Theoretical and Experimental Study of a Dye-Sensitized Solar Cell

Article pubs.acs.org/IECR

Theoretical and Experimental Study of a Dye-Sensitized Solar Cell Mona Bavarian,† Siamak Nejati,† Kenneth K. S. Lau,† Daeyeon Lee,‡ and Masoud Soroush*,† †

Department of Chemical and Biological Engineering, Drexel University, Philadelphia, Pennsylvania 19104, United States Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States

‡

ABSTRACT: Theoretical and experimental analyses of the performance of a dye-sensitized solar cell (DSSC) are presented. Using a macroscopic first-principles mathematical model of the DSSC, the effective electron diffusion coefficient, recombination rate constant, and difference between the conduction band and formal redox potentials are estimated from current−voltage (I−V) measurements. The mathematical modeling indicates that (i) diffusion is the dominant driving force for the transport of electrons and holes, and thus electric-field-induced migration can be neglected; (ii) the type of recombination rate equation has little effect on the estimates of the effective electron diffusion coefficient and the difference between the conduction band and formal redox potentials; (iii) the recombination rate constant affects both the cell open-circuit voltage and short-circuit current; (iv) the conduction band edge movement affects mostly the cell open-circuit voltage; (v) as expected, the I−V performance of the cell changes very little with operating temperature variations; and (vi) the effects of different light absorbers on the cell I−V performance is through the absorption coefficient and displacement of the conduction band. The transient behavior of the cell from the dark equilibrium conditions to short circuit conditions and the cell transient response to a step change in the external load are investigated theoretically. Experimental I−V results from a DSSC, under different light intensities and with two different dyes, are used to validate the model.

1. INTRODUCTION Among different photovoltaic technologies, dye-sensitized solar cell (DSSC) technology has evolved quite rapidly, to the point that it is currently undergoing commercialization.1 The DSSC technology was first introduced by Michael Grätzel and Brian O’Regan,2 and subsequent efforts have resulted in an increase in the efficiency of this design of up to 11%. More recently, laboratory-scale devices with efficiencies as high as 12.3% have been fabricated.3 To ensure successful commercialization of this technology and establishment of a viable market share, with the present cell efficiency1,4 the total cost of the cell must be reduced or the cell efficiency should be increased from the present level.5 Figure 1 shows a schematic of a typical DSSC. A DSSC mainly consists of a transparent conductive oxide (TCO) layer, a wideband-gap semiconductor (which is the electron acceptor), a layer of a chromophore sensitizer (molecular dye) that absorbs light, a hole conductor medium (which can be a solid-state or a liquidbased redox solution), a hole blocking layer, a scattering layer, and a platinum-coated TCO layer (as the cathode). The wideband-gap semiconductor along with the TCO and the sensitizer chromophore layers forms the photoanode of the cell. The conductor medium is sandwiched between the photoanode and the platinum-based cathode. The liquid conductor medium is typically an I3−/I− redox couple in a liquid electrolyte (a common one being lithium iodide and iodine in acetonitrile6). The transparent conductive oxide has a typical transparency of 85%− 95%.7 Fluorine-doped tin oxide (SnO2:F) is the most widely used TCO.8 Among different oxides, the mesoporous TiO2 in the anatase form is the most widely used anode material. This oxide layer is decorated with a monolayer of the sensitizer chromophore for efficient optical absorption. A highly efficient DSSC requires a large surface area that can accommodate a significant number of dye molecules within the same working volume. This requirement has led researchers to produce © XXXX American Chemical Society

Figure 1. Schematic of a dye-sensitized solar cell (DSSC).

interconnected networks of oxide layer with high internal surface area and mesoporosity.2,9 Upon light illumination on the cell, sunlight passes through the transparent electrode into the dye layer. The dye molecules absorb the photons and reach their excited state. The excitedstate molecules then either go through relaxation back to their Special Issue: David Himmelblau and Gary Powers Memorial Received: May 27, 2013 Revised: September 17, 2013 Accepted: September 26, 2013

A

dx.doi.org/10.1021/ie4016914 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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transient behavior of a DSSC. In this model, two electron densities were defined to account for electrons in the trap states and in the conduction band. The time-dependent simulation results were compared with small and large transient photovoltage and photocurrent measurements. Andrade et al.20 presented a one-dimensional transient first-principles model of a DSSC. In their model, the recombination rate was assumed to be first order. They also pointed to the advantages of a transient model such as allowing one to estimate the relevant kinetic parameters with higher accuracy. Using parameter values provided by Wang et al.,21 Andrade et al.20 showed that their simulation results agreed well with electrochemical impedance spectroscopy (EIS) measurements. DSSC models are highly nonlinear due to the nonlinear dependence of the I−V characteristics of DSSCs on temperature and irradiance level.22 In this work, we first present a first-principles mathematical model of a DSSC. The model development is along the work of Ferber et al.12 The widely used assumption that the electric field can be ignored is theoretically investigated. We show that including Poisson’s equation in the model changes the distribution of the redox species very slightly, and this slight change does not alter cell characteristics. To this end, we compare predictions by two models: Model I (Ferber et al.12) and Model II. Model II does not account for transport of charged species by the drift (migration). Several recombination rate equations based on different recombination mechanisms are evaluated. Critical parameters of the model, including the effective electron diffusion coefficient, recombination rate constant, and difference between the conduction band and formal redox potentials, are estimated from our own I−V curve data and I−V curve data given in ref 23. The effects of the conduction band energy level and the recombination rate constant on the cell performance are investigated. Experimental I−V results from a DSSC, under different light intensities and with two different dyes, are used to validate the model. Contributions of this study are as follows. First, it is shown that the impact of the internal electric field on the model predictions of the cell electrons and redox species distributions and the I−V curve is insignificant. Second, the effect of recombination rate constant and conduction edge band movement are investigated theoretically, and the influence of adding tert-butylpyridine on the cell performance studied. Third, the effects of two different dyes on the I−V performance of the cell are shown experimentally and theoretically. Fourth, the effect of cell temperature on the cell performance is investigated theoretically, and the results are compared qualitatively with the existing experimental data. Fifth, the transient response of the cell to load variations is shown. Organization of the rest of this paper is as follows. Section 2 describes a mathematical model of a DSSC. Section 3 investigates the effect of electric field on the cell performance and the species profiles in the cell. Section 4 evaluates the effect of the charge recombination mechanisms on the cell performance. Section 5 studies the effects of recombination rate constant and conduction band edge on the cell performance. Section 6 investigates the effects of irradiance and dye type on the cell performance, and Section 7 examines the effect of temperature. Section 8 discusses the transient behavior of the cell. Finally, section 9 presents some concluding remarks.

