Article pubs.acs.org/JPCB
Verification of Nonequilibrium Mechanism of Ultrafast Charge Recombination in Excited Donor−Acceptor Complexes Tatyana V. Mikhailova, Valentina A. Mikhailova, and Anatoly I. Ivanov* Volgograd State University, University Avenue 100, Volgograd 400062, Russia ABSTRACT: Control of charge transfer requires knowledge of its detailed mechanism. Due to the large number of known mechanisms, the identification of the mechanism in specific systems is a challenge so far. In this article we propose the idea of how to distinguish between thermal and nonequilibrium modes of charge recombination in excited donor−acceptor complexes. Simulations of the effect of solvent relaxation time scale on ultrafast charge recombination kinetics in photoexcited donor−acceptor complexes within the framework of the multichannel stochastic model have shown that a series of regularities inherent to the thermal and nonequilibrium charge transfer can strongly differ. Among them there are opposite regularities, for example, the dependence of the dynamic solvent effect on the free energy gap. In particular, theory predicts that in ultrafast charge recombination of excited donor−acceptor complexes the dynamic solvent effect is weak in the area of weak exergonicity and becomes stronger in the area of stronger exergonicity whereas for the thermal reactions an opposite trend is expected. Comparison of such trends with experimental data implemented in this article allowed establishing the regime in which the reaction proceeds. It is shown that observation of dynamic solvent effect in the region of strong exergonicity for ultrafast charge recombination is decisive evidence in favor of nonequilibrium mechanism.
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INTRODUCTION Photoinduced electron transfer being important for fundamental science and technological applications1−7 attracts much attention of the scientific community over the past decades.8−11 Despite the fact that there is a huge amount of information on charge transfer kinetic regularities and many details of charge transfer mechanism are known, the control of ultrafast charge transfer kinetics and the product yields remains a challenge. For example, low efficiency of modern photovoltaic devices (dyesensitized solar cells) caused by ultrafast recombination of the charges immediately following primary photoinduced charge separation is one of such challenges.7 Studies of geminate charge recombination (CR) kinetics in donor−acceptor complexes (DAC) excited into a charge transfer band have shown for the first time that the role of the nonequilibrium of the nuclear subsystem in such reactions is important.12−14 The rate constant of CR in excited DACs, being monotonically decreasing function of the reaction exergonicity, does not show the Marcus normal region.15−17 The non-Marcus rate constant behavior can be accounted for if one accepts that excitation of a DAC by a short laser pulse initially creates a nonequilibrium population on the excited free energy surface far away from its minimum.12−14 It appeared that the CR rate constant dependence on free energy gap (FEG) calculated within the multichannel stochastic pointtransition (MCSPT) model including the reorganization of the solvent and several intramolecular high-frequency vibrational modes can be well fitted13 to the experimental data.15−17 © XXXX American Chemical Society
At the same time, the very linear dependence of the CR rate constant on FEG does not prove a nonequilibrium mechanism, because such a dependence can be reproduced within the framework of both thermal and nonequilibrium theories.12,13,18−21 Therefore, the question of the ultrafast CR mechanism in the excited DACs is still open. To differentiate between the thermal and nonequilibrium modes, additional regularities should be exploited. The regularities showing opposite dependencies for different modes are the best suited. Indeed, MCSPT model simulations of the ultrafast CR kinetics in excited DACs have shown that the kinetic regularities of the charge transfer reactions occurring in nonequilibrium and equilibrium regimes can be strongly different.14 In the present article such a regularity has been found and used for the first time as a strong evidence of the nonequilibrium mechanism of ultrafast CR. This regularity is associated with the dependence of the CR rate constant on dynamic properties of solvent, namely, on the solvent relaxation time. This dependence is known as the dynamic solvent effect (DSE).22 In this article DSE is quantified as the relative change in the reaction rate constant with a variation in the solvent relaxation time, provided that the solvent polarity variation is minor. DSE is well pronounced in the solvent controlled regime when the solvent dynamics limits the electron transfer rate. In this regime the electronic coupling is sufficiently large and the electron Received: March 17, 2017 Revised: April 14, 2017 Published: April 14, 2017 A
