Modelling Ordered Nanopourous Structures by Anodization via ...

Modelling Ordered Nanopourous Structures by Anodization via Cellular Automata Bartosik L ukasz1 , Stafiej Janusz1 , and Di Caprio Dung2 1 2

Institute of Physical Chemistry Kasprzaka 44/52 , Warsaw, Poland Institut de Recherche de Chimie Paris, CNRS – Chimie ParisTech, 11 rue P. et M. Curie, 75005 Paris, France

Abstract. A cellular automata model to simulate nanostructured alumina creation via anodization is proposed. The model mimics the Field Assisted Dissolution theoretical approach of the anodization process. The key parameters influencing the structures of the layer are identified and an attempt to recreate the two step anodization procedure in simulation conditions is made. The effect of dissolution rate and surface diffusion are considered. Simulation have been run on NVIDIA Tesla cards using the techniques of parallel programming to increase speed of the simulations. Keywords: Anodization, Cellular Automata, Parallel programing, aluminum oxide growth.

1

Introduction

Anodization is an electrochemical process of coating a metal with a layer of its oxide. The process is known since the 1920’s and has been used in metal protection, staining and decoration. Anodization is a simple and versatile process than can be performed for many valve metals including titanium [1], aluminum [2] and hafnium [3]. To conduct the process one needs to apply voltage to a piece of metal submerged in a proper electrolyte. In 1953 Keller, et. al. [4] have first revealed the structure of anodic aluminum oxide using transition electron microscopy. In place of an expected barrier layer hexagonally ordered pores were discovered. At the time the discovery went rather unnoticed and little work was done to continue the research. However with the advent of nanotechnology ordered layers of oxides have found new applications, namely they are perfect scaffolds in synthesis of nanostructures [5]. In 1995 Matsuda et. al. [6] discovered the experimental conditions sufficient for synthesis of nanoporous anodic aluminum oxide. Additional work soon followed with papers by Li [7], Jessensky [8] and Matsuda [9]. A common problem in anodization is that the initial layer is very disorganized due to surface defects of the metal. Various procedures such as polishing, electropolishing and etching are used to pretreat the metal, one of the most common ones is the so called two step anodization suggested by Matsuda [10]. In two step anodization the metal layer undergoes two separate anodizations. The first anodization removes defects from the surface and establishes a J. Was, G.C. Sirakoulis, and S. Bandini (Eds.): ACRI 2014, LNCS 8751, pp. 176–186, 2014. c Springer International Publishing Switzerland 2014

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pore network, the oxide layer is then washed away and the second anodization takes place upon the already cleaned and organized metal surface. Despite the huge amount of experimental work on the subject fairly little theoretical work to explain the mechanism of pore formation has been done. Among proposed theories two are most widely considered: Field Assisted Dissolution (FAD) [11] and Field Assisted Flow (FAF) [12]. In this work we follow the FAD using the cellular automata approach.

2

Model

We employ a probabilistic, asynchronous cellular automaton on a cubic lattice with Moore connectivity. Furthermore we assume that the difference in distance between the facet neighbor sites and the corner neighbor sites is negligible. Additionally we use periodic boundary conditions along two of lattice axes and fixed boundary conditions along the remaining axis. There are a total of six states in the model and eight rules for updating lattice site states. In order to make simulation times more manageable we decided to use the computational power of GPU. The simulation program is designed to take advantage of multiple cores available on NVIDIA Tesla cards. In a CUDA setup the workload is divided between threads organized in blocks. Upon creation these blocks cannot interact with one another other than through updating the global memory on the card. Our goal was to statistically allow every site to have one chance of undergoing an update per one time step. The amount of information necessary to determine if a site update is to occur is equivalent to knowing exactly what are the states of the extended neighborhood of a given site. The extended neighborhood is 2 sites in each directions netting a 5x5x5 cube. All the sites in this cube have to remain in their states unless the central site fulfills the conditions for an update of states. This also means two adjacent cubes contain sites that do not influence each other in that particular step in any way. This allows us to organize threads to work on such cubes in parallel and thus improve efficiency over a conventional program. One iteration of this procedure however covers only 0.8% of sites, to account for this we repeat the update procedures 125 times per time step each time randomly choosing a central point for the starting cube and subsequently partitioning the whole space into appropriate cubes. This approach allows to combine parallelization with efficient information flow through the system. The states in our model are labeled and their physical meaning discussed as follows. Let us label M the state corresponding to the metal, OX - to the oxide, EF - to the walker for electric field in the oxide, A - to anion vacancy in the oxide S - to solvent. The metal, oxide and solvent states represent what their names suggest while the anion and especially electric field states require further explanation. These three states have one feature in common. They are all oxide-like sites containing specific additional molecules or information. The anion states represents a portion of the oxide layer that contains anions formed

