Dye Regeneration Kinetics in Dye-Sensitized Solar Cell: Long-Range

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Dye Regeneration Kinetics in Dye-Sensitized Solar Cell: Long-Range Charge-Transfer Effects Mehdi Ansari-Rad J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b02114 • Publication Date (Web): 06 Apr 2016 Downloaded from http://pubs.acs.org on April 9, 2016

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The Journal of Physical Chemistry

Dye Regeneration Kinetics in Dye-Sensitized Solar Cell: Long-Range Charge-Transfer Eects Mehdi Ansari-Rad

∗

Department of Physics, Shahrood University of Technology, Shahrood, Iran E-mail: [email protected]

Phone: +98 2332392204

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Abstract We study dye regeneration dynamics in dye-sensitized solar cells using extensive kinetic Monte-Carlo simulations of transient absorption experiments. Using the longrange charge transfer model as the regeneration mechanisms, the dependency of the regeneration kinetics on concentration and diusion speed of the reducing species is investigated. At initial times, the kinetics is exponential, and mainly determined by the intrinsic regeneration rate coecient, irrespective of the diusion coecient. At later times, however, depending on the concentration of the reducing species, the diusion starts to play a role in the regeneration process, leading to a stretched tail. We discuss that this concentration dependence cannot be described by the approaches based on the Smoluchowski model. We also compare dierent methods used in experiments for estimating the rate coecient from the decay curve, and it is shown that dierent methods may lead to dierent conclusions about the regeneration mechanisms. These results can also provide insight into the interpretation of the decay kinetics in similar processes as quenching and relaxation phenomena.

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Introduction Dye-sensitized solar cells (DSCs) are considered as promising candidates for low-cost photovoltaic applications. 1 In a DSC, dye molecules anchored to a wide band-gap semiconductor (typically, a porous electrode of TiO 2 nanoparticles) absorb light from the sun. After injecting their excited electrons into semiconductor conduction band (CB), the dye molecules are regenerated by redox species in the surrounding electrolyte. Fast dye regeneration is, therefore, an important factor in determining the overall solar cell performance since it prevents back electron transfer from the semiconductor CB to the oxidized dye. 2 Recently, one-electron redox couples based on cobalt and ferrocene, in combination with organic dyes, have attracted attention due to their advantages over the traditional I 3 − /I− redox shuttle. 35 In fact, the highest power conversion eciency reported for DSC (13%) is obtained by porphyrin sensitizers with [Co(bpy) 3 ]2+/3+ redox shuttle. 6 An important feature of these new redox mediators and dyes is that their energy levels and architectures can be tuned by molecular engineering in order to optimize various processes in DSC. To reveal the precise nature of the regeneration process, several groups studied the eect of regeneration driving force, dye architecture, and concentration of the redox species on the regeneration process. 712 Transient absorption spectroscopy (TAS) is often used to study the regeneration kinetics in DSC. In this method, a low-intensity laser pulse (pump) is used to excite a few dyes per nanoparticle. The population of the oxidized dyes is then monitored by an appropriate wavelength (probe), and the regeneration rate coecient is estimated from the transient absorption kinetics. Very recently TAS has been used by Daeneke et al. to study the regeneration kinetics of a series of organic carbazole-based dyes with ferrocene derivatives. 8 Interestingly they found that at high driving forces, the regeneration is diusion-controlled. In another work, however, 9 with a series of cobalt-based electrolytes, it was found that the regeneration kinetics can completely be explained by Marcus theory and the diusioncontrolled regeneration is not reached. An important dierence between two works was that the concentration of the reducing species was very low (5 mM) in the former report. In 3



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fact, the eect of the concentration of the redox species on the regeneration kinetics has been a question of interest for various kinds of electrolytes. It has been discussed that the interaction between the dye sensitizer, the redox species, and the cations in the electrolyte, as well as the delocalization of the charge on the redox species, can aect the concentration dependence of the regeneration kinetics. 1315 Stretched exponential decay (relaxation or quenching) has been observed in many dierent physical systems and is usually attributed to a source of disorder in that system. 16 For TAS measurement in DSC also, a stretched exponential decay has been reported in several works. Gaussian distribution of the regeneration activation energy has been proposed to be a possible disorder source that may account for the stretched decay. 7 A width of ∼ 0.1 -

0.2 eV was needed to obtain a moderate t to the experimental data. But, such broadening seems not to be consistent with the ne resolution of activation energy typically accessible in the experiments. In this work we address the interpretation of TAS measurement in DSC, using the kinetic Monte-Carlo simulations. We systemically study the eect of the concentration and the diusion coecient of the redox species as well as the regeneration driving force on the regeneration kinetics. Long range charge-transfer eect is also investigated and is proposed to be partly responsible for the stretched transient absorption decay. We also compare dierent methods usually used to calculate the regeneration rate coecient from the TAS measurement in experimental works. By comparing the rate coecients calculated from these analyzes, it is shown that dierent methods may lead to dierent conclusions about the regeneration mechanisms.

