Theoretical Kinetic Analysis of Heterogeneous Photocatalysis by TiO2

Article pubs.acs.org/JPCC

Theoretical Kinetic Analysis of Heterogeneous Photocatalysis by TiO2 Nanotube Arrays: the Effects of Nanotube Geometry on Photocatalytic Activity Baoshun Liu,† Kazuya Nakata,*,†,‡,§ Shanhu Liu,† Munetoshi Sakai,†,‡ Tsuyoshi Ochiai,†,§ Taketoshi Murakami,† Katsuhiko Takagi,‡ and Akira Fujishima*,†,‡,§ †

Photocatalyst Group, Kanagawa Academy of Science and Technology, KSP Building East 412, 3-2-1 Sakado, Takatsu-ku, Kawasaki, Kanagawa 213-0012, Japan ‡ Organic Solar Cell Assessment Project, Kanagawa Academy of Science and Technology, KSP Building East 308, 3-2-1 Sakado, Takatsu-ku, Kawasaki, Kanagawa 213-0012, Japan § Research Institute for Science and Technology, Energy and Environment Photocatalyst Research Division, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan ABSTRACT: TiO2 nanotube arrays are important functional materials in photocatalysis. Compared with other TiO2 materials, the geometrical parameters of the nanotubes in an array significantly affect photocatalytic activity, but how they do this remains unclear. In the present work, a simple theoretical kinetic model to study the effects of nanotube diameter, wall thickness, and length on the photocatalytic activity of TiO2 nanotube arrays is developed, in which reactant (O2) transport is considered. The photocatalytic activity first increases and then decreases as the diameter and wall thickness of the nanotubes increase because of changes in light absorption, surface area, and reactant transport. The photocatalytic activity increases, and then reaches saturation as the nanotube length increases, which is mainly influenced by the change of light absorption along the nanotube. The present kinetic model agrees well with experimental results and clearly explains the photocatalytic activity of TiO2 nanotubes, helping us to understand nanotube photocatalysis.

1. INTRODUCTION TiO2 has become an important functional material that is used in many photoelectrochemical applications.1−8 In particular, nanostructured TiO2 with different structures and morphologies including spheres, fibers, tubes, and sheets9−13 has drawn much attention. TiO2 nanomaterials generally have high surface area and pore volume, which make them attractive for application in photocatalysis and dye-sensitized solar cells.14−16 TiO2 nanotubes are functional materials used in photocatalysis and are one of the most intensively studied TiO2 nanomaterials. A number of methods have been used to prepare TiO2 nanotubes such as hydrothermal treatment of P25 in NaOH solution, anodization of titanium sheets, template methods, and electrospinning.17−22 For example, Jiang et al. used cojetting electrospinning to prepare multichannel TiO2 tubes and found that the number of pores in each tube has an obvious effect on its photocatalytic properties.23 Anodic oxidation is always used to prepare TiO2 nanotube arrays, which have been widely used in photocatalytic oxidation of pollutants and water splitting.24−28 The wall thickness, length, and diameter of the nanotubes in the array can be changed by controlling the experimental conditions. For example, Chanmanee et al. used pulsed anodic oxidation to prepare TiO2 nanotube arrays with different wall thickness and inner diameter.29 Paulose et al. prepared a highly ordered array of © 2012 American Chemical Society

ultralong TiO2 nanotubes with tunable diameter by simple anodic oxidation.30 The nanotubes can easily be lengthened by extending the anodic oxidation time; Liu et al. obtained TiO2 nanotubes with lengths of 0.2−17 μm.31 It has also been reported that highly transparent TiO2 nanotubes can be prepared on glass substrates, which is attractive for many applications.32 Doping other elements in the TiO2 nanotubes can produce photocatalysts with visible response.33−35 Overall, TiO2 nanotube arrays have become increasingly important in photocatalysis. Compared to other materials, the geometrical parameters of the nanotubes in an array, including their length, inner diameter, and wall thickness, have a significant effect on photocatalytic activity. Although the effect of these geometrical parameters has been studied experimentally, the physical mechanism remains unclear, which limits future photocatalytic studies of TiO2 nanotube arrays. Because the complete study of these effects from only experiments is difficult, a systematic theoretical study is needed. This research provides a theoretical kinetic model to analyze photocatalysis by TiO2 nanotube arrays. The effects of nanotube length, inner diameter, and wall thickness on Received: January 15, 2012 Revised: March 8, 2012 Published: March 22, 2012 7471

