Electro-chemo-mechanical properties of anodic oxide (passive) films formed on Cu, Ni and Fe

2.1 Oxygen adsorption from gas phase

Figure 1 shows the typical surface stress variation due to oxygen adsorption on single crystal Ni (111) from gas phase: the value of surface stress g is plotted as a function of oxygen coverage θ at 77 °C (Grossmann et al. 1995; Ibach 1997). It is noted that g in the ordinate of Figure 1 is referred to zero at θ = 0 since an absolute value of g cannot be measured by the cantilever bending method (Ibach 1997; Láng and Barbero 2012; Seo 2020). The surface stress g decreases almost linearly toward compressive direction with increasing θ up to about 0.25. Single oxygen atom occupies the face-centered cubic (fcc) three-fold hollow site on Ni (111) to form a p (2 × 2)structure of the oxygen overlayer at θ = 0.25. The g versus θ curve of Figure 1 deviates downward from the linearity at about θ = 0.3, which is responsible for the phase transition of the oxygen overlayer from a p(2 × 2) to a (3×3)R30° structure. The oxygen coverage corresponding to the (3×3)R30° structure is θ = 0.33.

Figure 1:

Typical surface stress variation due to oxygen adsorption on single crystal Ni (111) from gas phase: the value of g is plotted as a function of oxygen coverage θ at 77 °C for oxygen adsorbed on single crystal Ni (111) (Grossmann et al. 1995; Ibach 1997). The value of g in the ordinate of Figure 1 is referred to zero at θ = 0. Copyright ©1997. Reproduced with permission from Elsevier.

The fcc three-fold hollow site on Ni (111) is the most stable for single oxygen atom adsorption. The oxygen binding energy (−2.18 eV) of the hexagonal closest packing (hcp) three-fold hollow site on Ni (111) is less than the oxygen binding energy (−2.26 eV) of the fcc three-fold hollow site (Olatunji-Ojo and Taylor 2013). The structural analysis of the oxygen overlayer on Ni (111) (Schwennicke and Pfnür 1996), however, has indicated that the oxygen atoms partially reside in the hcp three-fold hollow sites at oxygen coverages between θ = 0.25 and 0.33. The downward deviation of the g versus θ curve from the linearity at about θ = 0.3 in Figure 1 is attributed to the oxygen atoms residing in the hcp three-fold hollow sites (Ibach 1997). It is deduced that the oxygen atoms in the hcp three-fold hollow sites contribute largely to the surface stress variation toward compressive direction as compared to the oxygen atoms in fcc three-fold hollow sites (Ibach 1997).

The surface stress varies toward compressive direction due to oxygen adsorption on Ni (100) as well as on Ni (111) (Grossmann et al. 1995; Ibach 1997). Single oxygen atom occupies the four-fold hollow site on Ni (100) to form a c (2 × 2) structure of the oxygen overlayer at θ = 0.5. The surface stress variation measured at 77 °C for the c (2 × 2) oxygen overlayer (θ = 0.5) on Ni (100) is Δg = −5.35 J/m² which is significantly larger than that (∆g = −1.15 J/m²) for the (3×3)R30° oxygen overlayer (θ = 0.33) on Ni (111) (Grossmann et al. 1995; Ibach 1997). The surface stress variation measured at 300 K for the c (2 × 2) oxygen overlayer (θ = 0.5) on Cu (100) is ∆g = −1.0 J/m² (Harrison et al. 2006), which is much less than that for the c (2 × 2) oxygen overlayer on Ni (100).

Although the cantilever bending method is unable to measure an absolute value of the surface stress of oxygen-overlayered surface as well as clean metal surface, its absolute value can be estimated from first-principles calculations based on density-functional theory (DFT) (Feibelman 1997; Harrison et al. 2006; Hong et al. 2004). Table 1 shows the comparison between experimental and theoretical values of the surface stress variations Δg due to the formation of oxygen overlayers on single crystals Cu, Ni and Pt from gas phase. The values of Δg in Table 1 are taken as the difference between the oxygen-overlayered and clean metal surfaces. For Cu (100), the theoretical value of ∆g is larger than the experimental value, while for Ni (100), the situation is reverse. At present, the causes listed for the above reverse situation have not been made clear (Harrison et al. 2006; Hong et al. 2004). Nevertheless, it is clear from Table 1 that the formation of oxygen overlayers on single crystal metals induces the surface stress variation toward compressive direction.

Since oxygen is electronegative with respect to the substrate metal, the oxygen adsorption relaxes the tensile stress and leads to the surface stress variation toward compressive direction. Carbon is a strong adsorbate for Ni (100) as compared with oxygen. It has been observed that the carbon adsorption on Ni (100) induces the surface reconstruction from a c (2 × 2) to a (2 × 2) p4g structure, in which the nearest neighbor distance of Ni atoms becomes larger in the reconstructed phase (Hong et al. 2004; Ibach 1997). The surface stress variation toward compressive direction due to the carbon adsorption on Ni (100) is initially much steeper than that due to the oxygen adsorption on Ni (100) (Ibach 1997). The surface stress variation, however, levels off to nearly constant at a coverage of θ ≈ 0.3, corresponding to the onset of the reconstruction (Ibach 1997). The level-off of the surface stress variation means the relaxation of the compressive stress. The increase in the nearest neighbor distance of Ni atoms due to the surface reconstruction induces the surface stress variation toward tensile direction and contributes to the relaxation of the compressive stress. Any surface reconstructions, however, have not been observed for the oxygen adsorption on Ni (100) and Ni (111) from gas phase at room temperature (Harrison et al. 2006; Ibach 1997).

By contrast, it has been reported that for the oxygen adsorption on Cu (100) from gas phase, the surface reconstruction of the oxygen overlayer from a c (2 × 2) to a (2×22)R45° structure takes place at room or elevated temperature (Harrison et al. 2006; Jensen et al. 1990; Kittel et al. 2001; Robinson et al. 1990). In the (2×22)R45° structure, every fourth [010] atomic row of the surface layer Cu atoms is missing and the nearest neighbor distance of the surface Cu atoms perpendicular to the missing rows (i.e. along the 22 direction) becomes large as compared with that in the unreconstructed phase (Harrison et al. 2006). As shown in Table 1, for the oxygen overlayers on Cu (100), both the experimental and theoretical values of Δg for the (2×22)R45° structure are less than those for the c (2 × 2) structure (Harrison et al. 2006), indicating that the missing row-surface reconstruction induces the surface stress variation toward tensile direction to relax the compressive stress, which is brought by the increase in the nearest neighbor distance of the surface layer metal atoms as well as the (2 × 2) p4g surface reconstruction for the carbon adsorption on Ni (100).

2.2 OH adsorption from aqueous solution

Figure 2 shows the voltammogram (A) and the volt-stressogram (g vs. E curve) (B) measured during a potentiodynamic polarization at a potential sweep rate of 10 mV/s in deaerated pH 8.4 borate buffer solution at 25 °C for the Cu thin film electrode with a thickness of 300 nm (prepared on a glass plate by magnetron sputtering) (Kim and Seo unpublished data). The value of g in the ordinate of Figure 2B is referred to zero at a cathodic limit potential of −0.8 V. The characteristic current peak positions in the voltammogram of Figure 2A are very close to those in the voltammogram of the single crystal Cu (111) electrode (Kunze et al. 2001, 2003a; Marcus and Maurice 2017; Maurice et al. 2000; Strehblow et al. 2001) measured in 0.1 M NaOH solution if the potential shift by 0.27 V due to the pH difference between two solutions is corrected. The in-situ STM study (Maurice et al. 2000) has revealed that OH adsorption, i.e. OH-underpotential deposition (UPD) on Cu (111) in 0.1 M NaOH solution proceeds, accompanying a surface reconstruction, in the potential region between −0.65 and −0.25 V, more negative than the oxide formation potential. On the other hand, Kunze et al. (2003b) have reported that OH-UPD on Cu (100) in 0.1 M NaOH solution proceeds in the potential region between −0.81 and −0.21 V.

