Journal of Electron Devices, Vol. 11, 2011, pp JED [ISSN: ] DYE SENSITIZED SOLAR CELL: PARAMETERS CALCULATION AND MODEL INTEGRATION

Transcription

1 ournal of Electron Devices, ol. 11, 11, pp. 1-4 ED [ISSN: ] DYE SENSITIZED SOLAR CELL: PARAMETERS CALCULATION AND MODEL INTEGRATION Ahmed. A. El Tayyan Physics Department, Al Azhar University, Gaza, Palestine ahmedtayyan@yahoo.com Received /11/11,, online /11//11 Astract-- This article proposes an accurate approach to calculate the internal parameters of a dye sensitized solar cell DSSC (L, α, m, D, n, τ). This approach is ased on the electron diffusion differential model and the values of the short circuit current density, the open circuit voltage oc, and the current density and the voltage at the maximum power point i.e. and, respectively. Assuming, that the charge transfer at the counter electrode is potential controlled, the Butler- olmer equation is adequately integrated with oth the electron diffusion differential model, and the Schottky arrier model to account for the interfacial effect at counter electrode/electrolyte and TiO /TCO interfaces on the - characteristics. Some parametric analyses were conducted to study the effect of teerature, and electrode thickness on various DSSC parameters. Keywords: DSSC, solar cell, photovoltaic I. INTRODUCTION In the last two decades, the dye sensitized solar cell (DSSC) has received much attention [1-8]. DSSCs have certain advantages over conventional silicon and thin film photovoltaic devices, due to the silicity of the manufacturing process and the cost-effectiveness of most of the cell materials, which makes this type of solar cells feasile for mass production. Currently, DSSC can achieve 11.1% energy conversion efficiency [9]. Further irovement is highly relied on etter understanding of the energy conversion mechanisms and precise modeling for design optimization. In general, a DSSC corises a nanocrystalline titanium dioxide (TiO ) or zinc oxide (ZnO) electrode modified with a dye deposited on a transparent conducting oxide (TCO). A counter electrode is another TCO coated with a platinum (Pt) thin layer. An electrolyte solution with a dissolved iodide ion/triiodide ion redox couple ( I / I 3 ) is added etween the electrodes. Efficient electron injection from excited dye to TiO (or ZnO) plays an iortant role in DSSC [1,11]. The flow of the injected electrons through the porous TiO film to the TCO depends on the incident intensity and trapping detrapping effect [1]. Susequently, the electrons flow to the counter electrode via an external load. The oxidized dye molecules are regenerated y redox mediators ( I / I 3 ). Finally, for a colete DSSC operation cycle, the oxidized redox mediators ( I ) are transported to the counter electrode where 3 regeneration of redox mediators occurs [13]. Due to the very small nanoparticle size (aout nm) together with the strong reening effect of electrolyte, there is no significant macroopic electric field in most of the porous thin films. Therefore, oth electrons and redox mediators are transported mainly y diffusion. The photoinjection of electrons from excited dye to TiO conduction and increases the electron density in the porous electrode thin film, resulting in shifting the quasi-fermi level E F closer to the conduction and edge E c. Theoretical models assume that the difference etween E F of TiO and the electrolyte redox potential E redox e the induced photovoltage [14-1]. That is true for the open-circuit condition in which there is no current flow through the TiO /TCO interface and electrolyte/counter electrode interface. However, under maximum power conditions, additional potential differences across these two interfaces should e considered. This article deries a method to calculate the internal parameters of DSSC (L, α, m, D, n, τ). This approach provides a tool for researchers to estimate theoretically the internal parameters instead of digging through literature for their experimental value. Also, the effect of voltage loss at the counter electrode/electrolyte interface on the total voltage and the ehavior of the cell are theoretically investigated. II. MODELING Due to the wide spread in DSSC research, the ientific community is interested in experimentally and theoretically investigation of the photoelectrochemical ehavior of nanostructured electrodes of DSSC. Södrgren et al. [14] has suggested a sile model which is asically ased on two equations. The first one is a

