12170
J. Phys. Chem. B 2004, 108, 12170-12174
Transient Behaviors of Anodizing Titanium Analyzed by an Ion-Hopping Conduction Model M. Saitou† Department of Mechanical Systems Engineering, UniVersity of the Ryukyus, 1 Senbaru Nishihara-cho, Okinawa, 903-0213 Japan ReceiVed: April 9, 2004; In Final Form: May 18, 2004
An interpretation of the potential-time curves for the anodic oxidation of titanium measured under a fixed current density is presented on the basis of an ion-hopping conduction model proposed by Cabrera and Mott (Cabrera, N.; Mott, N. F. Rep. Prog. Phys. 1948-49, 12, 163). The transient potential-time curves exhibit three continuous growth regions that correspond to the following transport modes related to the growth of the anodic oxidized titanium layers: ion diffusion in the titanium oxide layers, ion migration by the high electric field, and the dielectric breakdown in the titanium oxide layers. The analysis based on the ion-hopping conduction model extended to this study allows us to determine fundamental parameters in anodic oxidation.
Introduction Because of the scientific interest and technological importance in items such as dimensionally stable anodes (DSA)1 of TiO2based dye-sensitized photoelectrochemical cells2 and nanostructured (nanoporous) titanium oxides,3 great effort has been devoted to understanding the anodic oxidation of titanium4 and to the electrical conductivity of anodic oxidized titanium films.5-7 In general, both ion and electron transport contribute mainly to the electrical conductivity of oxide films. Specially, electron transport in an anodic oxidized titanium film, which is affected by the presence of trap-recombination sites, electroactive sites, and surface electron states, has been investigated in detail by electrochemical techniques. For example, Basame and White reported electroactive sites that extremely enhance local current densities and greatly affect electrochemical reactions on titanium oxide surfaces. Ion transport4 has been recognized that in anodic oxidation, mobile ions form additional oxide layers at metal/oxide or oxide/ electrolyte interfaces. Several models8-10 for anodic oxidation based on ion transport, which can be related to experimental parameters, have been proposed. For example, a point defect model proposed by Macdonald allows us to describe a steadystate anodic oxidation process, especially the experimental results measured with electrochemical impedance spectroscopy, but has not been applied to transient behaviors of the oxidation process. On the other hand, as oxide layers growing on anode electrodes behave as insulators across which the potential drop takes place, electrical migration of ions due to the high electric field is caused. Thus, an ion-hopping conduction model has been proposed and accepted because of the clear and comprehensive description of the model that enables one to relate the behaviors of mobile ions to macroscopic parameters. For thin film thickness of oxides or high electric field strengths, the ionhopping conduction model derives a formula of the current density i through oxides described by i ) A exp(Bη) where A and B are coefficients dependent on the physical properties of the oxides and the experimental conditions. A great number of experiments in anodic oxidation have been performed under a fixed potential condition (potentiostat condi†
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tion)11 and compared with the relation i ) A exp(Bη). This is because the consecutive changes of ion transport modes related to the growth of oxide layers in anodic oxidation have not been observed. This paper aims at reporting the anodizing processes under a fixed current density, which processes comprise three consecutive growth regions, (1) first, ion transport by diffusion, (2) second, ion migration by high electric fields, and (3) dielectric breakdown12,13 of the oxide layers at the final stage and at analyzing the transient anodizing process based on the ion-hopping conduction model. Experimental Setup Titanium square plates 7.0 mm wide on each side and 1.0 mm thick, which have a purity of 99.5 wt %, were prepared for the anode electrodes. The titanium surfaces were polished using 5, 1, and 0.1 µm diamond powder pastes and had mirrorlike appearances. Titanium sheets 65 mm long, 60 mm wide, and 1 mm thick were prepared for the cathode electrodes and polished using abrasive sheets. Those electrodes cleaned by a wet process were located parallel in a cell containing 1 mol/L solutions of sulfuric acid or phosphoric acid. The electrochemical cell was maintained at a temperature of 300 K. One-step current was supplied by a power supply, and a fixed current density was maintained during anodic oxidation. The area of the cathode electrode was 80 times as large as that of the anode electrode. Hence, the resistance and capacitance in series of the cathode electrode in the electrolyte compared with those of the anode electrodes can be neglected. A digital oscilloscope that enables us to store data within a time range from a millisecond to several hundred seconds measured the transient potential-time curves between the anode and cathode electrodes. Results and Discussion Figure 1a shows a typical plot of the cell voltage η versus time t for the titanium anode in the sulfuric acid electrolyte at a current density of 1.2 mA/cm2. The η-t curves are measured within a time range from a millisecond to several hundred seconds. Figure 1b shows a log-log plot of Figure 1a, from
10.1021/jp0484276 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/13/2004
Ion-Hopping Conduction Model of Titanium Oxidation
J. Phys. Chem. B, Vol. 108, No. 32, 2004 12171
N Aw b ) W + U - RdzFE a ) W + U + (1 - R)dzFE N Aw
(2)
where NA is Avogadro’s number, W is the activation energy for hopping, U is the energy for solution into oxides, R is a transfer coefficient, z is the charge number of the ions, F is Faraday’s constant, and E is the electric field strength. The Taylor expansion of n around x, if the series higher than second order can be ignored, is applied to eq 1, and the result becomes Figure 1. Typical cell voltage changes in anodic titanium oxidation in a 1 mol/L solution of sulfuric acid at a current density of 1.2 mA/ cm2. Figure 1b, which depicts Figure 1a on a log-log scale, makes clear the presence of three growth regions.