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during the dissociation of the solvent at the solvent-oxide layer interface. The electric field particles or rather their flux represent the information on the electric field at a given location. It is common to refer to the states as species or particles that occupy lattice sites. Another element of the CA approach are the rules to update the states of the lattice sites in the time step basing on the states of the sites in the preceding time step. In the following we explain the rules, how we write them and what they actually mean. The rules of reaction or diffusion follow the same scheme as presented below: P article1(position A) + P article2(position B) → P article3(position A) + P article4(position B) Position A and B are a pair of nearest neighbor sites for a given lattice connectivity. Let us note here that the position in the equation is important and related to the actual position in the grid, so reaction: M + OX → M + EF

(1)

M + OX → EF + M

(2)

is not the same as because the states of sites after each of these reaction would be different. The difference comes from the fact that in the case of the second reaction the metal particle moves from site A to site B while in the first one it stays at the same site. The rules to change the states are as follows: Reaction-type rules: M + S → OX + OX M + OX → M + EF EF + S → S + S

(3) (4) (5)

EF + S → A + S M + A → OX + OX

(6) (7)

EF + OX → OX + EF

(8)

EF + A → A + EF

(9)

Diffusion-type rules:

Each one of these rules is probabilistic with its probability given in a data file at the start of the simulation. Rule (3) describes the passivation of a metal surface in a corrosive environment. Rule (4) creates electric field walkers that simulate the presence of electric field. The sequence of rules (5), (6) give us the possible outcomes of what happens under influence of the electric field. Rules (5) and (6) mimic two possible processes occurring on the interface. The later rule describes water dissolution and inserting an anion into the oxide layer, while


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the former describes dissolving of the oxide layer itself. It is important to note that reactions (5) and (6) are checked in sequence during the execution of the simulation program. It is therefore of critical importance to properly adjust the data file probabilities to match the desired values. Typically this is achieved by setting the probability of reaction (6) to 1. In such a case reaction (5) occurs with probability P set in the data file and reaction (6) occurs with probability 1-P due to the sequencial nature of the code. There is only one rule governing the shape change of the metal-oxide interface, rule (7) which mimics the formation of metal oxide in a reaction of anions and the metal itself. The evolution of the porous oxide layer is thus essentially ruled by a subtle balance between its growth at the metal-oxide interface and its dissolution at the oxide-solvent interface controlled by the distribution of the electric field. The process described by the following equations: S + OX → OX + S S+A→A+S

(10) (11)

mimics surface diffusion and it is governed by a different mechanism than the previously described processes. Such a swap is always allowed if the number of pairs of oxide-like neighbors (bonds) increases or stays the same. Oxide-like neighbors are oxides, electric field and anions, all of which can exist within the oxide layer. If number of bond decreases the probability of the swap is given by exp(−Eb ΔNb ) where Eb is bond energy and Nb is the decrease in number of bonds (metropolis Monte Carlo algorithm [13]). Surface diffusion hence is described by two parameters: the number of surface diffusion steps per reaction step and the probability of a swap that decreases bond number by one (Pbond ). The latter scales with the power of the number of neighbors missing. In our CA model Eb is given as a probability of a particle to go to a site that has 1 bond less. The number of bonds is the number of non-solvent particles in the neares neighborhood of a given particle oxide-like particle. In the case of counting neighbors for the site containing a solvent the number of bonds is decreased by 1 to account for the fact that in the case of a swap a site that previously held an anion or an oxide will now hold a solvent and thus form no bonds. We assume that the system is in a steady state and that the oxide layer is a conductor. To determine the electric field at a given point we use the Poisson equation: ρf (12) Δϕ = Inside the conductor the number of charges is ρf = 0 and the Poisson equation is reduced to a Laplace equation Δϕ = 0

(13)

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This equation has identical form as the diffusion equation in steady state: Δc = 0

(14)

Hence the same methodology can be used to solve both problems. It is well known that the diffusion equation can be solved via Random Walk. We use the Random Walk approach to solve the Laplace equation and find the potential and electric field distribution on the oxide-solvent interface. The Random Walk is solved via swapping sites that are neighbors according to the diffusion rules. The simulation program has three main kernels: the reaction kernel, the diffusion kernel and the surface diffusion kernel. These kernels correspond to the rule types stated above. In each of the 125 iterations that make one full time step each kernels receives its own starting point on the lattice, the kernels are then executed in the order given previously. The reasons for such an organization are flexibility and computational effectivness. The main goal of designing this cellular automaton was to simulate various models of nanostructure creation. Hence the main design features were easy incorporation of new particles and new CA rules thus the separation into the three kernels by the kind of rules. This sepration also carries computational consequences since diffusion rules all have a probability of 1, so the step of generating a random number and checking if a given process should occur is redundant and can be omitted in order to increase simulation speed. In much the same manor only in surface diffusion explicit knowledge of all the extended neigbourhood in a given direction is required while for reaction and diffusion processes knowledge of the states of two cells per cube would be sufficient. Iteration of the three kernels also intermixes steps of diffusion, reaction and surface diffusion thus negating any artificial global order of the processes. Additional kernels include the initial condition kernel which gives the starting states of all the cells in the system, a termination kernel that ends a simulation once a given type of particles appears in the vicinity of one of the fixed boundaries of the system. In this paper we discuss the influence of two parameters on the resulting interface. These are the dissolution probability described by rule (5) and the number of surface diffusion steps per reaction step.