dx.doi.org/10.1021/jp300481a | J. Phys. Chem. C 2012, 116, 7471−7479

The Journal of Physical Chemistry C

Article

Figure 1. Diagram of the physical model of photocatalysis by a TiO2 nanotube array. A tightly packed TiO2 nanotube array with a unit area of 1 cm2 is considered in this model.

photoinduced holes can be related according to the following equation:

photocatalytic activity are analyzed in detail, and the theoretical model agrees well with reported results. Based on this model, we are able to formulate a detailed explanation of photocatalysis by TiO2 nanotube arrays. This research also shows that mathematical-physical analysis is not only feasible, but also important in studies of photocatalysis, because it can summarize experimental studies and provide deeper physical understanding than experimental results alone. By revising this theoretical model, we can study the photocatalysis by other types of materials such as nanoparticles, thin films, and nanorod arrays. In addition, on the basis of this theoretical study, we can easily distinguish the major and minor factors that affect photocatalytic activity, which is a powerful tool to direct future work in photocatalysis.

Dp

d2p(x) dx 2

+ αI0e−αl −

p(x ) =0 τp

(1)

where p(x) is the hole density along the radial direction, Dp is the hole diffusion coefficient, τp is the hole lifetime, α is the absorption coefficient, I0 is the initial light intensity, and l is the distance from a point in the nanotube to its surface, as shown in Figure 1b. Equation 1 is a simple linear differential equation, and its general solution is −1 −1 p(x) = C1e L p x + C2e−L p x + αI0 τpe−αl

(2)

where Lp is the square root of Dp × τp, which is the diffusion length of photoinduced holes. Photocatalytic reactions can occur on the both inner and outer walls of nanotubes at the same speed. For simplification, the boundary condition of this formula is treated as

2. THEORETICAL MODEL 2.1. Model Description. Figure 1 shows a diagram of the theoretical kinetic model, in which an array (1 cm2) of tightly packed, uniform TiO2 nanotubes is considered. Each nanotube has a wall thickness of d, length of L, and inner diameter of R. As shown in Figure 1b, the nanotube array is irradiated along the normal direction. Under irradiation with UV light (Figure 1c), photoinduced electrons and holes will be produced in the conduction band (CB) and valence band (VB), respectively. The photoinduced electrons and holes in the nanotube wall will diffuse to the nanotube surface in the radial and axial direction, which is accompanied by bulk recombination. Because the nanotube wall is much thinner than its length, diffusion along the axial direction is not considered. The transfer of electrons from TiO2 to O2 is considered to be the rate-determining step of photocatalysis.36,37 Photocatalysis by nanotubes is different from that by dispersed TiO2 nanoparticles, as we have shown in previous papers.38,39 The transport of O2 (or other reactants) from the outside into the inside of TiO2 nanotubes should play an important role in photocatalysis by the nanotubes. The interfacial transfer of photoinduced electrons and holes can take place on the inner and outer walls, top, and bottom of TiO2 nanotubes, and is also accompanied by surface recombination. Overall, photocatalysis by nanotube arrays includes photogeneration, diffusion, recombination, and interfacial transfer of photoinduced electrons and holes, as well as reactant transport. All of these processes are explained in detail in the following sections. 2.2. Photogeneration of Holes on Nanotube Surfaces. The photogeneration, diffusion, and bulk recombination of

