- Email: [email protected]
Analysis of Titanium Oxide Nanotubes System Formation Current
Available online at www.sciencedirect.com
ScienceDirect Physics Procedia 86 (2017) 44 – 49
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Analysis of titanium oxide nanotubes system formation current P.L.Titov*, S.A.Schegoleva, N.B.Kondrikov Far Eastern Federal University, Sukhanova str. 8, Vladivostok, 690950, Russia
Abstract Analysis of formation of highly-ordered titanium oxide nanotubes array at the level of currents is carried out. It has been found that marking impulse possesses fine structure characterized by uncommon behavior. It is a series of small steps alternated by sharp overshoots that can be identified as Levy flights. Attractor obtained from current realizations points at the existence of quasistochastic in terms of phase and strictly periodic mode of partial Levy flights. It corresponds to trigger mode of system behavior with two stable states peculiar to auto-oscillating process. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ©2017 2016The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015.
Keywords: titanium oxide nanotubes; anodic oxidation; Levy flights; trigger mode; self-organization
1. Introduction Nano- and microcrystalline titanium dioxide finds a wide application in the form of catalytic agents, ceramics, construction and textile materials, paints, etc. Nanostructured TiO2 (Berger (2009), Macak et al. (2007), Reinhardt et al. (2006)) is the basis for photonic devices, membranes, bio-compatible materials (Park et al. (2010)), sensors, electrochromic displays (Nanotechnologies in electronics (2013), Ghicov et al. (2006)).Porous TiO2 are used in environmental facilities, including water electrolysis, catalytic and photocatalytic applications (Elezovic et al. (2009), Macak et al. (2005), Sakai et al. (2001), Meyer et al. (2004), Fujishima and Honda (1972)). From the point of view of photoelectric converters designing (Belov et al. (2011)), possibility of TiO2 morphology control at nanoscale level (Macak et al. (2007), Lozovaya et al. (2011), Fang et al. (2011)) determines the perspective of
*
Corresponding author. Tel.: +7-902-489-66-75. E-mail address: [email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.016
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
development of commercialization of solar batteries on dyes and more stable solid-state structures with extremely thin absorption layer. A large surface area will allow decreasing thickness of absorption layer to the size comparable with diffusion length of photogenerated charge carriers. As a result a possibility to use materials with low diffusion length and/or high unsoundness, that is, less expensive materials or technically simpler processes, in photoelectric converters becomes available. 2. Experimental part. The structures at issue were obtained in non-aqueous electrolyte on the basis of ammonium fluoride in glycerin by the method of anodizing on the source of stabilized direct current voltage. 99.9% purity titanium foil was used as the electrode material. Titanium foil samples were preliminary treated in the mixture of concentrated HF+HNO3 (1:3) within 10 seconds and flushed by distilled water. Formation of nanostructures on the surface of titanium was carried out in non-aqueous glycerin and semi-aqueous electrolytes on the basis of glycerin containing 50% H2O and NH4F 0.2M. Oxidation was carried out within 3 hours under 14 V on the potentiostat “Solartron 1287”. Pt electrode was used as counter electrode. The samples were obtained during stirring with a magnetic agitator in thermostatically-controlled cell. Then the samples were flushed by distilled H2O and dried in the air.
Fig. 1. Surface SEM-image of nanotubes array in longitudinal direction.
Morphological characteristics of the samples were examined by SEM method on scanning electron microscope “Hitachi S-5500”. As far as TiO2 possesses semi-conducting properties and is rather stable, the research was carried out without preliminary spraying of conducting layers with accelerating voltage up to 20 kV. As a result, tubular oxide nanostructured formations on the surface were obtained (Fig. 1). During the process of electrochemical oxidation and growth of nanotubes the electrodes had been being de-energized. Upon the structure of such electrical impulse, one may determine the character of the process taking place during titanium oxide nanotubes formation. 3. Results and discussion In the work (Zaichenko et al. (2011)) the impedance characteristics of nanostructures on the basis of aluminum and titanium were examined. This paper analyzes the mission on identification of current impulse removed during formation of ordered system of titanium oxide nanotubes (Fig. 1, 2).
