Modeling Growth of Organized Nanoporous Structures by Anodic

Article pubs.acs.org/Langmuir

Modeling Growth of Organized Nanoporous Structures by Anodic Oxidation Fábio D. A. Aaraõ Reis,*,† J. P. Badiali,‡ and Dung di Caprio‡ †

Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói, RJ, Brazil Chimie Paris Tech, ENSCP, CNRS, 4, Pl. Jussieu, 75252 Paris Cedex 05, France

‡

ABSTRACT: Nanostructured porous oxides are produced by anodic dissolution of several metals. A scaling approach is introduced to explain pattern nucleation in an oxide layer, and a related microscopic model shows oxide growth with long nanopores. The scaling approach matches the time of ion transport across the thin oxide layer, which is related to metal corrosion, and the time of diffusion along the oxide/solution (OS) interface, which represents the extension of oxide dissolution. The selected pattern size is of order (dDS/vO)1/2, where d is the oxide thickness, vO is the migration velocity of oxygen ions across the oxide, and Ds is the diffusion coefficient of H+ ions along the oxide/solution interface. This result is consistent with available experimental data for those quantities, predicts the increase of pore size with the external voltage, and suggests the independence of pore size with the solution pH. Subsequently, we propose a microscopic model that expresses the main physicochemical processes as a set of characteristic lengths for diffusion and surface relaxation. It shows a randomly perturbed OS interface at short times, its evolution to pore nucleation and to stable growth of very long pores, in agreement with the mechanistic scenario suggested by two experimental groups. The decrease of the size of the walls between the pores with the interface tension is consistent with arguments for formation of titania nanotube arrays instead of nanopores. These models show that pattern nucleation and growth depend on matching a small number of physicochemical parameters, which is probably the reason for the production of nanostructured porous oxides from various materials under suitable electrochemical conditions. migration are controlled by the electric field and stress gradients are related to volume changes were also recently proposed,33−35 with some results consistent with experimental data for porous alumina and titania (formation efficiency and proportionality of pattern size and layer thickness).35 On the other hand, the mechanistic aspects of the growth of organized titania nanotubes were discussed by Grimes and coworkers36 and by Schmuki and co-workers,37 with support of a large amount of experimental data, particularly their thorough microscopy works in the initial stages of the anodization. The scenario emerging from those works is an important basis for comparison with dynamic models for the related problem of organized nanoporous oxide growth. The present work proposes two alternative approaches to explain the formation of organized nanoporous structures by anodization. Instead of focusing on a particular application, our aim is to understand the essential microscopic processes to produce morphologically similar patterns in a variety of materials and chemical conditions. For these reasons, some results also help understanding formation of the nanotubular structures. First, we use a scaling approach to explain pattern nucleation when metal corrosion is coupled with erosion of the thin oxide layer subject to a high field, which drives ion migration. The

I. INTRODUCTION Anodic oxidation in neutral electrolyte forms a compact barrier film above the metal and was introduced to protect that material from aggressive environments. On the other hand, in acidic electrolyte, a porous layer with an approximately hexagonal pattern of nanosized cylinders can be formed, as first observed in alumina.1−4 Morphologically similar structures were subsequently produced by titanium anodization,5 and advances in these techniques led to the production of ordered nanotubular titania structures.6−8 More recently, related nanostructures were produced with several valve metals, such as Zr, Hf, W, Nb, Ta, Zn, Fe, Sn, and some of their alloys.9−15 Due to these results, anodic oxidation is now viewed as a route to produce new materials, particularly when they grow as arrays of long parallel nanotubes. For instance, applications of TiO2 nanotube arrays include gas sensors, adhesive bonding, solar cells, filtration membranes, electrocatalysts, photocatalysts, and formation of templates.16−24 The anodization of aluminum was widely studied experimentally1 and theoretical models explained porous alumina formation as an unstable growth of coupled metal/oxide (MO) and oxide/solution (OS) interfaces.25−32 Recent models represent a series of physicochemical processes to determine conditions for unstable growth and pattern features.30−32 Despite those efforts, systematic experiments on alumina show discrepancies from the theoretical results of several groups and stress the need to account for pH effects.4 Instability growth models where oxide dissolution and ion © XXXX American Chemical Society

