Article pubs.acs.org/Langmuir
Model for Anodic Film Growth on Aluminum with Coupled Bulk Transport and Interfacial Reactions Stephen DeWitt† and Katsuyo Thornton*,†,‡ †
Applied Physics Program and ‡Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *
ABSTRACT: Films grown through the anodic oxidation of metal substrates are promising for applications ranging from solar cells to medical devices, but the underlying mechanisms of anodic growth are not fully understood. To provide a better understanding of these mechanisms, we present a new 1D model for the anodization of aluminum. In this model, a thin space charge region at the oxide/electrolyte interface couples the bulk ionic transport and the interfacial reactions. Charge builds up in this region, which alters the surface overpotential until the reaction and bulk fluxes are equal. The model reactions at the oxide/electrolyte interface are derived from the Våland−Heusler model, with modifications to allow for deviations from stoichiometry at the interface and the saturation of adsorption sites. The rate equations and equilibrium concentrations of adsorbed species at the oxide/electrolyte interface are obtained from the reactions using Butler−Volmer kinetics, whereas transport-limited reaction kinetics are utilized at the metal/oxide interface. The ionic transport through the bulk oxide is modeled using a newly proposed cooperative transport process, the counter-site defect mechanism. The model equations are evolved numerically. The model is parametrized and validated using experimental data in the literature for the rate of ejection of aluminum species into the electrolyte, embedded charge at the oxide/electrolyte interface, and the barrier thickness and growth rate of porous films. The parametrized model predicts that the embedded charge at the oxide/electrolyte interface decreases monotonically for increasing electrolyte pH at constant current density. The parametrized model also predicts that the embedded charge during potentiostatic anodization is at its steady-state value; the embedded charge at any given time is equal to the embedded charge during galvanostatic anodization at the same current. In addition to simulations of anodized barrier films, this model can be extended to multiple dimensions to simulate anodic nanostructure growth.
1. INTRODUCTION Through the process of anodization, a metallic substrate is electrochemically oxidized to create a film with a number of possible morphologies, including compact barrier films, nanoporous films, and nanotubular films. Traditionally, anodic films have been applied in electrolytic capacitors and in metal finishing for corrosion resistance, abrasion resistance, and improved dye absorption.1,2 With the advent of self-ordering anodic nanostructures, anodic films now have a wide range of promising applications including solar energy conversion,3,4 Liion microbatteries,5,6 sensors,6,7 high-density magnetic memory,7 and medical devices.8 To understand the mechanisms underlying the growth and self-organization of anodic nanostructured films, the growth process of barrier films must be understood. Even after decades of study, a comprehensive description of anodic barrier film growth has remained elusive. To elucidate these underlying mechanisms better, we present a new model of the anodization of aluminum. We choose aluminum as our model system because it is the most studied substrate in the literature.9 For © 2014 American Chemical Society
summaries of these studies, we refer readers to several reviews.1,2,10,11 Similar fundamental mechanisms govern anodization for all substrates; therefore, advances in the understanding of the anodization of aluminum are relevant to other material systems. Several experimental studies have shed light on the mechanisms governing anodization. Våland and Heusler12 measured the flux of aluminum species ejected from the oxide/electrolyte interface as a function of electrolyte pH and the applied current density. They fit this data using expressions for the partial current densities of Al3+ and O2− species. From the fitting parameters they determined a set of probable reactions at the oxide/electrolyte interface. However, their reaction expressions do not account for the possible saturation of ion sites in the adsorbed layer or for changes in the concentration of oxygen and aluminum ions within the oxide at Received: February 27, 2014 Revised: April 11, 2014 Published: April 16, 2014 5314
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electric double-layer in the electrolyte. The model also provides a framework that easily extends to multidimensional simulations of nanostructure ordering and could include other phenomena (e.g., impurities from the electrolyte, plastic flow). We parametrize and validate the model using experimental measurements of the aluminum-ejection current, the embedded charge density at the oxide/electrolyte interface, the steady-state pore barrier thickness, and the pore growth rate.
