A semi-empirical analysis of dye adsorption and electron transport in dye sensitized solar cells (DSSCs)

Abstract

In order to provide a comprehensive understanding of dye adsorption parameters and their relation with the structural and transport properties of dye sensitized solar cells (DSSCs), a combination of experiments and modeling of the dye adsorption and electron transport characteristics with respect to the photoanode thickness was performed. The obtained experimental data include scanning electron microscopy (SEM) images, UV-Vis data, steady state current–voltage (J–V) characteristics and open circuit voltage decay (OCVD) data. By monitoring the time evolution of the bulk dye solution and applying the fitting model, the dye loading is reproduced to determine the adsorption parameters, such as the optimum time and amount of dye loading. Additionally, the photoabsorption coefficient of the DSSCs for various active layer thicknesses was determined. In addition, the presented analysis approach builds up a relation between the dye adsorption dynamics and the structure of the nanoparticle matrix to estimate the aggregation parameters. The current–voltage characteristics are investigated for different photoanode thicknesses and the optimum thickness is determined. The discussion also highlights the importance of the localized state distribution in electron transport analysis regarding the active layer thickness by building up an analytical formalism for the OCVD, lifetime, diffusion length and nonlinearity recombination parameter (β). It is found that considering a constant or variable β can provide fundamental interpretations for recombination pathways. The proposed integrated strategy can provide a powerful tool to study the microscopic processes and parameters governing dye-sensitized solar cell (DSSC) behavior. The presented methodology is especially applicable for the investigation of photoanodes with different morphological features such as nanotubes or nanorods, as well as different sensitizers.

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1. Introduction

Soon after the invention of dye sensitized solar cells (DSSCs),¹ their unique features such as low cost and high efficiency qualified them as a promising alternative to conventional Si-based solar cells. Despite substantial research, their efficiency has recently reached only 13%.² In a typical DSSC there are three main components working in a cycle: a sensitizer, a metal oxide nanostructure semiconductor and an electrolyte. A photosensitizing dye adsorbed onto the surface of the nanocrystalline semiconductor absorbs light. The photogenerated excitons rapidly split, with electrons transferred to the conduction band of the semiconductor, leaving the oxidized dye molecules behind. The electrons diffuse towards the transparent conducting oxide and travel to the counter electrode to reduce the iodide (I⁻) ions of the electrolyte. The iodide ions transport the electrons back to the dye to reduce it. Scheme 1 plots the different electron transfer pathways in a typical DSSC.

Scheme 1 The general working principles of a typical DSSC. The green arrows show the possible forward electron transfer routes and the red arrows show the possible backward electron transfer routes.

Two of the critical phenomena operating in tandem to enable DSSCs to operate are the dye adsorption on the nanoparticle matrix and the dynamics of the injected electron from the excited dye in the porous medium moving towards the substrate. The optimum amount of dye adsorbate on an adsorbant like a nanocrystalline semiconductor such as TiO₂ can enhance the carrier injection into its conduction band (CB) leading to a higher incident photon to current efficiency (IPCE) and short circuit current (J_sc).^3,4 However, dye aggregation or inadequate dye loading by the semiconductor matrix can deteriorate the DSSC performance. The dye adsorption behavior has a notable effect on DSSC electrical performances, which has been recently investigated in a few works.^5,6

In most of the experimental work the amount of adsorbed dye is measured by the desorption of the adsorbed dye from the semiconductor matrix by diluted NaOH or KOH followed by UV-Vis spectroscopy,^7–9 which is time consuming and might be accompanied by some inaccuracy. In addition, almost no more significant information can be extracted from the porous semiconductor matrix and dye intermixed system. On the other hand, simulation of the adsorption/desorption processes of dye species can be a powerful tool to interpret the experimental data. Despite the significance of such a process, few related works can be found in the literature. Onodera et al.¹⁰ studied the electrical properties of DSSCs with regards to the porous TiO₂ structure by a simulation strategy but they did not include the adsorption process in their model. Pugliese et al.¹¹ developed a model to study the effect of dye loading on the impedance of DSSCs. They found that for a very short adsorption time, the impedance is very large. Nevertheless, their investigation was limited to the impedance study and lacked cell performance parameter extraction. Park et al.¹² used a simple pseudo-second-order model for the adsorption of dye molecules on TiO₂ with respect to the film thickness without considering the structural characteristics of the porous medium. Despite the feasibility of the experimental estimation of both the dye adsorption and optical parameters, modeling and comparison with experimental data can enable us to determine some properties, such as the nanoparticle aggregation percentage, aggregation diameter and optimum dye loading, for various geometrical nanostructures with high accuracy. Thereby, these findings can be helpful in understanding of the active layer quality. In addition, fitting a realistic comprehensive model to time evolution adsorption data not only enables researchers to evaluate the rates of adsorption/desorption to and from surfaces with different nanostructure morphologies, but comparison of various nanocrystalline semiconductors’ (TiO₂, ZnO,…) adsorption properties also becomes possible.

Modeling DSSCs with regards to the nature of the carrier transport phenomena in nanocrystalline semiconductors, such as trapping/detrapping, recombination nonlinearity, is also an active area of research. Bisquert et al.¹³ presented a model-based analysis to connect the different trap distribution regimes in the semiconductor band gap to recombination processes in DSSCs. A modeling procedure was introduced by Anta et al.^14,15 to find the important inputs and outputs of DSSCs considering multitrapping in TiO₂. The report by Jennings et al.¹⁶ revealed that the steady state and dynamic behavior of titania nanotube cells appear to be influenced by non-ideal electron statistics. Andrade et al.¹⁷ discussed the optimized design of DSSCs with regards to transport characteristics but they did not include the trapping effect in their calculations. Some studies are mainly focused on photoanode thickness-related transport characteristics.^18–21

By incorporating the semi-empirical findings in dye adsorption into a proper transport model, one is able to simulate the current–voltage (J–V) characteristics and associated transport properties. To the best of our knowledge, no previous work has developed an integrated experimental and modeling methodology by correlating the dye adsorption, structural and transport properties in DSSCs to perform a fundamental analysis.