Figure 2:

Voltammogram (A) and volt-stressogram (g vs. E curve) (B) measured during a potentiodynamic polarization at a potential sweep rate of 10 mV/s in deaerated pH 8.4 borate buffer solution at 25 °C for the Cu thin film electrode with a thickness of 300 nm (prepared on a glass plate by magnetron sputtering) (Kim and Seo unpublished data). The value of g in the ordinate of Figure 2B is referred to zero at a cathodic limit potential of −0.8 V.

The onset potential of the OH-UPD on Cu (100) is more negative by 0.16 V than that of the OH-UPD on Cu (111). In addition, the potential window (0.6 V) of the OH-UPD on Cu (100) is much wider than that (0.4 V) of the OH-UPD on Cu (111) (Kunze et al. 2003a,b; Maurice et al. 2000). As shown in the volt-stressogram of Figure 2B, g decreases toward compressive direction in the anodic potential sweep from −0.80 V, exhibiting a small pause at −0.35 V corresponding to the onset of the OH-UPD, and subsequently decreases rapidly up to 0.15 V. When the anodic potential sweep exceeds 0.15 V, g increases conversely toward tensile direction. The increase in g in the potential region more positive than 0.15 V corresponds to the growth of anodic oxide films on Cu, which will be discussed in the next section. The onset potential (−0.35 V) of the OH-UPD in Figure 2A almost coincides with that (−0.38 V) obtained by the correction of the pH difference for the results of Cu (111) in 0.1 M NaOH solution (Maurice et al. 2000). Furthermore, the potential window (0.5 V) of the OH-UPD in Figure 2A is wider that that (0.4 V) for the results of Cu (111) (Maurice et al. 2000), while it is narrower that that (0.6 V) for the results of Cu (100) (Kunze et al. 2003b).

Okolo et al. (2005) have reported that the magnetron-sputter-deposited Cu thin film (with a thickness of 500 nm) on Si₃N₄ substrate consists of a columnar microstructure with a more pronounced {111} fiber texture. The above similarity between Figure 2A and the results of Cu (111) (Maurice et al. 2000) suggests that the Cu thin film used for the experiments of Figure 2 is mainly oriented to {111} although no structural investigation was not made for the Cu thin film. The surface stress variation Δg = −0.5 J/m² in the potential range between −0.35 and 0.15 V in Figure 2B means that the surface stress varies toward compressive direction due to the OH-UPD on Cu (111). The clear STM images of the OH-UPD structure on Cu (111) in 0.1 M NaOH solution (Kunze et al. 2001, 2003a; Marcus and Maurice 2017; Maurice et al. 2000; Strehblow et al. 2001) have indicated that the OH-UPD reconstructs the topmost Cu plane: the adsorbed OH groups occupy the three-fold hollow sites of the reconstructed Cu plane to form a p (2 × 2) superstructure which has a hexagonal lattice with a unit cell parameter of 0.6 ± 0.02 nm and mimics the Cu and O sublattices in Cu₂O (111). The nearest neighbor distance between Cu atoms in the sub-lattice of this superstructure is 0.3 ± 0.01 nm, which is larger than that (0.256 nm) in the unreconstructed Cu (111) plane. The chemical bond of OH with surface metal atom induces the surface stress variation toward compressive direction if OH is electronegative with respect to the substrate metal. The increase in the nearest neighbor distance between Cu atoms due to the surface reconstruction, however, induces the surface stress variation toward tensile direction. As a result, the final direction of the surface stress variation due to the OH-UPD will be determined by the competition between the component (compressive direction) due to the chemical bond of OH with surface Cu atom and the component (tensile direction) due to the surface reconstruction. The discussion on magnitude of the above two components is valuable for better understanding of the surface reconstruction.

Figure 3 shows the atomic models of (A) the reconstructed p (2 × 2) superlattice structure of the OH-UPD layer on Cu (111) in 0.1 M NaOH solution and (B) the unreconstructed structure of the bulk Cu (111) plane (Kunze et al. 2003a). Let us estimate roughly the two components of the surface stress variation on basis of the atomic model in Figure 3A. The net surface stress variation Δg_net may be expressed by

where Δg_ad is the component caused by the change in adsorption energy of OH with an elastic deformation of the substrate Cu (111) surface (Levia et al. 2000), and Δg_rec is the component caused by an elastic stretching of the substrate surface due to the reconstruction. The sign of the first term Δg_ad in the right-hand side of Equation (3) should be negative because an elastic stretching of the substrate metal surface enhances an adsorption energy (Gsell et al. 1998; Jakob et al. 2001; Mavrikakis et al. 1998).

Figure 3:

Atomic models of (A) the reconstructed p (2 × 2) superlattice structure of the OH-UPD layer on Cu (111) in 0.1 M NaOH solution and of (B) the unreconstructed structure of the bulk Cu (111) plane (Kunze et al. 2003a). Copyright ©2003. Reproduced with permission from Elsevier. All rights reserved.

Assuming linear elasticity, the second term Δg_recin the right-hand side of Equation (3) is formulated as follows:

where Y is the biaxial elastic modulus, E is Young’s modulus, ν is Poisson’s ratio, ε is the elastic strain, and d_mono is the monolayer thickness (corresponding to the diameter of Cu atom, i.e. d_mono = 0.256 nm) for the Cu (111) plane. The values of E = 130 GPa and ν = 0.5 (and thus, Y = 260 GPa) of the Cu (111) plane can be calculated by using the transformation relationships (Brantley 1973; Ibach 2006) between the compliance values (Brandes and Brook 1992) published for the bulk cubic crystal Cu.

The elastic strain ε of the Cu (111) plane is given by

where a_n, r is the nearest neighbor distance between Cu atoms in the sub-lattice of the reconstructed phase and a_n,o is that in the unreconstructed Cu (111) plane. The tensile elastic strain ε = 0.172 is obtained from the values of a_n, r = 0.3 nm and a_{n, o} = 0.265 nm by using Equation (5). The substitution of Y = 260 GPa, ε = 0.172, and d_mono = 0.256 nm into Equation (4) leads to an extraordinarily large value of Δg_rec = 11.4 J/m². In Equation (4), it is supposed that the biaxial elastic stretching of the Cu (111) plane due to the reconstruction proceeds without any ejection of Cu atoms from the lattice sites followed by migration to the step edges.

Atomically resolved STM images in the OH-UPD potential region for Cu (111) in 0.1 M NaOH solution (Kunze et al. 2001, 2003a; Marcus and Maurice 2017; Maurice et al. 2000; Strehblow et al. 2001) have revealed that Cu atoms are ejected during the reconstruction and they are accumulated at the step edges to form Cu ad-islands on the terraces. Such dynamic processes of Cu atoms would reduce the biaxial modulus or induce the downward deviation from the linear elasticity in Equation (4). Molecular dynamic (MD) simulations performed for a Cu (111) plane using embedded atom method potentials (Trimble et al. 2005) have indicated that the surface stress variation associated with the biaxial stretching of a Cu (111) monolayer deviates downward from a linearity when ε exceeds 0.01. The extrapolation of the simulated Δg versus ε curve (Trimble et al. 2005) to ε = 0.172 leads to a value of Δg_rec = 3.0 J/m², which is only one fourth of that (Δg_rec = 11.4 J/m²) obtained from the linear elasticity in Equation (4).