Received: May 5, 2012 Revised: August 15, 2012

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dx.doi.org/10.1021/la301840c | Langmuir XXXX, XXX, XXX−XXX

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approach predicts growth of fluctuations of a selected wavelength related to the oxide layer thickness, to the characteristic time of oxygen ion migration across the layer and to the diffusion coefficient of H+ ions along the OS interface (which is connected to the spread of pH fluctuations). Large external potentials and low pH solutions are necessary for consistency of that approach, which in turn depend on control of the electrochemical cell. Second, we propose a nonequilibrium statistical growth model38−41 in which the basic physicochemical mechanisms are represented by characteristic lengths related to ion velocities, ion diffusion coefficients, and the interface tension of the OS interface. Simulations show the growth of an organized porous film, with pattern size depending on the relation of ion velocities and diffusion coefficients and with pore diameter increasing with the interface tension. The evolving patterns provided by this dynamic model resemble the microscopy images of growing titania nanotubes.17,36,37 The effect of interface energy on pore walls is consistent with the nanotubular form of anodized titania, as suggested by experimental works.36 The rest of this paper is organized as follows. In section II, the main features of experiments and basic concepts of our models are presented. In section III, a scaling approach is used to explain the nucleation of a regular pattern and consistency with experimental data is discussed. In section IV, the statistical growth model is introduced, and its simulations showing growth of organized nanopores and relations to experimental findings are discussed. In section V, a summary of results and conclusions is presented.

Figure 1. (a) Schematic view of the oxide layer separating metal and solution and the interfaces with those media. (b) The most relevant migration processes of ions in the oxide layer and along its interfaces, with the corresponding characteristic times. An oscillatory perturbation of wavelength λ is present in the OS interface and the MO interface has low roughness.

Many chemical and electrochemical reactions are involved in this process, but here we discuss only the main reactions for alumina growth. At the MO interface, the expected anodic reactions are1 2Al + 3OH− → Al 2O3 + 3H+ + 6e−

(1)

2Al + 3O2 − → Al 2O3 + 6e−

(2)

or

II. MAIN EXPERIMENTAL RESULTS AND MODEL CONCEPTS Experiments on formation of nanoporous alumina1,2 and titania nanotubes8,42 by anodization show an initial rapid decrease of the electric current, followed by a slow increase toward a steady state value. The initial decrease corresponds to the formation of a noncrystallized compact barrier layer with some breakdown sites where accelerated corrosion may occur. Recent microscopy (SEM, AFM, STM) and XPS study also show a hexagonal oxide pattern with an amorphous structure at the surface of anodized stainless steel.43 The oxide layer thickness is typically between 20 and 50 nm, depending on the applied voltage and material. When the current reaches a minimum value, electron microscopy shows that pore initiation begins, with randomly distributed positions. Finally, the steady state regime is related to the growth of pores or tubes (depending on the material), with diameters usually ranging from 50 to 100 nm (mild etching conditions provide titania tube diameters as low as 15 nm44 and control of fluoride concentration and anodization time provides diameters up to 350 nm45). The as-deposited titania nanotubes are amorphous and may crystallize upon annealing.7 The thickness of the solid MO interface is typically of 1 nm or less; thus, it is much narrower than the oxide layer. The OS interface is certainly wider because it includes a large solution region with concentration gradients, but as a first approximation we will not consider its detailed structure. For these reasons, we represent the initial oxide layer as a film of average thickness d located between the MO and OS interfaces, as shown in Figure 1a.