This paper is organized as follows. In Section 2, the experimental procedures and measurements are briefly explained. In Section 3, we present a semi-empirical modeling strategy to find the adsorption, optical and structural parameters. Moreover, a modeling procedure is applied to simulate the steady state (J–V) characteristics by entering the results of the adsorption model. Finally, the voltage transient response is analytically modeled to extract the carrier lifetime and nonlinearity recombination parameter (β). β can determine the type of local states (electron traps) which contribute to the recombination process. In Section 4, we consider the applicability of our models to experimental data to discuss the results. Parameters such as the optimum amount and time of dye loading, aggregation radius, and surface area loss are quantified indirectly via adsorption dynamic analysis. To confirm the results, scanning electron microscopy (SEM) images, UV-Vis data, steady state (J–V) characteristics and open circuit voltage decay (OCVD) are used. Similar methodology is implemented to interpret the current–voltage characteristics concerning the trapping/detrapping effect with regards to the active layer thickness. To perform a realistic analysis, the model outputs are compared to the experimental data. Finally, the transport properties of the DSSCs are obtained by combining the model and experiments. To provide a solid interpretation of the voltage transient response, we discuss the lifetime and β vs. the Fermi level regarding the photoanode thickness. The consideration of constant or variable values for β is also discussed and interpreted. Finally, the diffusion length vs. Fermi energy of the electrons is obtained by combining the parameters extracted from the steady state and transient response. The details of the adsorption model and current voltage characteristics simulation are also provided in Appendices 1 and 2.

2. Experimental

2.1. DSSC preparation

The TiO₂ paste was prepared according to the method reported in the previous literature.²² Briefly, an amount of commercial TiO₂ (average size ∼ 25 nm, P25 Degussa) nanoparticles was ground in a mortar with DI water, ethanol, acetic acid, ethyl cellulose, and finally, terpineol was added. The mixture was ultrasonicated and stirred repeatedly for 3 days to evaporate the solvent and avoid cracks. To prepare the DSSC working electrodes, the FTO glasses (Solaronix) were first cleaned in a detergent solution, acetone, methanol and diluted hydrochloric acid using an ultrasonic bath and then rinsed with DI water. The substrates were immersed into a 40 mM aqueous TiCl₄ (97%, Aldrich) solution at 70 °C for 15 min, washed with water and ethanol and sintered at 450 °C. TiO₂ paste was deposited on the substrate by the doctor blade method. In order to prepare DSSCs with different active area thicknesses, four cells were doctor bladed with 3, 5, 7 and 8 blade cycles and named C1, C2, C3 and C4, respectively. Their thicknesses and active areas were controlled with adhesive tape and to repeat the deposition the number of tape strips was increased. After each deposition, the cells were left horizontally to relax in a clean box and dried at 125 °C in air. Finally, all the cells were again immersed in a 40 mM aqueous TiCl₄ solution at 70 °C for 15 min, washed with DI water and ethanol and sintered at 500 °C in a programmable furnace to make the films porous. The electrodes were immersed in a 0.4 mM N719 dye (ruthenium (2,2′-bipyridyl-4,4′-dicarboxilate)₂(NCS)₂, Dyesol, Australia) solution in a mixture of acetonitrile and tert-butyl alcohol (volume ratio: 1 : 1) and kept at room temperature for 24 h to complete the dye uptake. To prepare the cathodes, pre-drilled FTO glasses were cleaned as explained above. The Pt catalyst was deposited on the FTO glass by spin coating with a drop of H₂PtCl₆ solution (2 mg Pt in 1 ml 2-propanol) and repeating the heat treatment at 500 °C for 15 min.

The dye-covered TiO₂ electrode and Pt counter electrode were assembled into a sandwich-type cell and sealed with a hot-melt 25 μm thick thermoplastic sealing film made of Surlyn (SX-1170-25, Solaronix). The electrolyte was prepared from a solution of 0.60 M PMII, 0.03 M I₂, 0.10 M LiI and 0.50 M 4-tert-butylpyridine (TBD) in a mixture of acetonitrile and valeronitrile (volume ratio: 75 : 25) and was introduced into the holes in the back of the counter electrode.

2.2. Absorbance measurement

To measure the absorbance spectra of the dye solution in the presence of cell C1, the sintered cell was immersed in a quartz cuvette containing the 0.133 mM N719 dye solution in a mixture of acetonitrile and tert-butyl alcohol (volume ratio: 1 : 1). To avoid evaporation of the solvent, the cuvette was tightly sealed and was placed in a UV-Visible spectrophotometer (Ocean Optics HR4000 spectrometer) at room temperature. The cell was laid on the bottom of the cuvette with the coated side facing upwards. This allowed monitoring of the depletion of the dye solution with time due to the continuous adsorption of dye molecules by the photoanode. The experiment was repeated for three similar samples to account for the possible effect of solution evaporation.

2.3. Current–voltage characteristics measurement

The J–V characteristics were measured using the linear sweep voltammetry mode of a potentiostat (Em Stat,² PalmSens BV, Netherlands) and solar simulator (Prova, Taiwan). The light intensity was adjusted using a built-in Si solar cell to AM1.5. The incident light intensity and active cell area were 100 mW cm⁻² (1 sun) and 0.25 cm², respectively.

2.4. The open circuit voltage decay (OCVD) measurement

The OCVD versus time was measured by switching off the light source at the steady state for 30 seconds.

2.5. Structural characterization of the photoanodes

In order to characterize the structure and surface of the electrodes, scanning electron microscopy (SEM) was employed (VEGA TESCAN scanning electron microscope).

3. Modeling

3.1. Dye adsorption modeling

Here, we develop a model for the time evolution of dye loading to extract the dye adsorption parameters and their relationship with the structural properties of the porous TiO₂ medium. Additionally, the obtained photoabsorption coefficient will be used as an input in the J–V curve simulation.

The adsorption from the dye solution by the solid nanoparticle adsorbant is studied using the kinetic form of the time dependent Langmuir formalism framework. Here, it is supposed that a quantity of adsorbant is mixed all at once with a quantity of solution for a period of time to set up a typical batch operation. In the present microscopic model it is assumed that the bulk density of the dye molecules is adsorbed only by the nanoparticle network by constant adsorption and desorption rates. The interconnection of the nanoparticles is also influenced by particle necking which leads to partial aggregation of the nanoparticles. Scheme 2 shows the details of the implemented model.