⎧ p(0) = p(d) = p , x = 0; x = d s ⎪ ⎨ dp(x) ⎪ = 0, x = d /2 ⎩ dx

(3)

where ps is the hole density on the nanotube surface, which directly affects the photocatalytic activity, as well as the electron concentration on the surface. According to this boundary condition C1 L p−1d /2 C2 −L p−1d /2 − =0 e e Lp Lp

(4)

where C1 and C2 are two linear-unrelated integration constants. According to eq 4, we can determine the relation between C1 and C2 −1 C2 = C1e L p d

(5)

C1 + C2 + αI0 τpe−αl = ps

(6)

Finally, it can be determined that C1 = 7472

ps − αI0 τpe−αl −1 e Lp d + 1

(7)



C2 =

Article

ps − αI0 τpe−αl −1 1 + e−L p d

where ns is the electron concentration on the nanotube surface, and [O2] is the concentration of oxygen. This electron transfer is widely accepted as the interfacial electron transfer mechanism if the TiO 2 surface has insufficient electron trapping centers.38,40 In steady state

(8)

The rate of photogeneration of photoinduced holes on a nanotube surface is equal to the diffusion flux of photoinduced holes ygenerate that is directed to a surface37,38 dp(x) ygenerate = −Dp |x = 0 = −υ p(C1 − C2) dx

υred = 2υox,1 (9)

Based on this formula, we can obtain the relation between ps and ns

where υp is defined as the diffusion speed of photoinduced holes to the TiO2 surface, which is equal to the square root of Dp/τp. Obviously, the rate of photogeneration of holes on a TiO2 nanotube surface determines the overall photocatalytic efficiency. υ1

OHs− + hs+ → OHs•,

υ1 = k1ps [OHs−]

ns =

υr

υr = k r[OHs•]ns

(10)

(11)

υox,2

(13)

JO = −DO2

(14)

2

(15)

The interfacial transfer rate of photoinduced holes is therefore 2υox,1 =

(21)

d[O2 ] dl

(22)

where JO2 is the oxygen diffusion flux from outside to inside a nanotube sheet, DO2 is the diffusion coefficient of oxygen, and [O2] is the O2 concentration at different positions in the nanotube array. As shown in Figure 2, O2 molecules that diffuse to the inside of nanotubes are consumed by photocatalytic reactions, i.e.,

k1ps [OHs−] 2kox,1[RH2,aq ] + k rns

(20)

Using the numerical method, the hole and electron concentrations on the surface of a nanotube can be calculated. Based on this, we can calculate the speed of interfacial transfer of photoinduced electrons and holes. 2.4. Transport of Oxygen. As illustrated above, it is widely accepted that the transfer of electrons to O2 is the ratedetermining step of photocatalysis. In the present case, because the water (or air) within the nanotube is almost static, the transport of oxygen molecules inside TiO2 nanotubes is different from that to a dispersed powder photocatalyst (when stirred). Therefore, the speed of O2 transport to the inside of the nanotubes and the voids between nanotubes (Figure 1a) should have an important effect on photocatalysis by the TiO2 nanotube array. Here, O2 transport is mainly dominated by diffusion, so it follows that

As a result, we can determine the value of OHs• as [OHs•] =

⎧ a = k rk red[O2] ⎪ ⎪ ⎨ b = 2kox,1k red[O2 ][RHaq,2] ⎪ ⎪ c = −2kox,1k1[OHs][RH2,aq ] ⎩

ygenerate = υr + 2υox,1

2.3. Surface Recombination and Interfacial Transfer of Holes. For Holes. According to the widely accepted photocatalytic mechanism, photoinduced holes are first trapped by a hydroxyl group on the TiO2 surface (OHs−), resulting in the formation of free hydroxyl groups (OHs•) according to eq 10. The organic substances in solution (RH2,aq) can be oxidized by the free hydroxyl groups according to eq 11. The resulting RHaq• can be further oxidized to Raq by the hydroxyl groups according to eq 13.36,37 The holes trapped on the hydroxyl groups can recombine with electrons in the CB (eq 12). According to refs 36 and 37, it can be considered that υox,2 = υox,1. In steady state, this means d[OHs•] = υ1 − 2υox,1 − υr dt

