cambridge, UK). The area exposed to the solution was 4 cm² (for current–potential and impedance measurements) and 0.25 cm² (for spectro-electrochemical studies). Their pretreatment included degreasing, chemical polishing for 60 s in 0.1 M KOH, and drying. Measurements of current–potential curves and impedance spectra were conducted in a standard three-electrode cell, with platinum mesh (99.9%, Goodfellow, Cambridge, UK) as the counter-electrode and a 3 M KCl/AgCl/Ag reference electrode (LL-type, Metrohm, Herisau, Switzerland, E = 0.2 V vs. SHE). Anodic oxidation was carried out in electrolytes containing 1 M (NH₄)₂SO₄ and different concentrations of NH₄F (0, 0.010, 0.025, 0.050, or 0.075 M; pH ranging from 5.0 to 5.6) at a temperature of 22 ± 1 °C in naturally aerated conditions. The electrolytes were prepared from pro-analysis chemicals and de-ionized water.

Current–potential curves were measured in potentiostatic mode, stepwise in the positive direction, after the variation of the current at a specific potential fell below 2%. A potential range from 0 to 0.8 V was explored using an Autolab PGSTAT 302N potentiostat (Metrohm, Herisau, Switzerland). Impedance spectra were registered with a similar potentiostat featuring an FRA32M impedance module after reaching stationary current density at each potential (criterion: variation in the current during the experiment < ±2%). Reproducibility of impedance spectra was ±1% (by impedance size) and ±2° (by phase shift). The frequency interval was 10 mHz–22 kHz, with an ac signal amplitude of 15 mV (rms). The linearity of the spectra was checked by measuring spectra at selected potentials with amplitudes of 5 ÷ 20 mV, and the causal conditioning was confirmed by the built-in measurement software test of comparability with the Kramers–Kronig transformation. Measurement points that failed the test (usually at both ends of the frequency range) were rejected. Complex non-linear least squares fitting of experimental data to the transfer function of the proposed kinetic model was performed using the Levenberg–Marquardt algorithm on an OriginPro 9.8 platform (Originlab).

Dissolved tungsten concentration during potentiodynamic sweeps in the positive direction was measured in situ by spectro-electrochemical measurements in a special three-electrode cuvette-like cell with a volume of 1 cm³ through which light of a set wavelength passed. For this purpose, the PGSTAT 302N was coupled to a UV/VIS/NIR spectrophotometer (Metrohm, Herisau, Switzerland). The wavelength range was 200–300 nm, and the range of potentials was 0.0–1.0 V at a scan rate of 0.1 mV s⁻¹.

The surface chemical composition of the anodic films was estimated by XPS using an ESCALAB II (MKVG SCIENTIFIC) electron spectrometer with a base pressure of 10⁻⁷ Pa. The angle of the X-ray beam with respect to of the surface was 90°. Photoelectrons were excited with an Al Kα X-ray source (1486.6 eV), and the band-pass energy of the analyzer was 20 eV. Detailed spectra of W4f, O1s, C1s, N1s, F1s, and S2p were analyzed by XPSPeak 4.1 using Gaussian–Lorentzian peaks following a Shirley background subtraction.

3. Results

3.1. Voltammetry and Spectro-Electrochemistry

Current–potential curves in the interval 0–0.5 V in 1 M (NH₄)₂SO₄ containing different concentrations of NH₄F are presented in Figure 1a. The maximum of the current is observed at 0.2–0.25 V, which is an indication of primary passivation. Figure 1b shows dependencies of steady-state current densities at potentials in the zone of active dissolution (0.10–0.20 V) and primary passivation (0.30–0.50 V) on the concentration of fluoride ions. It can be concluded that the order of the rate-determining stage of the process with respect to fluoride ions in these two zones is different.