initial state or inject electrons directly into the conduction band of the TiO 2 semiconductor. The electron injection is accompanied by the oxidation of the dye molecules. The electrons injected into the semiconductor then move by diffusion (driven by an electron concentration gradient) to the anode TCO layer, where they are collected for powering an external load. It is worth mentioning that not all of the injected electrons reach the electron collector of the cell, because some may, instead, react with I3− in the electrolyte at the electrode/ electrolyte interface. This charge recombination loss process results in the reduction of the incident photon-to-current conversion efficiency. Electrons that successfully flow out through the external circuit then return to the cell cathode (the platinum-coated TCO layer). The electrolyte then transports the electrons back to the oxidized dye molecules that had lost electrons to regenerate the stable neutral form of the dye. The mechanism of electron transport in the electrolyte is as follows. At the cathode, each triiodide ion in the electrolyte receives two of the electrons returned to the cell cathode forming three I− ions, which diffuse to the anode side of the electrolyte. At the anode, the oxidized dye molecules then strip electrons from I− ions in the electrolyte, oxidizing the I− ions back into I3− ions, which then diffuse to the cathode to complete the redox loop. One of the main reasons why a DSSC works is because this dye regeneration reaction occurs much faster than the recombination reaction of the injected electrons with the oxidized dye molecules. The recombination reaction, however, cannot be avoided completely and is one major source of inefficiency, because it leads to the internal short-circuiting of the cell. Different approaches including equivalent circuit modeling and continuum modeling have been used to effectively predict the cell system behavior.10−13 Södergren et al.14 developed a model accounting for electron diffusion in the TiO2 layer. To simplify their model, the effect of the electrolyte was not taken into account, and an analytical solution for the distribution of electron density was presented. Ferber et al.12 developed a DSSC model by assuming a pseudohomogeneous effective medium, containing the nanoporous TiO2 semiconductor, dye, and redox electrolyte. Their model included the continuity and transport equations for all charge carriers, including the electrons in the TiO2 layer, and the iodide, triiodide, and cations in the electrolyte. They also included the effect of electric field in their model to account for the unbalanced charge-carrier distribution under illumination. An extension to this model was presented by Oda et al.;15 in their model, an additional bulk electrolyte layer on top of the effective medium was taken into account. Topic et al.16 developed a model by considering an optical model based on the one-dimensional semicoherent optical simulator SunShine, in which the nanoporous active layer was modeled as the effective medium layer with an effective scattering of light at its front and back surface interface, using some effective roughness. It has been reported that in DSSCs, the electric field can be ignored, because the electron charge is screened by the electrolyte, eliminating the internal field.17 Gentilini et al.18 also investigated the effect of the drift term in a DSSC model but did not show how the drift term affects the cell electron density and redox species profiles or the current−voltage (I−V) curve; by comparing estimated values of parameters of their model including and excluding the drift term, Gentilini et al.18 concluded that the drift term can be ignored. Most studies on modeling of DSSCs have been for steady-state conditions, while a few have involved transient conditions. Barnes et al.19 developed a model to study the steady-state and

2. MATHEMATICAL MODEL We consider a DSSC with an iodide/triiodide redox electrolyte. To develop a mathematical model of the cell, as Figure 2 shows, we consider the mesoporous TiO2 semiconductor, the charge-transfer B



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electron in the oxide layer and the redox potential of the electrolyte.24 This voltage is the potential difference between the TCO−TiO2 and the TCO−Pt layers and is given by12 Vint =

1 n (E F (x = 0) − E Redox ) q

(3)

EnF

where is the electron quasi-Fermi level, q is the elementary charge, and ERedox is the redox energy. The electron quasi-Fermi level depends on the density of electrons in the conduction band; it is given by the relation19 ⎛ n ⎞ E Fn = ECB + KBT ln⎜ e ⎟ ⎝ NCB ⎠

(4)

where ECB is the energy of the conduction band edge, and NCB is the effective density of electrons in the conduction band, which is defined by

Figure 2. One-dimensional (1D), pseudo-homogeneous medium representing the DSSC.

NCB

dye, and the redox electrolyte as a pseudohomogeneous medium with a thickness of L. The mobile charged particles in this medium are the electrons e− in the TiO2 conduction band, the iodide (I−), and the triiodide (I3−) redox counterparts in the electrolyte solution, and cations (e.g., Li+ cations). The cations ensure charge neutrality of the cell. The continuity and transport equations for all charged species in the cell form the backbone of the model. Following the work of Ferber et al.,12 we develop a dynamic model of the cell and then simplify the model. 2.1. Equivalent Circuit Model of the DSSC. Figure 3 shows an equivalent circuit of the cell.12 RTCO, RS, and Rext are

⎛ 2πm*K T ⎞3/2 e B ⎟ = 2⎜ h2 ⎠ ⎝

(5)

Here, me* is the effective electron mass, and h is Planck’s constant. In the dark, electrons in the oxide layer are in equilibrium with the redox species, and their equilibrium density, according to eq 4, is given by n ⎞ ⎛ ECB − E F,Redox ne̅ = NCB exp⎜ − ⎟ KBT ⎠ ⎝