(19)

Steady state photocatalysis requires the holes generated on the nanotube surface to be completely consumed by interfacial transfer and surface recombination,38,40 so

(12)

OHs• + RHaq• ⎯⎯⎯⎯⎯→ OHs− + R aq + Hs•, υox,2 = kox,2[OHs•][RHaq•]

2a

with

υox,1

OHs• + es− → OHs−,

b2 − 4acps

−b +

OHs• + RH2,aq ⎯⎯⎯⎯⎯→ RHaq• + OHs− + Haq −, υox,1 = kox,1[OHs•][RH2,aq ]

(18)

2k1kox,1ps [OHs−][RH2,aq ] 2kox,1[RH2,aq ] + k rns

(16)

For Electrons. In the present research, we consider that photoinduced electrons are directly transferred to dissolved oxygen molecules (single electron transfer), as shown in the formula red

es− + O2 ⎯⎯⎯→ O2−,

υred = k redns[O2 ]

Figure 2. Diagram showing transport of oxygen into the interior of a TiO2 nanotube and the reduction of oxygen by electrons.

(17) 7473



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they will be reduced by photoinduced electrons. For the inner wall of a single nanotube, we have Jcon,inner (L) =

∫0

L

2πRk redns[O2 ]dl

N−1

≈

The rate of oxygen consumption is equal to the rate of interfacial transfer of electrons. For a single nanotube, the rate of interfacial transfer of electrons can be written as

∑ n=0

Je = Jcon,inner (L) + Jcon,outer (L)

2πRk redns[O2 ](n) N

In the present research, we consider a nanotube array with an area of 1 cm2 that is composed of uniform nanotubes with a wall thickness of d, and an inner radius of R. The number of nanotubes in the array is

(23)

where Jcon,inner(L) is the rate of oxygen consumption, and [O2](n) is the O2 concentration at position n, as shown in Figure 2. Similarly, on the outer wall of a nanotube, we have N−1

Jcon,outer (L) =

∑ n=0

Num = 1/(2 3 − π)(R + d)2 Je,wall = Num × Je

(24)

[O2 ](2) − [O1](1) L N

=

Jtopsurface = k redns,top[O2 ](0) × Num × π((R + d)2 − R2)

2πRL k redns[O2 ](1) N

Jbottom = k rednsbottom[O2 ](N ) × Num × πR2

⋮

2πRL k redns[O2 ](N − 1) N

Jtotal = Je,wall + Jtop,surface + Jbottom

(27)

Similarly, on the outer wall of a TiO2 nanotube, we have

=

The specific photocatalytic efficiency (QY) can be expressed

L N

QY = Jtotal/I0

[O2 ](2) − [O1](1) L N (29)

⋮ − (2 3 − π)(R + d)2 DO2 =

[O2 ](N ) − [O1](N − 1)

2π(R + d)L k redns[O2 ](N − 1) N

(37)

3. RESULTS AND DISCUSSION 3.1. Effect of Inner Radius. Figure 3 shows the dependence of the specific photocatalytic efficiency (QY) on the inner radius (R) of a TiO2 nanotube array for different O2 diffusion coefficients (DO2). QY first increases and then decreases as the inner radius increases. Moreover, the value of R for the TiO2 nanotube array to show the best QY decreases as DO2 increases, indicating that O2 transport has a significant effect on photocatalysis (similar to other reactants). QY is similar when R is greater than 150 nm independent of DO2, so O2 transport almost has no effect on QY for nanotube arrays with large R. This theoretical finding is in accordance with experimental results. For example, Zhuang et al.41 studied the effect of the inner radius of nanotubes on the photocatalytic activity of TiO2 nanotube arrays. They found that the photocatalytic activity depended strongly on R, increasing and