Figure 2. Typical cell voltage changes in anodic titanium oxidation in a 1 mol/L solution of phosphoric acid at a current density of 0.82 mA/cm2. Figure 2b, which depicts Figure2a on a log-log scale, makes clear the presence of three growth regions.
which it can be seen that the transient process of titanium oxidation comprises three continuous growth regions. In a similar way, Figure 2a shows a typical plot of the cell voltage versus time for the titanium anode in the phosphoric acid electrolyte at a current density of 0.82 mA/cm2. Figure 2b shows a log-log plot of Figure 2a, from which it can be seen that the transient process of titanium oxidation comprises three continuous growth regions similar to the anodic oxidation in the sulfuric acid electrolyte. The three regions of η, which are observed independently in two kinds of electrolyte and current densities, have not been reported anywhere. We make here an attempt to interpret the transient anodic oxidation process of titanium on the basis of an ion-hopping conduction model.9 Ion-Hopping Conduction Model. As the ion-hopping conduction model is well explained in ref 4, in this section we show how the ion-hopping conduction model is extended to this study. Let us consider the amount of ions hopping per unit time and unit area between atomic planes at x and x + d where x is an arbitrary site and d is the distance between the atomic planes.
dn ) nx b p - nx+d a p dt
(1)
where nx indicates the ion density at the atomic plane x, and b p and a p indicate the transition probabilities per ion described as
( )
b p ) ν exp -
w b kBT
and
)[ (
(
dn RdzFE W+U ) dν exp cx exp dt RT RT - (1 - R)dzFE dc cx + d exp dx RT
W + U dc dn ) -d2ν exp dt RT dx
(
(3)
)
(4)
This indicates a flux of ions per unit time and area. Equation 4 results in a partial differential diffusion equation such as
∂c ∂2c )D 2 ∂t ∂x
(5)
where D ) d2ν exp[- (W + U)/RT]. In general, anions and cations are mobile when an oxide layer is formed. There have been controversial discussions about the transport number of cations.4 The interpretation of potential changes in anodizing titanium electrodes measured with cyclic voltammetry seems to have been controversial.14 To avoid the subject of which sorts of ions move in the oxide layer, we consider here only an electrochemical reaction equation k
Ti + H2O 98 TiO2 + 4H+ + 4e where k is the rate constant. Taking into account a change of the equilibrium concentration of ions in the oxide layer caused by a sudden electric current applied across the electrochemical cell, we have
∂∆c ∂2∆c )D 2 ∂t ∂x
w a kBT
a is the activation energy of one ion respectively, where w b dependent on the direction of its movement and ν is the attempt frequency. The first term on the right-hand side in eq 1 gives the hopping rate of ions from the x plane to the x + d plane. The activation energy under electric fields is given by
)]
where cx ) nx/d is the concentration of ions per unit area at the x plane. This is a phenomenological equation that describes anodic oxidation in this study. Next, three special cases of interest are introduced. 1. Transient η-t Curves in Region I (Diffusion Dominant). During the time span of several hundred milliseconds in region I, oxide layer growth may not occur. The process in region I, in which ions that initially exist in the oxides are deleted and mobile ions are formed, proceeds. As is well-known, a titanium surface polished in air is covered with a titanium oxide layer whose thickness is on the order of nanometers. The resistance is so small that the electric field in the titanium oxide layer can be neglected. For ion transport in region I, diffusion is considered dominant. Instantly, eq 3 yields
( )
a p ) ν exp -
) (
(
)
k(c* - ∆c) ) -D
∂∆c ∂x
at the interface
∆c ) 0, at t ) 0
(6)
12172 J. Phys. Chem. B, Vol. 108, No. 32, 2004
Saitou
Figure 3. Transient behaviors of the cell voltage in the 1 mol/L sulfuric acid electrolyte in region I. Solid lines indicate values calculated from eq 10 and dashed lines indicate the experimental values. The values used for calculation are as follows: for im ) 0.4 mA/cm2, C1 ) 8 × 10-4 F cm-2, c* ) 1.0 × 10-6 cm-3, k/D1/2 ) 0.5 s-1/2, and for a current density of 1.6 mA/cm2, C1 ) 1.5 × 10-3 F cm-2, c* ) 1.75.0 × 10-6 cm-3, k/D1/2 ) 0.923 s-1/2.