Scheme 2 The schematic of the adsorption model.

Using k_a and k_d as the rate constants for adsorption and desorption, respectively, we can write the net rate of adsorption for solution phase species i as the difference between the rate of adsorption and the rate of desorption:

R_i,ads,net = k_aiC_isol(C_i-site,total − C_{i-sites,occupied by i}) − k_d,sC_{i-sites,occupied by i} (1)

which means that the rate of adsorption should be proportional to the concentration of dye molecules in the solution phase (C_isol) and to the number of sites available on the surface of particles. Additionally, the rate should be related at any time to the number of sites not covered at that time rather than to the total number of sites present per unit area. Conversely, and again by the principle of mass action, the rate of desorption should be proportional to the number of sites currently occupied at that time. It is assumed that all sites are identical, and the adsorption at one site has no effect on that at another site; that is, they interact with the solution phase independently. C_i-site,total is also a constant and is the number of sites available on the solid per unit area. If the dye molecules in the solution around the adsorbent solid particles occupy some fraction ε of a volume V, and if this volume contains an adsorbing dye molecule i, then the rate of adsorption of the dye molecules onto the adsorbent and the rate of depletion of that species from the solution phase are coupled batch processes.²³ The component mass balances for the solution and solid phases are as follows:

Solution phase

Solid phase

where A_s is the effective surface area of the nanoparticle matrix per unit of mass and ρ_s is the solid nanoparticle density. The important adsorption parameters included in the simulation are summarized in Table 1. The further derivation of the equations to calculate the optimum amount and time of dye loading and dye solution depletion can be found in Appendix 1.

Table 1 Adsorption model parameters

Parameter	Notation	Value	Unit
Net rate of adsorption of i on solid	Ri,ads,net	—	—
Concentration of dye molecules in the solution phase	Ci,sol	—	mol i per volume
Void fraction in the bed of solid and solution^23	ε	0.4	—
Concentration of molecules in the solid phase	Ci,s	—	mol i per area
Total volume occupied by solid and solution	V	—	m^3
Total surface area	Atot(= AsρsV)	—	m^2
Effective surface area of nanoparticle matrix per unit of mass	As	Fitted	cm^2 g^−1
Solid nanoparticle density	ρs	4.23	g cm^−3
Adsorption rate constant	ka	Fitted	s^−1
Desorption rate constant	kd	Fitted	s^−1
Concentration of molecules in the solution phase	Ci,sol	—	mol l^−1
Dye molecular weight	Mwi	1188.55	g mol^−1

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3.2. Steady state current–voltage characteristics simulation model

Our steady state simulation model considers the reactions and transport of three charged species in a porous photoanode in one dimension. The relationship between the competing processes involving electrons (e), iodide (I⁻) and tri-iodide (I₃⁻) ions can be seen in Scheme 3. First, the dynamic equations controlling each particle are established, then their density profiles are extracted and finally the total current density is calculated as a function of the applied voltage from zero to V_oc. Here the contribution of drift in electron and ion transport is neglected. Due to a very short Debye length the electrical field is not significant.^24,25 The main driving force for collecting electrons originates from the electron density gradient in the vicinity of x = 0; i.e. the diffusion current has the only significant role in electron transport. ​The parameters used in carrier transport simulation are summarized in Table 2.

Scheme 3 Schematics for the general transport phenomena of the DSSCs. The forward processes (green arrows), backward processes (red arrows) and trapping (black arrows)/detrapping of CB electrons to and from trap states. The CB electrons diffuse with an electron density dependent diffusion coefficient D(n).

Table 2 Input parameters in carrier transport simulation

Parameter	Notation	Value	Unit
Electron density at equilibrium	n^0	1 × 10^16	cm^−3
Initial iodide concentration^21	ni0	1 × 10^20	cm^−3
Initial triiodide concentration^21	ntri0	1 × 10^19	cm^−3
Electron diffusion constant	D0	Fitted	cm^2 s^−1
Iodide diffusion constant^21	Di	6 × 10^−6	cm^2 s^−1
Triiodide diffusion constant^21	Dtri	5 × 10^−6	cm^2 s^−1
TCO exchange current density	Jtco0	—	mA cm^−2
Constrictivity	Γ	—	—
Tortuosity	ξ	—	s^−1
The ratio of constrictivity to tortuosity^29		1.25	—
Recombination constant	kR	Fitted	s^−1
Electron charge	e	1.602 × 10^−19	C
Cathodic electron transfer parameter	σ	0.5	—
Trap parameter	α	Fitted	—
Recombination reaction order (nonlinearity recombination parameter)	β	Fitted	—
Porosity of the TiO2 film	Θ	0.5	—
Porosity available for the electrolyte once dye molecules are adsorbed^29	Θed	Θ/1.5	—
Boltzmann constant	Kb	1.38 × 10^−23	m^2 kg s^−2 K^−1

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3.2.1 Electron transport. The proposed transport model is based on well-known concepts for studying a nanocrystalline semiconductor. The first concept is that of a quasi-Fermi level (QFL) which states that when a semiconductor is under illumination, electrons are injected into the conduction band, raising the Fermi level, and a density gradient is established. The second is a multi-trap model (MTM) in which the electron transport is assumed to happen via extended states E_c.²⁶ However, the transport is slowed down by successive trapping and detrapping of electrons to and from localized trap centers. The result of this model is an effective diffusion coefficient and recombination constant that are quasi-Fermi level QFL (or electron density) dependent (Scheme 3). The last concept is the QSA which assumes that the trapping and detrapping processes are much faster than the transport and recombination processes.²⁶

The continuity equation for electrons in the photoanode is

withwhere D₀ is the diffusion coefficient and n, n_t and n_c are the total, trapped and free electron densities, respectively. Θ is the porosity of the TiO₂ film, and Γ and ξ are constrictivity and tortuosity, respectively.²⁷ Γ can narrow the carrier pathway and ξ makes them pass through a more indirect route. G_e(x, λ) and R_e(x, t) are the electron generation and recombination terms, respectively. The derivation of the electron continuity equation (eqn (4)) by the incorporation of MTM, QFL and QSA is given in Appendix 2. This also includes the electron generation and recombination relations in detail. The last term on the right hand side of eqn (4) is the recombination current from the TCO to the electrolyte which can be calculated from the Butler–Volmer equation:where J_tco0 and σ are the exchange current density and cathodic electron transfer. This describes how the electrical current on an electrode depends on the electrode potential, considering that both a cathodic and an anodic reaction occur on the same electrode. It should be emphasized here that eqn (6) is only valid in this simple form if the current is really kinetically controlled; i.e. if the diffusion of the redox species towards the electrode surface is sufficiently fast.²⁸ However, in the case of TiCl₄ treatment of the substrate, this term can be neglected.