the interface. Also, because their mechanistic description is limited to the partial currents at the oxide/electrolyte interface, they do not provide a model for ionic transport within the bulk film or a description of how the reactions lead to interfacial motion. The presence of embedded charges in anodic alumina barrier films has been detected experimentally, most recently through the use of electrostatic force microscopy13 and scanning Kelvin probe force microscopy.14 The embedded charges are believed to be concentrated at the interfaces, with positive charges resulting from an excess of Al3+ or a deficiency in O2− near the metal/oxide interface and negative charges resulting from a deficiency in Al3+, an excess of O2−, or stress-induced deep electron traps near the oxide/electrolyte interface. If the current is allowed to decay potentiostatically after a period of galvanostatic anodization, then the effective surface charge increases substantially in comparison to that of a purely galvanostatically grown film. Proposed mechanisms that would explain these observations include an increased concentration of impurities incorporated into the film from the electrolyte13 and a shift in the distribution of the embedded charges toward the interfaces.14 In acidic electrolytes, a nanoporous film, instead of a barrier film, can form. In these films, the pore geometry depends on the electrolyte and the applied current or potential. The pore diameter and interpore separation as well as the pore barrier thickness increase substantially as the anodizing potential increases.15,16 However, the aforementioned attributes are all approximately constant as the pH varies (at least for pH values between 0.7 and 1.9).15,16 On the other hand, the pH impacts the average growth rate of the pores, where increased pH decreases the growth rate.16 The choice of electrolyte also impacts the pore geometry,16 which may be due to electrolyte species being incorporated into the growing film.2 Many approaches have been applied to simulate the growth of anodic alumina films with notable successes in describing the pore geometry as a function of applied potential17 and in explaining the motion of a tracer element in porous films due to plastic flow.18 However, an accurate formulation of the interfacial reaction kinetics has yet to be developed. The role of interfacial reactions is often highly simplified18−21 or ignored entirely in favor of transport-limited behavior.11,22,23 Friedman, Brittain, and Menon16 compared the predictions from three models of porous film growth17,24−26 to their experimental data. Contrary to the experiments, which found no change in the pore feature size when the electrolyte pH was varied, the models predict that the pore features change substantially with electrolyte pH. These results indicate that an improved model of the reactions at the oxide/electrolyte interface is required to capture the effect of varying pH levels. Furthermore, none of these models provide a description for the charge embedded in the film at the interfaces. A promising approach is taken by Battaglia and Newman in their work modeling the oxidation of iron,27 where the reactions are handled using Butler−Volmer kinetics and transport through the film is described with a highfield transport equation. However, such an approach has not been applied to describe the anodization of aluminum. In this article, we present a new 1D model for anodic alumina growth, which couples bulk ionic transport to electrochemical reactions at the interfaces. The goal of this model is to provide an accurate description of the interfacial reaction kinetics while also capturing the effects of high-field transport within the film, embedded charge at the oxide/electrolyte interface, and the
2. MODEL The model presented here consists of three submodels: the submodel for the electric potential throughout the system, the submodel for ion transport within the film, and the submodel describing the chemical reactions at the oxide/electrolyte and metal/oxide interfaces. In this section, the three submodels are discussed in detail, followed by a discussion of the model parameters. In the interest of simplicity, we do not include effects due to mechanical stress or electrolyte species incorporated into the oxide. The impact of these effects is discussed in section 5. Before discussing the details of the model, we first define the terminology we use in describing the structure of the amorphous oxide. Oxygen and metal sites in section 2.2 refer to locations where the short-range bond interactions favor either oxygen or metal ions, respectively, not positions in a periodic lattice of a crystalline material. In section 2.3, pseudointerstitials refer to ions that lead to concentration values exceeding those expected from the average film density. Likewise, pseudovacancies refer to ions that lead to concentration values below those expected from the average film density. While true interstitials and vacancies are point defects localized to specific sites, the pseudointerstitials and pseudovacancies are not necessarily localized and represent an excess or deficiency of the species which may be spread over multiple ion spacings. 2.1. Submodel for the Electric Potential. The electric potential distribution within the growing anodic film can be calculated using Poisson’s equation and the Helmholtz doublelayer model at the oxide/electrolyte interface.27 During the anodization of aluminum, the substrate and bulk electrolyte can be approximated as ideal conductors in which the potential is constant; therefore, nearly all of the potential drop occurs across the oxide film.17 The distribution of the electric potential, ϕ, within the film is given by Poisson’s equation ∂ 2ϕ ρ =− ε ∂x 2
(1)
where the x axis is oriented from the oxide/electrolyte interface to the metal/oxide interface, ρ is the charge density, and ε is the permittivity of the oxide film. The potential within the bulk electrolyte is set to zero, but it drops across the interfacial double layer by an amount η, which is the surface overpotential. The potential of the aluminum substrate is set to the applied potential, ϕapplied. In concentrated electrolytes, including the electrolytes typically used during anodization, the interfacial double-layer behavior is dominated by the Helmholtz layer.28 Therefore, the double layer in the electrolyte can be modeled as a parallel-plate capacitor with capacitance CHelmholtz and a uniform electric displacement field given by DHelmholtz = −CHelmholtzη.29,30 When Gauss’s law is applied across the interface between the 5315
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Figure 1. Schematic diagram of the model system, with the value of ϕ marked at key locations.