Employing Δg_net = −0.5 J/m² in Figure 2B and Δg_rec = 3.0 J/m², Δg_ad = −3.5 J/m² is estimated from Equation (3). The positive value of Δg_rec means that the surface stress variation toward compressive direction caused by the OH adsorption (i.e. OH-UPD) on the Cu (111) plane is relaxed by the surface reconstruction. The monotonous decrease in g from −0.8 to −0.35 V prior to the OH-UPD in Figure 2B may result from the surface charging in the electric double layer region of Cu (111) electrode (Umeno et al. 2007; Weissmüller 2013) although not excluding the influence of hydrogen evolution on Cu at potentials more negative than −0.5 V. The potential of zero charge E_pzc of Cu (111) in pH 8.4 borate buffer solution would be −0.50 V as expected from E_pzc = −0.24 V of Cu (111) in KClO₄ solution of pH 4.0 (Lecoeur and Bellier 1985). A surface tension takes a maximum at E_pzc, while a surface stress of a solid electrode does not always take a maximum within the measured potential range (Kramer 2008; Schmickler and Levia 1998).

According to in situ AFM study of Cu (100) in 0.1 M H₂SO₄ and 0.1 M HClO₄ solutions (Cruickshank et al. 1993), chemisorbed oxygen or OH occupies the four-fold hollow site to form a c (2 × 2) superlattice, which is same as that for the oxygen adsorption on Cu (100) from gas phase. In contrast, in situ STM study of Cu (100) in 0.1 M NaOH solution (Kunze et al. 2003b) has shown that a surface reconstruction takes place in the OH-UPD potential region of Cu (100): Cu atoms are ejected from the Cu (100) plane to form dimers of superimposed Cu atoms which is stabilized by OH groups probably located in bridging positions, and finally c (2 × 6) and c (6 × 2) superstructure domains are formed as a result of zig-zag arrangement of the dimers. The adsorption energy of oxygen or OH groups on Cu (100) should be larger than that on Cu (111) (Kunze et al. 2003b), which may lead to a large negative value of the first term Δg_ad in the right-hand side of Equation (3). Although the mechanism of the surface reconstruction of Cu (100) is different from that that of Cu (111), the surface reconstruction of Cu (100) would play an important role in relaxing the surface stress variation toward compressive direction caused by adsorption of OH groups.

In situ STM study of Ni (111) in 10⁻³ M or 1 M NaOH solution (Marcus and Maurice 2017; Seyeux et al. 2005, 2006) has revealed the local formation of ordered p (2 × 2) islands of limited lateral extension (∼2 nm) on the Ni (111) terraces prior to OH adsorption-induced surface reconstruction followed by the formation of a two-dimensional passive layer at higher potential. The stability of the ordered p (2 × 2) islands on the Ni (111) terraces may be explained in terms of co-adsorption of mobile H₂O and OH^δ− (0 < δ < 1) species because of weak adsorption of OH^δ− on Ni (111) plane as compared with OH adsorption on Cu (111) plane (Seyeux et al. 2006). The weak adsorption of OH^δ− on Ni (111) plane and the local formation of p (2 × 2) islands on the Ni (111) terraces may decrease both magnitudes of the first and second terms in the right-hand side of Equation (3). The surface stress variation of (111)-textured Ni thin film in the potential region of OH adsorption has not been measured yet. In future, the experimental and theoretical investigations of the relationships between surface stress variation and OH-induced surface reconstruction on single metal surfaces will be indispensable for better-understanding of the stability of OH adlayer as a two-dimensional precursor for a three-dimensional passive film.

3.1 Anodic oxide films on Cu and Ni

The increase in g with the anodic potential sweep in the potential region more positive than 0.15 V in Figure 2B is associated with the growth of anodic oxide films on Cu. The broad anodic current peaks at 0.3–0.4 V and at 0.7–0.8 V in the voltammogram of Figure 2A are responsible for the formation of a duplex oxide film consisting of Cu₂O as an inner layer and CuO or Cu (OH)₂ as an outer layer, respectively. The volt-stressogram in Figure 2B deviates downward from the linearity at 0.5–0.6 V as shown by dotted lines, which results from the nucleation of CuO or Cu (OH)₂ at the Cu₂O/solution interface. The linearity in the volt-stressogram, however, is recovered in the potential region more positive than 0.6 V where the growth of inner and outer layers of the anodic oxide film proceeds simultaneously.

As shown in Figure 2B, the film growth in the anodic potential sweep from 0.2 to 1.2 V induces a stress variation of Δg = 1.7 J/m². A stress variation Δg during film growth is given by

where σ_f (J/m³ or Pa) is the internal stress generated in the film and d_f is the film thickness. Although σ_f may depend ond_f, Equation (6) is approximated by

where 〈σ_f〉 is the average internal stress in the film (i.e. film stress) and Δd_f is the increase in film thickness. The value of Δd_f for the anodic oxide film formed on Cu in the anodic potential sweep from 0.2 to 1.2 V can be estimated from the electric charge densities corresponding to the areas of the cathodic current peaks Q₁ and Q₂ (see Figure 2A) appeared for the film reduction in the cathodic potential sweep from 1.2 to −0.8 V. At Q₁, the outer layer CuO or Cu(OH)₂ is reduced to Cu₂O, while at Q₂, the inner layer Cu₂O in addition to Cu₂O reduced in the outer layer is reduced to metallic Cu. The electric charge densities are Q₁ = 10.0 C/m² and Q₂ = 22.4 C/m², respectively. The increases in thickness of the outer layer Δd_f,1 and of the inner layer Δd_f,2 are given by

and

where M₁ or M₂ is the molecular weight (g/mol) of the outer or inner layer, ρ₁ or ρ₂ is the density (g/cm³) of the outer or inner layer, and F is the Faraday constant (96,485 C/mol). Employing M₁ = 79.55 or 97.56 g/mol and ρ₁ = 6.31 or 3.37 g/cm³ for CuO or Cu(OH)₂ (Kunze et al. 2001), respectively, Δd_f,1 = 1.31 or 3.00 nm is estimated from Equation (8). Similarly, employing M₂ = 143.1 g/mol and ρ₂ = 6.04 g/cm³ for Cu₂O (Kunze et al. 2001), Δd_f,2 = 1.52 nm is estimated from Equation (9). If the outer layer is regarded as a mixture of CuO and Cu(OH)₂ (Seo et al. 1988), Δd_f,1 = 2.16 nm is pertinent as an average value. Substituting Δg = 1.7 J/m² and Δd_f = 3.68 nm into Equation (7), 〈σ_f〉 ≈ 460 MPa is approximately estimated as an average internal stress generated in the anodic oxide film on Cu.

Figure 4 shows the anodic current density i versus time t (A) and the stress g versus t (B) curves measured during anodic polarization at 0.8 V in deaerated pH 8.4 borate buffer solution at 25 °C for the electroplated Cu thin film electrode with a thickness of 800 nm (Seo and Hagioi 2007). As shown in Figure 4B, the stress evolves toward tensile direction and the stress variation during the film growth for 1 h is Δg = 1.8 J/m². Xiang et al. (2002) have reported that electroplated Cu thin films have a pronounced {111} texture. After the anodic oxidation for 1 h at 0.8 V in Figure 4A, the cathodic reduction of the anodic oxide film performed at a constant current density of i_c = 0.1 A/m² provided Q₁ = 9.92 C/m² and Q₂ = 20.8 C/m² from which Δd_f,1 = 2.13 nm as an outer layer consisting of CuO and Cu(OH)₂ mixture and Δd_f,2 = 1.34 nm as an inner Cu₂O layer were derived by using Equations (8) and (9). As a result, 〈σ_f〉 ≈ 520 MPa is eventually obtained as an average internal stress generated in the anodic oxide film on Cu, which is the same orders of magnitude as that (〈σ_f〉 ≈ 460 MPa) estimated from Figure 2B.