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P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
Fig. 2. Nanotubes arrays images in different scales (c), (d) and their Fraunhofer images (a), (b).
The tubes are closely packed on R2 surface, they possess well-defined mean diameter. That is why local environments (the first coordination sphere) of each tube upon the average possess rather high coordination ordering (Yudin et al. (2009), Yudin et al. (2010)). However, on large scales array structure is rather modulation one (Yudin (1987)), close to such for amorphous ordering; this is evidenced by relevant Fraunhofer images where halo is observed (Fig. 2). Current impulse analysis proved that main trend possesses complex form – it is a linear combination of exponent and hyperbolic curve. Impulse “tail” is long-ranging one (Yudin et al. (2009), Yudin et al. (2010)), with exponent factor less than 1 in absolute terms (Fig. 3ɚ).
Fig. 3. General current impulse waveform (a) and its fine structure (b)
To distinguish fine current structure (Fig. 3b), let us put it trigger, polynomial approximation, Fourier filtration (on the basis of discrete Fourier transformation) were successively used. The number of filtered harmonic curves was chosen experimentally to achieve the best result from the point of view of the form of trigger component itself
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
(Fig. 4ɚ) and to exclude low-frequency oscillations. Here it must be noted that the spectrum of this trigger component (Fig. 4b) visually looks much like spectrum of periodic sequence of square wave impulses, which by itself should point to probable regularities in position of switches (impulse edges and droops).
Fig. 4. Part of trigger component realization (ɚ) and its Fourier spectrum (b).
According to the form of current realization it is clear that they possess certain stochastic scaling (Mihalyuk et al. (2010)). If starting with the large overshoot, then similar one never follows it. But high-frequency (HF) fluctuation oscillations follow. Then, after a while they abruptly end with Levy flight (Nikitin et al. (2009), Romanovskiy (2009)). Rather convinced bimodality is observed on Fig. 4ɚ. It is clear that the amplitude of Levy flights possesses rather narrow spread function. In their turn, HF-fluctuations are several-fold less in amplitude than Levy overshoots and also possess spread function that is rather narrow in amplitude. These HF-fluctuations were smoothed with the help of median filter during system and attractor tracing. To analyze system dynamics system’s track in phase space was designed. One of its coordinates is the current, and the other – current derivative with respect to time. Attractor development dynamics for two moments of time is represented in Fig. 5.
Fig. 5. System behavior tracks for different number of points:(a) for 200 points, (b) for 2,000 points.
General track form corresponds to cyclic system option where closed loops are visible. At early formation stages (200 points that corresponds approximately to 30-50 s) one closed loop is observed. Besides stable states corresponding to the end points where the system in phase space spends most of the time are also observed. All cycles are fixed and peculiar to trigger switch mode. Several clusters are formed along the axis of currents, there are 2 of them for 200 points, and there are 5 of them for 2,000 points. Later they run into one cluster (Fig. 6a) in a landscape mode and corresponding to minimum values of derivative. Globally resulting attractor (Fig. 6ɚ) corresponds to “hard” mode of self-excited oscillator operation with two stable states, when transition from one end state to the another one takes place not smoothly, but as a step. Such behavior is typical for the system with strong feedback.
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P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
A significant number of attractor’s points locate near vanishing derivative (dense horizontal formation). It means that the system spends most of the time in relative “rest”. From time to time these stability sections alternate with sharp overshoots of both positive and negative polarity (diagonally stretched “cloud” of points). So, we arrive almost to classical determination of Levy flights (Nikitin et al. (2009), Romanovskiy (2009)) when minor changes may alternate with the transitions abnormal in terms of size. At the same time, in the case “jumps” amplitude is almost determinated, with minor variation, it again points to trigger character of the process. The points relevant to the moments of switch (Fig. 6ɚ) are grouped along a certain line that points to approximately constant speed of formation of impulse edges/droops. In whole, such uncommon behavior may point to high level of system organization.
Fig. 6. Attractor (ɚ) and statistics of time ranges between nearest switch moments (b).