Helmholtz layer and the oxide, η can be expressed in terms of the electric field at the interface on the oxide side:27 C Helmholtzη − ε
∂ϕ ∂x
i = A exp(BE)
where A and B are temperature-dependent parameters.32 Equation 6 is known as the high-field transport equation. Verwey33 proposed that the exponential dependence in eq 6 is due to the field-dependent motion of ions over a potential barrier for an ionic hop in the bulk oxide33,34 (although other explanations have also been proposed31,35). On the basis of this assumption, Fromhold developed the following continuum description of the ionic flux due to repeated ionic jumps driven by an applied electric field:36
=0 (2)
ox
On the oxide side of the oxide/electrolyte interface, the space charge region is also closely confined to the interface. Due to the large concentration of charged species within the oxide, the Debye length, which typically characterizes the width of the space charge region,27,31 is well below the atomic layer thickness. Therefore, similarly to the Helmholtz layer in the electrolyte, the space charge density in the oxide is confined to a single atomic layer at the interface, yielding a compact charge region. The ionic concentrations within this compact charge region are denoted by ccrcAl3+ and ccrcO2−, where subscripts on the left indicate the region of the variable (the compact charge region in this case). The electric field appearing in eq 2 is the field at the edge of the compact charge region abutting the Helmholtz layer. Due to the bulk charge neutrality and the approximately constant bulk oxide density, the ionic concentrations in the oxide outside of the compact charge region are also eq 2− 3+ approximately constant with values of eq oxcAl and oxcO . Outside the compact charge region, the right-hand side of eq 1 is zero and the electric field is given by E bulk = −
⎛ − qiai ∂ϕ ⎞ ⎛ − Wi ⎞⎡ ⎟⎢c sinh⎜ ⎟ Ji = − 4aiνi exp⎜ ⎝ kT ⎠⎣ i ⎝ kT ∂x ⎠ ⎛ q ai ∂ϕ ⎞⎤ ∂c ⎟⎥ + ai i cosh⎜ i ⎝ kT ∂x ⎠⎦ ∂x
(3)
where, as can be seen in Figure 1, ϕccr is the value of ϕ at the boundary between the oxide bulk and the compact charge region, L is the thickness of the oxide, and lccr is the thickness of the compact charge region. To find expressions for η and ϕccr, eq 1 is integrated across the compact charge region, with eqs 2 and 3 providing boundary conditions η=
ccr
ϕccr =
(
ρ lccr L −
lccr 2
) + εϕ
applied
ε + C HelmholtzL
(4)
2η(ε + C Helmholtzlccr) − ccr ρ lccr 2ε
(7)
Ji is the ionic flux for the ith species, ai is half of the jump distance, νi is the jump attempt frequency related to atomic vibrations in the film, ci is the concentration of species i, and Wi is the potential barrier height. For large applied fields and constant ionic concentrations, eq 7 simplifies to eq 6. Although eq 7 was originally conceived as a description of the transport of a single ion, experimental observations have provided evidence for a correlated ion-transport mechanism. Tracer experiments have shown that the current is carried by both metal and oxygen ions during the anodization of aluminum, niobium, tantalum, and tungsten.10,37−40 Fromhold41 noted that these results would be unlikely for a noncorrelated transport mechanism. Due to the exponential dependence of the ionic fluxes on the potential barrier height in eq 7, even a small difference in Wi would cause transport dominated by a single species.41 Experimental measurements of B, the field coefficient in eq 6, provide further evidence of correlated ion transport. Harkness and Young42 measured the average electric field during anodization as a function of the applied current and determined B to be 35 nm/V. According to eq 7, for typical anodization conditions under which the applied electric field is large and the concentration gradient across the film is low, B is approximately equal to qiai/kT. Assuming an average Al−Al spacing of 0.31 nm (ai = 0.16 nm) and an average O−O spacing of 0.28 nm (ai = 0.14 nm),43 qiai/kT for one Al3+ ion moving one Al−Al spacing is 19 nm/V and for one O2− ion moving one O−O spacing it is 11 nm/V. These field coefficients are much lower than the experimental value, implying that the primary transport mechanism is not individual ions moving single atomic spacings. Several correlated ion motion mechanisms have been proposed,41,44−46 including the hopon mechanism.41 A hopon is a mobile defect where a metal ion is located on an oxygen site
ϕapplied − ϕccr L − lccr
(6)
(5)
where ccrρ = qAl3+ccrcAl3+ + qO2−ccrcO2− is the charge density in the compact charge region and qi is the charge of the ion i. 2.2. Submodel for Ionic Transport within the Film. The ionic current during anodization is known empirically to have the following exponential dependence on the applied electric field, E, 5316