Two formulations are provided because the oxidation of the metal may be mediated by OH− or O2− ions. Both reactions consume oxygen ions. The first one produces H+ ions near the MO interface. The oxygen ions of both reactions originate from water, and thus, they are also correlated with production of H+ ions near the OS interface. These mechanisms lead to a strong coupling between those interfaces. Owing to the higher molar volume of the oxide, not all Al3+ ions change into oxide at the MO interface; instead, some of them migrate to the OS interface with the H+ ions. At the OS interface, there is a dissolution reaction, which can be globally summarized as46 Al 2O3 + 6H+ → 2Al+3 + 3H 2O

(3)

3+

in which Al is a soluble compound. Marker and tracer studies show that the reaction products formed at the MO interface migrate through the film and are dissolved in the solution, while oxygen ions migrate in the opposite direction.1 Prakasam et al.47 studied growth of titania nanotubes in electrolytes with water, NH4F, and ethylene glycol, at potentials up to 60 V, showing that the rate limiting step actually is migration of ionic species across the oxide, according to the so-called high field model.48,49 The electrochemical cell voltage is 20 V or more; thus, a difference of potential of this order is applied to a nanometer sized oxide. The electric field is very high and we expect that ions move with average speed depending on that field. This is consistent with the approximately constant current during nanopore nucleation and growth (after the initial current decrease, which corresponds to oxide layer formation). B


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III. SCALING APPROACH TO PORE NUCLEATION A. Cooperative Corrosion−Dissolution Process. Fluctuations of the MO and OS interfaces can be written as superpositions of Fourier components. The roughness of the MO interface is negligible compared to the remarkable patterns formed in the OS interfaces. Indeed, the MO interface evolves due to reactions of oxygen ions that reach the most prominent metal atoms, reducing height fluctuations. Consequently, here we focus on a single component of wavelength λ, as illustrated in Figure 1b, which mainly corresponds to a fluctuation of the OS interface. In section IIIB, we will explain why the more general case of different wavelengths in MO and OS interfaces does not change our conclusions. Figure 1b also shows the main ions that participate in the chemical and electrochemical reactions and that migrate across the oxide layer and along the interfaces (possible migration of the metal ions is not shown in Figure 1b but is discussed below). The flow of oxygen ions (O2−, OH−) from the OS to the MO interface leads to continued corrosion of the metal. Their migration occurs in a characteristic time τO, which depends on the layer thickness and the details of the driving mechanism. If oxygen ions move with the average speed vO, then we have τO = d /vO

smallest thickness, but the lateral extent of the eroded region is limited by H+ lateral diffusion. For those reasons, the cooperative process with pattern formation of size λ0 is observed if the time for H+ ions to spread along that size matches the time of transversal ion migration. This leads to τO + τ⊥ ∼ min{τ , τs}

Diffusion in the OS interface is typically faster than diffusion in the solid MO interface and thus τs < τ∥. Migration of H+ and oxygen ions across the layer is certainly coupled. To simplify, due to the smaller mass of H+ ions, it is reasonable to assume that their migration is faster; thus, oxygen ion migration is the rate limiting process (although the qualitative conclusions are not modified elsewise). This gives τO + τ⊥ ≈ τO and thus τO ∼ τs ⇒ d /v0 ∼ (λ 0 /2)2 /Ds

λ 0 ∼ 2 d Ds /vO

(8)

Since Ds is large compared to diffusion coefficients in the oxide, high external voltages are necessary to improve ion migration across the layer (i.e., to increase vO). On the other hand, low pH solutions inhibit the flux of H+ ions toward the solution, which facilitates their longitudinal spread. Consequently, the scaling approach is qualitatively consistent with two important electrochemical conditions of the anodic dissolution: high voltages and low pH. In section IIIB, the quantitative effect of these parameters is discussed. If one starts again from the basic scaling relation 6 but assumes that dominant longitudinal diffusion is inside the oxide, the same results are obtained with Ds replaced by D∥. Moreover, the assumption that H+ ion migration is the rate limiting process in the transversal direction leads to the same results with vO replaced by v⊥. In all cases, the structure of the scaling relation 8 is preserved and its main consequences do not change. B. Consequences and Discussion. First, we analyze the dependence of the pattern size with the external potential. It was experimentally shown that the oxide thickness d is approximately proportional to the applied potential;1 for instance, in aluminum anodization, typically 1.2 nm per volt is obtained. This means that the electric field in the oxide reaches a value approximately independent of the applied potential. The microscopic physicochemical parameters vO and vH depend on that field, and thus, they do not vary when the external potential changes. Diffusion along the OS interface (Ds) and longitudinal diffusion in the oxide (D∥) are also independent of the applied potential. Thus, as the external potential changes, eq 8 suggests that λ0 varies as √d. The increase of pattern size (λ0) with the applied potential is actually observed in anodization of titanium,37,50 aluminum,4 tantalum,11 hafnium,10 TiAl alloys,14 and titanium in nonaqueous electrolytes51 (a weak dependence of the pore diameter on voltage is found in zirconium anodization9). However, deviations from the √d dependence are certainly expected, since the proportionality of d with the external voltage is an approximate relation not tested in all materials. For this reason, our conclusion is restricted to a qualitative trend. On the other hand, quantitatively relevant results also follow from eq 8. Experiments with aluminum or titanium suggest that