Figure 4. Transient behaviors of the cell voltage in the 1 mol/L phosphoric acid electrolyte in region I. Solid lines indicate values calculated from eq 10 and dashed lines indicate the experimental values. The values used for calculation are as follows: for im ) 0.4 mA/cm2, C1 ) 7 × 10-4 F cm-2, c* ) 1.0 × 10-6 cm-3, k/D1/2 ) 0.7 s-1/2, and for im ) 1.6 mA/cm2, C1 ) 1.5 × 10-3 F cm-2, c* ) 1.75 × 10-6 cm-3, k/D1/2 ) 0.866 s-1/2.
where ∆c denotes a deviation from the equilibrium concentration c*. Laplace transformation of eq 6 gives a solution
d∆cj | | )dx |x)0
(
kc*
Dxp xp +
)
(7)
k xD
where ∆cj is a concentration in Laplace space p. On the other hand, the anodic oxidation process can be described by an equivalent electrical circuit comprising the current density ii due to ion charge transfer and the current density ic through the oxide layer capacitance C1 caused by the impulse current density im.
∂∆c | | ∂x |x)0 dη ic ) C 1 dt
ii ) -zFD
ii + ic ) imu(t)
(8)
where z is the valence number of ions, η is the potential across the oxide, and u(t) is the step function (for t g 0, u(t) ) 1and for t < 0, u(t) ) 0). Combining eqs 7 and 8 leads to
η j)-
im C1 p2
zFc*k
+
(
C1 pxp xp +
)
k xD
(9)
where η j is the potential in Laplace space. Using the Laplace transform table, we have in real space
η)
[
( )]
im k 2zFc*xD zFc*D x 2 xt t+ 1 - e-(k/ D) t erfc t C1 C k xD 1 C1xπ (10)
Figure 3 shows the experimental values in region I in the sulfuric acid electrolyte and theoretical values calculated from eq 10 for two kinds of current density im, 0.41 and 2.0 mA/ cm2. Figure 4 shows the experimental values in region I in the phosphoric acid electrolyte and theoretical values for two kind of current density im, 0.41 and 2.0 mA/cm2. The oscillatory behaviors that can be seen in Figures 3 and 4 are caused by digital errors of the digital oscilloscope used in this study. The
Figure 5. Plot of k/D1/2 vs current density obtained from Figures 3 and 4.
calculated values are in good agreement with the experimental ones. However, the values of C1 used for the calculation are very high in comparison with the familiar Helmholtz capacitance. To avoid this difficulty, one interpretation may be that the sum of the impedance in the electrolyte adds the Helmholtz capacitance to the Warburg impedance dependent on the frequency. More experiments using electrochemical impedance spectroscopy will be required to ascertain the presence of Warburg impedance. Anyway, the reason for that is at present unclear. Figure 5 shows the dependence of k/xD on the current density for two kinds of electrolytes. It can be seen that k/xD increases with the current density. As the larger current density causes a larger potential drop across the oxide layer, the rate constant k increases and consequently k/xD increases. Thus, the transient behavior in region I in the sulfuric acid electrolyte is found to be similar to that in the phosphoric acid electrolyte. 2. Transient η-t Curves in Region II (High Electric Field). Since η in region II increases with time, the oxide layer is considered to grow and the electric field strength in the oxide layer increases under a fixed current density. The electrical migration by the electric field becomes so dominant that ion transport by diffusion can be ignored. Equation 3 can be rewritten as
(
)[ (
W+U dn RdzFE exp ) acxν exp dt RT RT (1 - R)dzFE exp RT
)
(
)]
(11)
The current density i through the growing oxide layer obeys
Ion-Hopping Conduction Model of Titanium Oxidation
J. Phys. Chem. B, Vol. 108, No. 32, 2004 12173
the Faraday law that gives the oxide layer thickness X
i)
zF dn NA dt
dX M ) i dt zFF
(12)
where M is the molecular weight of the oxide and F is the density of the oxide. Substituting eq 11 into eq 12 yields
)[ (
(
RdzFE dX W+U ) nΩν exp exp dt RT RT - (1 - R)dzFE exp RT
(
)
)]
(13)
where
F)
M ΩNA
Figure 6. Transient behaviors of the cell voltage in the sulfuric acid electrolyte in regions II and III for three kinds of current density. The gradients of the straight lines in region II give the values of c ) i ln(i/io)/B in eq 17: for i ) 0.16 mA/cm2, c ) 0.03076 mA/cm2, for i ) 0.82 mA/cm2, c ) 0.1693 mA/cm2, and for i ) 2.0 mA/cm2, c ) 0.5314 mA/cm2.