(6)

The redox energy (ERedox) is obtained by subtracting the Pt activation overpotential, ηa, from the redox energy under opencircuit conditions: OC E Redox = E Redox − qηa

(7)

where the open-circuit redox energy is calculated using the Nernst equation: OC 0 E Redox = E Redox −

OC KBT ⎛⎜ n I3 (x = L)/nstd ⎞⎟ ln⎜ OC 3⎟ 2 ⎝ (nI (x = L)/nstd) ⎠

(8)

Here, E0Redox

is the standard redox potential, and nstd corresponds to the standard reference of 1 mol/L. The Pt activation overpotential, ηa, is calculated using the Butler−Volmer equation: ⎧ OC ⎡ ⎛ q ⎞ ⎤ ⎪ nI3(x = L)nI (x = L) ⎢ je = j0 ⎨ exp (1 − β ) ⎜ ⎟η ⎥ ⎢⎣ ⎪ nIOC (x = L)nI(x = L) ⎝ KBT ⎠ a ⎥⎦ 3 ⎩ ⎤⎫ ⎡ ⎛ q ⎞ ⎥⎪ nI(x = L) − OC exp⎢⎢− β ⎜ ⎟ηa ⎥⎬ KT nI (x = L) ⎣ ⎝ B ⎠ ⎦⎪ ⎭

Figure 3. Equivalent circuit of the DSSC.12

TCO, shunt, and external resistances, respectively. The TCO resistance is the sum of the ohmic resistances (to electron movement) of the two TCO layers. The shunt resistance accounts for undesired short circuits between the front and back surface contacts of the cell, which is caused by defects. Vint, Vext, Iint, and Iext are the internal and external voltages, and the internal and external currents, respectively. According to the equivalent circuit model: Iext =

Vint RTCO + R ext

Iint =

Vint + Iext RP

Vext = IextR ext

(9)

where j0 is the exchange current density, β the symmetry factor, and je the internal cell current density, which is the rate of electron transfer at the Pt electrode (where x = L, shown in Figure 2). The internal cell current (Iint) is calculated using Iint = je A

(1)

2.3. Concentration Overpotential (Mass-Transfer Resistances). To calculate the species concentrations at the reaction sites, one can write shell mole balances on (continuity equations for) the four charged species in the cell, yielding the following partial differential equations (PDEs):

(2)

There are a couple of other internal cell resistances, such as charge and mass transfer resistances, which will be considered in the model in the next sections. 2.2. Electrochemical Processes. The internal cell voltage, Vint, depends on the difference between the Fermi level of the

∂ne ∂Ne = Ge − Ce − ∂t ∂x C

(10)


Industrial & Engineering Chemistry Research ∂nI ∂NI = GI − CI − ∂t ∂x ∂n I3

= G I3 − C I3 −

∂t

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This assumption together with the preceding two equilibrium relations leads to the recombination rate equation:

(11)

Ce = K 2f nIn I2 − K 2rn I−2 + k4nen I•− 2

∂NI3 (12)

∂x

∂nc ∂Nc = Gc − Cc − ∂t ∂x

= k4nen I•− 2 = k4ne(K 2nen I2)

(13)

⎛ nI ⎞ = k4K 2ne 2⎜K1 3 ⎟ ⎝ nI ⎠

where Ge, GI, GI3, and GC are the rates of generation of electrons, iodide ions, triiodide ions, and cations, respectively. Ce, CI, CI3, and CC are the rates of consumption/loss of the same four species, respectively. Ne, NI, NI3, and NC are the fluxes, and ne, nI, nI3, and nC are the respective concentrations. Electron Recombination Rate. Not all of the electrons created by light absorption can be collected and transferred to the external circuit, because some of the charge carriers recombine before they are transported via diffusion or drift to the external circuit.25 Recombination can occur through two different pathways that short circuit the cell: (a) reduction of the dye cations with the electrons that return to the dye from the TiO2 semiconductor, and (b) reduction of triiodide ions to iodide ions at the anode side of the electrolyte. Since the first pathway has been shown26 to be negligible in the presence of a sufficient concentration of iodide and a sufficiently conductive electrolyte, which result in a rapid regeneration of the oxidized dye molecules by iodide, only the second pathway is considered in this study. The overall iodide−triiodide reaction is

I−3 + 2e− → 3I−

where K2f/k2r = K2. In contrast, several other investigators have assumed that reaction 17 is much faster than reaction 18; reaction 17 is the rate-determining reaction of the two parallel reactions 17 and 18. This assumption together with the two equilibrium relations leads to the different recombination rate equation: + Ce = C I·−2 = − K 2f nInI2 + K 2rnI•− 2

K2

k3

k4

K5

nI•− and 2

(22)

K6

I−3 ↔ I 2 + I−

(23)

Since reactions 22 and 23 are much faster than reaction 21, one can write n I•2 = K5n I2

n I2nI = K 6n I3

(24)

From reaction 21, the rate of recombination of electrons is given by Ce = k f nen I• − k bnI

(18)

(25)

At equilibrium, the concentrations are governed by

where K1 and K2 are the equilibrium rate constants for the reaction steps described by reactions 15 and 16, and k3 and k4 are the rate constants for the forward reaction steps described by reactions 17 and 18. Reaction 15 represents that I−3 is in chemical equilibrium with molecular iodine associated with the TiO2 surface and iodide ions in solution. Reversible reaction 16 represents the reduction of I2 to the I•2 − radical anion by injected electrons from TiO2.18 Each two I•2 − radical anions then decompose into one I−3 and one I− (reaction 17) or are reduced to four I− (reaction 18). Because the reversible reactions 15 and 16 are much faster than reactions 17 and 18, the following two equilibrium relations hold: K1n I3 = nIn I2

(20)

(21)

I 2 ↔ 2I•

(17)