+

At the MO interface, Al and H ions are produced. These ions migrate through the oxide layer in transverse and longitudinal directions, and also diffuse along the OS interface (Figure 1b). We will focus on H+ ions because they are involved in the dissolution reaction 3 of the OS interface. Their effect is sustained because a low pH is maintained in the solution close to that interface. On the other hand, owing to the high electric field, Al3+ ions rapidly diffuse away when they reach the OS interface. Transverse migration of H+ ions occurs in a characteristic time τ⊥. Therefore, for migration with an average speed vH, we have τ⊥ = d /vH

(7)

This relation gives the selected pattern size λ0 as

(4) 3+

(6)

(5)

The lateral migration of H+ ions is certainly diffusive. It is assumed to occur with coefficient D∥ inside the oxide layer and with coefficient Ds along the OS interface. Diffusion through a length λ/2 (Figure 1b) takes place with characteristic times τ∥ ∼ (λ/2)2D∥ and τs ∼ (λ/2)2/Ds, respectively. The narrowest oxide regions are subject to higher fields and, consequently, to faster transverse migration which corresponds to higher corrosion rates. Transversal migration of H+ ions is also faster there, which facilitates erosion of the OS interface (reaction 3). Thus, those regions are expected to have increased H+ concentration. The growth of an organized nanostructure is understood as a cooperative process between MO and OS interfaces. Their coupling depends on the rapid migration of ions across the oxide; that is, it is facilitated by small values of τO and τ⊥. On the other hand, longitudinal ion diffusion (in the oxide and in the solution) homogeneizes ion concentrations, working against pattern formation. In a pattern of wavelength λ, regions of smallest and largest thicknesses are separated by a typical distance λ/2 (Figure 1b). If λ is small, longitudinal diffusion rapidly suppresses the fluctuations of H+ concentration. If λ is large, the cooperative process of corrosion and erosion is possible around the point of C


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Figure 2. (a) Lattice structure of the oxide layer in its two-dimensional version. Green squares are the highest oxide molecules at each column (interfacial oxide). (b) Production of H+ ion (black dot) at the MO interface and possible targets after migration to the OS interface in a horizontal radius R0 (local mininum A or vertical migration B). (c) Erosion of the interfacial oxide at the target point, indicated by a crossed square. (d) Inside a circle of radius RI centered at the eroded point, the highest oxide (crossed square) is moved to the erosion point. The process clearly reduces surface area, mimicking interface tension.