(28)

2π(R + d)L k redns[O2 ](1) N

(36)

as

[O2 ](1) − [O1](0)

2π(R + d)L k redns[O2 ](0) N

− (2 3 − π)(R + d)2 DO2

(35)

where ns,bottom is the concentration of electrons on the bottom of a nanotube and [O2](N) is the concentration of oxygen near the bottom of a nanotube. The total value of electron interfacial transfer in a TiO2 nanotube array with a unit area of 1 cm2 is

[O ](N ) − [O1](N − 1) − πR2DO2 2 L N

=

(34)

where ns,top is the electron concentration on the top surface of a TiO2 nanotube array, which is the same as the electron concentration within a TiO2 nanotube when n = 0, as shown in Figure 2. The photocatalytic interfacial transfer of photoinduced electrons and holes is

(26)

− (2 3 − π)(R + d)2 DO2

(33)

2.5. Photocatalytic Reactions on the Bottom and Surfaces of TiO2 Nanotubes. Besides photocatalysis on the inner and outer walls of TiO2 nanotubes, photocatalysis on the top and bottom surfaces of the TiO2 array also need to be considered. Similarly, the photocatalytic interfacial transfer of photoinduced electrons is

[O ](1) − [O1](0) 2πRL − πR2DO2 2 = k redns[O2 ](0) L N N (25)

=

(32)

so the overall rate of interfacial transfer of electrons is 2π(R + d)L k redns[O2 ](n) N

where Jcon,outer(L) is the rate of oxygen consumption on the outer wall of a nanotube. When n is 0, [O2](0) is the initial oxygen concentration in the reaction solution. On the surface of the inner wall, the consumption of O2 is equal to the diffusion flux of oxygen molecules. Therefore, eq 23 can be written as the following equations:

− πR2DO2

(31)

L N (30) 7474



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Figure 3. Dependence of photocatalytic efficiency (QY) on nanotube inner radius for oxygen diffusion coefficients (DO2) of (A) 1.0 × 10−7 cm2 s−1, (B) 5.0 × 10−7 cm2 s−1, (C) 1.0 × 10−6 cm2 s−1, (D) 3.0 × 10−6 cm2 s−1, (E) 5.0 × 10−6 cm2 s−1, and (F) 1.0 × 10−5 cm2 s−1 (other parameters: L = 10 μm, d = 10 nm, α = 5.0 × 103 cm−1, I0 = 106 cm2 s−1, [O2](0) = 1.0 × 1017 cm3, kox,1 = 1.0 × 10−18 cm3 s−1, kred = 1.0 × 10−14 cm3 s−1, kr1 = 1.0 × 10−9 cm3 s−1, τp = 1.0 × 10−9 s, [RH2,aq] = 1.0 × 1019 cm3, [OHs−] = 3.0 × 1014 cm2, Dp = 0.01 cm2 s−1, k1 = 1.0 × 10−8 cm2 s−1).

then decreasing as R increased. However, it is difficult to clearly understand the physical reason for this dependence. Figure 4 shows the dependence of the surface areas of inner and outer walls, and top and bottom surfaces on the inner Figure 5. (A) Dependence of [O2](n)/[O2](0) on l in nanotubes with different inner radii. (B) Dependence of [O2](n)/[O2](0) on l in the voids between nanotubes for different R (calculation parameters were the same as those described in Figure 3).