2 A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 continuity equation that deries the transport, recomination, and generation of electrons within the nanoporous film: n( x) n( x) n D exp( x) n x t where n(x) is the excess concentration of the photogenerated electrons at position x within the film measured from the TiO /transparent conducting oxide (TCO) interface, n is the concentration of electrons under equilirium conditions in dark, τ is the conduction and free electrons life time, D is the diffusion coefficient of electrons, Φ is the illumination intensity (incident photon flux, cm - s -1 ), and α is the light asorption coefficient of the porous film. Under a steadystate condition of an irradiated DSSC, Eq. (1) ecomes n( x) n( x) n D x exp( x) (1). () The possiility of trapping-detrapping of electrons is not included in Eq. () since it is only iortant under nonsteady-state conditions [17]. Under short-circuit conditions, electrons are easily extracted as photocurrent and none of the electrons are drawn directly to the counter electrode. Therefore, the two oundary conditions are: n( ) n, (3) and dn dx xd, (4) where d is the porous electrode thickness. The second equation coleting the model links the excess concentration of photogenerated electrons at the ack contact, n contact, with the photovoltage, ph, through the following expression [14, 18] m q n contact ln n ph, (5) where k is the Boltzmann constant, q is the elementary charge, T is the asolute teerature and m is an ideality factor. Equation () can e solved using the oundary conditions in Eq. (3), and Eq. (4). The short circuit current density can e otained as [14, 18]: ql d L d exp( ) L tanh, () 1 L L d cosh L where L is the electron diffusion length given y: L D. (7) If the DSSC operates under a potential difference etween the Fermi level of the TiO and the redox potential of the electrolyte, the density of the electrons at the TiO /TCO interface (x = ) increases to n giving a new oundary condition: n( ) n. (8) The second oundary condition at x = d remains unchanged as shown in Eq. (4). Solving Eq. () and using Eq. (5) yields the relation ship etween the current density and [14,18] : m L( ) ln 1 q d qdn tanh L. (9) This equation can e re-arranged to have the form: qdn d q tanh exp 1 L L m. (1) III. CALCULATING THE INTERNAL PARAMETERS OF DSSC To determine the internal parameters of DSSC (Φ, L, α, m, D), one needs five pieces of information. These pieces of information are the short circuit current ( ) given y Eq. (), open circuit voltage ( oc ), current and voltage at the maximum power point ( and ), respectively. The fourth and the fifth pieces of information required for the calculation of the five parameters can e otained y realizing that the slope of the power at the maximum power point (dp/d ) and (dp/d ) are equal to zero. Thus, at the open circuit voltage and at the maximum power point Eq.(9) ecomes: m L ln 1 q d qdn tanh L oc, (11) m L( ) ln 1 q d qdn tanh L. (1) Additional two equations can e derived using the fact that on the P- and P- characteristics of a DSSC at the maximum power point, the derivatives of power with respect to voltage and with respect to current are equal to zero, that is; dp d d( ) d d d. (13) 17

3 Power (W/cm ) Current density (A/cm - ) A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 dp d d( ) d d d. (14) The five internal parameters of DSSC (L, α, m, D, n ) can e otained y simultaneously solving Eq. (), and Eq. (11) through Eq. (14) using the Mathcad solve lock (Given. Find) and y ilementing the method of Conjugate Gradients. The value of the conduction and free electrons life time τ is otained y sustituting the values of the electron diffusion length L and the diffusion coefficient of electrons D into Eq. (7), then, the - characteristics can e generated using Eq. () and Eq. (9). Calculations were done using light intensity Φ = 1. x 1 17 cm - s -1 which is equivalent to 1 sun condition (1 mw/cm ) [15,18,19], teerature T = 3 K, and porous layer thickness d = 1 x 1-4 cm. The values of, oc,, and are determined from the characteristics otained y sustituting the pulished values of the parameters into Eq. (1) (see tale 1). Then, the otained values of, oc,, and are ilemented into Eq. (), and Eq.(11) through Eq. (14) to otain the internal parameters of the DSSC, see Tale I. Notice that the calculated values are of the same order as the pulished values. TABLE I: A coarison etween the calculated values and the pulished values of DSSC internal parameters. Parameter Pulished value Calculated value L (cm -1 s -1 ).31x x1-3 α (cm -1 ) Refs. [15,18,19] M 4.5 Refs. [15,18,19] D (cm - s -1 ) 5.x1-4 N 1 1 Ref. [18] Refs[ 18,,1] τ ms 1 Refs. [15,18,] x x A good match etween the current-voltage and the power-voltage characteristics otained using the pulished values of the parameters [18], and the theoretically fitted current density-voltage and powervoltage curves otained using the present method is illustrated in Fig. 1 and Fig., respectively oltage (olt) Pulished alues Calculated alues Fig.1: - Characteristics. Points are pulished data [18], line is calculated data oltage (olt) Pulished data Calculated data Fig. : Power-voltage characteristics. Points are pulished data [18]. Line is calculated data. I. INTEGRATING THE MODEL Figure 3 illustrates the and energy levels of DSSC. In the dark, the potentials of TiO, TCO, and the counter electrode are equal to the redox potential of the electrolyte as illustrated in Fig.3.a. Upon irradiation of the DSSC, the quasi-fermi levels of TiO are raised due to photoinjection as illustrated in Fig.3.. Therefore the reference photovoltage can e expressed as:, (15) 1 where is the potential difference etween the TiO Fermi level (E F ) and the redox potential of the electrolyte (E redox ). Here, represents the same voltage given in Eq. (9) and (1). On the other hand, 1 is the voltage loss at 18