Figure 4:

Anodic current density i versus time t (A) and stress g versus t (B) curves measured during anodic polarization at 0.8 V in deaerated pH 8.4 borate buffer solution at 25 °C for the electroplated Cu thin film electrode with a thickness of 800 nm (Seo and Hagioi 2007). Copyright ©2006. Reproduced with permission from Elsevier. All rights reserved.

It is known that metal thin films prepared by evaporation or sputtering are usually subjected to residual stresses (D’Heurle and Harper 1989). A residual stress σ_mof the substrate metal is released due to a decrease in substrate thickness (i.e. consumption of the substrate metal) caused by active dissolution of metal and growth of anodic oxide film during anodic polarization. If σ_m is tensile, a decrease in substrate thickness leads to an apparent decrease in 〈σ_f〉, while if σ_m is compressive, it leads to an apparent increase in 〈σ_f〉. It has been reported that the residual stresses of Cu thin films electroplated on a Si wafer are tensile: σ_m ≈ 70 MPa (Hwang et al. 2008) for a thickness of 1 μm and σ_m = 19.3 MPa (Zhou et al. 2004) for a thickness of 9.4 μm, while the residual stresses of Cu thin films sputtered onto an amorphous SiO₂ are compressive: σ_m = −94 to −134 MPa (Okolo et al. 2005), depending on argon gas pressure for a thickness of 0.5 μm. The effect of σ_m on 〈σ_f〉 may be minor because the value of 〈σ_f〉 for the sputter-deposited Cu thin film is the same orders of magnitude as that for the electroplated Cu thin film, irrespective of the residual stress in the Cu substrate being tensile or compressive.

The growth of a barrier type of anodic oxide films on metals (Cabrera and Mott 1948–1949) is promoted by a high electric field with an order of 10⁸ V/m across the film. The high electric field across the film exerts a stress component normal to the film plane due to coulombic attraction between charges of opposite sign located on both sides of the film. The stress component normal to the film plane is converted to the in-plane stress σfel parallel to the film plane, i.e. the electrostriction stress in the film since the film is mechanically constrained by the metal surface. Consequently, the electrostriction stress σfel is included in 〈σ_f〉 as far as the high electric field is sustained by the film. The intrinsic film stress 〈σfi〉 in the absence of the electric field can be obtained by subtracting σfel from 〈σ_f〉:

If dealing with an electrostatic force based on a simple-parallel capacitor model (Butler and Ginley 1988; Sahu et al. 1990), σfel is given by

where ν_f is Poisson’s ratio of the film, ϵ₀ (8.854 × 10⁻¹² F/m) is the vacuum permittivity, ϵ_f is the relative dielectric constant of the film, and Δϕ_f is the potential difference across the film. The term (Δϕfdf) in the right-hand side of Equation (11) corresponds to the electric field E‾ perpendicular to the film plane. It is reminded that the film thickness d_f in Equation (11) is referred to zero at an onset-potential of film formation. It is assumed that E‾ is uniform through the film. The minus sign of Equation (11) indicates that σfel is compressive in the film geometry due to coulometric attraction along the film thickness. Sato (1971) has proposed that a local breakdown of passive films (i.e. initiation of pitting corrosion) results from mechanical deformation induced by combination of an electrostriction stress with a decrease in surface tension of the film due to chloride adsorption on the film. In Equation (11), however, a dielectrostriction term is neglected. Vanhumbeeck and Proost (2007, 2008b) have pointed out that the large difference between σfel values calculated by using Equation (11) and obtained experimentally for the anodic oxide film on Ti is caused by neglect of a dielectrostriction term in Equation (11). Dielectrostriction is defined as changes of dielectric properties of a material with deformation (Peng et al. 2005). Deformation of the anodic oxide film due to an applied electric field affects the dielectric constant of the film since the diploes in the film are aligned along the direction of the applied electric field.

If the contribution of dielectrostriction is taken into consideration (Lee et al. 2005; Shkel and Klingenberg 1998), σfel can be expressed by

where α₁ and α₂ are the electrostriction parameters due to changes in film thickness and its volume, respectively. Furthermore, in the case of an isotropic film, α₁ and α₂ can be expressed as a function of ϵ_f (Lee et al. 2005; Shkel and Klingenberg 1998):

and

Substituting Equations (13) and (14) into Equation (12), σfel (Vanhumbeeck and Proost 2007, 2008b) is eventually represented by

The value of ϵ_f = 54 ± 4 calculated from Equation (15) by using the experimental values of σfel for the anodic oxide film (ν_f = 0.25) on Ti was statistically equal to that (55 ± 6) obtained independently by impedance measurements (Vanhumbeeck and Proost 2008b).

Let us calculate σfel by using Equation (15) in order to evaluate the contribution of σfel to 〈σ_f〉 for the anodic oxide film on Cu. For the calculation of σfel, ν_f = 0.454 (Hallberg and Hanson 1970) and ϵ_f = 7.11 (Switzer et al. 1998) of Cu₂O are employed as a representative of the anodic oxide film on Cu since ν_f and ϵ_f of CuO or Cu(OH)₂ are unknown. The electric field E‾=Δϕfdf is 2.85 × 10⁸ V/m for the anodic oxide film on sputter-deposited Cu in Figure 2 and it is 1.87 × 10⁸ V/m for the anodic oxide film on electroplated Cu in Figure 4. If E‾ = 2.4 × 10⁸ V/m is employed as an average value, σfel = −1.5 and −7.6 MPa are obtained, respectively, from Equations (11) and (15), indicating that the contribution of σfel to 〈σ_f〉 for the anodic oxide film on Cu is less than 2 % of 〈σ_f〉, even if the effect of dielectrostriction on σfel is taken into consideration. From the above discussion, it is drawn that the intrinsic film stress 〈σfi〉 of the anodic oxide film on Cu is around 500 MPa.

Figure 5 shows the anodic current density i versus time t (A) and the stress g versus t (B) curves measured during anodic oxidation at 0.8 V in deaerated pH 8.4 borate buffer solution at 25 °C for the Ni thin film electrode with a thickness of 300 nm (prepared on a glass plate by magnetron sputtering) (Kim and Seo 2003). As shown in Figure 5B, the stress evolves toward tensile direction and the stress variation during anodic oxidation for 1 h is Δg = 1.3 J/m². The comparison between Figures 4A and 5A indicates that the passivation of Ni is very fast as compared with that of Cu in deaerated pH 8.4 borate buffer solution at 25 °C. According to the ellipsometric study of anodic oxide films on Ni in deaerated pH 8.4 borate buffer solution at 25 °C (Sato and Kudo 1974), the thickness of the anodic oxide film on Ni after 1 h-anodic oxidation at 0.8 V is about 1 nm. If Δd_f = 1 nm is employed, 〈σ_f〉 ≈ 1.3 GPa for the anodic oxide film on Ni is estimated from Equation (7), which is larger by a factor of 2.6 than that for the anodic oxide film on Cu. The residual stress σ_mof the Ni thin film electrode was not measured in the experiments of Figure 5 (Kim and Seo 2003). Nevertheless, Windischmann (1987) has reported that the residual stress of the Ni thin film prepared with magnetron sputtering is σ_m ≈ −300 MPa.

Figure 5:

Anodic current density i versus time t (A) and stress g versus t (B) curves measured during anodic oxidation at 0.8 V in deaerated pH 8.4 borate buffer solution at 25 °C for the Ni thin film electrode with a thickness of 300 nm (prepared on a glass plate by magnetron sputtering) (Kim and Seo 2003). Copyright ©2003. Reproduced with permission from IOP Publishing Ltd. All rights reserved.