DOI: 10.1021/acs.jpcb.7b02537 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
the term UCS (see blue bell in Figure 1). The pump is assumed to occur close to the red edge of the charge transfer band, so
transfer rate weakly depends on it. In the opposite case of weak electronic coupling the nonadiabatic regime is realized and the rate constant is proportional to the square of the electronic coupling and is independent of the solvent relaxation time constants. For nonequilibrium CR the DSE was shown to be weak in the area of weak exergonicity and to be strong in the area of stronger exergonicity.14,23,24 For the thermal reactions the trend is opposite. Experimental confirmation of one of the trends would be strong evidence in favor of a certain mechanism of the CR in excited DACs. It should be emphasized that the exploration of such regularities is important not only for control of the CR in excited DACs but also for management of the CR in intramolecular photoinduced charge transfer reactions because there is a deep analogy between them and the last plays a paramount role in organic photovoltaic and other molecular devices.7,25 As a result, the significancy of the validation of the mechanism is greatly increased. Such experimental data on CR in excited DACs for a series of solvents of different viscosity was reported in refs 11, 17, 26. In the article the MCSPT model is exploited for simulation of nonequilibrium ultrafast CR kinetics in excited DACs. To indicate the interrelation of the MCSPT model with widely known Bixon−Jortner, Sumi−Marcus, and Barbara hybrid models,19,27−29 we point out that it is a generalization of the Bixon−Jortner and the hybrid models to account for reorganization of several high-frequency vibrational modes and to overcome the Golden Rule limit. The MCSPT model in contrast to the Sumi−Marcus model operates with several classical relaxation modes with arbitrary ratio of their relaxation time constants. The hybrid model19 was shown to qualitatively well describe the free energy, the solvent and the temperature dependence of the CR dynamics of excited DACs.17 However, it predicted the nonexponential decay of the excited state population that was not really observed.17 In the same time, the kinetics simulated with the MCSPT model involving reorganization of several high-frequency vibrational modes are close to exponential in full accord with the experiment.13,14 The main aims of the article are (i) to simulate the FEG law for the ultrafast CR kinetics of the excited DACs proceeding in the nonequilibrium regime, (ii) to simulate the influence of the solvent relaxation time scale on the FEG law, (iii) to fit the simulated FEG law for the CR in a single solvent, (iv) to compare the calculated FEG law with the experimental data for a series of the solvents with different viscosity by using no variable parameters and so to get strong evidence of the nonequilibrium mechanism realization in the excited DAC recombination, and as a result (v) to present strong arguments supporting implementation of the nonequilibrium mechanism in the ultrafast CR of excited DACs.
Figure 1. Multichannel nonequilibrium CR in DACs resulting in excitation of intramolecular high-frequency vibrations. The free energy curves corresponding to excited vibrational sublevels of the ground electronic state, U(n) Gr , are shown by dashed lines. The vertical blue arrow indicates photoexcitation and red arrow shows its red edge. The initial excited state distribution for nonequilibrium CR is pictured by a blue bell. The relaxation of the solvent and the high-frequency intramolecular modes is shown schematically by the blue solid and black wavy arrows, respectively. ΔGCR is the CR free energy change, Erm is the solvent reorganization energy.