4 A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 the TiO /TCO interface, and is the voltage loss at the counter electrode/electrolyte interface. where is Schottky arrier height; h is Planck s constant equal to.x1-34 m kg s -1, m * is equal to 5. times the free electron mass m e for TiO, and A * is Richardson s constant of TiO equal to.71 x 1 Am - K -. Equation (1) can e rearranged to have the form: 1 ln 1 q * A T q exp( ). (18) In order to find an expression for, one needs to the following assution that the charge transfer at the counter electrode is potential controlled, thus the current over the interface is independent of the concentrations of the redox ions at the interface. It is only dependent on the potential. Thus, the current can e deried y the Butler-olmer equation: q q exp exp ( (19) 1 where is the exchange current density and β is a symmetry parameter with a value ranges etween and 1. A symmetry parameter of.5 deries a charge transfer reaction, which is symmetric forward and reverse potential. In this article we assume symmetric reaction β =.5. Expanding the two exponentials in Eq.(19), taking into account, only, the first three terms of the expansion of each exponential and silifying terms yields: q, () Fig. 3: Potential diagrams of DSSC: (a) in dark and () under illumination. Meng et al [3] have suggested that highly doped and highly conductive TCO can e considered as a metal, and the TiO /TCO interface can e simulated y Schottky arrier model. Under illumination, the flow of electrons through the TiO /TCO interface causes a voltage loss 1. This voltage loss can e correlated y the following expression referring to the thermionic emission theory: * q q1 A T exp exp 1 and A * 4 * m qk 3 h, (1), (17) or, q. (1) A charge transfer resistance R CT of dimension Ω cm - thus can e defined as shown in Eq. (). The dimension of R CT is the same as for a contact resistance. 1 q. () R CT 19

5 Current density (ma cm - ) Charge transfer Resistance ( cm - ) Current density (ma cm - ) A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp T=3 K = = = Teerature (K) oltage (volt) Fig. 4: The variation of the exchange resistance with teerature. The ack electrode of a DSSC is covered with some platinum, acting as a catalyst for the redox reaction. Provided the charge transfer resistance is not too high, it is a good approximation to derie electrolyte/platinized TCO contact y an ohmic charge transfer resistance and the Nernest equation [1]. A low R CT corresponds to a high exchange current density, which is essential for a DSSC with good performance [1]. The variation of R CT with teerature is illustrated in Fig. 4. The exchange resistance increases with increasing teerature. The diffusion controlled charge transfer reaction is not considered in this article since it is effective at higher voltage. In contrast, a Butler ollmerian ehavior is oserved at low voltages. Meng et al. [3] showed that at the TiO /TCO interface there exists a critical value of, elow which 1 is negligile. At value higher than the critical value, 1 increases with increasing. In this article, a value of equal to.5 e which is elow the critical value is chosen, in order to focus on the contriution of on the overall cell ehavior. Fig. 5: A coarison etween the theoretically calculated - characteristics showing the effect of including the voltages loss, - 1, and at 3 K. Figure 5 illustrates a coarison etween the calculated - characteristics using Eq. (9),, and the effect of including the voltage loss at the TiO /TCO interface 1 ; and the voltage loss at the counter electrode at 3 K. It is clear that inclusion of 1 term doesn t affect the - characteristics. On the other hand, the inclusion of the term affects the maximum power point location, ut it doesn t affect the short circuit current and the open circuit voltage points. The same ehavior is found at higher teerature values. At lower teerature values, the inclusion of the voltage loss 1 shows some deviation y lowering the maximum power point as shown in Fig = 1 T= 73 K = 1 - = oltage (olt) Fig. : A coarison etween the theoretically calculated - characteristics showing the effect of including the voltages loss, - 1, and at 73 K.