3.2.2 Electrolyte species transport. The transport of the redox species has a great impact on DSSC performance. The generation of two electrons always gives rise to the generation of one tri-iodide ion and the loss of three iodide ions; i.e.:where G_i(x, λ), G_tri(x, λ), R_i(x, λ) and R_tri(x, λ) are the iodide and tri-iodide generation and recombination rates, respectively. Therefore the continuity equations for mobile charged species in the electrolyte are:where D_i and D_tri are the iodide and tri-iodide diffusion coefficients of the photoanode. Θ_ed is the porosity available for the electrolyte once the dye molecules are adsorbed onto the surface.²⁹

3.2.3 Initial & boundary conditions. The system of partial differential equations, eqn (4), (8) and (9), can be solved simultaneously by applying appropriate initial and boundary conditions (BCs) for the electrons and iodide and tri-iodide species as follows.

Electron

The first BC indicates that the initial electron density at the beginning of the photoanode is voltage dependent and the second assures zero current density at the end of the photoanode.

For an electrolyte species we have

Iodide

n_i(x, 0) = n_i0 (11c)

Tri-iodide

n_tri(x, 0) = n_tri0 (12c)

The first two BCs in eqn (11a) and (12a) show that the electrolyte ion current at the beginning of the photoanode is associated with the TCO/electrolyte back reaction. The second BCs (eqn (11b) and (12b)) assure particle conservation through the photoanode thickness. Finally the diffusive current is calculated at x = 0 by

The power conversion efficiency of a solar cell is determined as the fraction of incident power which is converted to electricity and is defined as:

where P_MP = V_ocJ_scFF and J_sc and FF (= V_MPJ_MP/V_ocJ_sc) and P_in are the short-circuit current, fill factor and input power, respectively.

The simulation code is written in Mathematica 9 applying the method of lines to solve the set of partial differential equations. This explicit numerical method replaces all spatial derivatives with finite differences but does not alter the time derivatives. It is then possible to use a stiff ordinary differential equation solver on the time derivatives in the resulting system.

3.3. Transient voltage response & electron lifetime

There are two ways to elucidate the electron density and the electron lifetime. One way is “transient LASER spectroscopy”, which can show the density of injected electrons directly.³⁰ This method also requires an expensive experimental setup and demanding technique. An alternative way is to analyse the transient current and/or voltage arising from a changing light or voltage impulse. The major amplitude methods are electrical impedance spectroscopy (EIS),^31–33 intensity modulated photovoltage spectroscopy (IMVS)⁴³ and stepped light-induced transient measurement.³⁴ The lifetime from EIS is, so to say, “a voltage-dependent time constant”, which requires extra interpretation in order to obtain an exact lifetime. On the other hand, the lifetime definition form OCVD and extracted electron density is straightforward, and so, should be free of interpretation. The interpretation of the OCVD can provide key information for the diagnosis and optimization of DSSCs. Under the open circuit conditions, the injected electrons recombine with I₃⁻. To understand the loss mechanisms in DSSCs for further improvement of the photovoltaic performance, the electron density and the electron lifetime in the TiO₂ CB under the open circuit conditions are thoroughly addressed here. OCVD measurements were collected by holding the cell at open circuit under illumination until the voltage reached a steady state, then removing the light source and monitoring the decay in the cell voltage as the electrons in the anode were intercepted by the redox shuttle. The OCVD technique has certain advantages over frequency or steady-state based methods. The first is that this technique provides a continuous reading of the lifetime as a function of V_oc at high-voltage resolution; the second is being experimentally much simpler; and finally, the data treatment is outstandingly simple (basically, it consists of two derivatives) for obtaining the main quantities that provide information on the recombination mechanisms. Here we present an independent model-based interpretation of experimental OCVD data assuming a nonlinear recombination process to estimate the carrier lifetime.

According to Walker et al.³⁵ the OCVD should show a ln(t) dependence at longer times if the recombination involves C.B electrons that are detrapped from an exponential distribution of trapping states and a quasi-static state is maintained between free and trapped electrons. Under the OCVD conditions, diffusion and generation terms vanish. As a result, the continuity equation which controls the dynamic response of the electron density profile becomes

k here is the recombination rate. Assuming that and inserting the Boltzmann approximation into eqn (15) under open circuit voltage, we arrive at

The last term on the right is negligible and by integration we obtain

So the nonlinearity recombination parameter is

By inserting the modeled V_oc from eqn (17) into the Boltzmann approximation, the total electron density profile is calculated which enables us to find the carrier lifetime τ by a simple relation derived from eqn (15) and (17),

4. Results & discussion

4.1. Photoanode structural–optical absorption analysis

In order to validate the model and determine the N719 dye loading properties, the absorbance spectra of the dye solution kept in a sealed cuvette with cell C1 on the bottom were recorded for about 24 h, and the results are shown in Fig. 1C. According to the Beer–Lambert law, the absorbance is linearly related to the concentration of dye in solution, so by knowing the initial concentration one can find the molar extinction coefficient. Therefore, the ratio of the absorbance maxima around 537 nm can be used to compute the concentration of the bulk dye solution at different time intervals over 24 hours. Referring to the implemented adsorption model and using a spherical particle approximation, we take ε to be 0.4, which is a reasonable number for a packed bed of particles. The active area/volume is assumed to be an intermixed layer of dye solution and microscopic TiO₂ network. The effective nanoparticle surface area A available for dye molecules to be adsorbed on is proportional to the total surface area A_tot by a parameter which we introduce as the aggregation parameter χ (= A/A_tot). This parameter allows us to estimate the total surface area loss caused by aggregation. The total number of particles for dye molecules to be adsorbed on is calculated by dividing the total volume of active area by the single particle (TiO₂ P25) volume. Therefore, the ideal total surface area can be easily calculated in the volume. The initial amount of dye available in solution is 4.7 mg (0.4 mM). The correct value for C_i,s,tot is of the order of 10⁻⁹ mole sites cm⁻² which means that the number of sites per unit area is on the order of 10⁻⁹ per cm².²³

Fig. 1 (A) Top-view and (B) cross-sectional SEM image of the C1 deposited film. (C) Absorbance spectra of the bulk N719 dye solution for cell C1 measured at different time intervals over 24 hours.