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Theory The regeneration of an oxidized dye by the redox couples (donors) in the electrolyte can, in the rst approximation, be described using the continuity equation

∂C = DO2 C ∂t

(1)

with the so-called partially reecting boundary condition (also known as Collins-Kimball model) 17 C(r → ∞, t) = C0 ∂ C 2 4πR D = K 0 C(R, t) ∂r

(2) (3)

r=R

Here C and D are concentration and diusion coecient of the donors in the electrolyte, respectively, and C 0 is the concentration in bulk. R is the encounter distance at which charge transfer to the oxidized dye can occur, and is determined by the steric eects. K 0 is the second-order rate constant for the transfer process when the dye and redox molecule are almost at distance R from each other. K 0 has the units of m3 s−1 . Intrinsic transfer rate for the charge transfer process, K , depends both on the regeneration activation energy as exp(−Ea /kB T ), and on the spatial separation of the donor and acceptor states as exp(−r/α), where kB T is the thermal energy and α is a measure of (de)localization of the charge wavefunction. Based on the Marcus theory for outer-sphere electron transfer, intrinsic transfer rate K is given by 18

r Ea exp − K = K0 exp − α kB T

(4)

with Ea = (4G + λ)2 /4λ, where K0 is the intrinsic rate constant, 4G = Ered − Edye is the reaction free energy, and λ is the reorganization energy. K has the units of s−1 and is related to K 0 as K 0 ≈ 4πR2 αK . In this work, λ is treated as a constant 19 although one can also 5



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consider λ = λ(r). 20 The tunneling factor, exp(−r/α), implies that we deal with a long-range charge transfer for the regeneration process, in the sense that the transfer process can occur at distances larger than the encounter distance R. The continuity equation, therefore, should be replaced by 21

∂C = DO2 C − K(r)C(r, t) ∂t

(5)

What we are nally interested in is the regeneration rate coecient k that determines decay of the population of oxidized dye as

d[S+ ] = −k[S+ ] dt

(6)

The rate coecient k is in fact time dependent, and it has been shown that can be written as the sum of two rates: 22 diusive rate coecient, kD , related to the regeneration at encounter distance R, and a tunneling rate coecient, kT , for the regeneration from distances greater than R, respectively given by

∂ C kD (t) = 4πR D ∂r r=R Z ∞ K(r)C(r, t)r2 dr kT (t) = 4π 2

(7) (8)

R

At short times kD ∼ 0, and it is found

k ≈ kT ∼ C0 K(R)

(9)

Therefore, it is expected that the diusion rate does not enter into the problem at short times and the decay process is governed by a static charge transfer regime. Eq.(9) also suggests that with a higher initial concentration of the redox species, the diusion starts to play a

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Figure 1: (a) Illustration showing the eect of long-range charge transfer and diusion on the

regeneration kinetics. At short times, regeneration of a dye cation (red lled circles; +) may be accomplished by redox species already present at distances ∼ α around the dye (with time constant t ∼ 1/kT ), without needing any diusion. This leads to the same rate coecient kT for the regeneration, irrespective of the diusion coecient of the redox species, as shown in the schematic diagram (b). When the average population of the redox species around the dye cations is reduced, the diusion speed becomes important; the lower the diusion coecient, the slower the regeneration process. In (b), D1 > D2 > D3 . (c) Potential diagram of the regeneration process, showing the potentials used in the simulations. role in the decay process at later times. In other words, that the decay process is limited by the diusion or not, is not only determined by competition between intrinsic parameters

D and K , but also by the donor concentration. The proportionality factor in Eq.(9) is 4π(αR2 + 2α2 R + 2α3 ). As a consequence, at least for short times, a greater delocalization can also weaken the role of the diusion in the decay process. As time goes on, the average population of the redox species around the dye cations is reduced (Fig.1(a)). Although this depletion may already be compensated by the diusion of the redox species towards the dyes, but for low and intermediate diusion rates, the compensation is not complete, and consequently, the decay process is slowed down. In other words, the initial exponential decay becomes

stretched ; see Fig.1(b). Based on this

discussion, one can suggest that change of the regeneration dynamics, from the long-range