6 (ma) (olt) P max (mw cm - ) oc (olt) (ma cm - ) Effeciency (%) Fill factor Current density (ma cm - ) A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp = oltage (olt) T = 73 K T = 3 K T = 35 K T = 35 K T = 375 K Fig.7: The calculated - characteristics at various teeratures. The - characteristics at various teerature values are depicted in Fig.7. The open circuit voltage oc as well as the maximum power increase with increasing teerature, and the short circuit current is not affected as expected. higher series resistance contriutes to the decrease in oc and [18]. The short circuit current and the maximum power current increase aruptly with increasing thickness as depicted in Fig. 8, then reach a peak, and decrease gradually afterward. The variations of and with thickness can e easily explained y electron photogeneration. For a given porosity and pore size, any increase in electrode thickness is directly increases the internal surface area of the semiconductor, resulting in a higher dye loading. Therefore, a thicker electrode can asor more photons, leading to higher and higher. However, if the thickness of the electrode is greater than the penetration depth, the numer of photons useful for photogeneration of electrons will reach a limit causing no further increase in and. Instead, an increase in the thickness eyond the light penetration depth yields more recomination centers that cause more electron loss and resulting in a gradual decrease in and Thickness (m) Thickness (m) Thickness (m) Fig. 8: The variations of oc,,, and with porous film thickness. Figure 8 illustrates the variations of oc,,, and with porous film thickness. The results show that oc and decrease monotonically with increasing film thickness. This phenomenon can e explained y the electron dilution effect [15]. As light is transmitted through the porous electrode, the intensity gradually decreases. Therefore, as the thickness increases, the excessive electron density ecomes lower resulting in a decrease in oc and. Also, a thicker electrode with a Fig. 9: The variations of fill factor, efficiency, and maximum power with electrode thickness. The variations of fill factor, efficiency, and maximum power with electrode thickness is depicted in Fig. 9. The fill factor decreases with increasing electrode thickness, indicating an increase in the cell internal resistance. It is clear from the figure that the efficiency and the maximum power relation with thickness follow the same variation since the energy conversion efficiency of DSSC is the ratio of maximum output power to the incident light power. An optimal electrode thickness of 1

7 Open circuit voltage oc (olt) Short circuit current density (ma/cm - ) A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 aout 5 μm is inferred from Fig. 9 which is consistent with the theoretical finding of Ming. et al [18] COMPARISON WITH PUBLISHED EXPERIMENTAL DATA Calculated data Experimental pulished data.4. Calculated data Pulished data Electrode thickness (m) Fig. 1: Coarison etween the calculated data and pulished experimental data from ref. [4] of oc vs. electrode thickness for Φ= 1.x1 17 cm - s -1, α=5 cm -1, D=5.x1-4 cm s -1, m=4.5, and τ=1ms. Although much work on DSSC has een pulished in literature, studies of the effect of electrode thickness are limited. In this section some relevant previous experimental work are presented for coarison with the modeling results of this investigation. Mercurochrome sensitized solar cells have een studied y Hara et al. [4] who investigated the effect of different photoelectrode materials, i.e. TiO, N O 5, ZnO, SnO, and In O 3. They found that as the thickness of TiO increased from 4.5 to 8 μm, the oc decreased from.58 to.48. The predicted oc y the current investigation agreed well with Hara et al. experimental data as shown in Fig Thickness (m) Fig. 11: Coarison etween the calculated data and pulished experimental data from ref. [5] of vs. electrode