Figure 2a shows linear voltammograms measured in the spectrophotometric cuvette/electrochemical cell for spectro-electrochemical measurements with a sweep rate of 0.1 mV s⁻¹. The curves are qualitatively and quantitatively analogous to the steady-state current–potential dependences, indicating that the experimental conditions in the two configurations were closely similar. In addition, rotating disk measurements (not shown for the sake of brevity) did not show any influence of the electrode rotation rate on the current densities, indicating that mass transfer in the electrolyte plays a minor role in the whole oxidation process. Quantity of charge–potential curves obtained by numerical integration of the voltammograms are presented in Figure 2b. They are characterized by two slopes: a larger one in the interval 0–0.3 V that probably corresponds to active dissolution and a smaller one for higher potential, indicating primary passivation.

Figure 3a–d show the absorption spectra of the electrolyte measured at a number of potentials during the registration of the linear voltammograms in the solutions studied. The spectra are characterized by a peak at 245–250 nm corresponding to an ionic form of W(VI) [32], its intensity increasing both with the potential at a constant fluoride concentration and with the concentration of fluoride ions at constant potential. The A_peak–E curves at all fluoride ion concentrations (Figure 3e) exhibit two slopes analogous to the charge–potential curves, i.e., there is good agreement between the electrochemical and spectrophotometric data in the investigated range of potentials. In summary, both types of data indicate that there is a change in the anodic dissolution of tungsten at ca. 0.25–0.3 V, which can be identified as primary passivation since the rate of dissolution at potentials higher than that value is lower than that in the 0–0.25 V interval.

3.2. Electrochemical Impedance Spectroscopy

To identify the individual steps of the oxidation process that are distinguishable by electrochemical means, electrochemical impedance spectroscopy was employed. Impedance spectra in 1 M (NH₄)₂SO₄ + xM NH₄F (x = 0, 0.01, 0.025, 0.050) in the potential interval 0.05–0.50 V are collected in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

In agreement with current–potential curves, impedance values at frequencies tending to zero, which can be assumed to be inversely proportional to the rate of the limiting stage of the process near its steady state, decrease significantly with increasing applied potential at c(F⁻) = const and increasing fluoride ion concentration at E = const. This fact once again illustrates the accelerating effect of fluoride ions on the active dissolution and primary passivation processes of tungsten, with the effect of sulphate addition being of secondary importance.

At low potentials in the active dissolution region (0–0.25 V), three time constants are observed in the spectra: two capacitive and one pseudo-inductive. The capacitive time constant at high frequencies (above 1 Hz) corresponds to a parallel combination of the capacitance of the interface (characteristic of the double layer and/or a thin oxide film) and the charge-transfer resistance at that interface, while the two time constants at low frequencies (below 1 Hz) are most likely due to the relaxation of surface concentrations of intermediate products of the dissolution reaction.

In this line of reasoning, the pseudo-inductive time constant expresses the relaxation of the surface concentration of an intermediate accelerating the process rate in the non-stationary state, while the capacitive time constant at low frequencies most probably indicates the presence of an intermediate slowing down that rate.

In the region of primary passivation (0.3–0.5 V), the low-frequency pseudo-inductive time constant gradually disappears, and the low-frequency capacitive time constant becomes characterized by a negative real part of the impedance. This is in agreement with the negative slopes of the current–potential curves in this region. Moreover, as the potential in the primary passivation zone increases, a new pseudo-inductive time constant at relatively high frequencies (about 1 Hz) begins to occur which is most probably due to point-defect recombination at the passive film/electrolyte boundary, leading to an acceleration of the process speed in the non-stationary state [33,34].

In general, it can be concluded that impedance spectra indicate the presence of at least two intermediate products of the oxidation reaction at the film/electrolyte interface as well as interactions between defects in this film. A quantitative interpretation of these observations will be made on the basis of the kinetic model proposed below.