At the next stage distances between switch moments (impulses’ fronts and falls) were analyzed. The first indication to the order was obtained earlier in the form of Fourier spectrum of fine component (Fig. 4b). And accumulation of statistics of intervals between switches gave very even “grid” of modes (Fig. 6b) with multiple values of time: 10, 20, 30 s, …. To the first approximation the mode statistics envelope can be taken as exponential. From the characteristics obtained it can be inferred that behavior of high-frequency component of current possesses high ordering. 4. Conclusion Nanotubular titanium dioxide films are formed by the method of anodic oxidation in non-aqueous, semi-aqueous and aqueous fluorinated electrolytes. The form of current for nanotubes formation is identified. The main hyperbolic trend of current impulse is complicated by stable “trigger” component. To analyze behavior of such component the phase space was established, system behavior trajectory and attractor were calculated and plotted. This attractor corresponds to “hard” mode of system operation with positive feedback and quasirandom phase. On trigger component in time domain long sections of static behavior alternate with rare steps of significant, but stable amplitude. It allows classifying this process as a certain analogue of Levy flights. Statistics of intervals between the moments of steps, switches possesses a stable modal structure with exponential envelope. It may be that the processes taking place during nanotubes formation possess some features of Turing and Belousov–Zhabotinsky auto-oscillating reactions going beyond the scope of simple physico-chemical process. Such behavior scenarios are typical for chaotic self-organizing systems. References Belov A., Gavrilin I., Gavrilov S., Dronov A., Shulyat’ev A., 2011. Highly ordered nanotube arrays of TiO2 in the photovoltaic cells on a flexible carrier. Izvestiya Vuzov. Electronika. 88, 39–40.
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
Berger S., 2009. Selbstorganisierten anostrukturierte anodische Oxidschichten auf Titan und TiAl-Legierungen: Morphologie, Wachstum und Dünnschichtanodisation. Dissertation, Erlangen, pp. 212. Chaplygin Yu., 2013. Nanotechnologies in electronics (Ed.). Tekhnosphera, Moscow,pp.688. Elezovic N., Babic B., Radmilovic V., Vracar Lj., Krstajic N., 2009. Nb–TiO2 supported platinum nanocatalyst for oxygen reduction reaction in alkaline solutions. Electrochim. Acta54, 2404–2406. Fang D., Luo Z., Huang K., Lagoudas D., 2011. Effect of heat treatment on morphology, crystalline structure and photocatalysis properties of TiO2 nanotubes on Ti substrate and freestanding membrane. App. Surface S. 257, 6451–6461. Fujishima A., Honda K., 1972. Electrochemical Photolysis of Water at a Semi-conductor Electrode. Nature 238, 37–38. Ghicov A., Tsuchiya H., Hahn R., Macak J., Munoz A., Schmuki P., 2006. TiO2 nanotubes: H+insertion and strong electrochromic effects. Electrochem. Comm.8. 528–534. Lozovaya O., Tarasevich M., Bogdanovskaya V., Kasatkina I., Scherbakov A., 2011. Electrochemical synthesis, research and modification of nanotubes TiO2. Fizikokhimiya poverkhnosti i zaschita materialov. 