The model fitting result for the amount of bulk dye concentration for cell C1 and the measured values are shown in Fig. 2A. It should be noted that we have diluted the dye concentration to 0.133 mM to avoid the experimental measurement noises and enhance the accuracy. Finally, the parameters are adjusted to a 0.4 mM dye solution. As it can be seen for times shorter than 80 min of immersion, the agreement between the theoretical predictions of the model and the experimental data is quiet poor. However, a better fit is obtained for longer times. The best fit returns values of 65 and 10⁻³ s⁻¹ for k_a and k_d, respectively, and χ is about 0.7. The latter indicates a surface area loss of about 30 percent and the significance of particle aggregation in the dye loading process. The ideal case for C1 (χ = 1) is also shown by the dashed line to confirm the result. It is predicted well by the model that for ideal conditions, i.e. without any aggregation (χ = 1), the simulated values reach the equilibrium point at a later time compared to the cell with aggregation (χ < 1). This is due to more available surface area for the dye molecules to be adsorbed, which delays the equilibrium point being reached. It is also implied from the time evolution of the concentration of the bulk N719 dye solution that, at the time the solution concentration reaches the equilibrium point, it does not show any noticeable decrease. As a result of that, dye aggregation is very unlikely to occur on the porous layer. This significant outcome from the presented experiment–modeling method can even enable us to indirectly estimate the dye aggregation. The obtained fitting parameter can be used to simulate the dye loading of layers C2, C3 and C4. As the layer thicknesses increase, it takes a longer time for the cell dye-loading to saturate. Fig. 2B shows the bulk dye molar evolution for all solutions in the presence of the layers. The amounts of adsorbed dye on the effective surface area of the nanoparticle matrix are shown for all layers in Fig. 3. The presented simulation results are summarized in Table 3.

Fig. 2 Concentration of the bulk N719 dye solution (A) measured and fitted to the model for cell C1. The green dashed line represents the concentration of the dye solution with maximum surface area and dye loading (ideal). (B) Simulated evolution of the concentration of the bulk N719 dye solution for C1, C2, C3 and C4 at different time intervals over 24 hours. The dashed vertical lines mark the 24 hour limit.

Fig. 3 Simulated evolution of the N719 dye loading with time using a spherical particle approximation for C1, C2, C3 and C4 at different time intervals over 24 hours. The inset shows the evolution of the amount of the adsorbate in grams on the nanoparticle network. The dashed vertical line marks the 24 hour limit.

Table 3 The simulated results of the adsorption parameters of N719 DSSCs

Cell	Thickness (μm)	Da (mg)	Ca (mol cm^−2)	Cbulk (mmol l^−1)
C1	9.0	0.866	8.00 × 10^−10	0.325
C2	15.7	1.45	1.37 × 10^−9	0.274
C3	21.7	1.99	1.86 × 10^−9	0.23
C4	23.4	2.19	2.03 × 10^−9	0.21

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The photoabsorption coefficient of the dye with the molar extinction coefficient ε(λ) is then given by³⁶

where the dye concentration in the cell is found by C_dye = (σ_mlφ/d). σ_ml is the concentration of the dye adsorbed on a flat surface of monolayer TiO₂ and calculated by the adsorbed dye concentration per unit area (D_a) and φ is the roughness factor and is related to the porosity, Θ, and aggregation diameter a by³⁶

The ideal value of φ for the almost non-aggregated layer would be 10³. Here we put a more realistic value of 10² which yields the approximate value of around a few hundred nanometer for the aggregation radius as can be seen from the cross-sectional SEM image in Fig. 1B. Putting all the values in eqn (20) we obtain 0.62 × 10⁵ cm⁻¹ for the photoabsorption coefficient which can be used as an input parameter in the simulation of current–voltage characteristics.

4.1.1 Dye adsorption and quantum yield. The quantum efficiency (Φ), defined as the ratio of the number of electrons transferred across the interface to the number of photons absorbed by the adsorbed dye layer, is not easy to determine. This is on account of the problems of measuring the light absorption by one monolayer or separating it from the absorption by the dye solution. Therefore, we prefer to give values of an injection quantum yield. Using the results from the adsorption model for the values of the extinction coefficient (ε) and the amount of dye loading (D_a), we can simply estimate the injection quantum yield of the cells defined as the number of injected electrons per incident photon by³⁰

Φ_inj = Φ(1 − 10^−εD_a) (22)

If we put Φ = 1, then we obtain Φ_inj values of 0.07, 0.118, 0.153 and 0.17 for C1, C2, C3 and C4, respectively. The estimated collection efficiency and injection quantum yield strongly suggest that the limiting performance factor for DSSCs is the inefficient injection efficiency of the cells. However, low sensitizer regeneration efficiency can lead to poor cell performance.³⁷