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corresponds to ion motion toward the metal/oxide interface. Equation 8 is fitted with J0csd as a fitting parameter, and the calculated E values are within 4% of Harkness and Young’s experimental results. The aluminum transport number, the fraction of the current carried by Al3+, for this mechanism is a constant value of 0.6 across all current densities, which matches experimental results which showed that the transport number was 0.58 ± 0.05 for current densities of 0.1−10 mA/cm2.37 The transport number is also independent of the applied potential and electrolyte pH because it solely depends on the fraction of the defect charge due to an Al ion, +3e, out of the total defect charge, +5e (including an O vacancy). Because eq 8 is consistent with both the experimental current/field relation and the experimental transport numbers, it is used in our model. The number of ions of each species within the system must be conserved. The evolution of the ion concentrations in the oxide is described by a continuity equation taking the form
and an oxygen ion is located on a metal site. This defect travels by causing adjacent metal−oxygen pairs to exchange places as well, and one hop results in an effective charge of qMn+ − qO2− moving one metal−oxygen spacing. If two hopons are coupled and share a central ion (Fromhold’s two-hopon process41), then the net effect is that a metal ion travels between adjacent oxygen sites, as shown in Figure 2a. Conversely, an oxygen ion
∂J ∂ci =− i ∂t ∂x
In the oxide bulk, the ionic flux is spatially constant and the enforcement of eq 9 is trivial. In the compact charge region, eq 9 governs the evolution of ccrcAl3+ and ccrcO2−. For multidimensional simulations of anodic growth, the electric field is no longer uniform in the oxide bulk, and eq 9 must be solved within the entire domain to describe the formation and evolution of the bulk space charge. As the oxide/electrolyte interface moves due to oxidation or dissolution, the compact charge region moves at the same velocity, vo/e. To correct for this moving frame of reference, an advective term must be added to eq 9. The evolution of the ion concentrations in the compact charge region are then given by
Figure 2. Schematic illustrations of (a) the paired hopon and (b) the counter-site defect transport mechanisms in two dimensions. In both mechanisms, an Al3+ ion on an O2− site effectively exchanges locations with the O2− ion to its right, propagating the excess charge to the right. The ions that are located on a site belonging to the other species are circled in black.
could travel between adjacent metal sites, but the oxygen ion’s significantly larger ionic radius makes this case less likely. The coupled hopons are most visible in the second step of Figure 2a, where two Al3+ ions on O2− sites form hopons with an O2− on an Al3+ site. Assuming an average Al−O spacing of 0.18 nm in the oxide,43 qiai/kT for each of these steps is approximately 17 nm/V, which is well below the experimental value. Alternatively, if the hopons are coupled tightly enough, then the combined motion of the hopons could face a single potential barrier, with the jump distance corresponding to the distance the defect travels after both steps. If such a tightly coupled hopon pair is traveling between oxygen sites in anodic alumina, then qiai/kT is 27 nm/V. This value is much closer to the experimental value than are the single-ion mechanisms. We propose a related but more direct transport mechanism than the two-hopon mechanism, the counter-site defect mechanism. Instead of two sequential aluminum−oxygen place exchanges, the Al3+ ion on the O2− site directly exchanges positions with an adjacent O2− ion, as depicted schematically in Figure 2b. Because the counter-site defect mechanism is a single-step mechanism, the single potential barrier framework can be applied directly. Unlike the coupled hopon mechanism, the counter-site defect mechanism does not require that O2− ions be located on the smaller Al3+ ion sites. Since the motion of a multi-ion defect can also take the form of eq 7,41 the bulk ionic flux due to the counter-site defect mechanism is given by ⎛ q acsdE bulk ⎞ 0 sinh⎜ csd JAl3+ = −JO2− = −Jcsd ⎟ kT ⎝ ⎠
(9)
∂ccrci = ∂t
J − Ji
o/e i
lccr
−
vo/e eq ox ci lccr
(10)
where o/eJi is the flux into the compact charge region due to interfacial chemical reactions. 2.3. Submodel for Chemical Reactions at the Oxide/ Electrolyte Interface. The goal of this subsection is to present the reaction mechanisms considered in our model and to derive equations for the oxidation/dissolution reaction rate, the reaction flux between the electrolyte and the compact charge region, the aluminum species ejection flux, and the velocity of the oxide/electrolyte interface. In the process of these derivations, we obtain a set of equations that specifies the concentration of adsorbed species at the oxide/electrolyte interface. The approach taken here extends that of Våland and Heusler12 for pH