3.3. Composition of Oxide Films

The chemical composition of the primary passive layer was characterized by XPS depending on the fluoride ion concentration (0–0.05 M) at two potentials (0.4 and 1.0 V). Detailed spectra of W4f and O1s after 1 h oxidation at 0.4 V in 1 M sulfate electrolytes with fluoride ion concentrations of 0, 0.01, 0.025, and 0.05 M are presented in Figure 9. The layers at 0.4 V are very thin, since in the W4f spectra, the metallic state (W(0)) either predominates or has a significant presence. Moreover, the films are nonstoichiometric since W(VI) and W(IV) states are present in the W4f spectra independently of the fluoride concentration, and in the O1s spectra, a peak is observed at about 533 eV (O_n-st), characteristic of a nonstoichiometric oxide. Nitrogen at the surface of the oxide most probably represents adsorbed ammonium ions (predominant peak in the N1s spectra) and ammonia (minor peak). The amount of adsorbed fluoride at the surface is around the detection limit, and adsorbed sulphate is found on the surface only for the specimens oxidized at 1.0 V.

The composition of the oxides is summarized in

Table 1. Surface chemical composition of oxides formed at 0.4 and 1.0 V in solutions with different fluoride concentrations as estimated by XPS, n.d.—not determined.

E/V	F− Concentration/mol dm−3	L/nm	W(IV)/(W(VI) + W(IV))/%	On-st/(Oox + On-st)/%	O/W
0.4	0	1.4	43	33	2.2
0.4	0.010	3.0	22	21	2.7
0.4	0.025	2.0	35	40	2.2
0.4	0.050	1.1	35	35	2.7
1.0	0.010	6.6	8.0	n.d.	3.1
1.0	0.025	5.2	8.0	n.d.	3.2

. It can be concluded that the layer thickness grows with the applied potential and is generally smaller after oxidation in solutions containing fluoride ions. The non-stoichiometry of the oxide (i.e., the concentration of low-valent tungsten cations) generally decreases with potential, with the non-stoichiometric oxide peak at 1.0 V O1s not detected. This most probably indicates that the thin oxide through which dissolution proceeds is heavily non-stoichiometric and contains significant amounts of W(IV), whereas the oxide in the passivation range more closely resembles WO₃. A tentative conclusion that the dissolution process is mediated by W(IV) at the surface can be drawn from these observations. The influence of the fluoride ion concentration on the non-stoichiometry of the oxide at constant potential is weak, with the non-stoichiometry in the presence of fluoride being lower at 0.4 V. In general, it can be concluded that in the domain of the primary passivation of tungsten in the studied solutions, a thin nonstoichiometric oxide is formed, through which intensive dissolution of tungsten occurs. A quantitative interpretation of these observations will be presented based on the kinetic model proposed below.

4. Discussion

4.1. Kinetic Model

For the interpretation of the experimental data, a recently proposed generalized quantitative model of the metal/film/electrolyte system was adapted and further developed [33,34,35,36,37]. According to the model, it is suggested that the growth of a thin nonstoichiometric oxide layer on tungsten in the primary passivation region is represented by a sequence of processes of generation (at the metal/film, M/F interface), transport, and consumption (at the film/solution, F/S) interface of oxygen vacancies (${\mathrm{V}}_{\mathrm{O}}^{••}$):

$$\begin{array}{l}{(\mathrm{M}/\mathrm{F})\mathrm{W}}_{\mathrm{m}}\stackrel{{\mathrm{k}}_{\mathrm{O}}}{\to }\left[{\mathrm{WO}}_3\right]\left({3\mathrm{V}}_{\mathrm{O}}^{••}\right){+6\mathrm{e}}_{\mathrm{m}}^{-}\\ (\mathrm{F}/\mathrm{S})\left[{\mathrm{WO}}_3\right]\left({3\mathrm{V}}_{\mathrm{O}}^{••}\right){+3\mathrm{H}}_2\mathrm{O}\stackrel{{\mathrm{k}}_{2\mathrm{O}}}{\to }{\mathrm{WO}}_3{+6\mathrm{H}}_{\mathrm{aq}}^{+}\\ {(\mathrm{F}/\mathrm{S})\mathrm{WO}}_3{+6\mathrm{H}}_{\mathrm{aq}}^{+}\stackrel{{\mathrm{k}}_{\mathrm{d}}}{\to }{\mathrm{W}}_{aq}^{6+}{+3\mathrm{H}}_2\mathrm{O}\end{array}$$

(1)

A stationary oxide thickness at a given potential is reached when the rates of layer formation at the M/F and dissolution at the F/S interface become equal. The nature of the dissolution products is ionic, and it can be assumed that they represent fluorotungsten ions of the type WO₃F₂²⁻.