4.2. Steady state current–voltage analysis

Fig. 5 shows the simulated fit and measured J–V curves for the four fabricated cells with definite thicknesses (Fig. 4). The highest PCE belongs to cell C3. As can be seen, no specific behavior is observed from the film thickness dependence behavior of open circuit voltages. The fitted values of V_oc strongly depend on the input values of k_R. The best fit to the experimental J–V curves returns 5 × 10⁻¹⁰ and 1 × 10⁻⁹ s⁻¹ for C1 and C3, respectively, and 3 × 10⁻⁹ s⁻¹ for C3 and C4. The highest measured V_oc belongs to cell C1 with a thickness of 9.02 μm and it is almost decreasing for thicker films while there is a slight increase for C3 with a thickness of 21.7 μm. The other two films, C2 and C4 with thicknesses of 15.7 μm and 23.4 μm, respectively, show the least open circuit voltages. The V_oc behavior is fully discussed for DSSCs C3 and C4 in the next section. The V_oc decrease means that the CB edge of TiO₂ shifts positively. The positive shift with respect to the dye energy narrows the energy difference between TiO₂ and the dye which allows low lying excited states of the dye to inject electrons, leading to a rise in the short circuit current.^38,39 The decrease in V_oc by varying the thickness can be related to back electron transfer to tri-iodide ions. Increasing the surface area of the electrode as a result of increasing film thickness leads to an increase in surface trapping states which can cause back electron transfer. Time dependent V_oc data can reveal some underlying factors related to traps and carrier relaxation times which are attended to in the next section. On the other hand, J_sc represents a more common trend. The J_sc increases by increasing the active layer thickness and suddenly experiences a drop for C4 with a thickness of 23.44 μm. The J_sc increase is a result of IPCE improvement implying a uniform charge generation throughout the film.⁴⁰ In addition, the improvement of J_sc is related to the increase in the injection current from the excited dyes to the CB of TiO₂ arising from the increased surface area and more dye adsorption. Another reason for the J_sc increase can be related to the V_oc drop allowing low lying excited states of the dye to inject electrons.

Fig. 4 The cross-sectional SEM images of fabricated DSSCs active layer with different thickness: (A) 9.02 μm (3 blade cycles, C1), (B) 15.7 μm (5 blade cycles, C2), (C) 21.7 μm (7 blade cycles, C3) and (D) 23.4 μm (8 blade cycles, C4).

Fig. 5 The experimental and fitted model results of the fabricated DSSCs. The inset shows the variations of PCEs obtained from experimental data for four different active layer thicknesses.

Thicker films give a higher total amount of traps and hence a larger fraction of electrons being trapped/detrapped, and a longer time for the electron transport gives a larger probability for the electron to be lost to the electrolyte. The experimental values of current densities for C1 and C3 at low voltages (<200 mV) show a non-ideal behavior which can be related to low charge-collection efficiency or low dye regeneration efficiency.⁴¹ The simulated values of J_sc are underestimated for C1 and C3 which can be the result of their non-ideal response.⁴¹ To compare the applied model and the experimental measurements, Table 4 shows the simulated and measured performance parameters of the DSSCs with various active layer thicknesses.

Table 4 Simulated and experimental performance parameters of the DSSCs

Cell	Jsc (mA cm^−2)		Voc (mV)		Pmpp (Wm^−2)		FF		η (%)
	Exp.	Sim.	Exp.	Sim.	Exp.	Sim.	Exp.	Sim.	Exp.	Sim.
C1	6.96	6.25	790	793	30.37	30.50	0.55	0.61	3.37	3.5
C2	8.04	8.00	755	751	39.90	39.97	0.65	0.66	3.99	3.97
C3	10.5	9.95	780	774	50.07	50.17	0.61	0.65	5.07	5.17
C4	8.47	8.39	756	775	42.40	42.6	0.67	0.66	4.24	4.26

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4.2.1 Charged carriers density profile. The behavior of the electron density profile under different operation conditions plays the most important role in determining DSSC performance parameters. Fig. 6A shows the spatial dependent behavior of the electron density profile under short circuit conditions throughout the photoactive layer under working electrode side illumination for all DSSCs. At each spatial point, the electron density profile increases with time and reaches an almost constant value. The electron density increases by reaching deeper into the photoanode. It is clearly seen that the electron density gradient is much larger at points close to x = 0. It implies that the first micrometers of the photoanode, where most electrons are generated, mainly contribute to the total current density. As x increases, the electron density increases but its gradient decreases. The trap parameter (α) and diffusion coefficient (D₀) are adjusted to obtain the best fit for all cells under short circuit conditions. The electron density profile for C3 gives the largest values by choosing α = 0.35 and D₀ = 2.2 × 10⁻⁴ cm² s⁻¹ to obtain the best fit and a convergent current density for the voltages within the zero to V_oc range. The fitted values of the diffusion coefficient for DSSCs C1, C2 and C4 are 1.3 × 10⁻⁴, 1.3 × 10⁻⁴ and 1.2 × 10⁻⁴ cm² s⁻¹, respectively, and the continuity equation produces convergent values for current density values within the open circuit voltage by inserting α = 0.29. The larger trap parameter means a greater contribution of shallower surface traps with respect to the CB edge. The dependence of the diffusion coefficient on the thickness arises from the change in trap densities. At the same light intensity, electrons experience more trapping/detrapping in thicker films, so that the charge transport rate is faster. The dependence of the diffusion coefficient on the light intensity arises from the change in trap occupancy with intensity. At low light intensity, electrons are exchanged between deep traps and the conduction band, and the transport rate is slower. At high light intensity, deep traps are filled, and electron trapping/detrapping involves shallower levels, so that the electron experiences a faster transport rate.⁴² At short circuit and for homogeneously absorbed light, the electron diffusion coefficient can be estimated from the relation D₀ = d²/2.35τ_tr, where d is the layer thickness of the active layer. The calculated transport times (τ_tr) for C1, C2, C3 and C4 are 2.6, 8, 9 and 19 ms. Thicker films give a higher total amount of traps and hence a larger fraction of electrons being trapped/detrapped and a longer time for the electron transport gives a larger probability for the electrons to be lost to the electrolyte.¹⁶ In experiments, IMPS measurements or dark current decay are used to estimate the electron transport time τ_tr, influenced by the total number of surface states in the film, the position of the Fermi level, and the electrical connectivity of the nanoparticle network.⁴³

Fig. 6 The results of the simulated (A) electron, (B) iodide, and (C) tri-iodide density profiles under short circuit conditions as a function of distance from the conducting substrate.