3

Results and Discussion

Presented below are the results for simulations:


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Table 1. Parameters used in the simulations Simulation parameters X-axis size 100 Y-axis size 100 Z-axis size 200 Surface steps 1 Diffusion steps 1

Reaction probabilities M + S → OX + OX 1 M + OX → M + EF 0.005 EF + S → S + S Pdis EF + S → A + S 1 − Pdis M + A → OX + OX 1

The parameter Pbond is found to have far less influence than the number of diffusion steps and its value was fixed at 0.02.

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Dissolution probability 0.5 Dissolution probability 0.4 Dissolution probability 0.3 Dissolution probability 0.2 Dissolution probability 0.1

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Fig. 1. Oxide layer distribution with respect to dissolution chance

The plots for various dissolution values share common features. From bottom to top in the system (left to right on figure 1) one can see an initial peak which corresponds to the pore bottoms, then a nearly flat layer at a given level which corresponds to the pore walls and a peak at the top of the system which corresponds to remnants of the initial interface and the beginning of pore formation. The smallest dissolution probability corresponds to the situation when oxide growth exceeds the rate of dissolution due to oxygen anion creation and thus no pores are formed. In figure 2 one can see that for a dissolution rate of 0.5 the amount of material is insufficient to form a regular porous structure. The pores are interconnected and full of side holes. In contrast a well defined porous structure can be seen for a dissolution chance of 0.3. Further evidence of a porous structure of the material can be seen in the cross section of the layer in figure 3, the Fourier transform of this cross section reveals hexagonal symmetry in the system.

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Fig. 2. Oxide solvent interface for a dissolution chance 0.5 left and 0.3 right

Fig. 3. Left cross section at the 102 level, right Fourier transform of the interface


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We also mimic the process of surface diffusion at the oxide layer surface with the solvent. The oxide sites “prefer”sites on the lattice that have many oxide neighbors and will move towards sites with more neighbors if they get a chance. The parameter we use to control this process is the number of times per reaction step that oxide particles search for another location to which they could move. If a site with more neighbors is found then a swap of particles takes place. Results of changing the rate of surface diffusion are presented in figures 4-7.

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Surface Diffusion steps 1 Surface Diffusion steps 0.25 Surface Diffusion steps 0.166 Surface Diffusion steps 0.125 Surface Diffusion steps 0.1

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Fig. 4. Oxide layer distribution with respect to surface diffusion rates

Fig. 5. Oxide solvent interface for surface diffusion rate 1 left and 0.1 right

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Fig. 6. Left cross section of the interface, right Fourier transform of the interface for surface diffusion rate 1

Fig. 7. Left cross section of the interface, right Fourier transform of the interface for surface diffusion rate 0.1


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All the profiles have a similar outline with only the fastest surface diffusion rate simulation having a porosity vastly different from the other results. The reason for this behavior is over liquefaction of a layer making oxide particles amass in regions where there is already many of them. This causes the peak at the “top”of the system to broaden noticeably. The side views of the layer do not reveal at first site the profound change in structure that has actually taken place. However the cross sections leave little room for doubt. The cross sections reveal that surface diffusion plays a crucial role in the formation of nanopores. In the case of a lower surface diffusion rate the pores have a noticeably smaller diameter, there are more of them and they are not interconnected as much as in the case of a higher surface diffusion rate. On the Fourier transform figures 6 and 7 it is visible that the hexagonal symmetry of the pores is retained when changing the surface diffusion rate.

4

Conclusions

We succeed to create a relatively simple cellular automata model that mimics nanostructured oxide formation on aluminum. Our simulation results are in qualitative agreement with experimental data. The resulting structures show hexagonal symmetry which brakes the quadratic symmetry imposed on the system via the choice of the Moore neighborhood on a cubic lattice. We identified key parameters influencing the porosity of the layer, pore size and distribution and managed to successfully replicate the two step anodization procedure in our simulations. Acknowledgments. The authors would like to acknowledge the Foundation for Polish Science for financial support of the research.

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