The dissolution of tungsten through the oxide is limited by a complex reaction at the F/S interface featuring W(IV) and W(V) intermediate species:

$$\begin{array}{l}{{(\mathrm{F}/\mathrm{S})\mathrm{W}}_{\mathrm{W}}^{\mathrm{IV}}}^{″}\stackrel{{\mathrm{k}}_{21}}{\to }{{\mathrm{W}}_{\mathrm{W}}^{\mathrm{V}}}^{\prime }+\mathrm{e}\prime \\ {{(\mathrm{F}/\mathrm{S})\mathrm{W}}_{\mathrm{W}}^{\mathrm{V}}}^{\prime }\stackrel{{\mathrm{k}}_{22}}{\to }{\mathrm{W}}_{\mathrm{aq}}^{6+}{+\mathrm{e}\prime +\mathrm{V}}_{\mathrm{W}}^{6\prime }\end{array}$$

(2)

$$\begin{array}{l}{{(\mathrm{F}/\mathrm{S})\mathrm{W}}_{\mathrm{W}}^{\mathrm{V}}}^{\prime }\stackrel{{\mathrm{k}}_{31}}{\to }{\mathrm{W}}_{\mathrm{W}}^{\mathrm{VI}\ast }+\mathrm{e}\prime \\ {(\mathrm{F}/\mathrm{S})\mathrm{W}}_{\mathrm{W}}^{\mathrm{VI}\ast }\stackrel{{\mathrm{k}}_{32}}{\to }{\mathrm{W}}_{\mathrm{aq}}^{6+}{+\mathrm{V}}_{\mathrm{W}}^{6\prime }\end{array}$$

(3)

The tungsten cation vacancies (${\mathrm{V}}_{\mathrm{W}}^{6\prime }$), generated by reactions (2) and (3), are transported through the oxide by high field migration and consumed at the M/F interface via the formation of normal cation positions:

$${(\mathrm{M}/\mathrm{F})\mathrm{V}}_{\mathrm{W}}^{6\prime }+{\mathrm{W}}_{\mathrm{m}}\stackrel{k_W}{\to }{{\mathrm{W}}_{\mathrm{W}}^{\mathrm{IV}}}^{″}+4e^{\prime }$$

(4)

Furthermore, cation vacancies recombine with their oxygen counterparts according to the inverse Schottky reaction:

$${{(\mathrm{F}/\mathrm{S})3\mathrm{V}}_{\mathrm{O}}}^{••}{+\mathrm{V}}_{\mathrm{W}}^{6\prime }\to \mathrm{null}$$

(5)

Kroger–Vink notations for the point defects and substituted ions are used in the above equations.

To reduce the number of parameters in the model, it is assumed that the reactions at the M/F interface are not rate limiting; instead, their rates are adjusted by the transport of cations and oxygen vacancies in the forming oxide.

The material balance of cation vacancies at the F/S interface can be expressed by the variation in their surface charge (q_n) with time:

$${\displaystyle \frac{d{q}_n}{dt}}=I_{M,F/S}-I_M-I_{O}S{q}_n=I_{O}S\left({\displaystyle \frac{I_{M,F/S}-I_M}{I_{O}S}}-q_n\right),S={\displaystyle \frac{1}{F{\beta }_O}}$$

(6)

In that equation, I_{M, F/S,} and I_M are the instantaneous cation current densities at the interface and in the oxide; I_O is the oxygen current density; and β_O is the surface concentration of oxygen vacancies.