By provisionally employing σ_m ≈ −300 MPa, let us estimate the contribution of σ_m to 〈σ_f〉 due to the decrease in thickness (consumption) of the substrate metal caused by the dissolution and film formation during anodic oxidation of Ni. The relationship between σ_m and 〈σ_f〉 is represented by

where 〈σ_f,c〉 is the film stress after the correction of the residual stress and Δd_m (<0) is the decrease in thickness of the substrate metal. In Equation (16), it is assumed that σ_m is uniform in depth-direction of the substrate metal. Furthermore, Δd_m is expressed by

where ∆W_m (g/cm²) is the dissolution amount of the substrate metal, ρ_m (g/cm³) is the density of the substrate metal and α_PB is the Pilling–Bedworth ratio (Pilling and Bedworth 1923) that represents the relative volume change per the substrate metal atom due to oxidation of the metal. Furthermore, α_PB is given by

where x is the stoichiometric number of the metal component in the film (Me_xO_y), V_f is the molar volume of the film, V_m is the molar volume of the substrate metal, M_f is the molecular weight of the film, M_m is the atomic weight of the substrate metal and ρ_f is the density of the film.

The second term (−ΔdfαPB) in the right-hand side of Equation (17) has been taken into consideration for the stress evolution during anodic oxidation of Ti thin film (Vanhumbeeck and Proost 2008). The first term, however, was not taken into consideration under the assumption of negligibly small dissolution of the substrate Ti (Vanhumbeeck and Proost 2008). In the case of passivation of Ni as well as Fe, the first term (−∆Wmρm) in addition to the second term in the right-hand side of Equation (17) must be taken into consideration for the correction of the residual stress since Ni is subjected to active dissolution prior to passivation and subsequently to slight dissolution through the passive film. If ∆W_m = 3.2 × 10⁻⁷ g/cm² and ρ_m = 8.845 g/cm³ are employed from the results of anodic oxidation of Ni at 0.8 V for 1 h in pH 8.4 borate buffer solution (Sato and Kudo 1974), the first term in Equation (17) is estimated to be −∆Wmρm = −0.36 nm. In addition, α_PB = 1.65 is obtained from Equation (18) by using ρ_f = 6.83 g/cm³ and ρ_m = 8.845 g/cm³ (Sato and Kudo 1974) for the anodic oxide film on Ni, M_f = 74.69 g for NiO (x = 1), and M_m = 58.69 g for Ni, respectively: the second term in Equation (17), therefore, is estimated be −ΔdfαPB = −0.61 nm from α_PB = 1.65 and Δd_f = 1 nm. The summation of the first and second terms in Equation (17) is Δd_m = −0.97 nm, which leads to σ_mΔd_m ≈ 0.3 J/m². As a result, 〈σ_f,c〉 ≈ 1.0 GPa for the anodic oxide film on Ni is derived from Equation (16), suggesting that the contribution of σ_m to 〈σ_f〉 is 23 % of 〈σ_f〉. Nevertheless, the more exact estimation of the contribution of σ_m to 〈σ_f〉 needs the preceding measurement of the residual stress of the metal thin film electrode.

Next, let us estimate the magnitude of the electrostriction stress σfel for the growth of the anodic oxide film on Ni in pH 8.4 borate buffer solution. The electric field across the anodic oxide film on Ni is E‾ = 1.6 × 10⁹ V/m (Sato and Kudo 1974), which is larger by a factor of about seven than that for the anodic oxide film on Cu, indicating that the anodic oxide film on Ni can sustain very high electric field as compared with that on Cu. In addition, ν_f = 0.32 (Robertson and Manning 1990) and ϵ_f = 13 (Fuschillo et al. 1974) are known for a NiO film. The substitution of the above parameters into Equations (11) and (15) leads to σfel = −60.4 and −520 MPa, respectively, indicating that the effect of dielectrostriction on σfel cannot be neglected for the growth of the anodic oxide film on Ni. It is noted that σfel = −520 MPa comes up to 50 % of 〈σ_f,c〉 ≈ 1.0 GPa. As a result, the intrinsic film stress of the anodic oxide film on Ni derived from the values of 〈σ_f,c〉 ≈ 1.0 GPa and σfel = −520 MPa is 〈σfi〉 ≈ 1.5 GPa, which is three times as much as that of the anodic oxide film on Cu.

3.2 Anodic oxide films on Fe

Figure 6 shows the successive potential steps in the passive potential region (A), the corresponding i versus t (B) and g versus t (C) curves in deaerated pH 8.4 borate buffer solution at 25 °C for the Fe thin film electrode with a thickness of 100 nm (evaporated on a glass plate) (Seo and Ueno 2020). The residual stress σ_m = 1.17 GPa of the evaporated Fe thin film was estimated from the linear relation between the anodic charge density and stress variation in the potential region between −0.6 and −0.3 V where the active dissolution of Fe proceeds (Seo and Ueno 2020) and it is close to that (1.35 GPa) of the Fe thin film (100 nm) evaporated on MgF₂ (Thurner and Aberman 1990). In Figure 6A, after an air-formed oxide film on Fe was cathodically reduced at −0.6 V, the successive potential steps (0.2 V → 0.3 V → 0.5 V → 0.7 V → 0.9 V) with each 1 h-interval are applied to grow the anodic oxide film on Fe (Seo and Ueno 2020). The value of g in Figure 6C is referred to zero at −0.6 V. As shown in Figure 6C, at the first potential step (0.2 V), g decreases abruptly toward compressive direction (although not shown in Figure 6C) and then increases toward tensile direction with a decay of the anodic current to attain a steady state after 1 h.

Figure 6:

Successive potential steps in the passive potential region (A), the corresponding i versus t (B) and g versus t (C) curves in deaerated pH 8.4 borate buffer solution at 25 °C for the Fe thin film electrode with a thickness of 100 nm (evaporated on a glass plate) (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

The abrupt decrease in g results from the relaxation of the tensile residual stress of the substrate Fe due to the active dissolution (i.e. consumption) of Fe (Seo and Ueno 2020). On the other hand, the subsequent increase in g is associated with the anodic oxidation of ferrous ions once dissolved: the hydroxide layer mainly consisting of Fe(OH)₃ or FeOOH⋅nH₂O (1 ≥ n > 0) is deposited on Fe by the anodic oxidation and simultaneously dehydration of the deposited layer proceeds to induce the volume shrinkage, which leads to the stress variation toward tensile direction (Seo and Ueno 2020). In contrast, g begins to decrease gradually toward compressive direction at the second potential step (0.3 V) and it decreases stepwise from the third potential step (0.5 V). Figure 7 shows the steady state-values of g after 1 h at each potential step in Figure 6C and the thickness values (Sato et al. 1976) of the inner and outer layers measured with ellipsometry as a function of potential for the anodic oxide films formed on Fe for 1 h in deaerated pH 8.4 borate buffer solution at 25 °C.

Figure 7:

Steady state-values of g after 1 h at each potential step in Figure 6C and the thickness values (Sato et al. 1976) of the inner and outer layers measured with ellipsometry as a function of potential for the anodic oxide films formed on Fe for 1 h in deaerated pH 8.4 borate buffer solution at 25 °C (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

Sato et al. (1976) have reported that the anodic oxide films formed on Fe at potentials above 0.1 V in deaerated pH 8.4 borate buffer solution at 25 °C consist of the inner barrier layer (γ-Fe₂O₃) and the outer deposit layer (γ-FeOOH). As shown in Figure 7, the thickness of the inner layer increases with increasing potential, while that of the outer layer keeps nearly constant, indicating that the stepwise decrease in g toward compressive direction caused by the successive potential steps from 0.3 V is associated with the growth of the inner layer. In Figure 7, however, g must be corrected for the residual stress relaxation due to the film growth and dissolution of metal ions through the film. For the estimation of the residual stress relaxation due to the film growth of the inner layer on Fe, it is necessary to know the relationship between Δd_m and Δd_f in Equation (17).