that only the ground state of intramolecular high-frequency vibrational mode is supposed to be populated. Photoexcitation triggers a series of relaxation processes shown schematically in Figure 1. The motion of the wave packet toward the excited state term minimum visualizes the solvent relaxation (the blue solid arrow). Electronic transitions occur in parallel with the wave packet motion at each (n) intersection point of the terms UGr and UCS.38,39 Such transitions are called nonequilibrium CR (the red curved arrows). The nonequilibrium CR results in population of vibrationally excited sublevels of the ground electronic state (the black dashed parabolas). Figure 1 shows several vibrationally excited sublevels of the ground state in the case of single intramolecular vibrational DAC mode with the frequency Ω. Here ℏ is the Planck constant. In simulations the model with 5 modes is used and the density of vibrational sublevels quickly increases with increasing the free energy. Each vibrationally α) excited state relaxes with a rate constant, 1/τ(n vα (α = 1, ..., 5), according to single-quantum mechanism, nα →nα − 1 (the 40 (1) α) black wavy arrows). It is further assumed that τ(n vα = τvα /nα. The intramolecular vibrational redistribution is well-known to proceed on the time scale of τvα ∼ 100 fs.41 In simulations the value τ(1) vα = 150 fs is set for all high-frequency vibrational modes.42 The solvent reorganization dynamics play a key role in the kinetics of charge transfer reactions and are fully characterized by the solvent relaxation function, X(t).30 Direct information on the relaxation function of a solvent can be obtained from measurements of the dynamic Stokes shift.43−48 Some solvents are characterized by a single Debye relaxation time, whereas most of polar solvents are described by a sum of several Debye functions43−48
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THEORY AND COMPUTATIONAL DETAILS The MCSPT model is exploited in this study to simulate CR dynamics.13,23,24,30−37 The model was recently described in detail elsewhere14 so that we only briefly outline main ideas. The model involves two electronic states: the ground (Gr) and excited charge separated (CS) states. A short laser pulse, which duration is much less than the relaxation time of the solvent, transfers a DAC from the ground to the excited CS state. Immediately after the excitation, the nuclear subsystems of the DAC and surrounding medium are strongly nonequilibrium. Such an excited state with a nonequilibrium nuclear configuration can be visualized as a wave packet located on
N
X (t ) =
∑ xie−t /τ
i
i=1
B
(1) DOI: 10.1021/acs.jpcb.7b02537 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B ∂ρCS
where N is the number of the Debye relaxation modes, xi is the weight of ith relaxation mode. Each solvent relaxation mode with the reorganization energy Eri = xiErm, and relaxation time, τi, corresponds to the separate reaction coordinate Qi (i = 1, 2, ..., N).49 The simulations of the CR dynamics are performed for four solvents (valeronitrile (VaCN), acetonitrile (ACN), acetone (ACE), and ethyl acetate (EtAC)) at room temperature, kBT = 0.025 eV. Typically a solvent manifests two or more relaxation time scales. The shorter time, τ1, is connected with so-called inertial relaxation mode. The second time scale, τ2, is associated with diffusional rotation of the solvent molecules and it correlates with the solvent microscopic viscosity. Information on inertial relaxation is scant and often the data of different authors vary greatly. In the literature there is experimental data on influence of the solvent viscosity and, hence, τ2 on charge transfer rate but there is no such information on influence of τ1. This is the main reason why the effect of τ1 on charge transfer kinetics is not considered. In principle, the magnitude of the faster relaxation time scale, τ1, can considerably affect the ultrafast CR dynamics but only when the initial wave packet passes at least a part of the reaction zone at the stage of fast solvent relaxation. Such a situation is implemented if the CR occurs in low exergonic region. In this article this region is not considered and the effect of τ1 variation on the results obtained is minor. Here the weights xi and relaxation time constants τi are borrowed from the experimental data on solvent relaxation. The dynamic parameters of the solvents used are those reported in refs 50 and17 (see Table 1).
∂t
= L̂CSρCS −
ϵ050 ϵ∞50 τ2, ps
VaCN
EtAC
37.5 1.81 0.519
20.07 1.85 0.8319
19.7 19.5 4.748
6.0 1.88 2.6319
(2)
(n ⃗) (n ⃗) = L̂Gr ρGr − k n(⃗ Q 1 , Q 2)(ρGr − ρGr )
∂t
+
∑ α
1 ρ(n⃗ ′α ) (nα + 1) Gr τvα
−
∑ α
1 (n⃗) ρ (nα) Gr τvα