values of λ0 (pattern size) and 2d are close, with λ0 of order 100 nm. Typical current values and nanotube growth rates suggest an average speed of order 10−8 cm/s, since one layer of oxide molecules is produced in nearly 10 s (note also that 3 oxygen ions participate in reactions 1 and 2). Thus, eq 8 gives Ds ∼ 10−13 cm2/s. At room temperature, this diffusion coefficient is typical of physisorbed species with activation energy near 0.5 eV (considering an adatom oscillation frequency 1013 Hz).41 It is much smaller than usual values of diffusion coefficient in solution (10−5 to 10−7 cm2/s), which discards this possibility (also, due to the electric field, the trend of H+ ions in solution is to move away from the OS interface, not contributing to the interface reaction). Our estimate of Ds is also larger than typical diffusion coefficients in solids (∼ 10−17 cm2/s). This analysis supports the hypothesis that H+ ions are adsorbed and diffuse along the OS interface. Our results are also consistent with the finding of ref 4 that the pH has no effect on the pore diameter and interpore spacing of anodized alumina. Indeed, while the effect of pH on transport of H+ ions in solution is significant, it is not expected to affect transport of the ions adsorbed in the oxide. This may be viewed as additional support to our approach. Two subtle aspects of this approach should now be addressed: (1) the possibility of a surface fluctuation of size λ > λ0 that could modify the cooperative corrosion-erosion process; (2) the possibility of fluctuations with different wavelengths in MO and OS interfaces. In the following, kinetic roughening concepts are used to show that these points do not affect our conclusions. In kinetic roughening38,52 of an initially flat interface, the maximal excited wavelength is the lateral correlation length ξKR, which increases in time as ξKR ∼ t 1/ z

1.5 and 438). Moreover, the excitation of longitudinal size ξKR is also the one of largest transversal amplitudes, and thus, it dominates the interface roughness. For short times, only small wavelengths are excited; thus, all fluctuations of H+ concentration are suppressed by longitudinal diffusion. For times in which ξKR is of order λ0, the cooperative process is possible because longitudinal inhomogeneities are not suppressed anymore. However, modes with λ > λ0 have not been excited at that time. Thus, the unstable growth is restricted to size λ0 ∼ ξKR and larger wavelengths do not develop in the system. This explains pattern selection. Moreover, the amplitudes of fluctuations with λ < ξKR are smaller than the amplitude of the fluctuation of size λ0 ∼ ξKR.38,52 The latter corresponds to maximal depths of the oxide layer because the OS interface is rougher than the MO front. Thus, the narrowest oxide regions are separated by a distance of order λ0 ∼ ξKR corresponding to the fluctuations of the OS interface, independently of the small fluctuations of the MO interface.

IV. MICROSCOPIC MODEL FOR PORE GROWTH In the previous section, we have addressed the question of the pore nucleation with scaling arguments and proposed an expression of the characteristic wavelength for the pore formation. In this section, we study the complete process of pore growth using a nonequilibrium statistical mechanics model. A. Model Definition. First we describe and justify the geometry of the oxide layer and of the interfaces. The oxide layer has a square lattice structure and is initially bounded by flat interfaces MO and OS, as illustrated in Figure 2a. This twodimensional version facilitates simulations of large systems, avoiding effects of lateral boundaries, but the extension to a three-dimensional structure is possible. The lattice constant corresponds to typical sizes of oxide molecules (around 0.3 nm for titania or alumina), and all model parameters are given in this unit.

(9)

(here, z is the dynamic exponent, which depends on basic symmetries of the growth dynamics and usually ranges between D