the same position in a nanotube array, [O2] increases as R increases because more O2 can diffuse into the nanotubes. Figure 5B shows the dependence of [O2] on l within the voids between nanotubes for different R. The [O2] in the voids decreases more sharply than inside the nanotubes because of the smaller relative area outside the nanotubes. For R = 10 nm, O2 is almost depleted when l is more than 3 μm. The [O2] in the voids between TiO2 nanotubes also increases as R increases. Therefore, the oxygen transport within the TiO2 nanotubes increases as R increases, which will lead to increased photocatalytic activity. In addition, the amount of TiO2 per unit area decreases as R increases, which will result in decreased light absorption and subsequently decreased generation of electrons and holes. Therefore, as R increases, the photocatalytic activity will decrease because of decreased electron− hole generation. The photocatalytic activity of the nanotube array is affected by increased O2 transport, decreased light absorption, and decreased surface area as R increases. Overall, the photocatalytic activity first increases and then decreases as the R of the TiO2 nanotubes in an array increases. 3.2. Effect of Wall Thickness. According the model in Figure 1, another geometrical parameter that can affect the photocatalytic activity of a TiO2 nanotube array is the wall thickness (d). Figure 6 shows the relation between QY and the wall thickness of a TiO2 nanotube array for different DO2. The photocatalytic activity first increases and then decreases as d increases, which

Figure 4. Relation between the surface areas of outer and inner walls, and top and bottom surfaces of nanotubes with a unit area of 1 cm2 and the nanotube inner radius (calculation parameters were the same as those described in Figure 3).

radius of a TiO2 nanotube array with a unit area of 1 cm2. The surface areas of the top and bottom surfaces are much smaller than those of the inner and outer walls, so their contribution to photocatalysis can be neglected. The sum of the surface areas of the inner and outer walls decreases as the inner radius increases. In other words, the area of active TiO2 surface decreases as the inner radius increases, which will lead to a decrease of photocatalytic activity with increasing R. Figure 5A shows the dependence of the oxygen concentration ([O2](n)) within a TiO2 nanotube on l for different R. It can be seen that [O2] decreases as l increases because of reduction by electrons. At 7475



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surface area, and decreased O2 transport. It is these factors that lead to a dependence of photocatalytic activity on d that first increases and then decreases as d increases. Many experimental studies concerning TiO2 nanotube arrays were reviewed; however, studies of the effect of wall thickness on photocatalytic activity have seldom been reported. It seems that the preparation of a series of TiO2 nanotube arrays with adjustable wall thickness is difficult. The present theoretical research gives a reasonable description of the effect of wall thickness on photocatalysis, and forms a good basis for future experimental studies. 3.3. Effect of Nanotube Length. The last geometrical parameter that affects the photocatalytic activity of TiO2 nanotube arrays is nanotube length L, which is the easiest parameter to adjust in experiments. Figure 8 shows the dependence of QY

Figure 6. Dependence of photocatalytic efficiency on the wall thickness of nanotubes for DO2 of (A) 1.0 × 10−7 cm2 s−1, (B) 5.0 × 10−7 cm2 s−1, (C) 1.0 × 10−6 cm2 s−1, (D) 5.0 × 10−6 cm2 s−1, and (E) 1.0 × 10−6 cm2 s−1 (R = 50 nm, L = 10 μm, other parameters are the same as those described in Figure 3).

is related to DO2. For the same wall thickness, the photocatalytic activity increases as DO2 increases. In addition, as DO2 increases, the d value for the nanotube array to show the highest photocatalytic activity increases. Figure 7 shows the dependence of the surface areas of inner and outer walls, and top and bottom surfaces of a 1 cm2 TiO2

Figure 8. Dependence of photocatalytic efficiency on nanotube length for different nanotube inner radii (DO2 = 3.0 × 10−7 cm2 s−1, other parameters are the same as those described in Figure 3).