Charged carriers in the electrolyte play a crucial role in the optimized design of a DSSC. The most commonly employed redox couple in DSSCs is I⁻/I^3− 44 but the [Co(bpy)₃]^3+/2+ (ref. 2) couple in particular has recently achieved success equal to I⁻/I³⁻. Fig. 6B and C show the spatial dependence behavior of the electrolyte species density profiles under short circuit conditions. As expected, under short circuit conditions the I⁻ density decreases at spatial points close to x = 0 and increases at points closer to the counter electrode where they are mainly generated. The I₃⁻ density increases in the vicinity of the working electrode and decreases towards the counter electrode due to recombination with electrons. Under short circuit conditions the gradient of the I₃⁻ density profile increases at points closer to x = 0. As one can see, the I₃⁻ density increases with the electron concentration increase for thicker films in the extended states. It can be concluded that deeper traps in C4 lead to lower electron and I₃⁻ densities. On the other hand, a higher electron density, by increasing the film thickness, means there is more oxidized dye to regenerate which brings about a lower I⁻ density along the photoactive layer.

4.3. Interpretation of the OCVD, electron lifetime and density profile with the assistance of the nonlinear recombination parameter β with regards to the photoanode thickness

The experimental data and the model fitted to the OCVD data of C3 and C4 vs. ln(t) are plotted in Fig. 7. The normal exponentially modeled fit and measured OCVD data are also shown on the top right. The best fit gives the values of α as 0.48 and 0.45 and of β as 0.87 and 0.85 for C3 and C4, respectively. The fitted values for both cells are assumed to be constant while we obtained variable β values vs. the Fermi level from eqn (18). The explanation for the latter might arise from the different effective electron population regions of the two cells contributing in recombination. In other words, the dominant contribution of the electron population in recombining with the electrolyte species happens at about 0.4 and 0.15 eV for C3 and C4, respectively (Fig. 8A inset). This is why a single fit value for β can explain the recombination nonlinearity in both cells. In the case of C3, for voltage values between 0.45 to 0.8, and for C4 if the values are less than 0.4, the V_oc decay vs. ln(t) deviates from a simple linear variation or a simple exponential decay as can be seen in Fig. 7 (inset). These observations suggest a direct reaction of trapped electrons with I₃⁻.⁴⁵ The TCO/electrolyte interception can be neglected due to the TiCl₄ treatment of the substrate. Thus, the nonlinearity can be mainly attributed to the direct transfer of the trapped electrons to I₃⁻.

Fig. 7 Open circuit voltage decay (OCVD) experimental data as a function of logarithmic time variations and the corresponding fitting model for C3 and C4 with the highest efficiencies. The inset shows the V_oc variation with a linear variation of time.

Fig. 8 Logarithmic variations of electron lifetime versus (A) Fermi level variations and (C) electron density profile under OCVD conditions. The variations of the recombination nonlinearity parameter as a function of Fermi level variations is shown in the inset of (A). The variations of the electron density profile versus Fermi level is depicted in (B) which can determine the bottom of the CB edge of the cells.

The simulated values for electron lifetimes (τ) vs. Fermi level variations are plotted in Fig. 8A. According to Bisquert et al.¹³ the exponential and parabolic shape of the logarithmic τ vs. Fermi level variations can be interpreted as different electron transfer routes from the CB to the electrolyte accompanying the change of the rate of trapping/detrapping and from surface states to electrolyte, respectively. The obtained τ vs. Fermi level represents different regimes of behavior. At high Fermi levels one observes the lifetime of the electrons in the CB without trap involvement. At a low Fermi level, charge transfer is still dominated by CB states. However, it is governed by trapping/detrapping processes and at low Fermi level the charge transfer is governed by the distribution of surface traps. The mentioned classified regions can be obviously distinguished for C3 and C4. For both of them, the lifetime below 0.7 eV increases exponentially, corresponding to the second region, and the conduction band capacitance is dominant. For C3 with a Fermi level below 0.41 eV the lifetime displays a parabolic shape which implies the dominant transfer through surface states at the Fermi level. The same parabolic shape is seen below 0.45 eV for C4 which can be interpreted as a longer tail of trap states. This evidence leads to the conclusion that the lower trap density in C3 is the main reason for the higher transport rate of electrons to substrate, compared to C4.

As can be seen from Fig. 8A (inset), β tends towards unity for higher Fermi levels, which means that the uplift of the Fermi level leads to a more linear recombination rate. In this case the voltage decay satisfies the relation

Different values of 0.7–1,²⁴ 0.8 (ref. 46) and 1.3–1.5 (ref. 47) are found for β in the literature. Our results show that this parameter can take values between 0.8 and 1. A constant β implies that the diffusion length is a strictly increasing function of the Fermi level but in some experimental works it has been observed that the diffusion length increases with the Fermi level at low and intermediate Fermi levels and approaches a limit value when sufficiently high.^46,48

Fig. 8B shows the calculated electron density profile for C3 and C4 under OCVD conditions. C3 presents a more linear behavior than C4 that can be assigned to the higher trap parameter of C3 which can be a sign of a shallower distribution of trap states compared to C4. Noticeably, there are inflection points for C3 and C4 at 0.5 and 0.53 eV, respectively, which indicate the lower edge of the CB of both cells. Therefore, the electron density is very small below the mentioned Fermi levels which indicates a slower recombination rate, as shown in Fig. 8C.

4.4. Semi-empirical diffusion length

The electron diffusion length (L) can be obtained by combining the calculated electron density dependent diffusion constant values from solution of continuity equations (eqn (4), (8) and (9)) and lifetime values under OCVD conditions using L = (Dτ)^1/2 at coincident quasi Fermi levels. The estimated semi-empirical diffusion length values vs Fermi level are plotted in Fig. 9. The diffusion length increases with the Fermi level which is expected for a nonlinear recombination process.^49,50 The diffusion length values are sufficiently longer than the C3 and C4 thicknesses, which can be interpreted as an almost 100% collection efficiency. However, larger values of L for C3 compared to C4 implies a higher collection efficiency for C3. This result is consistent with our findings in Section 4.1.1.

Fig. 9 Variations of electron diffusion coefficients calculated from a simultaneous solution of continuity equations (eqn (4), (8) and (9)) (inset) and the corresponding diffusion length versus Fermi level variations for C3 and C4.

Based on the presented analysis, by comparing the lifetime and diffusion constant data of C3 and C4, it can be concluded that the different trap distributions, as explained before, have led to a higher PCE in C3 with a higher electron density (Fig. 6 and 8B). For the optimized cell, C3, the distribution of traps is closer to the CB as explained before, so that they contribute more to the thermal activation of electrons compared to C4 with a deeper trap distribution. The latter leads to a higher diffusion constant (Fig. 9 inset) which yields a longer diffusion length for C3. There have been some reports^51,52 associating the drop in PCE with the photoanode thickness increase to a higher trap density which, here, is proven not to be the only reason and the distribution of the traps throughout the entire band gap may even be a more significant determining factor.