47, 45–50. Macak J., Tsuchiya H., Bauer S., Schmuki P., Barczuk P., Nowakowska M., Chojak M., Kulesza P., 2005. Self-organized nanotubular TiO2 matrix as support for dispersed Pt/Ru nanoparticles: Enhancement of the electrocatalytic oxidation of methanol. Electrochem. Comm.7, 1417–1422. Macak, J., Tsuchiya, H., Ghicov, A., Yasuda,K., Hahn,R., BauerS., SchmukiP.,2007. TiO2 nanotubes: Self-organized electrochemical formation, properties and applications. Current opinion in Solid State & Materials Sci. 11, 3–8. Meyer R., Gorges G., Kreisel S., 2004. Preparation and characterisation of titanium dioxide films for catalytic applications generated by anodic spark deposition. Thin Solid Films 450, 276–281. Mihalyuk A., Titov P., Yudin V., 2010. Penrose tiling fractality in coordination Cayley’s tree graphs representation. Physica A 389, 4127–4139. Nikitin A., Chernavskaya O., Chernavskiy D., 2009. Pareto distribution in the dynamical systems in noise field. Trudy instituta obschey fiziki im. A.M. Prokhorova. 65, 107–123. Park H., Park S., Kim K., Jeon W., Park B., Kim H., Bae T., Lee M., 2010. Bioactive and electrochemical characterization of TiO2 nanotubes on titanium via anodic oxidation. Electrochim. Acta 55, 6109–6115. Reinhardt D., Krieck S., Meyer S., 2006. Special titanium dioxide layers and their electrochemical behavior. Electrochim. Acta52, 825–830. Romanovskiy M., 2009. Levy functional walks. Trudy instituta obschey fiziki im. A.M. Prokhorova. 65, 20–28. Sakai N., Fujishima A., Watanabe T., Hashimoto K., 2001. TiO2photocatalysis and related surface phenomena. J. Electrochem. Soc.148, 395–401. Yudin V., 1987. Stochastic magnetic structure of films with microporous structure.Nauka, Moscow, pp. 236. Yudin V., Titov P., Mikhalyuk A., 2009. Percolation of entropy functionals on Cayley tree graphs as a method of order–disorder character diagnostics of complex structures. Bulletin of the RAS: Physics 73, 1269–1276. Yudin V., Titov P., Mikhalyuk A., 2010. Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs. Theoretical and Mathematical Physics 164, 905–919. Zaichenko A., Tsarev S., Titov P., Kirillov A., Shegoleva S., Kuryaviy V., Kondrikov N., 2011. The investigation of impedance parameters, texture and self-organization of oxide formations on titanium and aluminum surfaces, 19th Int. Symp. “Nanostructures: Physics and Technology”. Ekaterinburg, Russia, 241–242.
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ScienceDirect Physics Procedia 86 (2017) 44 – 49
International Conference on Photonics of Nano- and Bio-Structures, PNBS-2015, 19-20 June 2015, Vladivostok, Russia and the International Conference on Photonics of Nano- and MicroStructures, PNMS-2015, 7-11 September 2015, Tomsk, Russia
Analysis of titanium oxide nanotubes system formation current P.L.Titov*, S.A.Schegoleva, N.B.Kondrikov Far Eastern Federal University, Sukhanova str. 8, Vladivostok, 690950, Russia
Abstract Analysis of formation of highly-ordered titanium oxide nanotubes array at the level of currents is carried out. It has been found that marking impulse possesses fine structure characterized by uncommon behavior. It is a series of small steps alternated by sharp overshoots that can be identified as Levy flights. Attractor obtained from current realizations points at the existence of quasistochastic in terms of phase and strictly periodic mode of partial Levy flights. It corresponds to trigger mode of system behavior with two stable states peculiar to auto-oscillating process. © Published by Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license ©2017 2016The TheAuthors. Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015.