Model for Anodic Film Growth on Aluminum with Coupled Bulk Transport and Interfacial Reactions Stephen DeWitt† and Katsuyo Thornton*,†,‡ †
Applied Physics Program and ‡Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *
ABSTRACT: Films grown through the anodic oxidation of metal substrates are promising for applications ranging from solar cells to medical devices, but the underlying mechanisms of anodic growth are not fully understood. To provide a better understanding of these mechanisms, we present a new 1D model for the anodization of aluminum. In this model, a thin space charge region at the oxide/electrolyte interface couples the bulk ionic transport and the interfacial reactions. Charge builds up in this region, which alters the surface overpotential until the reaction and bulk fluxes are equal. The model reactions at the oxide/electrolyte interface are derived from the Våland−Heusler model, with modifications to allow for deviations from stoichiometry at the interface and the saturation of adsorption sites. The rate equations and equilibrium concentrations of adsorbed species at the oxide/electrolyte interface are obtained from the reactions using Butler−Volmer kinetics, whereas transport-limited reaction kinetics are utilized at the metal/oxide interface. The ionic transport through the bulk oxide is modeled using a newly proposed cooperative transport process, the counter-site defect mechanism. The model equations are evolved numerically. The model is parametrized and validated using experimental data in the literature for the rate of ejection of aluminum species into the electrolyte, embedded charge at the oxide/electrolyte interface, and the barrier thickness and growth rate of porous films. The parametrized model predicts that the embedded charge at the oxide/electrolyte interface decreases monotonically for increasing electrolyte pH at constant current density. The parametrized model also predicts that the embedded charge during potentiostatic anodization is at its steady-state value; the embedded charge at any given time is equal to the embedded charge during galvanostatic anodization at the same current. In addition to simulations of anodized barrier films, this model can be extended to multiple dimensions to simulate anodic nanostructure growth.
1. INTRODUCTION Through the process of anodization, a metallic substrate is electrochemically oxidized to create a film with a number of possible morphologies, including compact barrier films, nanoporous films, and nanotubular films. Traditionally, anodic films have been applied in electrolytic capacitors and in metal finishing for corrosion resistance, abrasion resistance, and improved dye absorption.1,2 With the advent of self-ordering anodic nanostructures, anodic films now have a wide range of promising applications including solar energy conversion,3,4 Liion microbatteries,5,6 sensors,6,7 high-density magnetic memory,7 and medical devices.8 To understand the mechanisms underlying the growth and self-organization of anodic nanostructured films, the growth process of barrier films must be understood. Even after decades of study, a comprehensive description of anodic barrier film growth has remained elusive. To elucidate these underlying mechanisms better, we present a new model of the anodization of aluminum. We choose aluminum as our model system because it is the most studied substrate in the literature.9 For © 2014 American Chemical Society
summaries of these studies, we refer readers to several reviews.1,2,10,11 Similar fundamental mechanisms govern anodization for all substrates; therefore, advances in the understanding of the anodization of aluminum are relevant to other material systems. Several experimental studies have shed light on the mechanisms governing anodization. Våland and Heusler12 measured the flux of aluminum species ejected from the oxide/electrolyte interface as a function of electrolyte pH and the applied current density. They fit this data using expressions for the partial current densities of Al3+ and O2− species. From the fitting parameters they determined a set of probable reactions at the oxide/electrolyte interface. However, their reaction expressions do not account for the possible saturation of ion sites in the adsorbed layer or for changes in the concentration of oxygen and aluminum ions within the oxide at Received: February 27, 2014 Revised: April 11, 2014 Published: April 16, 2014 5314
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electric double-layer in the electrolyte. The model also provides a framework that easily extends to multidimensional simulations of nanostructure ordering and could include other phenomena (e.g., impurities from the electrolyte, plastic flow). We parametrize and validate the model using experimental measurements of the aluminum-ejection current, the embedded charge density at the oxide/electrolyte interface, the steady-state pore barrier thickness, and the pore growth rate.