− − I•− 2 + e → 2I

⎛ n I ⎞2 1 2 2 K1 K 2 k 3ne 2⎜ 3 ⎟ 2 ⎝ nI ⎠

kb

(16)

− − 2I•− 2 → I3 + I

=

kf

(15)

I 2 + e− ↔ I•− 2

1 2 k 3nI•− 2 2

I• + e− ⇄ I−

To calculate the recombination rate, a reaction mechanism consisting of intermediate steps should be postulated.26 Two different reaction mechanisms have been suggested for the overall reaction of 14.17,26,27 One (Mechanism A) consists of the following elementary reaction steps:27 K1

≈

1 2 + k4nenI•− k 3nI•− 2 2 2

•− where C•− I2 is the rate of consumption/loss of I2 . The second reaction mechanism (Mechanism B), which has been proposed in ref 12, is

(14)

I−3 ↔ I 2 + I−

(19) 27,31

k f ne̅ n I̅ • = k bnI̅

(26)

and n I̅ •2 = K5n I̅ 2

n I̅ 2 nI̅ = K 6n I̅ 3

where ne̅ is the dark equilibrium electron density, and nI̅ and n̅I• are equilibrium concentrations of I− and I•, respectively. Using eqs 24−26, the rate of recombination is obtained as follows: ⎛ n I3 n I̅ 3 ⎞ Ce = k f K5K 6 ⎜⎜ne − nI ne̅ ⎟⎟ nI nI̅ 3 ⎠ ⎝

K 2nen I2 = n I•− 2

I2• −,

(27)

If the rate of the backward reaction is very slow, the rate equation shown in eq 27 reduces to27

where nI2 are the concentrations of and I2, respectively. Several investigators28−30 have assumed that reaction 18 is much faster than reaction 17; that is, reaction 18 is the rate-determining reaction of the two parallel reactions 17 and 18.

Ce = k f K5K 6 D

n I3 nI

ne

(28)



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where kf(K5K6)1/2 is the so-called electron relaxation rate constant, denoted by ke. Because the initial iodide and triiodide concentrations are usually orders of magnitude higher than that of electrons,12 equilibrium concentrations of the redox couple are not very different from initial concentrations of the redox couple. Electron Generation Rate. As in Barnes et al.,19 the electron generation rate is described using a single dye absorption coefficient: Ge = αI0 exp( −αx)

2.4. Initial and Boundary Conditions. The equilibrium concentrations of the species are considered as initial conditions

ne(x , t = 0) = ne̅ n I3(x , t = 0) = n I̅ 3

nI(x , t = 0) = nI̅

(29)

nc(x , t = 0) = nc̅

where α is the absorption coefficient and I0 is the incident photon flux. Equation 29 is based on the common assumption that illumination is monochromatic. Other Generation and Consumption Rates. According to the stoichiometry of the triiodide−iodide reaction of 23, the generation of two electrons leads to the production of one triiodide ion and the consumption of three iodide ions. Since the cations are not involved in the electrochemical reaction, they are neither consumed nor generated. Thus, the rates of redox species generation and consumption are related to each other, according to G I3 =

1 Ge , 2

GI =

3 Ge , 2

C I3 =

Gc = R c = 0

1 Ce , 2

CI =

For the electrons, at x = 0 (where TCO and TiO2 are in contact), the electron flux is correlated to the flow of electrons (cell current) through the external circuit to the counter electrode (where the reduction occurs), which is calculated using the Butler−Volmer equation and is given by ⎛ ∂n (x , t ) q⎜⎜ − De e ∂x ⎝

3 Ce , 2

∂ne(x , t ) ∂x

(30)

∫0

∫0

∂n ∂⌀ NI = −DI I − nIz IuI ∂x ∂x

NI3 = −DI3

Nc = −Dc

∂x

− n I3z I3u I3

(31)

∫0

where the mobility of each species is related to its diffusion coefficient through the Einstein relation: q ui = i = e, I, I3, c Di KBT

(35)

L

L

nc dx = nc̅ L

(36)

⎛ ⎛ 1 ⎞ 1 ⎞ ⎜n + nI⎟ dx = ⎜n I̅ 3 + nI̅ ⎟L ⎝ I3 ⎝ 3 ⎠ 3 ⎠

(37)

L

⎛1 ⎛1 1 ⎞ 1 ⎞ ⎜ n + nI⎟ dx = ⎜ ne̅ + nI̅ ⎟L ⎝2 e ⎠ ⎝ 3 2 3 ⎠

(38)

The no-flux boundary conditions are applied to the iodide ions, triiodide ions, and cations at the TCO−TiO2 contact (x = 0). NI(x = 0, t ) = NI3(x = 0, t ) = Nc(x = 0, t ) = 0

where KB is the Boltzmann constant. The drift term is due to the electric field and is described by Poisson’s equation:12 q dE = (nc − ne − nI − n I3) dx εε0

=0 x=L

These two boundary conditions represent the fact that the number of cations and the number of I atoms in the electrolyte are always constant.11 Also, since the generation of two electrons is accompanied by the consumption of three iodide ions, one can write

∂⌀ ∂x

∂nc ∂⌀ + nczcuc ∂x ∂x

(34)

From the conservation of particles in the electrolyte, one can write the following two integral boundary conditions:

∂ne ∂⌀ − nezeue ∂x ∂x

∂n I3

⎞ ⎟⎟ = j e x=0 ⎠

The flux of electrons at x = L is zero, because there is no contact between TiO2 and platinized TCO. Therefore, the boundary condition is given by

Electron and Ionic Species Transport. The driving force for the transport of species in the DSSC comes from the gradient of the electrochemical potential. Accounting for drift (migration) and diffusion, the fluxes of the species are given by Ne = −De

(33)

(39)

To solve the Poisson’s equation, the macroscopic electric field is assumed12 to be zero at x = 0:

(32)

E*(0) = 0

where ε0 and ε are the permittivity of free space and the effective dielectric constant of the electrolyte medium, respectively, and E is the electric field. The algebraic and differential equations presented in this section represent a dynamic mathematical model of the DSSC. Solving the equations with appropriate initial and boundary conditions allows one to describe the cell performance in terms of cell parameters, such as the effective electron diffusion coefficient, conduction band energy level and recombination rate constant, and solar radiation intensity.