on L for different R. It can be seen that QY increases as L increases, and finally reaches saturation. For the same L, the photocatalytic activity increases as R decreases because of increased light absorption. In addition, R almost has no effect on the relation between QY and L. There are many papers that report this result. For example, Liu et al.31 used an anodic oxidation method to fabricate TiO2 nanotube arrays and studied the effect of L on the photocatalytic activity for the degradation of phenol. They found that the photocatalytic activity increases with L until it reaches saturation. Liu et al. also studied the photocatalytic oxidation of gaseous acetaldehyde using TiO2 nanotube arrays prepared by anodic oxidation.42 They found that the photocatalytic activity also increases as L increases and then reaches a saturated value. Kontos et al. studied the photocatalysis of toluene and benzene by TiO2 nanotube arrays prepared by anodic oxidation. They observed the same dependence of photocatalytic activity on L.43 They considered that the photocatalytic activity first increases as L increased because of increased harvesting of photons. However, nanotubes that are beyond a certain length will exceed the diffusion length of reactants, so the photocatalytic activity will reach a saturated value versus L. This explanation is just one possible reason for L influencing photocatalytic activity and has not been verified; the main factor affecting the dependence of photocatalytic activity on L is unknown.

Figure 7. Relation between the surface area of the inner and outer walls, and top and bottom surfaces of a TiO2 nanotube array and wall thickness (calculation parameters are the same as those described in Figure 5).

nanotube array on d. Similar to the above, the surface area of the top and bottom surfaces is much smaller than that of the inner and outer walls, so their contribution to photocatalysis was also omitted. In addition, the amount of TiO2 will increase as d increases, resulting in increased light absorption and electron and hole generation. The sum of the surface area of inner and outer walls decreases as d increases, so the active surface area decreases as d increase. In addition, the overall amount of O2 that can be transported into the nanotube array will decrease as d increases. Therefore, unlike the effect of R, an increase in d will lead to increased light absorption, decreased 7476



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According to experiments, transport of the reactant (O2 in the present case) may be an important factor affecting photocatalytic activity. Figure 9 shows the dependence of QY on L

Figure 10. Dependence of photocatalytic efficiency on nanotube length for different α (R = 50 nm, d = 10 nm, DO2 = 1.0 × 10−6 cm2 s−1, other calculated parameters were the same as those described in Figure 3). Figure 9. Dependence of photocatalytic efficiency on nanotube length for DO2 of (A) 1.0 × 10−5 cm2 s−1, (B) 3.0 × 10−6 cm2 s−1, (C) 1.0 × 10−6 cm2 s−1, (D) 5.0 × 10−8 cm2 s−1, and (E) 3.0 × 10−8 cm2 s−1 (R = 50 nm, d = 10 nm, α = 1.0 × 104 cm−1, other parameters are the same as those described in Figure 3).

surface and the value of l along the nanotube is shown in Figure 11. It can be seen that ns and ps become very small when the nanotube

for different DO2. The photocatalytic activity decreases as DO2 decreases, which may be understood by considering that [O2] in a nanotube array will decrease as DO2 decreases. DO2 has almost no effect on the relation between QY and L, indicating that O2 transport may be not an important factor influencing the dependence of QY on L. To clearly understand this, we investigated the dependence of [O2] on nanotube length in TiO2 nanotubes and in the voids between nanotubes for different R. Because the R of TiO2 nanotubes used in photocatalysis is generally 30−50 nm, as shown in Figure 5, O2 cannot be consumed completely even for an array of nanotubes with a length of 10 μm. When L is more than 5 μm, [O2] is almost unchanged, indicating that O2 is not effectively used in the interior of long nanotubes. Therefore, this calculation shows that O2 diffusion is not an important factor controlling the dependence of QY on L, which should also be same for other reactants. Another parameter that can affect the dependence of QY on L is light absorption. The effect of the absorption coefficient (α) of TiO2 was studied. α can affect the penetration length of light in a TiO2 nanotube arrays, and is related to the wavelength of light. In experiments, we can use light of different wavelength to realize different α. Figure 10 shows the relation between QY and nanotube length for different α. α has an obvious effect on the relation between QY and L. For low α (1.0 × 103 cm−1), QY increases with L in an almost linear fashion. Saturation is not observed because light of this wavelength has a sufficiently long penetration length (>10 μm). For high α (1.0 × 104 cm−1), QY first increases with increasing L, and then reaches a saturated value when the nanotube is longer than 5 μm. Compared with O2 transport, α has a more important role in the dependence of QY on L because the intensity of light will decrease much faster than the reactant concentration. To clearly illustrate this, the relation between the concentration of electrons and holes (ns and ps, respectively) on the nanotube