The dissolution scheme expressed by Equations (2) and (3) leads to the following equation for the current density of cationic vacancies at the film/electrolyte interface:

$${\displaystyle \frac{I_M}{6F}}=k_{22}{\gamma }_5+k_{32}{{\gamma }^{\ast }}_6$$

(7)

γ₅ and γ₆* are the surface fractions occupied by W(V) and W(VI) * reported for the cation sublattice. It is assumed that the surface fraction of cation vacancies is negligible compared to γ₅ and γ₆*. The material balances in the outermost cation layer are:

$${\displaystyle \frac{{\beta }_{M}d{\gamma }_4}{dt}}={\displaystyle \frac{I_M}{6F}}-k_{21}{\gamma }_4$$

(8)

$${\displaystyle \frac{{\beta }_{M}d{\gamma }_5}{dt}}=k_{21}{\gamma }_4-k_2{\gamma }_5-k_{31}{\gamma }_5$$

(9)

$${\displaystyle \frac{{\beta }_{M}d{{\gamma }_6}^{\ast }}{dt}}=k_{31}{\gamma }_5-k_{32}{{\gamma }_6}^{\ast }$$

(10)

At steady-state, (8), (9), and (10) transform into:

$${\displaystyle \frac{{\overline{I}}_{M,F/S}}{6F}}-{\overline{k}}_{21}{\overline{\gamma }}_4=0$$

(11)

$${\overline{k}}_{21}{\overline{\gamma }}_4-{\overline{k}}_{22}{\overline{\gamma }}_5-{\overline{k}}_{31}{\overline{\gamma }}_5=0$$

(12)

$$k_{32}{\overline{\gamma }}_5-{\overline{k}}_{31}{{\overline{\gamma }}_6}^{\ast }=0$$

(13)

$${\overline{\gamma }}_4+{\overline{\gamma }}_5+{\overline{\gamma }}_6^{\ast }+{\overline{\gamma }}_6=1$$

(14)

Here, ${\overline{\gamma }}_6$ is the surface fraction of WO₃, regular W(VI) sites at which chemical dissolution of the oxide takes place. To calculate the steady-state current density ${\overline{I}}_{M,F/S}$, Equations (11)–(14) are employed:

$${\overline{I}}_{M,F/S}={\displaystyle \frac{6F{k}_{21}{k}_{32}\left({\overline{k}}_{22}+{\overline{k}}_{31}\right)}{{\overline{k}}_{21}{\overline{k}}_{31}+{\overline{k}}_{32}\left({\overline{k}}_{21}+{\overline{k}}_{22}+{\overline{k}}_{31}\right)}}\left(1-{\overline{\gamma }}_6\right)={\overline{k}}_M\left(1-{\overline{\gamma }}_6\right)$$

(15)

The rate constants k₂₁, k₂₂, and k₃₁ are exponentially dependent on the potential drop at the film/solution interface:

$$k_{21}={k_{21}}^0\exp \left({\displaystyle \frac{{\alpha }_{21}\alpha F}{RT}}E\right),k_{22}={k_{22}}^0\exp \left({\displaystyle \frac{{\alpha }_{22}\alpha F}{RT}}E\right),k_{31}={k_{31}}^0\exp \left({\displaystyle \frac{{\alpha }_{31}\alpha F}{RT}}E\right)$$

(16)

where k_i⁰ (i = 21, 22, 31) are the rate constants at E = 0 V, α_i (i = 21, 22, 31) are the transfer coefficients of the electrochemical steps, and α is the part of applied potential consumed as a drop at the F/S interface.

At steady-state, the current density of cation vacancies in the oxide ${\overline{I}}_M$ is assumed proportional to $\left(1-{\overline{\gamma }}_6\right)$, i.e., the part of the surface that is not occupied by regular W(VI); whereas the oxygen-vacancy current density ${\overline{I}}_O$ flows at the fraction of the surface ${\overline{\gamma }}_6$:

$${\overline{I}}_O=2F{k}_d{c}_{H^{+}}^n{\overline{\gamma }}_6=2F{k_d}^{\prime }{\overline{\gamma }}_6=k_{W{O}_3}{\overline{\gamma }}_6$$

(17)

where n is the order of the chemical dissolution reaction with respect to the H⁺ concentration. If ${\overline{\gamma }}_6$ is a function of the rates of the consumption of anion and the generation of cation vacancies:

$${\overline{\gamma }}_6={\displaystyle \frac{k_{W{O}_3}}{{\overline{k}}_M+k_{W{O}_3}}}$$