Keywords: titanium oxide nanotubes; anodic oxidation; Levy flights; trigger mode; self-organization
1. Introduction Nano- and microcrystalline titanium dioxide finds a wide application in the form of catalytic agents, ceramics, construction and textile materials, paints, etc. Nanostructured TiO2 (Berger (2009), Macak et al. (2007), Reinhardt et al. (2006)) is the basis for photonic devices, membranes, bio-compatible materials (Park et al. (2010)), sensors, electrochromic displays (Nanotechnologies in electronics (2013), Ghicov et al. (2006)).Porous TiO2 are used in environmental facilities, including water electrolysis, catalytic and photocatalytic applications (Elezovic et al. (2009), Macak et al. (2005), Sakai et al. (2001), Meyer et al. (2004), Fujishima and Honda (1972)). From the point of view of photoelectric converters designing (Belov et al. (2011)), possibility of TiO2 morphology control at nanoscale level (Macak et al. (2007), Lozovaya et al. (2011), Fang et al. (2011)) determines the perspective of
*
Corresponding author. Tel.: +7-902-489-66-75. E-mail address: [email protected]
1875-3892 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of PNBS-2015 and PNMS-2015. doi:10.1016/j.phpro.2017.01.016
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
development of commercialization of solar batteries on dyes and more stable solid-state structures with extremely thin absorption layer. A large surface area will allow decreasing thickness of absorption layer to the size comparable with diffusion length of photogenerated charge carriers. As a result a possibility to use materials with low diffusion length and/or high unsoundness, that is, less expensive materials or technically simpler processes, in photoelectric converters becomes available. 2. Experimental part. The structures at issue were obtained in non-aqueous electrolyte on the basis of ammonium fluoride in glycerin by the method of anodizing on the source of stabilized direct current voltage. 99.9% purity titanium foil was used as the electrode material. Titanium foil samples were preliminary treated in the mixture of concentrated HF+HNO3 (1:3) within 10 seconds and flushed by distilled water. Formation of nanostructures on the surface of titanium was carried out in non-aqueous glycerin and semi-aqueous electrolytes on the basis of glycerin containing 50% H2O and NH4F 0.2M. Oxidation was carried out within 3 hours under 14 V on the potentiostat “Solartron 1287”. Pt electrode was used as counter electrode. The samples were obtained during stirring with a magnetic agitator in thermostatically-controlled cell. Then the samples were flushed by distilled H2O and dried in the air.
Fig. 1. Surface SEM-image of nanotubes array in longitudinal direction.
Morphological characteristics of the samples were examined by SEM method on scanning electron microscope “Hitachi S-5500”. As far as TiO2 possesses semi-conducting properties and is rather stable, the research was carried out without preliminary spraying of conducting layers with accelerating voltage up to 20 kV. As a result, tubular oxide nanostructured formations on the surface were obtained (Fig. 1). During the process of electrochemical oxidation and growth of nanotubes the electrodes had been being de-energized. Upon the structure of such electrical impulse, one may determine the character of the process taking place during titanium oxide nanotubes formation. 3. Results and discussion In the work (Zaichenko et al. (2011)) the impedance characteristics of nanostructures on the basis of aluminum and titanium were examined. This paper analyzes the mission on identification of current impulse removed during formation of ordered system of titanium oxide nanotubes (Fig. 1, 2).
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P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
Fig. 2. Nanotubes arrays images in different scales (c), (d) and their Fraunhofer images (a), (b).
The tubes are closely packed on R2 surface, they possess well-defined mean diameter. That is why local environments (the first coordination sphere) of each tube upon the average possess rather high coordination ordering (Yudin et al. (2009), Yudin et al. (2010)). However, on large scales array structure is rather modulation one (Yudin (1987)), close to such for amorphous ordering; this is evidenced by relevant Fraunhofer images where halo is observed (Fig. 2). Current impulse analysis proved that main trend possesses complex form – it is a linear combination of exponent and hyperbolic curve. Impulse “tail” is long-ranging one (Yudin et al. (2009), Yudin et al. (2010)), with exponent factor less than 1 in absolute terms (Fig. 3ɚ).
Fig. 3. General current impulse waveform (a) and its fine structure (b)
To distinguish fine current structure (Fig. 3b), let us put it trigger, polynomial approximation, Fourier filtration (on the basis of discrete Fourier transformation) were successively used. The number of filtered harmonic curves was chosen experimentally to achieve the best result from the point of view of the form of trigger component itself
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
(Fig. 4ɚ) and to exclude low-frequency oscillations. Here it must be noted that the spectrum of this trigger component (Fig. 4b) visually looks much like spectrum of periodic sequence of square wave impulses, which by itself should point to probable regularities in position of switches (impulse edges and droops).
Fig. 4. Part of trigger component realization (ɚ) and its Fourier spectrum (b).