The ratio of Δd_m to Δd_f corresponding to reciprocal of α_PB (see the second term of the right-hand side of Equation (17)) can be obtained from the linear potential dependencies (Figure 6 in ref. Sato et al. 1971) of the anodic charge density q_f for the film formation and of the film thickness d_f in the passive potential region between 0.3 and 0.9 V for the anodic oxidation of Fe in deaerated pH 8.4 borate buffer solution at 25 °C:

where M_m and ρ_m are the atomic weight and density of Fe, respectively. In Equation (19), it is assumed that the iron component in the anodic oxide film on Fe is Fe³⁺. The value of ΔdmΔdf = −0.395 (i.e. α_PB = 2.53) (Seo and Ueno 2020) was obtained from the values of ΔqfΔdf = 1.61 × 10 C/m³ (Sato et al. 1971) and Mm3Fρm = 2.45 × 10⁻¹¹ m³/C. As a result, the residual stress relaxation σ_mΔd_m (the second term in the right-hand side of Equation (16)) of the substrate Fe for the film growth of Δd_f (nm) is given by

where σ_m = 1.17 GPa is the residual stress of the substrate Fe. Equation (20) indicates that the residual stress relaxation is −0.46 J/m³ per 1 nm-film growth.

Furthermore, the residual stress relaxation due to the dissolution of ferric ions through the anodic oxide film can be estimated from the increase in anodic charge density of dissolved ferric ions Δq_d:

Nevertheless, it is difficult to estimate accurately Δq_d from the time-variation of anodic current density during the film growth at each potential step with 1 h-interval since the anodic charge is almost consumed for the film growth. The solution analysis for the Fe component dissolved into solution during anodic oxidation of Fe for 1 h in pH 8.4 borate buffer solution has indicated that no dissolution of Fe takes place in the potential range between 0 and 1.1 V within a detection limit of the solution analysis (Sato and Kudo 1971). The anodic current density after 1 h-anodic oxidation of Fe becomes less than 10⁻³ A/m² (Seo and Ueno 2020). If an average dissolution current density of ferric ions is equal to 10⁻³ A/m², Δq_d = 3.6 C/m² is estimated for 1 h at each potential step. The substitution of Δq_d = 3.6 C/m² into Equation (21) leads to the residual stress relaxation of −0.1 J/m² for each potential step with 1 h-interval.

Figure 8 shows the values of g corrected for the residual stress relaxation of the substrate Fe due to the film growth of the inner layer and due to the film growth plus the dissolution which are plotted versus potential for comparison with the uncorrected values. In Figure 8, the corrections were made for g at potentials more positive than 0.3 V by referring to g at 0.3 V in Figure 7. Although the corrected values of g shift upward (toward tensile direction), the decreasing tendency (toward compressive direction) of g with increasing potential does not change. The changes in film stress Δσ_f due to the growth of the inner layer may be roughly estimated from Figure 8:

where Δg′ and df′ are referred to g and d_f at 0.3 V, respectively, e.g. Δg′ = g (0.9 V) − g (0.3 V) and df′ = d_f (0.9 V) − d_f (0.3 V) are taken at 0.9 V.

Figure 8:

Values of g corrected for the residual stress relaxation of the substrate Fe due to the film growth of the inner layer (solid circles) and due to the film growth plus the dissolution (open triangles) which are plotted versus potential for comparison with the uncorrected values (open circles). The corrections were made for g at potentials more positive than 0.3 V by referring to g at 0.3 V in Figure 7 (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

Figure 9 shows the estimated values of Δσ_f as a function of potential, indicating that Δσ_f decreases toward compressive direction with increasing potential. The estimated values of Δσ_f at 0.9 V are −2.6 GPa for no correction, −2.2 GPa for the correction of the residual stress relaxation due to the film growth, and −1.9 GPa for that due to the film growth plus the dissolution. The magnitude of Δσ_f at 0.9 V for the correction of the residual stress relaxation due to the film growth or the film growth plus the dissolution is almost same as that of the film stress 〈σ_f〉 = −1.88 GPa for the anodic oxide (TiO₂) film on evaporated Ti film electrode (Vanhumbeeck and Proost 2008b). It is reminded that Δσ_f differs from the average film stress 〈σ_f〉 for the whole thickness of the anodic oxide film on Fe. A tensile stress is generated for the formation of outer (deposit) layer to contribute to 〈σ_f〉 for the whole thickness of the anodic oxide film on Fe (Seo and Ueno 2020). Nevertheless, the contribution of the tensile stress in the outer layer to 〈σ_f〉 for the anodic oxide film formed on Fe at 0.9 V seems to be minor because the increase in thickness of the outer layer with potential is insignificant in the potential region between 0.3 and 0.9 V (Sato et al. 1976). The composite film stress of the anodic oxide film on Fe could be estimated by assuming a linear combination of the compressive stress in the inner layer and the tensile stress in the outer layer. At present, however, it seems difficult to estimate the composite film stress of the anodic oxide film on Fe from the stress data obtained experimentally so far (Seo and Ueno 2020).

Figure 9:

Changes in film stress Δσ_f estimated as a function of potential for no correction (open circles) and the corrections of the residual stress due to the film growth (solid circles) and due to the film growth plus the dissolution (open triangles) (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

In order to investigate the contribution of the electrostriction stress σfel to Δσ_f in the inner layer at 0.9 V, the cathodic polarization with a potential sweep rate of 1 mV/s was performed after 1 h-anodic oxidation at 0.9 V in deaerated pH 8.4 borate buffer solution at 25 °C for the Fe thin film electrode. Figure 10 shows the voltammogram (A) and the corresponding volt-stressogram (B) measured during the cathodic potential sweep for the Fe thin film electrode (Seo and Ueno 2020). In Figure 10B, a small maximum peak of g appears at about 0.5 V where the current density changes from anodic to cathodic although it is difficult to see clearly the maximum of g and the polarity change of the current density from Figures 10A and 10B. The part of Figure 10B in the potential range between 0.25 and 0.9 V is magnified to Figure 11 to confirm the maximum of g at about 0.5 V. The stress variation toward tensile direction due to the cathodic potential sweep from 0.9 to 0.5 V in Figure 11 is Δg^el ≈ 0.42 J/m², which is caused by the relaxation of the electrostriction stress in the inner layer since the electric field in the inner layer decreases with the cathodic potential sweep. Therefore, the electrostriction stress component σfel ≈ −350 MPa in the inner layer at 0.9 V is estimated from Δg^el ≈ 0.42 J/m² and df′ = d_f (0.9 V) − d_f (0.3 V) = 1.2 nm:

Figure 10:

Voltammogram (A) and the corresponding volt-stressogram (B) measured during the cathodic polarization at a sweep rate of 1 mV/s from 0.9 V after 1h-anodic oxidation at 0.9 V in deaerated pH 8.4 borate solution at 25 °C for the Fe thin film electrode (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

Figure 11:

Magnification of Figure 10B in the potential range between 0.25 and 0.9 V (Seo and Ueno 2020). Copyright ©2020. Reproduced with permission from Springer Nature.

The value of σfel ≈ −350 MPa is 16 % of Δσ_f = −2.2 GPa for the correction of residual stress relaxation due to the film growth or 18 % of Δσ_f = −1.9 GPa for the correction of the residual stress relaxation due to the film growth plus the dissolution.