(40)

2.5. Dimensionless Variables. To facilitate numerical solution of the governing equations, we nondimensionalize the model equations derived in this section. The following main dimensionless variables are used: nI n n n x ne* = e , nI* = I , n I*3 = 3 , nc* = c , x* = L ne̅ nI̅ n I̅ 3 nc̅ For the sake of brevity, the dimensionless model equations are not given here. E



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3. EFFECT OF THE ELECTRIC FIELD In this section, we present our theoretical investigation of the validity of the assumption that diffusion is the dominant driving force for the transport of species in DSSCs. To this end, we consider two models and compare their predictions. One is Model I, which was described in section 2, and the other is Model II, which does not include the drift terms in eqs 10−13. Note that the recombination rate equation of (27) is used in Models I and II. As a consequence of this assumption, since there is no source or sink for the cations, in Model II, Nc = 0, which implies constant density of the cations in the absence of an electric field. Under steady-state conditions, Model I and Model II equations represent two nonlinear boundary value problems. Steady-state Model I has four second-order and one first-order ordinary differential equations (ODEs), while Model II has three second-order ODEs. After converting the system of ODEs to a set of first-order ODEs, the ODEs with the appropriate boundary conditions are solved using the MATLAB routine bvp5c to investigate the effect of electric field on the I−V performance of the cell and species profiles. We initially use the same model parameter values reported by Ferber et al.13 and listed in Table 1 in the

Figure 5 depicts the calculated profiles of electron density, current density, and triiodide and iodide ion concentration deviations from their equilibrium values under the two limiting conditions of short circuit and open circuit. It shows that the profiles of each variable predicted by Models I and II are very similar. The largest difference in the predictions by the two models is in the iodide ion concentration. However, the I−V curves predicted by the two models, shown in Figure 6, are not distinguishable. This theoretical study confirms the validity of the assumption that the effect of electric field is negligible and diffusion (not drift) dominates the transport of electron and charged species in the cell.

4. EFFECT OF CHARGE RECOMBINATION In the previous section, it was shown that the reduced model (Model II) can describe the DSSC behavior accurately. In this section, we simulate the steady-state behavior of Model II with the four different recombination rate eqs 19, 20, 27, and 28. We will refer to these models as Models IIa, IIb, IIc, and IId, respectively. Indeed, Model IIc is exactly Model II. The recombination rate equations of 19 and 20 are second order, with respect to electron density. However, the recombination rate equations of eqs 27 and 28 are both first order, with respect to electron density. Several experimental studies have been conducted to understand the mechanism of reaction 14,26 using techniques such as intensity modulated photovoltage spectroscopy, 17,32−34 intensity modulated absorbance,32,33 and transient charge extraction.35 Some of these studies concluded that the recombination rate is second order with respect to electron density,28,34,36 while some other studies reported a first-order recombination reaction.37,38 Using each of the four models, we estimate four cell parameters: Ke (the recombination rate constant), De (the effective electron diffusion coefficient), j0 (the exchange current density), and RP (the shunt resistance) by fitting the model to experimental data reported by Nazeeruddin et al.23 Nazeeruddin et al.’s cell had an efficiency of 11.1%, which is one of the highest efficiencies achieved for DSSCs. The photoanode of this cell was composed of a 12-μm-thick mesoporous titanium dioxide electrode layer. The electrolyte solution contained 0.60 M butylmethylimidazolium iodide (BMII), 0.03 M I2, 0.10 M guanidinium thiocyanate, and 0.50 M tert-butylpyridine in a mixture of acetonitrile and valeronitrile. In this case, the cation is butylmethylimidazolium. This cell was masked with a black plastic with a hole (0.158 cm2) and tested under 1 sun (1.5 AM) irradiance. For parameter estimation, the MATLAB function fminsearch, which is based on the Nelder−Mead simplex algorithm, is used to calculate the parameter values that minimize the following objective function:

Table 1. Model Parameter Values13 Initially Used description

symbol

value 4 −1

electron recombination rate constant electron mobility iodide diffusion coefficient triiodide diffusion coefficient

Ke ue DI− DI−3

10 s 0.3 cm2/(V s) 8.5 × 10−6 cm2/s 8.5 × 10−6 cm2/s

initial concentration of iodide initial concentration of triiodide

c0I− cI−3 0

0.45 M 0.05 M

effective mass of electron exchange current density at the platinum electrode symmetry parameter effective relative dielectric constant difference of TiO2 conduction band and standard redox energy TCO resistance shunt resistance thickness of the cell cell area light intensity light absorption coefficient

m*e j0

5.6 me 0.1 A/cm2

B ε ECB − E0Redox RTCO RP L A I0 α

0.78 50 0.93 eV 6Ω 10 kΩ 10 μm 1 cm2 1 × 1017 cm−2 s−1 5000 cm−1

simulations. However, we will later estimate four of these parameters from experimental I−V data. The I−V curve of the cell is constructed using the continuation method; the external load resistance is varied from open circuit (Rext = ∞) to short circuit (Rext = 0) stepwise with a step size of 1 Ω, and for each value of Rext, the algebraic-differential equations are solved with an initial guess equal to the solution of the equations for the previous value of the load resistance. The simulation results for Model I are shown in Figure 4. They show the concentrations of the different species in the cell, the current density, and the electric field. As can be seen, the magnitude of the electric field is less than 0.05 mV/μm, which is insignificant. Furthermore, the electric field in the case of the open-circuit condition is less than the one in the short-circuit case, because of the smaller spatial variation of the densities of the charged species in the open-circuit case.

i=n

J(Δ⃗) =

∑ (Imod ,i(Δ⃗) − Imea,i)2 + (Vmod ,i(Δ⃗) − Vmea,i)2 i=1

where Δ⃗ = (Ke , De , j0 , RP)

Table 2 lists the values of the estimated parameters of Models IIa, IIb, IIc, and IId. Based on the I−V curve shown in Figure 7, the best fit corresponds to Model IIc. The sum of F



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Figure 4. Model I predictions: (a) electron density, (b) electron current density, (c) triiodide ion concentration (deviation from the equilibrium value), (d) iodide ion concentration (deviation from the equilibrium value), (e) electric field, and (f) lithium ion concentration (deviation from the equilibrium value), under short-circuit (SC) and open-circuit (OC) conditions.