(18)

then:

$${\overline{I}}_M={\displaystyle \frac{{{\overline{k}}^2}_M}{{\overline{k}}_M+k_{W{O}_3}}},{\overline{I}}_O={\displaystyle \frac{{k^2}_{W{O}_3}}{{\overline{k}}_M+k_{W{O}_3}}},\overline{I}={\overline{I}}_M+{\overline{I}}_O$$

(19)

The total impedance of the system is given by the transfer function:

$$Z=R_{el}+{\left[{\displaystyle \frac{1}{Z_e}}+{\displaystyle \frac{1}{Z_O}}+{\displaystyle \frac{1}{Z_{M,f}+Z_{M,F/S}}}\right]}^{-1}$$

(20)

The Faradaic impedance due to the generation of cation vacancies at the film/electrolyte interface is derived on the basis of Equations (7), (9) and (10) with β_M being the concentration of cation lattice sites at the F/S interface:

$$Z_{M,F/S}^{-1}=6F\left[{\tilde{k}}_{22}{\overline{\gamma }}_5+{\overline{k}}_{22}{\tilde{\gamma }}_5+k_{32}{\tilde{\gamma }}_6^{\ast }\right]$$

(21)

$$j\omega {\beta }_M{\tilde{\gamma }}_4={\displaystyle \frac{Z_{M,F/S}^{-1}}{6F}}-{\overline{k}}_{21}{\tilde{\gamma }}_4-{\tilde{k}}_{21}{\overline{\gamma }}_4$$

(22)

$$j\omega {\beta }_M{\tilde{\gamma }}_5={\overline{k}}_{21}{\tilde{\gamma }}_4+{\tilde{k}}_{21}{\overline{\gamma }}_4-{\overline{k}}_{22}{\tilde{\gamma }}_5-{\tilde{k}}_{22}{\overline{\gamma }}_5-{\tilde{k}}_{31}{\overline{\gamma }}_5-{\overline{k}}_{31}{\tilde{\gamma }}_5$$

(23)

$$j\omega {\beta }_M{\tilde{\gamma }}_6^{\ast }={\tilde{k}}_{31}{\overline{\gamma }}_5-{\overline{k}}_{31}{\tilde{\gamma }}_5-k_{32}{\tilde{\gamma }}_6^{\ast }$$

(24)

$$\begin{array}{l}{Z}_{M,F/S}^{-1}=6F{\displaystyle \frac{{\left(\right((\mathrm{X}}_4{-\mathrm{X}}_5{)\mathrm{k}}_{31}{+\mathrm{X}}_6{\mathrm{Z}}_5{)\mathrm{Z}}_4{-\mathrm{X}}_4{\mathrm{k}}_{21}{\mathrm{k}}_{31}{)\mathrm{k}}_{32}{+\left(\right((\mathrm{X}}_4{-\mathrm{X}}_5{)\mathrm{k}}_{22}+b_{22}{\mathrm{k}}_{22}{\overline{\gamma }}_5{\mathrm{Z}}_5{)\mathrm{Z}}_4{-\mathrm{X}}_4{\mathrm{k}}_{21}{\mathrm{k}}_{22}){\mathrm{Z}}_6}{{(\mathrm{Z}}_4{\mathrm{Z}}_5-{\mathrm{k}}_{21}{\mathrm{k}}_{22}{)\mathrm{Z}}_6{-\mathrm{k}}_{21}{\mathrm{k}}_{31}{\mathrm{k}}_{32}}}\\ {\mathrm{X}}_4=b_{21}{\mathrm{k}}_{21}{\overline{\gamma }}_4,{\mathrm{X}}_5=\left(b_{22}{\mathrm{k}}_{22}+b_{31}{\mathrm{k}}_{31}\right){\overline{\gamma }}_5,{\mathrm{X}}_6=b_{31}{\mathrm{k}}_{31}{\overline{\gamma }}_5,{\mathrm{Z}}_4=j\omega {\beta }_M+{\mathrm{k}}_{21},\\ {\mathrm{Z}}_5=j\omega {\beta }_M+{\mathrm{k}}_{22}+{\mathrm{k}}_{31},{\mathrm{Z}}_6=j\omega {\beta }_M+k_{32},{\overline{\gamma }}_4={\displaystyle \frac{{\overline{I}}_M}{6F{k}_{21}}},{\overline{\gamma }}_5={\displaystyle \frac{{\overline{I}}_M}{6F\left(k_{22}+k_{31}\right)}}\end{array}$$