In order to investigate the effect of dielectrostriction on σfel in the inner layer, it is necessary to know the relative dielectric constant ϵ_f of the inner layer. The substitution of ν_f = 0.3 as an average value of iron oxides (Chicot et al. 2011), E‾ = 5.7 × 10⁸ V/m (Sato et al. 1976) and σfel ≈ −350 MPa into Equation (15) leads to ϵ_f ≈ 25, which is close to a mean value of ϵ_f = 12 (Stimming and Schultze 1979) and 40 (Azumi et al. 1987) reported for the inner layer (γ-Fe₂O₃). By contrast, the substitution of the above parameters into Equation (11) leads to ϵ_f ≈ 570, which is extraordinarily large and unrealistic, indicating that the effect of dielectrostriction on σfel in the inner layer must be taken into consideration. Consequently, the intrinsic film stress Δσfi in the growing inner layer at 0.9 V is estimated to be −1.85 GPa for the correction of the residual stress relaxation due to the film growth or −1.55 GPa for the correction of the residual stress due to the film growth plus the dissolution.

The flat band potential E_fb of the anodic oxide film on Fe in deaerated pH 8.4 borate buffer solution at 25 °C is located around −0.1 V (Stimming and Schultze 1979), which means that the increase in g toward tensile stress due to the relaxation of the electrostriction stress in the cathodic potential sweep from 0.9 V should continue down to E_fb at which no electric field is present in the inner layer. Nevertheless, g begins to decrease toward compressive direction at potentials below 0.5 V. The decrease in g may result from the alteration of the outer deposit layer such as hydration although the cause is not clear. The outer layer would be gradually hydrated by a penetration of H₂O at potentials below 0.5 V to induce the volume expansion of the outer layer, which would lead to the decrease in g toward compressive direction (Seo and Ueno 2020). The above speculation may be supported by the ellipsometrical evidence (Azumi 1990) that the thickness of the anodic oxide film on Fe increases gradually after the anodic polarization was interrupted, and simultaneously, the refractive index of the film decreases gradually due to hydration. If the increase in g due to the relaxation of the electrostriction stress in the inner layer is masked with the decrease in g due to the volume expansion caused by hydration at potentials below 0.5 V, the real values of Δσfi in the growing inner layer may be less compressive as compared with the estimated values.

At present, the measurements of stress variations are limited to the anodic oxide films formed on Cu, Ni and Fe in deaerated pH 8.4 borate buffer solution at 25 °C. In future, it is necessary to measure the stress variations under various conditions (electrolyte composition, pH, and temperature) of anodic oxidation in order to accumulate the knowledge of the electro-chemo-mechanical properties of anodic oxide films.

3.3 Main factors controlling the sign of the intrinsic film stress

It seems that the sign of the intrinsic film stress in growing anodic oxide films on metals depends on semiconductive properties of the films. The anodic oxide films on Cu (Maurice et al. 1999; Strehblow et al. 2001) and Ni (Seyeux et al. 2006; Zuili et al. 2000) have p-type semiconductive properties. The major point defects in p-type semiconductive oxides are metal cation vacancies. In contrast, the anodic oxide films on Fe (Azumi et al. 1987; Stimming and Schultze 1979) have n-type semiconductive properties. The major point defects in n-type semiconductive oxides are oxygen vacancies and interstitial metal cations. Macdonald (1992, 1999, 2011 has proposed a point defect model (PDM) for the growth of passive films on metals. In the PDM model, it is assumed that the barrier (inner)layer of passive film is the stoichiometric oxide MO_χ/2 and the precipitated (outer) layer is formed by hydrolysis of metal cations transmitted through the barrier layer and by reaction with anions in solution.

The metal cation vacancy VMχ− with a negative charge is moved toward the metal/film interface by an applied electric field and reacts with the substrate metal atom m to form the metal cation M_M in the cation site of the film and the vacancy V_m in the metal phase:

Although VMχ− is consumed at the metal/film interface, the dissolution of M_M from the film into solution at the film/solution interface creates VMχ−:

where Maqδ+ is the dissolved metal cation species. In addition, the interstitial metal cation Miχ+ is created at the metal/film interface:

The created Miχ+ is moved toward the film/solution interface and dissolves as Maqδ+ into solution:

On the other hand, the oxygen vacancy VO∙∙ with a positive charge is created at the metal/film interface:

The created VO∙∙ is moved toward the film/solution interface by an applied electric field and is consumed by reaction with H₂O:

where O_O is the oxide ion in the anion site of the film. Furthermore, MO_χ/2 reacts with H⁺ to dissolve as Maqδ+ into solution:

In p-type semiconductive oxides such as anodic oxide films on Cu and Ni, VMχ− is continuously supplied to the metal/film interface since VMχ− is created at the film/solution interface during film growth. Consequently, the reaction of Equation (24) proceeds to bring the increase in V_m as far as VMχ− is supplied. If V_m does not sink sufficiently in the bulk metal, the increase in concentration of V_m leads to the formation of a void space (i.e. free space) in the metal side, by which the cantilever electrode is concaved to induce a tensile stress in the film (Macdonald 1992, 1999, 2011; Nelson and Oriani 1993). As a result, the intrinsic tensile stress in growing anodic oxide films on Cu and Ni may result from the void formation in the metal side. By the way, the PDM does not take the transport of metal cations via vacancy and interstitial sites, and of oxygen anions via vacancy sites into account (Seyeux et al. 2013). In p-type semiconductive oxides, the metal cations transport outward via cation vacancy sites and react with water at the film/solution interface to form a new oxide. The film/solution interface would not be mechanically constrained since the solution side is plastically deformed and thus the volume expansion due to the formation of the new oxide could not exert significantly the intrinsic film stress.

In n-type semiconductor oxides such as a barrier (inner) layer of the anodic oxide film on Fe, the oxygen anions transport inward via oxygen vacancy sites and react with metal atoms at the metal/film interface to form a new oxide in the metal side. The formation of the new oxide in the metal side leads to a volume expansion because of α_PB > 1 (e.g. α_PB = 2.53 for the inner layer (γ-Fe₂O₃) of the anodic oxide film on Fe). Since the metal/film interface is mechanically constrained, the bending of the cantilever electrode becomes convex due to the volume expansion, which induces a compressive stress in the film. On the other hand, interstitial metal cations Miχ+ in n-type semiconductor oxides transport outward via interstitial sites to react with water at the film/solution interface to form a new oxide. The formation of the new oxide at the film/solution interface could not exert significantly the intrinsic film stress since the film/solution interface would not be mechanically constrained. As a result, the compressive stress in the inner layer of the anodic oxide film on Fe may result mainly from the volume expansion due to the formation of the new oxide in the metal side at the metal/film interface (Seo and Ueno 2020).

3.4 Stress variations during cathodic reduction of anodic oxide films once formed on metals

As shown in Figures 2B and 10B, g changes toward compressive direction and then recovers toward tensile direction during cathodic reduction of anodic oxide films on Cu and Fe. In Figure 2B, two minima of g appear at about 0 V and −0.25 V corresponding to the cathodic current Q₁ and Q₂ peaks in Figure 2A, respectively. At the Q₁ peak, the reduction of the outer (CuO/Cu(OH)₂) layer to Cu₂O proceeds and at the Q₂ peak, the reduction of the converted Cu₂O and inner (Cu₂O) layers to metallic copper proceeds. The initial decrease in g results from the penetration of H⁺ into the film followed by its participation in the reduction of the film to produce H₂O in the film as an intermediate stage (Seo and Hagioi 2007), which induces the volume expansion of the reducing film leading to the temporary decrease in g toward compressive direction. The subsequent desorption of H₂O from the reducing film into solution brings the recovery of g toward tensile direction. The reduction reaction process of the outer layer of the anodic oxide film on Cu may be represented by

where the outer layer CuO_x(OH)_2−2x is partially hydrated (i.e. 1 > x > 0) (Seo et al. 1988) and [Cu2O∙(1+x)H2O]* means the intermediate species. In addition, the reduction reaction process of the produced Cu₂O and inner Cu₂O layers may be also represented by

where [2Cu∙H2O]* means the intermediate species. On the other hand, it seems that the reduction reaction process of the anodic oxide film on Fe is different from that of the anodic oxide film on Cu.