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transfer regime to a diusion-inuenced regime is a possible origin for the stretched decay observed in the transient absorption spectroscopy in DSCs. This mechanism can also be considered for other decay (or, relaxation) processes without needing to a source of disorder in the system. Finally it must be noted that from Eq.(1) with boundary conditions Eqs.(2) and (3), i.e. Collins-Kimball model, the population of the dye cations is, approximately, given by 23

2R0 t [S ] ≈ [S ]0 exp −4πR DC0 1 + √ πDt +

+

0

(10)

with R0 = RK 0 /(4πRD + K 0 ). Eq.(10) gives a stretched decay at short times as t1/2 and an exponential one at longer times. However, as mentioned above, a long-range transfer suggests an exponential decay at short times, independent of the diusion coecient. Eq.(10) also predicts a linear dependence of the regeneration rate on C 0 . This is in fact seen in all models based on the Smoluchowski approach. 24 We will see that a long-range transfer mechanism may not show this linear dependency.

Kinetic Monte-Carlo Simulation

The Model To simulate the porous electrode in a typical DSC, we generated networks with the size of

(101 nm)3 and average porosity of ∼ 56% from interconnected spheres with the radius of 10 nm, using the algorithm described in our previous work. 25,26 About 130 spheres are needed to be embedded into a network for above-mentioned porosity. Helical boundary conditions were imposed in three directions to eliminate the nite-size eects, and for each run, a new morphology was generated. Fig.2 shows a slice in the xy plane of a network generated in a simulation. Cubic lattice was used to generate the lattice points and lattice spacing, which determines the spatial resolution of the simulation, was set to 1 nm. 200 to 230 lattice points

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Figure 2: A slice in the xy plane of a 3D simulation box. Gray circles are slices of the interconnected

spheres that form a continuous porous network in the 3D box. A part of the slice has also been shown schematically in the gure with higher magnication. Red circles are dye molecules (oxidized or not). Small blue circles are the redox species, allowed to diuse in the pore regions. on the surface of each sphere are then randomly assigned as dye molecules. Redox species, with the concentration of C 0 , are also randomly placed on the empty lattice points outside the spheres. At the beginning of the simulation, ∼ 3 dyes from each sphere are randomly selected as oxidized dyes. This corresponds to a typical TAS measurement with pump pulse intensity of

∼ 0.1 mJ/cm2 , tuned to 600 nm and irradiated on an electrode with 4 µm thickness. 9 Also, the same number of electrons are randomly placed on the lattice points inside the spheres, playing the role of the electrons injected by the excited dyes into the electrode. Our kinetic Monte-Carlo code allows to implement dierent events in the simulation: random jump of an electron or a redox species to an empty lattice point, recombination of an electron with an oxidized dye, and regeneration of an oxidized dye by a redox species. At each step of the simulation, one of the events is randomly selected, with the probability that is proportional to the corresponding rate constant. We have used the so-called Random Selection Method to perform the selection, details of which can be found in Ref. 27 For the regeneration process, intrinsic transfer rate of Eq.(4) was used in the simulations with

α = 0.2, 0.3, 0.4 nm, λ = 0.7 eV, and −4G = 0.25 - 0.74 eV. 9 The range of the free energies used here is equivalent to Edye = 1.11 and Ered = 0.37 - 0.86 V

vs. NHE, as shown in

Fig.1(c). It must also be noted that a localization radius smaller than the lattice spacing 9