According to the form of current realization it is clear that they possess certain stochastic scaling (Mihalyuk et al. (2010)). If starting with the large overshoot, then similar one never follows it. But high-frequency (HF) fluctuation oscillations follow. Then, after a while they abruptly end with Levy flight (Nikitin et al. (2009), Romanovskiy (2009)). Rather convinced bimodality is observed on Fig. 4ɚ. It is clear that the amplitude of Levy flights possesses rather narrow spread function. In their turn, HF-fluctuations are several-fold less in amplitude than Levy overshoots and also possess spread function that is rather narrow in amplitude. These HF-fluctuations were smoothed with the help of median filter during system and attractor tracing. To analyze system dynamics system’s track in phase space was designed. One of its coordinates is the current, and the other – current derivative with respect to time. Attractor development dynamics for two moments of time is represented in Fig. 5.
Fig. 5. System behavior tracks for different number of points:(a) for 200 points, (b) for 2,000 points.
General track form corresponds to cyclic system option where closed loops are visible. At early formation stages (200 points that corresponds approximately to 30-50 s) one closed loop is observed. Besides stable states corresponding to the end points where the system in phase space spends most of the time are also observed. All cycles are fixed and peculiar to trigger switch mode. Several clusters are formed along the axis of currents, there are 2 of them for 200 points, and there are 5 of them for 2,000 points. Later they run into one cluster (Fig. 6a) in a landscape mode and corresponding to minimum values of derivative. Globally resulting attractor (Fig. 6ɚ) corresponds to “hard” mode of self-excited oscillator operation with two stable states, when transition from one end state to the another one takes place not smoothly, but as a step. Such behavior is typical for the system with strong feedback.
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P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
A significant number of attractor’s points locate near vanishing derivative (dense horizontal formation). It means that the system spends most of the time in relative “rest”. From time to time these stability sections alternate with sharp overshoots of both positive and negative polarity (diagonally stretched “cloud” of points). So, we arrive almost to classical determination of Levy flights (Nikitin et al. (2009), Romanovskiy (2009)) when minor changes may alternate with the transitions abnormal in terms of size. At the same time, in the case “jumps” amplitude is almost determinated, with minor variation, it again points to trigger character of the process. The points relevant to the moments of switch (Fig. 6ɚ) are grouped along a certain line that points to approximately constant speed of formation of impulse edges/droops. In whole, such uncommon behavior may point to high level of system organization.
Fig. 6. Attractor (ɚ) and statistics of time ranges between nearest switch moments (b).
At the next stage distances between switch moments (impulses’ fronts and falls) were analyzed. The first indication to the order was obtained earlier in the form of Fourier spectrum of fine component (Fig. 4b). And accumulation of statistics of intervals between switches gave very even “grid” of modes (Fig. 6b) with multiple values of time: 10, 20, 30 s, …. To the first approximation the mode statistics envelope can be taken as exponential. From the characteristics obtained it can be inferred that behavior of high-frequency component of current possesses high ordering. 4. Conclusion Nanotubular titanium dioxide films are formed by the method of anodic oxidation in non-aqueous, semi-aqueous and aqueous fluorinated electrolytes. The form of current for nanotubes formation is identified. The main hyperbolic trend of current impulse is complicated by stable “trigger” component. To analyze behavior of such component the phase space was established, system behavior trajectory and attractor were calculated and plotted. This attractor corresponds to “hard” mode of system operation with positive feedback and quasirandom phase. On trigger component in time domain long sections of static behavior alternate with rare steps of significant, but stable amplitude. It allows classifying this process as a certain analogue of Levy flights. Statistics of intervals between the moments of steps, switches possesses a stable modal structure with exponential envelope. It may be that the processes taking place during nanotubes formation possess some features of Turing and Belousov–Zhabotinsky auto-oscillating reactions going beyond the scope of simple physico-chemical process. Such behavior scenarios are typical for chaotic self-organizing systems. References Belov A., Gavrilin I., Gavrilov S., Dronov A., Shulyat’ev A., 2011. Highly ordered nanotube arrays of TiO2 in the photovoltaic cells on a flexible carrier. Izvestiya Vuzov. Electronika. 88, 39–40.