(3)
for the probability distribution functions in the space of solvent relaxation mode coordinates, Q1, Q2, for the electronic excited state, ρCS(Q1, Q2, t), and the ground neutral state, ρ(n⃗)Gr(Q1, Q2, t) is numerically solved. Here the vector n⃗ has 5 components (n1, ..., nα, ..., n5), nα= 0, 1, 2, ... is the number of vibrational quanta for αth high-frequency intramolecular vibrational mode. In the base of this model lies the concept of parabolic free energy surfaces constructed in the space of solvent polarization coordinates.49,52 Motion of particles along these surfaces reflects the reorganization of a solvent in the course of charge transfer and is determined by the solvent relaxation function ) (1). Diffusion on the free energy surfaces U(n⃗ Gr and UCS, 2 (n ⃗) UGr =
∑
2Eri )2
(Q i − 2
i=1 2
UCS =
∑ i=1
+
∑ nαℏΩα + ΔGCR , α
Q i2 2
(4)
are described by the Smoluchowski operators L̂ Gr and L̂ CS, ∂ ∂2 ⎞ 1⎛ ⎜⎜1 + Q i ⎟⎟ + ⟨Q i2⟩ τi ⎝ ∂Q i ∂Q i2 ⎠
2
L̂Gr =
∑ i=1
ACE
(n ⃗) k n(⃗ Q 1 , Q 2)(ρCS − ρGr )
n1, n2 ,..., nM
(n ⃗) ∂ρGr
Table 1. Optical, ϵ∞, and Stationary, ϵ0, Dielectric Constants and Diffusional Solvation Time, τ2, of the Solvents ACN
∑
2
L̂CS =
∑ i=1
1⎛ ⎜1 + (Q i − τi ⎜⎝
2Eri )
(5)
∂ ∂2 ⎞ ⎟⎟ + ⟨Q i2⟩ ∂Q i ∂Q i2 ⎠ (6)
⟨Qi2⟩
correspondingly. = kBT is the dispersion of the distribution along the ith reaction coordinate. Here kB is the Boltzmann constant, T is the temperature. In eqs 2 and 3 the ) parameters kn⃗ = 2πV2elFn⃗δ(U(n⃗ Gr − UCS) /ℏ describe the electron transitions between the ground neutral and the electronic excited states. Here Vel is the electronic coupling,
35
The CR rate was shown to weakly depend on the vibrational spectral density when the number of the highfrequency vibrational modes M ≥ 5. The number is fixed at M = 5. As universal spectral density we accept the high-frequency vibrational spectrum of the DAC consisting of phenylcyclopropane (PhCP) as the electron donor and tetracyanoethylene (TCNE) as the electron acceptor.35,51 The values of the spectral parameters (the Huang−Rhys factors Sα = Ervα/ ℏΩα, where Ervα and Ωα are the reorganization energy and frequency of αth high-frequency vibrational mode, respectively) of the complex PhCP-TCNE are given in Table 2. The total reorganization energy is Erv = ∑α5 = 1 Ervα. To simulate ultrafast nonequilibrium CR kinetics, a set of coupled Smoluchowski-type differential equations34
Fn⃗ =
α
α=1
2
3
4
5
0.1272 0.1245
0.1469 0.1211
0.1823 0.1143
0.1935 0.5150
0.1993 0.2302
Sαnαe−Sα nα!
is the Franck−Condon factor. The excited state is assumed to be formed by a short pump pulse. The pulse duration is supposed to be short enough so that the solvent is considered to be frozen at the excitation stage. In this extreme case, the excitation can be visualized as a partial vertical transfer of the distribution of the thermal population from the ground to the excited term (blue bell in Figure 1). This distribution is regarded as the initial condition for eqs 2 and 3. The Brownian simulation method37,53 is used for numerical solution of eqs 2 and 3. The detailed description of the code was reported in ref 54. The focus of the study is the influence of the solvent relaxation time scale on the free energy dependence of the effective CR rate constant, kCR(Δ GCR). The effective CR rate constant is determined as
Table 2. Parameters of High-Frequency Intramolecular Vibrational Modes for CR in Excited DAC Consisting of PhCP and TCNE51 ℏΩα, eV Sα
∏
C
DOI: 10.1021/acs.jpcb.7b02537 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B −1 k CR =
∫0
∞
PCS(t )dt
benzene derivatives (MoBs) as donors in VaCN solution.17 The term free energy gap is used to denote the absolute value of the net exergonicity of CR and differs from the term vertical free energy gap by the amount of the solvent reorganization energy. The same dependence for the above-mentioned DACs in ACN, ACE, and EtAC are calculated without any variable parameters. The time constant of the slow diffusion solvent relaxation mode, τ2, is the only changeable quantity. Its values are borrowed from independent experiment data for the solvents considered.17 Results of simulations are presented with red lines in Figures 2 and 3. The values of the parameters obtained
(7)
where PCS(t ) =
∫ ρCR (Q 1, Q 2 , t )dQ 1dQ 2
(8)
is the normalized population of the CS state of a DAC. In the simulations the values of dynamical solvent parameters, τi, xi (i = 1, 2), the vibrational spectral parameters, Sα, Ωα (Table 2), and the total reorganization energy of highfrequency vibrational modes, Erv, are fixed, the net solvent reorganization energy, Erm, and the electronic coupling, Vel, are considered to be variable.