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does not exclude regeneration events over 1nm. In fact, using a localization radius smaller than the lattice spacing is very common in the kinetic Monte-Carlo simulation of the charge transfer processes (for example recombination in organic solar cells or charge transport in disordered systems). Many experiments suggest a regeneration eciency as high as 99% for the optimized dyes. For this reason, we did not pay careful attention to the details of the electron transport (for example, considering a distribution of trap states in the simulation), and the electron-dye intrinsic recombination constant was chosen small enough to guarantee the high regeneration eciency. In order to examine the eect of diusion rate of the redox species on the regeneration kinetics, simulations were done at three dierent diusion coecients, 2.44×10−7 , 2.87×10−8 , and 3.09 × 10−9 cm2 s−1 , hereafter referred to as D1 , D2 , and D3 , respectively. These values were established by adjusting the jump rate in the simulations, and were computed from independent simulations with C 0 = 0.2 M, using the denition D = hr2 i/6t, where t is the simulation time and hr2 i is the mean square displacement of the redox species. In fact, this corresponds to a simple random walk simulation on a cubic lattice, upon which, the redox species can jump to their nearest neighbors, but double occupancy and jump to the electrode region are not allowed. Diusion coecients of the order of 10−6 - 10−8 cm2 s−1 have been reported for redox couples in DSC. 28,29 In experiments three methods are usually used to estimate the regeneration rate coefcient k from the TAS measurement, hereafter referred to as M1 , M2 , and M3 : inverse of the regeneration halftime t1/2 (M1 ), modeling transient curve by a stretched exponential decay (M2 ), or modeling by a biexponential decay ( M3 ); see Appendix for mathematical description. Accordingly, we use these methods to extract the regeneration rate coecient from our simulations. The results will then be interpreted in terms of the theory discussed in the previous section.

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Figure 3: Regeneration rate coecient

vs.

regeneration driving force (−∆G), calculated using methods M1 , M2 , and M3 , for diusion coecient D3 . The solid line represents a t based on Marcus model, leading to a λ ≈ 0.7 eV. For the rates calculated by methods M2 and M3 , such a tting was not possible and the dashed lines are only guides to eyes. Results are for simulations with C0 = 0.2 M and α = 0.2 nm.

Results and Discussion Fig.3 shows regeneration rate coecient k

vs. free energy ∆G, for simulations with diusion

coecient D3 (see SI for D1 and D2 ). Decay kinetics were analyzed with methods M1 , M2 , and M3 , results of which are shown in the gure. As seen, the rate coecients obtained by method M1 (i.e. from regeneration halftime t1/2 ) are higher than those estimated by

M2 and M3 . More importantly, a tting based on the Marcus model was possible for the results from M1 . This t, shown as solid line in the gure, led to the reorganization energy

λ ≈ 0.7 eV, in agreement with the value used in the simulations. One may, therefore, conclude that the regeneration process is reaction-limited and diusion-controlled regime is not reached. However, a consistent t based on the Marcus model was not possible for the rate coecients computed from methods M2 and M3 , and as can be seen in Fig.3, the regeneration rate reaches a plateau value for −∆G near λ. From these analyzes, therefore, one comes to the conclusion that the regeneration process is diusion-controlled. As discussed in the theory section, the initial kinetics of the regeneration process is independent of the diusion coecient and is mainly determined by the tunneling rate coecient, i.e. k ≈ kT ∝ C0 K(R), with K(R) being Marcus transfer rate, see Eq.(9). 11



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Figure 4: Decay kinetics of the oxidized dyes for two concentrations of the redox species (a)

C0 = 0.05 M and (b) 0.6 M, and diusion coecients D1 , D2 , and D3 . Filled circles show points of 50% regeneration. Results are for simulations with ∆G = −0.55 eV and α = 0.2 nm. The rates obtained by method M1 can be well tted by Marcus rate model because, during the time t1/2 , the dominant pathway for the regeneration is the long-range charge tunneling. On the other hand, methods M2 and M3 contain information about the diusion coecient because the tting is done over the whole time of the regeneration. This interpretation may explain the observation by Daeneke et al. of a diusion-controlled regeneration, concluded based on the rates estimated by method M2 , 8 as well as a reaction-controlled regeneration reported by Feldt et al. based on the analysis by method M1 . 9 Our discussion, however, is not complete, without considering the eect of concentration of the reducing species on the regeneration kinetics. Fig.4(a) and (b) shows the decay kinetics of the oxidized dyes for two dierent concentrations. Points at which half of the dye cations has been regenerated are also indicated on the curves. For the high concentration of the redox species, as can be seen in Fig.4(b), the regeneration halftime is independent of the diusion coecient, regardless of the fact that D1 /D3 ∼ 102 . For the low concentration, however, the diusion coecient has a strong eect on the decay kinetics. In fact, as mentioned in the theory section, since the tunneling rate coecient kT is proportional to C 0 , the diusion speed becomes unimportant at high C0 ; see also Fig.1(b). The result of Fig.4 clearly suggests that aside from the diusion coecient, the concentration of the donor must be taken into account when interpreting TAS measurements.