where Ω is the atomic volume of the oxide compound. For the high electric field (FE/RT . 1), eq 13 reduces a simplified form
()
X1 dX ) u exp dt X
(14)
where
(
u ) zΩν exp -
)
(
)
Figure 7. Dependence of (i/c) log i on i/c using values of c obtained from Figure 6. The straight lines yield io and β in eq 18: (a) H2SO4, io ) 7.20 × 10-7 A/cm2, β ) 1.58 × 10-6 cm V-1, (b) H2PO4, io ) 9.73 × 10-7 A/cm2, β ) 1.27 × 10-6 cm V-1.
W+U w+u ) u0 exp RT kBT
and
determine the constant values of io and B (or β). Setting the gradient i ln(i/io)/B in eq 17 as c which is dependent on the current density, we have
azFη azeη ) X1 ) 2RT 2kBT
i i ln i ) ln io + B c c
In a similar way, the current density i is given by
i)
( )
X1zFF zFF u exp M MQ
(15)
where
Q)
∫0 ti dt
io )
zFF u M
Setting
and
B)
zFF RdzF zFF ) β M RT M
we have
(BηQ )
i ) io exp
(16)
Under a fixed current density, eq 16 can be rewritten as
η - iRe )
i ln(i/io) t B
(17)
Here we use η - iRe instead of η to take into consideration the resistance due to the electrolyte. Equation 17 indicates how to
(18)
Figure 6 shows typical plots of η versus t in regions II and III in the sulfuric acid electrolyte for three kinds of current density. As eq 17 predicts that η in region II increases linearly with time, the gradient i ln(i/io)/B can be determined from Figure 6. Parts a and b of Figure 7, using the values of c obtained from Figure 6, show how the values of io and β for the sulfuric acid electrolyte and phosphoric acid electrolyte are determined. All the data fall on straight lines in Figure 7 as predicted by eq 18. Consequently, we obtain io ) 7.20 × 10-7 A/cm2, β ) 1.58 × 10-6 cm V-1 for the sulfuric acid electrolyte, io ) 9.73 × 10-7 A/cm2, β ) 1.27 × 10-6 cm V-1 for the phosphoric acid electrolyte. These values are consistent with those reported before.4 3. Transient V-t Curves in Region III (Breakdown Takes Place). The dielectric breakdown15 of titanium oxide films is known to occur at a high field strength. For the homogeneous breakdown that takes place all over the oxide layer, the oxide growth stops because no current by electrons caused by the breakdown contributes to the formation of oxides. However, as inhomogeneities6 associated with some defects generally exist in oxides, the local formation of the breakdown takes place and the oxide thickness continues to increase except at the sites where the breakdown takes place. In other experiments,3,12,16 using many kinds of instruments such as ellipsometry, SEM, and AFM, the film thickness of oxides has been determined. The titanium oxide film exhibits an interference color17 that allows us to estimate its film
12174 J. Phys. Chem. B, Vol. 108, No. 32, 2004
Saitou
Figure 8. Typical transient cell voltage in the sulfuric acid electrolyte in regions II and III. The straight line yields the value of VB + iR at a current density of 0.41 mA/cm2.