the interface. Also, because their mechanistic description is limited to the partial currents at the oxide/electrolyte interface, they do not provide a model for ionic transport within the bulk film or a description of how the reactions lead to interfacial motion. The presence of embedded charges in anodic alumina barrier films has been detected experimentally, most recently through the use of electrostatic force microscopy13 and scanning Kelvin probe force microscopy.14 The embedded charges are believed to be concentrated at the interfaces, with positive charges resulting from an excess of Al3+ or a deficiency in O2− near the metal/oxide interface and negative charges resulting from a deficiency in Al3+, an excess of O2−, or stress-induced deep electron traps near the oxide/electrolyte interface. If the current is allowed to decay potentiostatically after a period of galvanostatic anodization, then the effective surface charge increases substantially in comparison to that of a purely galvanostatically grown film. Proposed mechanisms that would explain these observations include an increased concentration of impurities incorporated into the film from the electrolyte13 and a shift in the distribution of the embedded charges toward the interfaces.14 In acidic electrolytes, a nanoporous film, instead of a barrier film, can form. In these films, the pore geometry depends on the electrolyte and the applied current or potential. The pore diameter and interpore separation as well as the pore barrier thickness increase substantially as the anodizing potential increases.15,16 However, the aforementioned attributes are all approximately constant as the pH varies (at least for pH values between 0.7 and 1.9).15,16 On the other hand, the pH impacts the average growth rate of the pores, where increased pH decreases the growth rate.16 The choice of electrolyte also impacts the pore geometry,16 which may be due to electrolyte species being incorporated into the growing film.2 Many approaches have been applied to simulate the growth of anodic alumina films with notable successes in describing the pore geometry as a function of applied potential17 and in explaining the motion of a tracer element in porous films due to plastic flow.18 However, an accurate formulation of the interfacial reaction kinetics has yet to be developed. The role of interfacial reactions is often highly simplified18−21 or ignored entirely in favor of transport-limited behavior.11,22,23 Friedman, Brittain, and Menon16 compared the predictions from three models of porous film growth17,24−26 to their experimental data. Contrary to the experiments, which found no change in the pore feature size when the electrolyte pH was varied, the models predict that the pore features change substantially with electrolyte pH. These results indicate that an improved model of the reactions at the oxide/electrolyte interface is required to capture the effect of varying pH levels. Furthermore, none of these models provide a description for the charge embedded in the film at the interfaces. A promising approach is taken by Battaglia and Newman in their work modeling the oxidation of iron,27 where the reactions are handled using Butler−Volmer kinetics and transport through the film is described with a highfield transport equation. However, such an approach has not been applied to describe the anodization of aluminum. In this article, we present a new 1D model for anodic alumina growth, which couples bulk ionic transport to electrochemical reactions at the interfaces. The goal of this model is to provide an accurate description of the interfacial reaction kinetics while also capturing the effects of high-field transport within the film, embedded charge at the oxide/electrolyte interface, and the
2. MODEL The model presented here consists of three submodels: the submodel for the electric potential throughout the system, the submodel for ion transport within the film, and the submodel describing the chemical reactions at the oxide/electrolyte and metal/oxide interfaces. In this section, the three submodels are discussed in detail, followed by a discussion of the model parameters. In the interest of simplicity, we do not include effects due to mechanical stress or electrolyte species incorporated into the oxide. The impact of these effects is discussed in section 5. Before discussing the details of the model, we first define the terminology we use in describing the structure of the amorphous oxide. Oxygen and metal sites in section 2.2 refer to locations where the short-range bond interactions favor either oxygen or metal ions, respectively, not positions in a periodic lattice of a crystalline material. In section 2.3, pseudointerstitials refer to ions that lead to concentration values exceeding those expected from the average film density. Likewise, pseudovacancies refer to ions that lead to concentration values below those expected from the average film density. While true interstitials and vacancies are point defects localized to specific sites, the pseudointerstitials and pseudovacancies are not necessarily localized and represent an excess or deficiency of the species which may be spread over multiple ion spacings. 2.1. Submodel for the Electric Potential. The electric potential distribution within the growing anodic film can be calculated using Poisson’s equation and the Helmholtz doublelayer model at the oxide/electrolyte interface.27 During the anodization of aluminum, the substrate and bulk electrolyte can be approximated as ideal conductors in which the potential is constant; therefore, nearly all of the potential drop occurs across the oxide film.17 The distribution of the electric potential, ϕ, within the film is given by Poisson’s equation ∂ 2ϕ ρ =− ε ∂x 2
(1)
where the x axis is oriented from the oxide/electrolyte interface to the metal/oxide interface, ρ is the charge density, and ε is the permittivity of the oxide film. The potential within the bulk electrolyte is set to zero, but it drops across the interfacial double layer by an amount η, which is the surface overpotential. The potential of the aluminum substrate is set to the applied potential, ϕapplied. In concentrated electrolytes, including the electrolytes typically used during anodization, the interfacial double-layer behavior is dominated by the Helmholtz layer.28 Therefore, the double layer in the electrolyte can be modeled as a parallel-plate capacitor with capacitance CHelmholtz and a uniform electric displacement field given by DHelmholtz = −CHelmholtzη.29,30 When Gauss’s law is applied across the interface between the 5315
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Figure 1. Schematic diagram of the model system, with the value of ϕ marked at key locations.