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Figure 5: Decay kinetics of oxidized dyes for two diusion coecients D1 and D3 , and three

concentrations of the redox species C0 : a = 0.6 M, b = 0.3 M, and c = 0.05 M. For each case, the time axis has been rescaled by the corresponding C0 . Straight line indicates similar slope of all curves at short times. Results are for simulations with ∆G = −0.55 eV and α = 0.2 nm. The prediction that the initial regeneration kinetics is given by the rate k ∝ C0 K(R), suggests that irrespective of the diusion coecient and the redox concentration, all the decay curves must show similar initial slope ( ∝ K(R)) when the time axis is rescaled by C 0 . Fig.5 conrms this idea, where the decay kinetics for simulations with D1 and D3 are plotted with the rescaled axis (for the sake of clarity, results for D2 are not shown). As indicated by the solid straight line, all the decay curves show a similar initial slope. However, at later times, depending on the diusion coecient and (or) concentration of the redox species, the initial exponential decays exhibit stretched tails. Our simulations with higher delocalization radii, α = 0.3 and 0.4 nm, have shown similar behavior, but, as expected, the decay curves retained their initial slope for longer times. As discussed in the introduction, a stretched exponential function, in the form exp[−(t/τ )β ], ts well relaxation phenomena in many physical systems, with β being the stretch parameter; see Appendix. Fig.6 shows the eect of the diusion coecient and concentration of the reducing species on the normalized rate coecient and the corresponding stretch parameter

β , for simulations with delocalization radius α = 0.4 nm. 30 As seen in Fig.6(a), the results for diusion coecient D1 are independent of the concentration, implying a reaction-controlled

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Figure 6: (a) Normalized regeneration rate coecient and (b) stretch parameter β as a function of

concentration of the redox species, computed using method M2 , for diusion coecients D1 , D2 , and D3 . Results are for simulations with ∆G = −0.55 eV and α = 0.4 nm. mechanisms for the regeneration process. For D2 and D3 , however, the normalized rate shows a dependency on the concentration, rising to a plateau at high concentration. A similar concentration dependence is also observed for the stretch parameter β , as seen in Fig.6(b). The reason for such a concentration dependence is change in the regeneration mechanisms from a diusion-inuenced regime (at low concentration), to a reaction-controlled one (at high concentration). The conventional meaning of a reaction-controlled reaction is that the rate of the diusion is much faster than the intrinsic reaction rate so that the overall reaction time is controlled by the intrinsic reaction rate. Here, however, the regeneration becomes reaction-controlled, interestingly even for a low diusion coecient as D3 , because, at high concentrations the dye cations can directly be regenerated by the donors via the long-range tunneling. Recently, for uorescence quenching in polymer-fullerene photovoltaic blends, a shift from partly diusive regime to the purely Förster regime has been observed to occur by increasing the quencher (fullerene) concentration in the blend. 31 The results of Fig.6(b) clearly shows that, in the relaxation phenomena, in addition to the diusion speed, the concentration of the donors also controls the stretch parameter. We must remark, however, that for the smaller delocalization radius α = 0.2 nm, the stretch parameter does not show concentration dependence (see SI). This is, in fact, expected, since for smaller α, change in

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Figure 7: Distribution of the regeneration distance, Y (r), for (a) α = 0.2 nm and (b) 0.4 nm, and

two dierent donor concentrations (∆G = −0.55 eV). The solid lines are guides to eyes. The inset shows the behavior of Y (r) when the geometrically-limited points are not considered. the surrounding donor concentration cannot be

seen by the dye cations.

To explore how the delocalization radius aects the regeneration process, distribution of the regeneration distance, Y (r), was extracted from the simulations which represents the number of electron transfer events from distance r, to the total number of the events. The results for two delocalization lengths α = 0.2 and 0.4 nm, and diusion coecient D3 are shown in Fig.7(a) and (b), respectively. (Note that the spatial resolution of our simulation box is 1 nm which corresponds to an encounter distance of R = 1 nm.) As is seen from the gure, for the higher α, the distribution is broader, and the larger distances are involved in the regeneration process (this was also seen for diusion coecient D1 and D2 ). This is quite reasonable since the long-range transfer becomes more ecient with a higher delocalization length. It should be noted, however, that, despite the broad distribution in the case of

α = 0.4 nm, the stretch parameter may be small or not (0.65 and 0.87 for C 0 = 0.05 and 0.6 M, respectively; see Fig.6(b)). As discussed throughout the paper and schematically illustrated in Fig.1, the regeneration kinetics becomes stretched when the donors around the acceptors are consumed and further regeneration is limited by the diusion of the donors toward the acceptors. For C 0 =0.6 M, the donor concentration around the dye cations is high and consequently the kinetics is less aected by the diusion, leading to β = 0.87 despite a