Figure 5. Comparison of OC and SC predictions by Models I and II: (a) electron current density, (b) electron density, (c) triiodide ion concentration (deviation from the equilibrium value), and (d) iodide ion concentration (deviation from the equilibrium value).

the squared errors (SSE) of the curve fittings are as follows: SSE = 1.88 for Model IIa, SSE = 2.45 for Model IIb, SSE = 8.5 × 10−4 for Model IIc, and SSE = 1.2 × 10−3 for Model IId. These values indicate that Model II with eqs 27 and 28 can fit the

I−V data reasonably well. The estimated parameter values in Table 2 show that, as expected, the estimates of (ECB − E0redox), α, and I0 are very weakly dependent on the rate equation type. G



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Figure 8. Effect of the recombination rate constant on the cell I−V curve. The value 1342.5 s−1 is the estimated value of the recombination rate constant that matches experimental data. The experimental data shown in this figure have been taken from ref 23.

Figure 6. The cell I−V curves predicted by Models I and II.

Table 2. Estimated Parameters of Models IIa, IIb, IIc, and IId (Model II with Four Different Recombination Rate Equations)

increase as the recombination rate constant decreases. The effect of the rate constant is stronger on the short-circuit current than on the open-circuit voltage. Figure 9 shows the effect of

Value parameter

Model IIa

Model IIb

Model IIc

Model IId

Ke (1/m3 s or 1/s) De (m2/s) j0 (A/m2) RP (Ω)

1.7 × 10−19 9.8 × 10−7 2.5 × 103 4.1 × 102

2.9 × 10−20 9.9 × 10−7 1.3 × 103 4.1 × 102

1.3 × 103 9.8 × 10−7 1.1 × 103 4.4 × 103

1.0 × 103 9.2 × 10−7 3.0 × 103 4.7 × 102

Figure 9. Effect of the conduction band position on the cell I−V curve. The value 1.05 eV is the estimated value of the conduction band position that matches experimental data. The experimental data shown in this figure have been taken from ref 23.

conduction-band edge movement; it indicates that the edge movement affects the open-circuit voltage but not the shortcircuit current. The open-circuit voltage (also VOC × ISC) increases as the band edge movement increases. These results are similar to the experimental observations of surface treatment of the cell by 4-tert-butylpyridine (TBP),39 which shows an ∼200 mV increase in the open-circuit voltage without a significant change in the short-circuit current when the electrode is dipped for 15 min in TBP. It has been pointed out that the addition of TBP shifts the electronic states of TiO2 toward higher energies,34 which mainly increases the open-circuit voltage of the cell.40 Nakade et al.40 showed that the large increase in VOC is accompanied by no significant change of the electron lifetime. Based on the experimental observations, Nakade et al. concluded that this increase in VOC cannot be justified with the change of the electron lifetime, since the electron lifetime must be at least 2 orders of magnitude larger to increase the VOC significantly.28,40

Figure 7. The cell I−V curves predicted by Models IIa (blue trace), IIb (red trace), IIc (black trace), and IId (green trace). The experimental data shown in this figure have been taken from ref 23.

5. EFFECT OF RECOMBINATION RATE CONSTANT AND CONDUCTION BAND EDGE To study the effects of the recombination rate constant and conduction band edge movement on the cell behavior, steadystate Model II including the recombination rate constant equation of eq 27 is simulated with different values of the two model parameters to evaluate the impacts of changes in the cell parameters on the I−V performance of the cell. Figure 8 depicts the effect of the changes in the recombination rate constant on the cell I−V curve. As expected, both the opencircuit voltage and short-circuit current (also their product) H



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ruthenium(II) (RuL2(CN)2) [Dye 1] and cis-bisdicyano-bis(2,2′-bipyridyl-4,4′-dicarboxylic acid) ruthenium (RuL2(SCN)2) [Dye 2], and at equal electrode thicknesses and dye loadings. The experimental data, shown in in Figure 11, indicate that the short

This shift in TiO2 conduction band with TBP surface treatment can be attributed to the fact that TBP is a weak base,41 which renders the surface charge more negative when adsorbed on TiO2.42 In summary, the presented model predictions agree well with experimental findings.

6. EFFECT OF IRRADIANCE AND DYE TYPE In this section, we first compare the model predictions and our measurements of the I−V behavior of a DSSC made in our group when the cell is operated under different light intensities. Solar Cell Preparation and Characterization. The titanium dioxide colloidal paste consisting of TiO2 nanoparticles P25 (Evonik) is prepared using a published procedure.39 The prepared paste is spin-coated at 700−1200 rpm onto fluorinedoped tin oxide (FTO) conductive glass (Hartford Glass) to obtain a film of the paste with a thickness of 2−9 μm. To create the mesoporous structure, the spin-coated TiO2 paste is heated up to 500 °C and annealed for 30 min, samples are cooled to 100 °C and immersed in a 3 × 10−4 M solution of photosensitizer cis-dicyano-bis(2,2′-bipyridyl-4,4′-dicarboxylic acid) ruthenium(II) (RuL2(CN)2) or cis-bisdicyano-bis(2,2′-bipyridyl-4,4′-dicarboxylic acid) ruthenium (RuL2(SCN)2) (in pure ethanol), and left in the dark for 20 h. We will refer to these two dyes as Dye I and Dye II, respectively. Photovoltaic measurements of each assembled DSSC is carried out in a custom solar simulator equipped with a 300 W Xe lamp filtered to AM 1.5 spectral conditions and changed with neutral density filters as needed. Data are taken using a Gamry Reference 600 system. The illuminated area is kept constant at 0.25 cm2 by placing a mask on the sample. 6.2. Model Predictions and Validation. As shown in Figure 10, as the light intensity increases for a 9 μm cell, the

Figure 11. I−V model predictions and measurements for two different dye types. The experimental data shown in this figure are ours.