Helmholtz layer and the oxide, η can be expressed in terms of the electric field at the interface on the oxide side:27 C Helmholtzη − ε
∂ϕ ∂x
i = A exp(BE)
where A and B are temperature-dependent parameters.32 Equation 6 is known as the high-field transport equation. Verwey33 proposed that the exponential dependence in eq 6 is due to the field-dependent motion of ions over a potential barrier for an ionic hop in the bulk oxide33,34 (although other explanations have also been proposed31,35). On the basis of this assumption, Fromhold developed the following continuum description of the ionic flux due to repeated ionic jumps driven by an applied electric field:36
=0 (2)
ox
On the oxide side of the oxide/electrolyte interface, the space charge region is also closely confined to the interface. Due to the large concentration of charged species within the oxide, the Debye length, which typically characterizes the width of the space charge region,27,31 is well below the atomic layer thickness. Therefore, similarly to the Helmholtz layer in the electrolyte, the space charge density in the oxide is confined to a single atomic layer at the interface, yielding a compact charge region. The ionic concentrations within this compact charge region are denoted by ccrcAl3+ and ccrcO2−, where subscripts on the left indicate the region of the variable (the compact charge region in this case). The electric field appearing in eq 2 is the field at the edge of the compact charge region abutting the Helmholtz layer. Due to the bulk charge neutrality and the approximately constant bulk oxide density, the ionic concentrations in the oxide outside of the compact charge region are also eq 2− 3+ approximately constant with values of eq oxcAl and oxcO . Outside the compact charge region, the right-hand side of eq 1 is zero and the electric field is given by E bulk = −
⎛ − qiai ∂ϕ ⎞ ⎛ − Wi ⎞⎡ ⎟⎢c sinh⎜ ⎟ Ji = − 4aiνi exp⎜ ⎝ kT ⎠⎣ i ⎝ kT ∂x ⎠ ⎛ q ai ∂ϕ ⎞⎤ ∂c ⎟⎥ + ai i cosh⎜ i ⎝ kT ∂x ⎠⎦ ∂x
(3)
where, as can be seen in Figure 1, ϕccr is the value of ϕ at the boundary between the oxide bulk and the compact charge region, L is the thickness of the oxide, and lccr is the thickness of the compact charge region. To find expressions for η and ϕccr, eq 1 is integrated across the compact charge region, with eqs 2 and 3 providing boundary conditions η=
ccr
ϕccr =
(
ρ lccr L −
lccr 2
) + εϕ
applied
ε + C HelmholtzL
(4)
2η(ε + C Helmholtzlccr) − ccr ρ lccr 2ε
(7)
Ji is the ionic flux for the ith species, ai is half of the jump distance, νi is the jump attempt frequency related to atomic vibrations in the film, ci is the concentration of species i, and Wi is the potential barrier height. For large applied fields and constant ionic concentrations, eq 7 simplifies to eq 6. Although eq 7 was originally conceived as a description of the transport of a single ion, experimental observations have provided evidence for a correlated ion-transport mechanism. Tracer experiments have shown that the current is carried by both metal and oxygen ions during the anodization of aluminum, niobium, tantalum, and tungsten.10,37−40 Fromhold41 noted that these results would be unlikely for a noncorrelated transport mechanism. Due to the exponential dependence of the ionic fluxes on the potential barrier height in eq 7, even a small difference in Wi would cause transport dominated by a single species.41 Experimental measurements of B, the field coefficient in eq 6, provide further evidence of correlated ion transport. Harkness and Young42 measured the average electric field during anodization as a function of the applied current and determined B to be 35 nm/V. According to eq 7, for typical anodization conditions under which the applied electric field is large and the concentration gradient across the film is low, B is approximately equal to qiai/kT. Assuming an average Al−Al spacing of 0.31 nm (ai = 0.16 nm) and an average O−O spacing of 0.28 nm (ai = 0.14 nm),43 qiai/kT for one Al3+ ion moving one Al−Al spacing is 19 nm/V and for one O2− ion moving one O−O spacing it is 11 nm/V. These field coefficients are much lower than the experimental value, implying that the primary transport mechanism is not individual ions moving single atomic spacings. Several correlated ion motion mechanisms have been proposed,41,44−46 including the hopon mechanism.41 A hopon is a mobile defect where a metal ion is located on an oxygen site
ϕapplied − ϕccr L − lccr
(6)
(5)
where ccrρ = qAl3+ccrcAl3+ + qO2−ccrcO2− is the charge density in the compact charge region and qi is the charge of the ion i. 2.2. Submodel for Ionic Transport within the Film. The ionic current during anodization is known empirically to have the following exponential dependence on the applied electric field, E, 5316
dx.doi.org/10.1021/la500782d | Langmuir 2014, 30, 5314−5325
Langmuir
Article