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The MO interface is assumed to be flat at all times. The very low roughness of the MO interface is consistent with the discussion in the beginning of section IIIA and is supported by results of recent models for growth of passive layers.53,54 The MO front is also assumed to move with velocity sufficiently large to maintain metal and solution always separated by a finite oxide layer, as shown in Figure 2a. The OS interface obeys the solid-on-solid (SOS) condition, with no overhangs. Thus, at a given horizontal position x, all lattice sites between the MO and OS interfaces are of oxide type. The OS site at each column is defined as the position of the top oxide molecule (highlighted squares in Figure 2a). Now we describe the main physicochemical processes to be modeled. Corrosion occurs at randomly distributed positions of the MO interface due to the flux of oxygen ions to that region. The corrosion reactions are correlated with the production of H+ ions: they may come from a corrosion reaction (eq 1) at the MO interface and subsequently migrate to the OS interface, or they may come from water decomposition at the OS interface, correlated with oxygen ion production (eq 2). Both processes are driven by the field, and thus, H+ ions are preferentially produced at the valleys of the OS interface rather than the hills. Those ions react with the oxide, eroding the OS interface (eq 3), and there may also be restructuring due to interface tension. Following these ideas, at each step of the model simulation, a random point of the MO interface is chosen for production of a H+ ion. Subsequently, a target site at the OS interface is chosen inside a horizontal radius R0 around the production point (Figure 2b). The procedure to choose this target site is described below. This site is removed to represent the oxide erosion (Figure 2c). Finally, interface tension is simulated by moving the molecule with maximal height inside a radius RI to the eroded point (Figure 2d). Figure 2b illustrates the choice of the target site to be eroded. First, one searches for the minimal OS point within a horizontal distance R0 from the production point. This is point A in Figure 2b, with the smallest distance dmin from the MO interface in that neighborhood. On the other hand, the distance between the production point and the OS point immediately above it (point B in Figure 2b) is dp. Point B is chosen as the target with probability p ≡ exp[−(dp − dmin)2/Δ2], otherwise the target point is A. After this choice, Figure 2c illustrates the oxide erosion at that site. If dp − dmin is small, p is large (near 1) and there are small height fluctuations inside the radius R0. Thus, H+ ions are vertically driven. Due to the randomness of the choice of the production point, the erosion of the OS interface is also random. On the other hand, if dp − dmin is large, p is small and OS valleys are deep. The field drives the H+ ions to those valleys and erosion occurs there. The parameter Δ is the typical vertical distance separating the regimes of small and large dp − dmin described above. Δ is related to the oxide layer thickness d. Flux of H+ ions to minimal heights depends on significant field inhomogeneities across the oxide layer (higher fields near the valleys of the OS interface, lower fields near the hills). For larger d, larger height fluctuations are necessary for those inhomogeneities to appear, and thus, Δ increases with d. For these reasons, this model represents the effect of the long-range electric forces in the system. The parameter R0 represents a typical horizontal distance traveled by H+ ions before reaching the OS interface. It is expected to increase as (Dsτ⊥)1/2, as discussed in section III.

After an erosion event, the OS interface is reorganized due to interface tension, as illustrated in Figure 2d. First, the OS site with maximal height inside a circle of radius RI centered at the position of the target site is found. If this maximal height is larger than the height of the eroded site, then the OS molecule at this maximal height moves to the eroded site. Otherwise, nothing occurs. The mechanism of moving particles to lower heights is similar to the Family model55 of thin film deposition and is known to represent interface tension effects.38,52 Note that the reorganization in Figure 2d clearly leads to reduction of the surface area in contact with the solution. The radius RI increases with the interface energy. B. Simulation Results. A lattice with lateral size L = 2048 was used in our simulations. One time unit corresponds to migration of L ions from the MO to the OS interface (and correspondingly L erosion events). In Figure 3, we show a sequence of configurations of the oxide layer obtained with R0 = 40, Δ = 7, and RI = 3. At very

Figure 3. Sequence of configurations of a section of the oxide layer with width 384 lattice units. The model parameters are R0 = 40, Δ = 7, and RI = 3, and times are (a) t = 5, (b) t = 10, (c) t = 20, and (d) t = 80. The blue horizontal lines are MO interfaces.

short times (Figure 3a), the OS interface is randomly eroded, with no evidence of pattern formation. However, a pattern with a selected wavelength begins to form as soon as that wavelength is excited (Figure 3b). This nucleation regime is rapidly followed by pore growth, illustrated in Figure 3c and d. A variety of model parameters under the conditions R0 ≫ Δ and R0 ≫ RI produce similar structures. Figure 4a and b shows the grown pores for two parameter sets different from those of Figure 3. Comparison of Figures 3d and 4a shows that the pattern wavelength is proportional to R0, as expected. Since R0 increases as (Dsτ⊥)1/2, this result matches the scaling picture of section III (eq 8). E


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dynamics. They have striking similarity to the illustrations of the mechanistic growth scenario for titania nanotube growth proposed after microscopy work in several stages of the anodization process.17,36,37 This is particularly true in the initial stages of the process, in which a rough oxide surface evolves to a patterned one. The main difference between nanopore formation (which is described here and is typical of alumina) and nanotube growth (typical of titania) seems to be the presence of voids between the pores of the latter. The growth of those voids leads to the formation of the parallel arrays of tubes. Mor and co-workers36 argue that this is related to the high Ti−O bond energy and high chemical solubility in the electrolyte, both favoring thin pore walls. Their arguments are completely consistent with our results: Figure 4 shows that the pore walls get thinner as the interface tension (or interface energy, both related to RI) increases. This suggests that our model actually represents the essential physicochemical mechanisms of the anodization process. Unfortunately, large values of RI lead to instability of pore walls and no reconstruction because there is no collective molecular dynamics; thus, a description of void formation would require more complex models.