P.L. Titov et al. / Physics Procedia 86 (2017) 44 – 49
Berger S., 2009. Selbstorganisierten anostrukturierte anodische Oxidschichten auf Titan und TiAl-Legierungen: Morphologie, Wachstum und Dünnschichtanodisation. Dissertation, Erlangen, pp. 212. Chaplygin Yu., 2013. Nanotechnologies in electronics (Ed.). Tekhnosphera, Moscow,pp.688. Elezovic N., Babic B., Radmilovic V., Vracar Lj., Krstajic N., 2009. Nb–TiO2 supported platinum nanocatalyst for oxygen reduction reaction in alkaline solutions. Electrochim. Acta54, 2404–2406. Fang D., Luo Z., Huang K., Lagoudas D., 2011. Effect of heat treatment on morphology, crystalline structure and photocatalysis properties of TiO2 nanotubes on Ti substrate and freestanding membrane. App. Surface S. 257, 6451–6461. Fujishima A., Honda K., 1972. Electrochemical Photolysis of Water at a Semi-conductor Electrode. Nature 238, 37–38. Ghicov A., Tsuchiya H., Hahn R., Macak J., Munoz A., Schmuki P., 2006. TiO2 nanotubes: H+insertion and strong electrochromic effects. Electrochem. Comm.8. 528–534. Lozovaya O., Tarasevich M., Bogdanovskaya V., Kasatkina I., Scherbakov A., 2011. Electrochemical synthesis, research and modification of nanotubes TiO2. Fizikokhimiya poverkhnosti i zaschita materialov. 47, 45–50. Macak J., Tsuchiya H., Bauer S., Schmuki P., Barczuk P., Nowakowska M., Chojak M., Kulesza P., 2005. Self-organized nanotubular TiO2 matrix as support for dispersed Pt/Ru nanoparticles: Enhancement of the electrocatalytic oxidation of methanol. Electrochem. Comm.7, 1417–1422. Macak, J., Tsuchiya, H., Ghicov, A., Yasuda,K., Hahn,R., BauerS., SchmukiP.,2007. TiO2 nanotubes: Self-organized electrochemical formation, properties and applications. Current opinion in Solid State & Materials Sci. 11, 3–8. Meyer R., Gorges G., Kreisel S., 2004. Preparation and characterisation of titanium dioxide films for catalytic applications generated by anodic spark deposition. Thin Solid Films 450, 276–281. Mihalyuk A., Titov P., Yudin V., 2010. Penrose tiling fractality in coordination Cayley’s tree graphs representation. Physica A 389, 4127–4139. Nikitin A., Chernavskaya O., Chernavskiy D., 2009. Pareto distribution in the dynamical systems in noise field. Trudy instituta obschey fiziki im. A.M. Prokhorova. 65, 107–123. Park H., Park S., Kim K., Jeon W., Park B., Kim H., Bae T., Lee M., 2010. Bioactive and electrochemical characterization of TiO2 nanotubes on titanium via anodic oxidation. Electrochim. Acta 55, 6109–6115. Reinhardt D., Krieck S., Meyer S., 2006. Special titanium dioxide layers and their electrochemical behavior. Electrochim. Acta52, 825–830. Romanovskiy M., 2009. Levy functional walks. Trudy instituta obschey fiziki im. A.M. Prokhorova. 65, 20–28. Sakai N., Fujishima A., Watanabe T., Hashimoto K., 2001. TiO2photocatalysis and related surface phenomena. J. Electrochem. Soc.148, 395–401. Yudin V., 1987. Stochastic magnetic structure of films with microporous structure.Nauka, Moscow, pp. 236. Yudin V., Titov P., Mikhalyuk A., 2009. Percolation of entropy functionals on Cayley tree graphs as a method of order–disorder character diagnostics of complex structures. Bulletin of the RAS: Physics 73, 1269–1276. Yudin V., Titov P., Mikhalyuk A., 2010. Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs. Theoretical and Mathematical Physics 164, 905–919. Zaichenko A., Tsarev S., Titov P., Kirillov A., Shegoleva S., Kuryaviy V., Kondrikov N., 2011. The investigation of impedance parameters, texture and self-organization of oxide formations on titanium and aluminum surfaces, 19th Int. Symp. “Nanostructures: Physics and Technology”. Ekaterinburg, Russia, 241–242.
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