Figure 10. I−V model predictions and measurements when the cell is under different illuminations. The experimental data shown in this figure are ours.

current density increases and the open-circuit voltage remains almost unchanged. We also used the model to estimate only the light intensity (I0), from each set of our I−V data. We keep the absorption coefficient constant, knowing that the cell has not changed. The calculated estimates of the light intensity for 0.6, 0.8, and 1 sun are 3.18 × 1021, 4.33 × 1021, and 5.24 × 1021 m−2 s−1, respectively. If we consider the light intensity under 1 sun as the reference, the estimation errors for 0.6 and 0.8 sun are 3.34% and 1.27%, respectively, which shows the high accuracy of the estimates. We then fabricated and tested DSSCs with two different types of dye, cis-dicyano-bis(2,2′-bipyridyl-4,4′-dicarboxylic acid)

Figure 12. Effect of temperature on the DSSC I−V performance. The experimental data have been obtained from a cell operated at 25 °C and are taken from ref 23.

circuit current density and open-circuit voltage of the DSSCs with the two dyes are different. In the macroscopic model, the impact of the dye type, RuL2(CN)2 or RuL2(SCN)2, on the cell performance is mainly through the light absorption of the dye,43−46 and the dye type in our case has minimal effect on the dye molecule adsorption on the surface of semiconductor oxide due to molecular similarity and identical anchoring sites. Thus, I



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Figure 13. Electron density and concentrations of iodide and triiodide redox species from dark equilibrium to short-circuit conditions when the cell is short-circuited externally.

we expect that the recombination rate constant of the electrons in TiO2 conduction band with the electrolyte species to be comparable for the two dyes. The only expected impact of the dye type on the cell is through the electron concentration in the conduction band of TiO2, which has an impact on the photocurrent density and open-circuit voltage of the cell. This has already been observed experimentally and reported.46 As can be seen in Figure 11, the model can predict the I−V performance of our 9-μm cells well. Using the model, we estimate the difference between the formal redox potential and conduction band energy (ECB − E0Redox) and the absorption coefficient for the two dyes from our measured I−V data. The estimates of (ECB − E0redox) for Dye 1 and Dye 2 are 0.72 and 0.69 eV, respectively, and the estimated absorption coefficients for Dye 1 and Dye 2 are 3.48 × 105 and 3.67 × 105 m−1, respectively. The model parameter estimation results suggest that the observed difference in the I−V performances of the cell with the two dyes is related to the absorption coefficient and displacement of the conduction band directly. Since there is no trivial way of measuring ECB in a working cell, we can state that these results are consistent with previously reported results. Also, in the case of Dye 1, the absorption coefficient estimate is consistent with the dye absorption spectrum in solution and its lower extinction coefficient.

Figure 12, as the operating temperature increases, the opencircuit voltage of the cell decreases, but the short-circuit current does not change. The experimental data shown in Figure 12 is from ref 23. This temperature-dependent behavior is qualitatively in agreement with the experimental results reported by others.47,48 The dependency of the open-circuit voltage on temperature has been attributed to the change in the electron quasi-Fermi level and the increased recombination rate at elevated temperature.47−49 In cases where the hole transport media composition was changed and an ionic liquid50,51 or 2,2′,7,7′-tetrakis N,N-dimethoxypheny-amine-9,9′-spirobifluorene (Spiro-MeOTAD)52 was used, a more pronounced change was observed. However, wherever these materials have been used, the observed change in open-circuit voltage is usually accompanied by a concomitant increase in the short-circuit current density of the cell, which is related to stronger dependency of the hole mobility of this medium to the operating temperature.50,51 From Figure 12, we can observe that the DSSC performance is not very sensitive to temperature for the standard liquid electrolyte cell. In contrast, other solar cell technologies such as crystalline silicon solar cells show a decline of 15%−25% in the power conversion efficiency when the temperature increases from 20 °C to 60 °C.53

7. EFFECT OF TEMPERATURE ON THE DSSC I−V PERFORMANCE The effect of temperature on the DSSC I−V performance is investigated by using the model to predict the cell I−V curve at different temperatures in the range of 0−70 °C. As can be seen in

8. TRANSIENT RESPONSE To simulate the cell under transient conditions, the Model II equations including the recombination rate equation described by eq 27 are discretized spatially using the pdepe function in MATLAB,54 and then the resulting ODEs are integrated, with J



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Figure 14. DSSC response to a step change in the external load resistance from 1 Ω to 100 Ω.