V. CONCLUSION We propose a simple picture for the nucleation of organized porous nanostructures by metal anodization. Corrosion at the metal/oxide interface is coupled with dissolution at the oxide/ solution interface due to ion migration across the oxide layer, while diffusion along the interfaces homogeneizes H + concentration, working against pattern formation. Matching characteristic times of the main migration processes leads to scaling relations involving the pattern size, diffusion coefficient, and average ion speed. The increase of pore diameter with the external applied voltage is explained by this approach. It also suggests that diffusion of H+ ions adsorbed in the oxide/ solution interface is the main mechanism for their longitudinal spreading. This is consistent with the independence of the pattern size with the solution pH. The scaling arguments proposed here advance over previous works, since they provide a simple physicochemical picture of the basic mechanisms of pore nucleation, which can be applied to a variety of materials. We also introduced a microscopic model with mechanisms that parallel the ones proposed in the scaling theory. It shows the remarkable growth of long and organized pore structures after an initial random perturbation of the oxide/solution interface and subsequent nucleation of a pattern. These structures are closely related to the illustrations of previous mechanistic models based on microscopy work. Simulations also confirm the dependence of the pattern size on the diffusion coefficient of H+ ions along the OS interface and highlight the role of interface tension to rearrange pore bottom and walls. The effect of high interface energy on thinning of pore walls is also consistent with the explanation for growth of titania nanotubes (instead of nanopores). The microscopic model also supports the central reasoning of the scaling approach, that anodization of different metals and alloys leads to nanopattern growth due to a suitable matching of a small number of physicochemical parameters and oxide layer thickness. Experimentally, this matching depends on careful control of the electrochemical cell conditions.

Figure 4. Configurations of a section of the oxide layer with width 384 lattice units and model parameters: (a) R0 = 80, Δ = 7, and RI = 3 at time t = 40; (b) R0 = 80, Δ = 7, and RI = 6 at time t = 70. The blue horizontal lines are MO interfaces.

Comparison of Figure 4a and b shows that the average size of the pores increases with RI; that is, it increases with the interface tension. The pattern size has no significant change when RI varies. Results in Figures 3d and 4a,b also show that larger values of RI produce less smooth pore walls. This occurs because the interface tension stretches the pore mouth at short times, while at long times it helps to maintain almost vertical pore walls. For this reason, the condition R0 ≫ RI is necessary to observe formation of long pores. We also performed simulations with values of Δ different of that in Figures 3 and 4, but there is no significant difference in the final structures. That parameter only affects the nucleation process, which becomes faster for small values of Δ (i.e., patterns form at very short times). On the other hand, pore formation becomes impossible for very large Δ due to large random fluctuations of the OS interface. The SOS condition for the OS interface, which does not allow overhangs, implicitely represents a strong interface tension along the pore walls. This is reasonable for a material with strong dipolar interactions in the direction parallel to the pore walls, which is expected in metal oxides subject to high electric fields during anodization. It is also important to stress that our model represents relaxation (interface tension) at short time scales, but neglects long time relaxation. This condition does not affect pore growth because it does not change the long-range forces responsible for that feature (represented by the mechanisms of Figures 2b and 2c). To our knowledge, this microscopic model is the first one to illustrate the growth of those organized porous structures, although previous ones have already shown instabilities and pattern formation during solid dissolution.56−58 C. Comparison with Experiments. The growing patterns of Figure 3 are produced by a growth model with atomistic F


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]ff.br. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.D.A.A.R. and J.P.B. acknowledge support from CNPq (Brazil) and CNRS (France) for their cooperation project.

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Modeling Growth of Organized Nanoporous Structures by Anodic

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