(25)

The impedances of transport of cationic and anionic vacancies, considering the influence of point-defect recombination [33,34], are given by:

$${Z_{M,f}}^{-1}={\displaystyle \frac{12Fa{I}_M}{RT\overline{L}}}\left[(1-\alpha )+{\displaystyle \frac{I_{0}S\alpha }{j\omega +I_{0}S}}\right],{Z_{O,f}}^{-1}={\displaystyle \frac{4Fa{I}_0}{RT\overline{L}}}\left[(1-\alpha )+{\displaystyle \frac{I_{0}S\alpha }{j\omega +I_{0}S}}\right]$$

(26)

where $\overline{L}$ is the steady-state thickness at a given potential and a is the half-jump distance of a moving defect.

The impedance of the electronic properties of the oxide is approximated by the capacitance of the oxide C_b:

$$Z_e={\displaystyle \frac{1}{j\omega C_b}},C_b^{-1}={\displaystyle \frac{\overline{L}}{\epsilon {\epsilon }_0}},\overline{L}={\overline{L}}_{E=0}+{\displaystyle \frac{(1-\alpha )}{\overrightarrow{E}}}E$$

(27)

In the above equation, ${\overline{L}}_{E=0}$ is the steady-state thickness at E = 0 V, and ε is the dielectric constant of the oxide.

4.2. Comparison to Experimental Data

A three-step calculation procedure was adopted to estimate the kinetic, transport, and structural parameters of the proposed model:

(1)

The current–potential curves in all the electrolytes studied were fitted by non-linear regression with respect to Equation (19) to estimate the parameters k₂₁⁰, α₂₁, k₂₂⁰, α₂₂, k₃₁⁰, α₃₁, k₃₂, and k_d’.

(2)

At constant values of these parameters, a complex non-linear least squares fit of the impedance data to the transfer function expressed with Equations (20)–(27) was performed. As a result, the parameters $\overrightarrow{E}, \alpha , a, {\beta }_O, {\beta }_M$ were estimated. The values of the dielectric constant of the oxide (ε = 54) and its molar volume (31 cm³ mol⁻¹) were adopted from the literature.

(3)

Using the mean values of the parameter estimates from (1) and (2), a global fit of the impedance spectra in the whole interval of potentials was performed for a specific electrolyte composition. This ensured enough degrees of freedom in the system so that the parameter estimates were statistically viable and could be used for further discussion.

The current–potential curves stemming from the described calculation procedure are shown in Figure 1, whereas the calculated impedance spectra are in Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 with solid lines and illustrate the ability of the model to reproduce both the steady-state and the impedance response of the studied system. The main discrepancies between experimental and calculated data are observed in the high-frequency semicircle of impedance spectra, the experimental semicircles being somewhat more flattened than the calculated ones. This issue can be addressed by introducing a constant-phase-element behavior of the high-frequency capacitance, but since this would increase the number of adjustable parameters, at this stage of the investigation, it was judged to be premature.

Since the model contains a significant number of parameters, a sensitivity analysis was performed to demonstrate their significance. For this purpose, the parameters were given values ±10% away from the optimal result and the respective impedance spectra were calculated. Examples of such analyses are presented in Figure 10 (for the rate constants at the film/solution interface) and Figure 11 (for the parameters related to film growth, i.e., part of the potential consumed at the film/solution interface α, field strength in the oxide, half-jump distance a, and surface concentration of oxygen vacancies β_O).

It can be concluded that all the parameters have a significant effect on the values of impedance, and the model can be considered a robust one with respect to its parameters.