concentration of I− ion near the anode and cathode. The figure also indicates that, under steady-state conditions, as x increases, the electron density and iodide ion concentration increase, but the triiodide ion concentration deceases. These concentration spatial variations are expected in such a cell. The response of the solar cell to a step change in the external load resistance from 1 Ω to 100 Ω at time t = 0.05 s is depicted in Figure 14. The increase in the voltage and decrease in the current can be explained by referring to the I−V curve of the cell; as the load increases, the operating point moves to the right; that is, the current decreases but the voltage increases. As can be seen, there is an abrupt increase in the electron density, exhibiting an overshoot and eventually a decrease due to the recombination process. The redox species transient responses show that the

respect to time, using an ODE solver. The transient model allows one to study the behavior of the cell under variations in ambient temperature, external load resistance, and irradiance. Here, the transient behavior of the cell from the dark equilibrium conditions to short-circuit conditions and the cell response to a step change in the external load are investigated. To this end, the discretized Model II initialized at dark equilibrium conditions is integrated under two scenarios: a constant zero Rext (short circuit) and a step-shape nonzero Rext. Figure 13 shows that the response of electron density is faster than those of the redox species. While the concentration of I− ion near the anode decreases with time, the concentration of I− ion near the cathode increases with time. Variations with time of the concentration of triiodide ion are opposite to those of the K



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EnF = quasi-Fermi energy (J) ERedox = redox energy (J) G = generation rate (m−3 s−1) h = Planck’s constant (J s) I0 = incident photon flux (m−2 s−1) Iext = external current (A) j0 = exchange current density at Pt electrode (A m−2) je = internal cell current density (A m−2) KB = Boltzmann constant (J K−1) m*e = effective electron mass (kg) n = species density (m−3) n̅ = equilibrium density (m−3) N = flux of species (m−2 s−1) NCB = effective density of electron in the conduction band (m−3) q = elementary charge (C = A s) Rext = external load resistance (Ω) RP = shunt resistance (Ω) RTCO = TCO resistance (Ω) T = temperature (K) Vext = external voltage (V) Vint = internal voltage (V) u = species mobility (m2 V−1 s−1) x = coordinate (m) z = charge valence

concentration profiles in the front and rear sides are affected mostly by the step change in the external load resistance. The gradient of electron density is lower at the lower external load, as less current is withdrawn from the cell at lower currents. Also, the gradients of the redox ions are lower after the step change in the external load than those before the step change. Again, this can be explained by the withdrawal of less current at the higher external load.

9. CONCLUDING REMARKS Theoretical and experimental analyses of the performance of a dye-sensitized solar cell (DSSC) were presented. The theoretical study indicated that diffusion is the dominant driving force for transport of electrons and holes, and thus electric field-induced migration can be neglected. The accuracy of the model predictions was evaluated under different light intensities and utilization of two different dyes. The use of different types of recombination rate equations obtained using different recombination mechanisms showed that the rate equation type has little effect on the estimates of the effective electron diffusion coefficient and the difference between the conduction band and formal redox potentials. The theoretical studies indicated that recombination rate constant affects both the cell opencircuit voltage and short-circuit current, while the conduction band edge movement mostly affects the cell open-circuit voltage. They also revealed that the effects of different light absorbers on the cell current−voltage (I−V) performance is through the absorption coefficient and displacement of the conduction band. The transient behavior of the cell from the dark equilibrium conditions to short-circuit conditions and the cell transient response to a step change in the external load were shown. The dynamic model can be used to study the dynamics and control of the energy integration systems, in which a solar cell is utilized in conjunction with other energy generation and storage devices. We have been using the macroscopic model to mathematically represent a polymer−electrolyte DSSC and arrive at optimal design specifications of the polymer−electrolyte DSSC. The results of this sequel study will be presented in a forthcoming publication.

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Greek Letters

α = absorption coefficient (m−1) β = symmetry factor ε = dielectric constant ε0 = permittivity of free space (F m−1) ηa = activation overpotential (V) ϕ = electric potential (V) Subscripts

e = electron I3 = triiodide ion I = iodide ion c = cation ref = reference Superscripts

AUTHOR INFORMATION

Corresponding Author

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* Tel.: (215) 895-1710. Fax: (215) 895-5837. E-mail: [email protected]. Notes

OC = open circuit SC = short circuit

REFERENCES

(1) Baxter, J. B. Commercialization of dye sensitized solar cells: Present status and future research needs to improve efficiency, stability, and manufacturing. J. Vac. Sci. Technol., A 2012, 30 (2), 020801−020819. (2) O’Regan, B.; Grätzel, M. A low-cost, high-efficiency solar cell based on dye-sensitized colloidal TiO2 films. Nature 1991, 353, 737−740. (3) Yella, A.; Lee, H.-W.; Tsao, H. N.; Yi, C.; Chandiran, A. K.; Nazeeruddin, M. K.; Diau, E. W.-G.; Yeh, C.-Y.; Zakeeruddin, S. M.; Grätzel, M. Porphyrin-Sensitized Solar Cells with Cobalt (II/III)− Based Redox Electrolyte Exceed 12% Efficiency. Science 2011, 334 (6056), 629−634. (4) Hardin, B. E.; Snaith, H. J.; McGehee, M. D. The renaissance of dye-sensitized solar cells. Nat. Photonics 2012, 6 (3), 162−169. (5) Snaith, H. J. Estimating the Maximum Attainable Efficiency in DyeSensitized Solar Cells. Adv. Funct. Mater. 2010, 20 (1), 13−19. (6) Han, L.; Koide, N.; Chiba, Y.; Islam, A.; Mitate, T. Modeling of an equivalent circuit for dye-sensitized solar cells: improvement of efficiency of dye-sensitized solar cells by reducing internal resistance. C. R. Chim. 2006, 9 (5−6), 645−651. (7) Liu, H.; Avrutin, V.; Izyumskaya, N.; Ö zgür, Ü .; Morkoç, H. Transparent conducting oxides for electrode applications in light

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based on work supported by the National Science Foundation (NSF) (under Grant Nos. CBET-1236180 and CBET-1234993). S.N. and K.K.S.L. acknowledge support from the NSF under Grant Nos. CBET-0820608 and CBET-0846245. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF.

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NOTATION A = cell area (m2) C = consumption or recombination rate (m−3 s−1) D = diffusion coefficient (m2 s−1) E = electric field (V m−1) ECB = conduction band energy (J) L



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Theoretical and Experimental Study of a Dye-Sensitized Solar Cell

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