thickness for Φ= 1.x1 17 cm - s -1, α=5 cm -1, D=5.x1-4 cm s -1, m=4.5, and τ=1ms. Fukai et al [5] have experimentally investigated the variation of with photoelectrode thickness using SnO as photoelectrode and using N719 dye as photon asorer. Figure 11 shows that the present modeling is consistent with the experimental work of Fukai et al [5] which shows that the measured increases significantly with electrode thickness and reaches the maximum at an electrode thickness of aout 11 μm. Further increase in photoelectrode thickness would cause a slight decrease in. The different values of optimal electrode thickness found y individual research groups may e attriuted to different dye molecules of various light asorption coefficients. The microstructure parameters, such as TiO particle size, pore size, porosity, and roughness factor, have significant effects on the electron diffusion coefficient, electron life time, light asorption coefficient, as well as specific surface area, resulting in apparent difference in the DSSC performance [-9]. Thus it is iortant to model the performance of a DSSC with respect to the microstructure of its photoelectrode [,9]. I. CONCLUSIONS An accurate approach to calculate the internal parameters of a DSSC (L, α, m, D, n, τ) is proposed in this article. This method provides a tool for researchers to estimate theoretically the internal parameters instead of digging through literature for their experimental value. This approach is ased on the electron diffusion differential model and the values of the short circuit current density, open circuit voltage oc, and the

8 A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 current density and voltage at the maximum power point i.e. and, respectively. By assuming that the charge transfer at the counter electrode is potential controlled, the Butler-olmer equation is integrated with the electron diffusion differential model, and the Schottky arrier model to account for the interfacial effect at counter electrode/electrolyte and TiO /TCO interfaces on the - characteristics. Parametric analyses were conducted to study the effect of teerature, and electrode thickness on various cell parameters. References [1] B. M. O Regan, and M. Grätzel, M., A low-cost, high-efficiency solar cell ased on dye-sensitized colloidal TiO films, Nature, 353, (1991). [] F. T. Kong, S. Y. Dai, and K. T. Wang, New Ahiphilic Polypyridyl Ruthenium(II) Sensitizer and Its Application in Dye-Sensitized Solar Cells, Sol. Energy Mater. Sol. Cells, 5, (7). [3] G. Smestad, C. Bignozzi, and R. Argazzi, Testing of dye sensitized TiO solar cells I: Experimental photocurrent output and conversion efficiencies, Sol. Energy Mater. Sol. Cells, 3, 59-7 (1994). [4] G. Smestad, Testing of dye sensitized TiO solar cells II: Theoretical voltage output and photolumineence efficiencies, Sol. Energy Mater. Sol. Cells, 3, (1994). [5] A. Kay, and M. Grätzel, Low cost photovoltaic modules ased on dye sensitized nanocrystalline titanium dioxide and caron powder, Solar Energy Mater. Solar Cells, 44, (199). [] S. Y. Dai,. Weng, Y. F. Sui, C. W. Shi, Y. Huang, S. H. Chen, X. Pan, X. Q. Fang, L. H. Hu, F. T. Kong, and K.. Wang, Dye-Sensitized Solar Cells, from Cell to Module, Sol. Energy Mater. Sol. Cells, 84, , (4). [7]. Halme, M. Toivola, A. Tolvanen, and P. Lund, Charge transfer resistance of spray deposited and coressed counter electrodes for dye-sensitized nanoparticle solar cells on plastic sustrates, Solar Energy Materials & Solar Cells, 9, (). [8]. Xia, N. Masaki, M. Lira-Cantu, Y. Kim, K. iang, and S. Yanagida, Influence of Doped Anions on Poly(3,4-ethylenedioxythiophene) as Hole Conductors for Iodine-Free Solid-State Dye-Sensitized Solar Cells,. Am. Chem. Soc., 13, (8). [9] Y. Chia, A. Islam, Y. Watanae, R. Komiya, N. Koide, and L. Han., Dye-Sensitized Solar Cell with Conversion Efficiency of 11.1% apanese ournal of Applied Physics, 45, 38-4 (). [1] G. Franco,. Gehring, L. M. Peter, E. A. Ponomarev, and