thickness. In addition, the formation factor for anodic titanium oxidation is dependent on experimental conditions but is known within a range of 1.4-3.35 nm/V. In this experiment, the light golden-brown oxide layers have a thickness below 30 nm. Hence, the average electric field across the titanium oxide film at the end of oxide growth in region II may attain the breakdown electric field strength. As shown in Figure 6, there exists a distinct crossover point at which the gradient of the η-t curves changes. Here, setting the breakdown voltage and current density as VB and iB, we have a formula that describes the i-η characteristic in region III from eq 16
i - iB ) io exp
[
]
B(η - VB - iR) Q
(19)
where i ) iB + ii and ii is the current density by ion charge transport. Equation 19 can be rewritten as
η ) VB + iR +
(i - iB) ln[(i - iB)/io] t B
(20)
Figure 8 shows a typical plot of the cell voltage versus t in the phosphoric acid electrolyte in region III. The straight line best fitted to the data yields the term in eq 20, VB + iR at a fixed current density. Figure 9 shows plots of VB + iR versus the current density i for the two kinds of the electrolyte. All the data fall on straight lines as predicted by eq 20. Consequently, we have VB ) 3.09 V and R ) 0.337 Ω cm2 for the sulfuric acid, VB ) 3.37 V, R ) 0.303 Ω cm2 for the phosphoric acid. This result shows that the oxide layer grown in phosphoric acid has a little larger breakdown voltage than that grown in sulfuric acid. If potential drops across the electrochemical cell are mainly due to the resistance of the oxide film, this result means that the electric field across the titanium oxide layer has a value large enough to cause the breakdown at some sites in the titanium oxide layer. Thus, the ion-hopping conduction model seems to explain well the transient anodic behaviors of the potential-time curves comprising the three continuous growth regions, but it cannot
Figure 9. Plots of VB + iR for various kinds of the current densities. The straight lines give the breakdown voltage VB and the electrolyte resistance R: (a) H2SO4, VB ) 3.09 V and R ) 0.337 Ω cm2, (b) H2PO4, VB ) 3.37, R ) 0.303 Ω cm2.
indicate how the movements of anions and cations contribute to anodic oxidation, which is at present left as a problem to be solved in the future. Conclusions We have investigated the anodic oxidation of titanium by measuring the transient potential-time curves in sulfuric or phosphoric acid electrolyte under a fixed current density. The oxide growth process is found to comprise the diffusioncontrolled region at the first stage, the electrical-migration dominant region at the second stage, and the dielectric breakdown region at the final stage, which are analyzed by the ionhopping conduction model extended to this study. The ionhopping conduction model explains the experimental results well. We gain the fundamental physical and chemical properties in anode oxidation. Acknowledgment. The author thanks Mr. Hideki Higa at the University of the Ryukyus for the experimental preparations. References and Notes (1) Chen, G.; Chen. X.; Yue, P. L. J. Phys. Chem. B 2002, 106, 4364. (2) O’Regan, B.; Gra¨tzel, M. Nature 1991, 353, 737. (3) Gong, D.; Grimes, C. A.; Varghese O. K. J. Mater. Res. 2001, 16, 3331. (4) Lohrengel, M. M. Mater. Sci. Eng. 1993, R11, 243. (5) Bisquer, J. J. Phys. Chem. B 2002, 106, 325. (6) Basame, S. B.; White, H. S. J. Phys. Chem. B 1998, 102, 9812. (7) Wang, H.; He, J.; Boschloo, G.; Lindstro¨m Hagfeldt, A.; Lindquist, S.-E. J. Phys. Chem. B 2001, 105, 2529. (8) Macdonald, D. D. J. Electrochem. Soc. 1992, 139, 3434. (9) Cabrera, N.; Mott, N. F. Rep. Prog. Phys. 1948-49, 12, 163. (10) Vetter, K. J. Electrochim. Acta 1971, 16, 1923. (11) Schultze, J. W.; Lohrengel, M. M. Electrochim. Acta 2000, 45, 2499. (12) Ohtsuka, T.; Matsuda, M.; Sato, N. J. Electrochem. Soc. 1985, 132, 787. (13) Climent, F.; Capellade, R. Electrochim. Acta 1988, 33, 433. (14) Marino, C. E. B.; de-Oliveria, E. M.; R-Filho, R. C.; Biaggio, S. R. Corros. Sci. 2001, 43, 1465. (15) Khail, N.; Leach, J. S. Electrochim. Acta 1986, 31, 1 279. (16) Schultze, J. W. Mater. Sci. Forum 1995, 18-188, 377. (17) Bernardi, M. I. B.; Lee, E. J. H.; L-Filho, P. N.; Leite, E. R.; Longo, E.; Varela, J. A. Mater. Res. 2001, 4, 223.