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broad distance distribution Y (r). Interestingly, at low donor concentration, the distribution is also broader, and in the case of α = 0.4 nm, Y (r) shows a maximum around 1.5 nm. The appearance of a maximum in

Y (r) has already been predicted theoretically by Murata and Tachiya. 20 Although it seems here that Y (r) exhibits several extrema, but one must note that since a cubic lattice was used in our simulations, there are geometrically excluded (or less dense) regions in the simulation box, leading to the minima seen in the Figure. The black curves in the inset of Fig.7(b) show the general behavior of Y (r) if these minima are neglected, corresponding to the continuum model of Ref. 20 We nally point out that the molecular dynamics simulation has shown that the redox ion distribution near the surface of the electrode (where the dye molecules are anchored) may really show peaks at some preferred distances. 13

Concluding Remarks Kinetic Monte-Carlo simulation has been used to simulate the dye regeneration process in DSCs. We have analyzed how a long-range regeneration mechanisms (tunneling) can aect dependency of the regeneration rate coecient on the concentration of the reducing species and on the diusion speed. We have interpreted our results using the continuity equation that includes the tunneling term, with the partially reecting boundary conditions. It has been found that irrespective of the diusion coecient of the reducing species, the initial kinetics of the regeneration is mainly determined by the intrinsic rate coecient. At the moderate and high concentration of the donors, more than half of the acceptors can be directly regenerated via this mechanisms. In other words, irrespective of the diusion coecient, the regeneration process is reaction-limited. However, the long-term kinetics may be aected by the diusion speed of the donors, especially at low donor concentrations. It is worthwhile to note that the analytical approach based on the Collins-Kimball model cannot predict the results obtained here. It is because the long-range regeneration reduces

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the role of diusion at high concentrations. Our simulations, in turn, shows the advantage of using the kinetic Monte-Carlo method in studying the regeneration process, especially for more complex situations in which the electron-dye recombination can compete with the regeneration. By comparing the dierent methods used in the experiment for estimating the regeneration rate coecient, it has been shown that the rates estimated using the decay halftime are generally a measure of the intrinsic regeneration rate. On the other hand, the rates estimated using the stretched exponential or biexponential decay tting are also aected by the value of diusion coecient. In case that the kinetics is described by the stretched exponential, it has been seen that the stretch parameter is not only determined by the diusion coecient, but also by the concentration of the donors. These results provide insight into interpreting decay kinetics in various relaxation processes, as dye regeneration and also uorescence quenching. However, eect of the morphology on the regeneration rate, competition between the recombination and the regeneration for less-ecient dye-electrolyte systems, and some other important issues remain to be seen. Since all the molecules involved in the regeneration process are charged particles, surface chemistry of the porous electrode also needs to be considered in more advanced models. It can be done by including a mean eld electric potential (repulsive or attractive) near the surface of the electrode that can aect concentration prole of the redox species near the surface. Simulation of the regeneration process to study these factors are currently in progress.

Supporting Information Available Regeneration rate coecient as a function of free energy for dierent diusion coecients; Stretch parameter as a function of free energy; Stretch parameter as a function of donor concentration for dierent delocalization radii. This material is available free of charge via the Internet at http://pubs.acs.org.

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Appendix To estimate the regeneration rate coecient from the TAS measurement, three methods are found in the literature.

Method M1 : The transient decay is approximated by a single exponential function, and the rate coecient is computed using the regeneration half-time t1/2 (the time to 50% decay), as

kM1 =

ln 2 t1/2

Method M2 : The stretched exponential function, e−(t/τ ) , is used to t the transient decay. β

The rate coecient is then given by ( 0 < β ≤ 1 is the stretch parameter, and Γ is the gamma function)

kM2 =

β τ Γ(1/β)

Method M3 : The transient decay is tted to the biexponential function, as A1 e(−t/τ1 ) + A2 e(−t/τ2 ) , from which the rate coecient is calculated by

kM3 =

A1 τ1 + A2 τ2 A1 τ12 + A2 τ22

Acknowledgement I would like to thank Juan Bisquert for his valuable insights.

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log (oxidized dye population)

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static regeneration

diffusion-assisted reg.

time Graphical TOC Entry

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Dye Regeneration Kinetics in Dye-Sensitized Solar Cell: Long-Range

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