The dependences of the estimated parameters on fluoride concentration are collected in Figure 12. The following points can be emphasized based on the estimated parameter values:

• The rate constants of the individual stages of the tungsten dissolution reaction at the layer/solution boundary depend significantly on the fluoride ion concentration, except for the oxidation step 31 of substituted W(V) to W(VI) *. The influence of the fluoride ion is most significant on the rate constants of the stages that lead to the formation of soluble tungsten ions and cationic vacancies (stages 22 and 32 and the chemical dissolution of tungsten oxide characterized by a rate constant k_d); the apparent orders of these reactions are between 1.2 and 1.7. As the surface fluoride concentration is at or below the detection limit of XPS, it can be stated that the interaction between tungsten and fluoride ions takes place in the double layer.
• The transition coefficients of the individual stages of the layer/electrolyte boundary dissolution reaction have values between 0.35 and 0.50, i.e., the energy barriers of these stages are close to symmetric.
• The thickness of the oxide layers is very small (0.5–3.0 nm), is in agreement with the data from XPS analyses, and increases linearly with the potential as first approximation, confirming the assumption of the independence of the electric field strength from the potential. The thickness of the layer at constant potential is practically independent of the concentration of fluoride ions, i.e., fluoride ions do not take part in oxide growth, according to the assumption of the model.
• The field strength increases with fluoride ion concentrations from 0.01 to 0.025 M and then slightly decreases, the total variation being < 15%. An analogous conclusion can be drawn for the half-jump distance, as for this parameter, the total change is below 18%. In general, there is some influence of the fluoride ion on the electrical properties of the oxide, but it cannot be deemed overtly significant.
• The surface concentration of anion vacancies increases with the concentration of the fluoride ion, reaching saturation at about 0.05 M. It is important to note that this concentration is always much lower than the concentration of cationic positions in the outermost layer of the oxide. The latter does not depend significantly on the fluoride ion content, which can be taken as an indirect confirmation of the hypothesis that the interaction between tungsten and fluoride ions is not at the oxide surface but in the double electrical layer in which the formation of fluoride-containing tungstate ions can be hypothesized.

The passive state of tungsten in similar electrolytes (range of potentials 4–10 V) under illumination of the electrode with ultraviolet light was studied in a companion paper [31]. In that paper, a simplified version of the model used in the present work (without the contribution from W(IV) species) was compared to EIS and intensity modulated photocurrent spectroscopic (IMPS) measurements complemented with optical gap determination using energy spectroscopy of the modulated photocurrent. Based on the results obtained in these two studies, the synthesis and characterization of thick, porous anodic oxides on W as prospective photo-absorbers and photo-catalysts will be attempted.

5. Conclusions

In the present paper, the first stages of the anodic oxidation of tungsten in sulfate–fluoride solutions formation of soluble ions (active dissolution) and oxide growth (primary passivation) are investigated using a combination of in situ electrochemical techniques (voltammetry, spectro-electrochemistry, and electrochemical impedance spectroscopy) coupled with surface composition estimation using ex situ XPS. Particularly, spectro-electrochemical measurements of W dissolution in the presence of a surface oxide are (to the best of the authors’ knowledge) reported for the first time in the literature. This combination of techniques that allows for the discrimination of the formation of soluble and solid products of anodic oxidation has enabled the development of a quantitative kinetic model of the processes. The model is validated by a quantitative comparison of all the obtained data and parameterized by a complex non-linear regression of the impedance spectra to the respective transfer function in a wide range of potentials. On the basis of the experimental results and supporting calculations, a hypothesis of the important role of a W(IV) species present at the oxide/solution interface in promoting active dissolution can be advanced, which represents a significant progress in the fundamental understanding of tungsten electrochemistry. In addition, based on the obtained results, it can be stated that the interaction between tungsten and fluoride ions takes place in the double layer without significant incorporation of these ions in the oxide. Of course, a fully quantitative interpretation of the spectro-electrochemical and XPS data has eluded us so far due to the lack of appropriate calibrations for different dissolved W(VI) species. Further investigations using scanning electrochemical microscopy in both dc and ac modes are in progress to further substantiate the ability of the model to predict the spatial separation of dissolution and passivation reactions. The results of these studies will be communicated shortly.