I. Uhlendorf, Frequency-Resolved Optical Detection of Photoinjected Electrons in Dye- Sensitized Nanocrystalline Photovoltaic Cells,. Phys. Chem. B, 13, 9-98, (1999). [11] N.. Cherepy, G. P. Smestad, M. Gratzel,. Z. and Zhang, Ultrafast Electron Injection: Ilications for a Photoelectrochemical Cell Utilizing an Anthocyanin Dye-Sensitized TiO Nanocrystalline Electrode,. Phys. Chem. B, 11, , (1997). [13] K. Imoto, K. Takahashi, T. Yamaguchi, T. Komura,. Nakamura, and K. Murata, High-performance caron counter electrode for dye-sensitized solar cells, Sol. Energy Mater. Sol. Cells, 79, (3). [14] S. Södergren, A. Hagfeldt,. Olsson, and S. E. Lindquist., Theoretical Models for the Action Spectrum and the Current-oltage Characteristics of Microporous Semiconductor-Films in Photoelectrochemical Cells,. Phys. Chem., 98, (1994). [15] R. Gómez, and P. Salvador: Photovoltage Dependence on Film Thickness and Type of Illumination in Nanoporous Thin Film Electrodes According to a Sile Diffusion Model, Solar Energy Materials & Solar Cells, 88, (5). [1]. Ferer, R. Stangl, and. Luther, An electrical model of the dye-sensitized solar cell, Sol. Energy Mater. Sol. Cells, 53, 9-54 (1998). [17]. Nelson, in: A.. Bard, M. Stratmann (Eds.), Encyclopedia of Electrochemistry, (Wiley-CH, Weinheim,, 43 (). [18] N. Meng, M. K. H. Leung, and D. Y. C. Leung, Theoretical Modelling of the Electrode Thickness on Maximum Power Point of Dye-Sensitized Solar Cell, The Canadian ournal of Chemical Engineering, 8, 35-4 (8). [19].. Lee, G. M. Coia, and N. S. Lewis, Current Density ersus Potential Characteristics of Dye- Sensitized Nanostructured Semiconductor Photoelectrodes.. Simulation,. Phys. Chem. B, 18, (4). [] G. Rothenerger, D. Fitzmaurice, and M. Grätzel, Optical Electrochemistry. 3. Spectroopy of Conduction-Band Electrons in Transparent Metal-Oxide Semiconductor-Films-Optical Determination of the Flat- Band Potential of Colloidal Titanium-Dioxide Films,. Phys. Chem, 9, (199). [1]. Ferer, and. Luther, Modeling of Photovoltage and Photocurrent in Dye-Sensitized Titanium Dioxide Solar Cells,. Phys. Chem. B, 15, (1). [] L. Dloczik, O. Ileperuma, I. Lauermann, L. M. Peter, E. A. Ponomarev, G. Redmond, N.. Shaw, and I. Uhlendorf, Dynamic Response of Dye-Sensitized Nanocrystalline Solar Cells: Characterization y Intensity Modulated Photocurrent Spectroopy,. Phys. Chem. B, 11, (1997). [3] N. Meng, M. K. H. Leung, D. Y. C. Leung, and K. Sumathy, Theoretical modeling of TiO/TCO interfacial effect on dye-sensitized solar cell performance, Solar Energy Materials & Solar Cells, 9, -9 (). 3

9 A. El Tayyan., ournal of Electron Devices, ol. 11, 11, pp. 1-4 [4] K. Hara, T. Horiguchi, T. Kinoshita, K. Sayama, H. Sugihara, and H. Arakawa, Highly Efficient Photonto-Electron Conversion with Mercurochrome-Sensitized Nanoporous Oxide Semiconductor Solar Cells, Sol. Energy Mater. Sol. Cells, 4, (). [5] Y. Fukai, Y. Kondo, S. Mori, and E. Suzuki, Highly Efficient Dye-Sensitized SnO Solar Cells having Sufficient Electron Diffusion Length, Electrochem. Commun., 9, (7). [] N. G. Park, G. Schlichthorl,. an de Lagemaat, H. M. Cheong, A. Maarenhas, and A.. Frank., Dye- Sensitized TiO Solar Cells: Structural and Photoelectrochemical Characterization of Nanocrystalline Electrodes Formed from the Hydrolysis of TiCl 4,. Phys. Chem. B, 13, (1999). [7]. van de Lagemaat, K. D. Benkstein, and A.. Frank: Relation etween Particle Coordination Numer and Porosity in Nanoparticle Films, Ilications to Dye-Sensitized Solar Cells,. Phys. Chem. B, 15, , (1). [8] M.. Cass, A. B. Walker, D. Martinez, and L. M. Peter, Grain Morphology and Trapping Effects on Electron Transport in Dye-Sensitized Nanocrystalline Solar Cells,. Phys. Chem. B, 19, (5). [9] S. Nakade, Y. Saito, W. Kuo, T. Kitamura, Y. Wada, and S. Yanagida., Influence of TiO Nanoparticle Size on Electron Diffusion and Recomination in Dye-Sensitized TiO Solar Cells,. Phys. Chem. B, 17, , (4). 4