corresponds to ion motion toward the metal/oxide interface. Equation 8 is fitted with J0csd as a fitting parameter, and the calculated E values are within 4% of Harkness and Young’s experimental results. The aluminum transport number, the fraction of the current carried by Al3+, for this mechanism is a constant value of 0.6 across all current densities, which matches experimental results which showed that the transport number was 0.58 ± 0.05 for current densities of 0.1−10 mA/cm2.37 The transport number is also independent of the applied potential and electrolyte pH because it solely depends on the fraction of the defect charge due to an Al ion, +3e, out of the total defect charge, +5e (including an O vacancy). Because eq 8 is consistent with both the experimental current/field relation and the experimental transport numbers, it is used in our model. The number of ions of each species within the system must be conserved. The evolution of the ion concentrations in the oxide is described by a continuity equation taking the form
and an oxygen ion is located on a metal site. This defect travels by causing adjacent metal−oxygen pairs to exchange places as well, and one hop results in an effective charge of qMn+ − qO2− moving one metal−oxygen spacing. If two hopons are coupled and share a central ion (Fromhold’s two-hopon process41), then the net effect is that a metal ion travels between adjacent oxygen sites, as shown in Figure 2a. Conversely, an oxygen ion
∂J ∂ci =− i ∂t ∂x
In the oxide bulk, the ionic flux is spatially constant and the enforcement of eq 9 is trivial. In the compact charge region, eq 9 governs the evolution of ccrcAl3+ and ccrcO2−. For multidimensional simulations of anodic growth, the electric field is no longer uniform in the oxide bulk, and eq 9 must be solved within the entire domain to describe the formation and evolution of the bulk space charge. As the oxide/electrolyte interface moves due to oxidation or dissolution, the compact charge region moves at the same velocity, vo/e. To correct for this moving frame of reference, an advective term must be added to eq 9. The evolution of the ion concentrations in the compact charge region are then given by
Figure 2. Schematic illustrations of (a) the paired hopon and (b) the counter-site defect transport mechanisms in two dimensions. In both mechanisms, an Al3+ ion on an O2− site effectively exchanges locations with the O2− ion to its right, propagating the excess charge to the right. The ions that are located on a site belonging to the other species are circled in black.
could travel between adjacent metal sites, but the oxygen ion’s significantly larger ionic radius makes this case less likely. The coupled hopons are most visible in the second step of Figure 2a, where two Al3+ ions on O2− sites form hopons with an O2− on an Al3+ site. Assuming an average Al−O spacing of 0.18 nm in the oxide,43 qiai/kT for each of these steps is approximately 17 nm/V, which is well below the experimental value. Alternatively, if the hopons are coupled tightly enough, then the combined motion of the hopons could face a single potential barrier, with the jump distance corresponding to the distance the defect travels after both steps. If such a tightly coupled hopon pair is traveling between oxygen sites in anodic alumina, then qiai/kT is 27 nm/V. This value is much closer to the experimental value than are the single-ion mechanisms. We propose a related but more direct transport mechanism than the two-hopon mechanism, the counter-site defect mechanism. Instead of two sequential aluminum−oxygen place exchanges, the Al3+ ion on the O2− site directly exchanges positions with an adjacent O2− ion, as depicted schematically in Figure 2b. Because the counter-site defect mechanism is a single-step mechanism, the single potential barrier framework can be applied directly. Unlike the coupled hopon mechanism, the counter-site defect mechanism does not require that O2− ions be located on the smaller Al3+ ion sites. Since the motion of a multi-ion defect can also take the form of eq 7,41 the bulk ionic flux due to the counter-site defect mechanism is given by ⎛ q acsdE bulk ⎞ 0 sinh⎜ csd JAl3+ = −JO2− = −Jcsd ⎟ kT ⎝ ⎠
(9)
∂ccrci = ∂t
J − Ji
o/e i
lccr
−
vo/e eq ox ci lccr
(10)
where o/eJi is the flux into the compact charge region due to interfacial chemical reactions. 2.3. Submodel for Chemical Reactions at the Oxide/ Electrolyte Interface. The goal of this subsection is to present the reaction mechanisms considered in our model and to derive equations for the oxidation/dissolution reaction rate, the reaction flux between the electrolyte and the compact charge region, the aluminum species ejection flux, and the velocity of the oxide/electrolyte interface. In the process of these derivations, we obtain a set of equations that specifies the concentration of adsorbed species at the oxide/electrolyte interface. The approach taken here extends that of Våland and Heusler12 for pH