5. Conclusion

The purpose of the present study was to present a model-based strategy for interpreting experimental data from adsorption and electron transport mechanisms with a correlation to structural and optical properties. To achieve a realistic analysis, flexible modeling frameworks were developed to extract the related parameters. The adsorption parameters and transport characteristics of the DSSCs were investigated for various active layer thicknesses. To validate the model results and analyze the structure of the porous anode, scanning electron microscopy (SEM) images, UV-Vis data, steady state current–voltage characteristics and transient voltage response measurements were also used.

In summary, (i) the optimum time and amount of dye loading was determined for various active layer thicknesses, fitting the developed model to the corresponding experimental data. The suggested method helps to avoid dye desorption from the layers, which can cause inaccuracy, and reduce the workload; (ii) the result of fitting to the overnight UV-Vis measurement indicates that around 30% of the surface area loss in the nanoparticle matrix is due to aggregation. The aggregation radius is also estimated; (iii) the optimum photoanode thickness is found to be around 21 μm with a power conversion efficiency of 5.07%; (iv) the model fitting of the J–V curves gives the diffusion coefficient with a varying Fermi level. It also yields the nonlinearity recombination parameter β and trap parameters; (v) the significance of considering whether a single fitted or variable value for the nonlinearity recombination parameter (β) vs. the Fermi level was discussed; (vi) combining the obtained values of the diffusion coefficient with the lifetime obtained from the OCVD data, the diffusion length of the cells with the highest power conversion efficiencies were compared from a trap distribution point of view. This suggests that how the traps are distributed in the band gap of the semiconductor is as important as their density of states in limiting the PCE.

The applied procedure can provide a flexible package including a minimal experiment workload combined with theory to measure and analyse the DSSC performance parameters for further device optimization. The results of the present and other major reports strongly suggest the significant role of the low injection efficiency in the deterioration of DSSCs’ performance. Therefore, our future concern is to find a similar trend to that reported here to study this effect in DSSCs.

Appendix 1: adsorption model development to calculate the amount of dye loading

By rearranging and writing with the explicit rate of adsorption, eqn (2) and (3) become:

Solution phase

Solid phase

We need to integrate eqn (A1.2), so,or,where m_i,s,tot = C_i,s,tot(1 − ∈)A_sρ_sVM_wi. By inserting the appropriate values as inputs, the solution of eqn (A1.5) can yield the amount of adsorbed dye on the effective surface area (m_i,s = D_a) of the nanoparticle matrix. Consequently, the remaining concentration of the bulk dye (C_bulk) can be determined.

Appendix 2: derivation of continuity equation for electrons

In a semiconductor the density of states, D_e, in the CB is the number of electron states per volume and per energy interval:and the density of electrons in the CB iswhere f(E_e) is the Fermi–Dirac function. When the Fermi level is separated from the CB by more than 3K_bT, where K_b is the Boltzmann constant, we have

If E_f0 is the Fermi level with no illumination, then the CB electron density n⁰_c can be written as

where is the effective density of states in the CB, is the electron effective mass and h is the Planck constant. According to the QFL approximation, under illumination electrons are injected to the CB and the Fermi level is raised, so

In a nanocrystalline semiconductor the trap energy distribution follows from

where N_t, α and T are the total trap density, trap parameters and temperature respectively.

Assuming that most electrons are trapped (n ≈ n_t) we can write:

Combining eqn (A2.7) and (A2.8) we arrive at:

where n and n_c are total and free electron densities. A similar relationship has been derived by van de Lagemaat.⁵³

The continuity equations for electrons in the photoanode is

where D₀ is the diffusion coefficient and n_t and n_c are trapped and free electron densities. Θ is the porosity of the TiO₂ film, and Γ and ξ are the constrictivity and tortuosity, respectively.²⁷ Γ can narrow the carrier pathway and ξ makes them pass through a more indirect route. G_e(x, λ) is the electron generation term for the working electrode side illumination and is calculated by applying the Beer–Lambert law for the absorption coefficient:where ϕ_inj is the quantum injection yield, I₀(λ) the solar irradiation spectrum for AM1.5 and α(λ) is the absorption coefficient of the N719 dye. The values of α(λ) are calculated using the procedure introduced in Section 4.1.

The charge reaction transfer between TiO₂ and the redox electrolyte is a single electron reaction

in which k_f and k_b are the forward and backward reaction rates. Applying a standard chemical approach, the reaction rates are proportional to the concentrations of the species involved. In the absence of illumination

k_fn_I0 = k_bn⁰n_i0 (A2.12)

where n_I is the iodine (I) density – the oxidized species of the charge-transfer reaction. Eqn (A2.12) implies that the forward reaction rate equals the backward reaction rate. Finally the net electron flow at the interface may be written as⁵⁴

R_e = k_bnn_I − k_fn_i (A2.13)

Taking the mass action law into account and considering non-linear chemical kinetics, the recombination term can be rewritten as:

where k₀ and β are the recombination rate and the nonlinearity recombination parameter, respectively. Recently, Ansari et al.⁵⁵ showed that β is less than or equal to unity. n_c(x, t), n_tri(x, t), n_i(x, t), n⁰, n_tri0 and n_i0 are the electron densities in the CB, the tri-iodide density, the iodide density, the initial electron density in the CB, the initial tri-iodide density and the initial iodide density respectively. Making use of the mathematical form of the QSA:and using eqn (A2.9) and assuming that most electrons are trapped (n ≈ n_t), one obtains:

By rewriting eqn (A2.10) using the equation above, we reach the continuity equation corresponding to an electron density dependent diffusion coefficient and an electron density dependent recombination constant.

whereand

Acknowledgements

The authors express their appreciation to Prof. Juan Antonio Anta, Universidad Pablo de Olavide, Spain, for helpful discussions. Special thanks to Dr Fatollahi at Laser Plasma research institute and Dr Pooya Tahay at the chemistry department of Shahid Beheshti University for experimental assistance.

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