Figure 11. Dependences of hole and electron concentrations on the l of a nanotube (α = 1.0 × 104 cm−1, R = 50 nm, d = 10 nm, L = 10 μm, other calculated parameters were the same as those described in Figure 3).

is longer than 7 μm. ns is so small that the O2 in the nanotube array cannot be efficiently consumed, so the O2 concentration remains almost unchanged for large l. Because the decrease of ns along the nanotube is mainly affected by the light absorption, the change of light absorption along the nanotube should be the main factor affecting the dependence of photocatalytic activity on L, which may not be decided by the transport of O2 and other reactants as was stated in an experimental report.43

4. CONCLUSIONS A kinetic theoretical model that takes reactant (O2) transport into the inside of nanotubes into consideration was established to analyze photocatalysis by TiO2 nanotube arrays. The effects of the inner radius, wall thickness, and length of the nanotubes on the photocatalytic activity of TiO2 nanotube arrays were investigated. The photocatalytic activity first increases and then decreases as the inner radius and wall thickness of nanotubes 7477



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increase, because of the combined effects of light absorption, surface area, and reactant transport. The photocatalytic activity increases until it reaches a saturated value as nanotube length increases, which is mainly affected by the change of light absorption along the nanotube. This simple theoretical kinetic model developed to study photocatalysis by nanotube arrays agrees well with experimental results and also clearly explains the physical reason for the effects of the geometrical parameters of the nanotubes on the photocatalytic activity of TiO2 nanotube arrays. These effects are difficult to study using experiments alone, so the present research provides support for experimental studies. This theoretical kinetic model can also be used to analyze photocatalysis by nanotube arrays of materials other than TiO2. Based on this model, we can predict the optimized values of R, d, and L for nanotube arrays to have the best photocatalytic activity. If the DO2 and α are 1.0 × 10−6 cm2 s−1 and 1.0 × 104 cm−1, the optimized R, d, and L are 10−20 nm, 20− 30 nm, and >5 μm.

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Corresponding Author

*E-mail: [email protected] (K.N.); president@admin. tus.ac.jp (A.F.). Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS

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ABBREVIATIONS

REFERENCES

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AUTHOR INFORMATION

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Article

B.L. acknowledges the Japan Society for the Promotion of Science (JSPS) for a Postdoctoral Fellowship for Foreign Researchers. K.N. and A.F. acknowledge JSPS for financial support from the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST) Program”.

I0=incident photonic flux (cm−2 s−1) [RH2,aq]=pollutant concentration (cm−3) Dp=hole diffusion coefficient (cm2 s−1) p(x)=hole concentration (cm−3) ps=hole concentration on TiO2 surface (cm−3) ns=electron concentration on TiO2 surface (cm−3) α=absorption coefficient (cm−1) τp=hole lifetime (s) R=inner radius of nanotube (nm) d=wall thickness of nanotube (nm) L=length of nanotube (μm) l=the distance from a point in a nanotube to its surface ygenerated=rate of hole photogeneration on surface (cm−3 s−1) υp=rate of hole transport in bulk TiO2 (cm s−1) C1, C2=integrated constants for carrier continuity equation υ1=trapping rate of surface holes by bridging oxygen (cm−3 s−1) υox,1=photooxidation rate of (RH2)aq (cm−3 s−1) υox,2=photooxidation rate of RHaq• (cm−3 s−1) υr=surface recombination rate(cm−3 s−1) υred=electron transfer rate from the impurities to oxygen molecule (cm−3 s−1) QY=apparent quantum yield (photocatalytic efficiency) for interfacial transfer [O2]=O2 concentration (cm−3) DO2=O2 diffusion coefficient (cm2 s−1) 7478



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Theoretical Kinetic Analysis of Heterogeneous Photocatalysis by TiO2

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