As shown in Figure 10A, the cathodic current peaks of the outer layer (γ-FeOOH) and inner layer (γ-Fe₂O₃) on Fe at −0.2 and −0.3 V are not completely separated, become doublet unlike those on Cu and the concomitant changes of g in Figure 10B exhibit one broad minimum. The one broad minimum of g observed during the cathodic reduction of the outer and inner layers on Fe may be explained in terms of the formation of Fe(OH)₂ as an intermediate stage followed by its dissolution as Feaq2+ into solution or its deposition as metallic Fe (Seo and Ueno 2020). The formation of the intermediate Fe(OH)₂ for the cathodic reduction of the outer and inner layers is represented by

and

where [Fe(OH)2]* means the intermediate species. The molar volume V_f = 2.64 × 10⁻⁵ m³/mol (West 1973–1974) for Fe(OH)₂ is larger than V_f = 2.08 × 10⁻⁵ m³/mol (West 1973–1974) for α-FeOOH and V_f = 1.67 × 10⁻⁵ m³/mol (Seo and Ueno 2020) for γ-Fe₂O₃, indicating that the formation of [Fe(OH)2]* induces the volume expansion leading to the decrease in g toward compressive direction.

Furthermore, [Fe(OH)2]* reacts with H⁺ to dissolve as Feaq2+ into solution or deposit as metallic Fe (Seo and Ueno 2020):

or

The reaction of Equations (35) or (36) leads to the increase in g, i.e. g recovers toward tensile direction. Sato et al. (1971) have reported that the reductive dissolution as Feaq2+ does not proceed with a current efficiency of 100 % for the cathodic reduction of the anodic oxide film on Fe in pH 8.4 borate buffer solution after its formation in the same pH borate buffer solution, suggesting that the partial contribution of the reductive deposition as metallic Fe is not excluded in pH 8.4 borate buffer solution. In Figure 10, at potentials lower than −0.5 V where hydrogen evolution proceeds on Fe, g decreases rapidly toward compressive direction with increasing cathodic current density, which is attributed to the volume expansion in the surface region of the Fe thin film electrode due to the hydrogen adsorption or absorption (Seo and Ueno 2020).

In contrast to the anodic oxide films on Cu and Fe, the anodic oxide films on Ti (Ohtsuka et al. 1987) and Ni (MacDougall and Cohen 1974, 1976a,b) are not completely reduced in neutral solutions. The following compositional changes of the anodic oxide film (TiO₂) on Ti during cathodic polarization in pH 6.9 phosphate solution have been proposed from the ellipsometrical results (Ohtsuka et al. 1987):

where x varies from 0 to 1 with changing cathodic potential from −0.66 to −1.31 V. Concomitantly, the complex refractive index of the film, N_f = n − ki (n: refractive index, k: extinction index) changes from N_f = 2.1 − 0.03i to N_f = 1.85 − 0.35i (Ohtsuka et al. 1987). If the charge neutrality during hydrogen ingress in the film is taken into consideration, Equation (37) can be written by introducing the valence change of Ti ions (Kim et al. 2003):

The stress variations (Kim et al. 2003) have been measured during cyclic potential steps between 0 and −1.0 V for the anodic oxide film formed on sputter-deposited Ti thin film electrode for 1 h at 5.2 V in pH 8.4 borate buffer solution: g decreases toward compressive direction at the downward potential step (0 V → −1.0 V), while it increases toward tensile direction at the upward potential step (−1.0 V → 0 V), and the changes in g during cyclic downward and upward potential steps are reversible, indicating that the reaction of Equation (37) or (38) proceeds reversibly in the potential region between 0 and −1.0 V.

The molar volume V_f = 29.42 cm³/mol of TiOOH (x = 1) (Michaelis et al. 1995) is larger than V_f = 20.8 cm³/mol of anatase type of TiO₂ (x = 0) (Samsonov 1973). Consequently, the changes in g toward compressive direction at the downward potential step result from the volume expansion due to the formation of TiO_2−x(OH)_x or [xTi3+(1−x)Ti4+]O2−x(OH)x, and the reversible changes in g toward tensile direction at the upward potential step are caused by the volume contraction due to the regeneration of TiO₂. The difference Δσfi = 320 MPa (Kim et al. 2003) in film stress between TiOOH and TiO₂ was calculated from the linear relationship between Δg and anodic charge Q_a (for hydrogen desorption) obtained during upward potential steps from various cathodic potentials (between −1.2 and −0.7 V) to 0 V after the formation of TiO₂ film (thickness: 1.8 nm) on Ti for 1 h at 5.2 V in pH 8.4 borate buffer solution.

MacDougall and Cohen (1976a) have reported that the anodic oxide films formed on Ni in Na₂SO₄ solutions of pH 2.0−8.4 cannot be completely reduced at pH > 2.8. Moreover, the multiple-angle-of-incidence reflectivity measurements (Ohtsuka and Heusler 1979) have indicated that the film thickness increases initially, passing through a maximum and then decreases during cathodic reduction in 1 M H₂SO₄ solution after the formation of the anodic oxide film on Ni for 1 h in 1 M H₂SO₄ solution: concomitantly the complex refractive index of the film changes from N_f ≈ 2.1 − 0.5i to N_f ≈1.6 − 0.1i. The above changes of the thickness and complex refractive index of the film suggest that the anodic oxide film on Ni is cathodically reduced in 1 M H₂SO₄ solution via the formation of Ni(OH)₂ as an intermediate stage followed by dissolution of Ni(OH)₂ as Ni²⁺ into solution (Ohtsuka and Heusler 1979). The formation of Ni(OH)₂ leads to the increase in thickness i.e. the volume expansion since the molar volume of Ni(OH)₂ is larger than that of NiO (Ohtsuka and Heusler 1979).

The stress variations (Kim and Seo 2003) have been measured during the downward (from 0.8 or 0 V to −0.75 V) and upward potential steps (from −0.75 V to 0 or 0.8 V) after the formation of the anodic oxide film on sputter-deposited Ni thin film electrode at 0.8 V in pH 8.4 borate buffer solution: the downward and upward potential steps induce the changes in g toward compressive and tensile directions, respectively. The changes in g toward compressive direction during the downward potential step were mainly attributed to the volume expansion due to the formation of Ni(OH)₂ (Kim and Seo 2003). The cathodic reduction of the anodic oxide film on Ni in pH 8.4 borate buffer solution may be represented by

where the anodic oxide film on Ni is regarded as a p-type of semiconductive oxide with cation vacancies, i.e. Ni_1−xO (1/3 >x > 0) and x depends on the film formation potential. In Equation (39), Ni(OH)₂ is stable at −0.75 V where hydrogen evolution proceeds in pH 8.4 borate solution. The changes in g toward tensile direction during the upward potential step were mainly attributed to the reverse reaction of Equation (39), i.e. the volume contraction due to the formation of Ni_1−xO. At −0.75 V, hydrogen atoms are absorbed into the substrate Ni through the produced Ni(OH)₂, which induces a compressive stress. Nevertheless, its contribution to the changes in g toward compressive direction was about one fourth (Kim and Seo 2003).