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RESULTS AND DISCUSSION Analysis of results of MCSPT model simulations of the ultrafast nonequilibrium CR kinetics in excited DACs has discovered several trends. It appears that the trends inherent to charge transfer reactions occurring in the nonequilibrium and equilibrium regimes may strongly differ.14 The most important difference is DSE is strong in the area of weak exergonicity and it is weak in the area of strong exergonicity for the thermal reactions23 while for the nonequilibrium reactions the regions of strong and weak DSE are in the opposite areas.14 To get an idea of the origin of the trends, we note that the DSE arises when the reaction rate is limited by the delivery of the particles to the reaction zone (term crossing points).38,39 According to the model, the pumping pulse instantly transfers an electron from the donor to the acceptor while the solvent keeping its state appears in a nonequilibrium state relative to the new charge distribution in the DAC. This solvent nonequilibrium state is visualized as a wave packet placed far away from the minimum of excited state term (see blue bell in Figure 1). Further the reaction and the solvent relaxation proceed in parallel. When the CR time scale is shorter than the solvent relaxation time the reaction has the single nonequilibrium stage. In opposite case when the CR time scale is longer than the solvent relaxation the reaction includes both nonequilibrium and thermal stages. For the thermal reactions the delivery of the particles is needed when the reaction occurs in the Marcus normal region whereas in the barrierless and inverted regions it proceeds through the sinks placed in the vicinity of the reactant term minimum and the particles can react without significant diffusional movement. For nonthermal reactions the wave packet is initially placed far from the bottom of the excited state surface (blue wave packet in Figure 1). For weakly exergonic reactions the nonequilibrium wave packet is initially localized in the region of strong sinks and no delivery required, whereas in the area of moderate exergonicity the wave packet initially appears in the region of weak sinks and delivery of particles to the reaction zone is required. It should be noted that in the region of very strong exergonicity the reaction becomes slow and proceeds in the thermal nonadiabatic regime. In this limit the DSE is vanished for both reactions with initial thermal and nonequilibrium distribution of the particles. To avoid misunderstandings, we would like to stress that these conclusions are obtained for the systems with typical values of reorganization energy of the high-frequency vibrational modes. The fitting of simulated results to experimental dependence of the CR rate constant, kCR, on the free energy gap is accomplished for CR in DACs composed of pyromellitic dianhydride (PMDA) as acceptor and methoxy substituted
Figure 2. Free energy dependence of the CR rate constant, kCR. The experimental data for the excited complex MoBs−PMDA borrowed from ref 17 are pictured by the symbols. In numerical simulations the high-frequency vibrational spectrum involving 5 modes is employed (see Table 2). The simulation data for the different solvents are shown by solid (ACN) and dashed (VaCN) lines. Parameters used: Vel = 0.12 eV, τ1 = 0.19 ps, x1=x2 = 0.5, Erv = 0.51 eV, τ(1) vα = 0.15 ps. The values of the solvent reorganization energy, Erm in eV, are shown near the lines: 0.5 (red lines), 0.57 (black line).
in the fitting are Vel = 0.12 eV and Erm = 0.5 eV. These values are rather typical for CR in excited DACs.17 At first sight the
Figure 3. Free energy dependence of the CR rate constant, kCR. The experimental data for the excited complex MoBs−PMDA borrowed from ref 17 are pictured by the symbols. In numerical simulations the high-frequency vibrational spectrum involving 5 modes is employed (see Table 2). The simulation data for the different solvents are shown by solid (ACE) and dashed (EtAC) lines. Parameters used: Vel = 0.12 eV, τ1 = 0.19 ps, x1=x2 = 0.5, Erv = 0.51 eV, τ(1) vα = 0.15 ps. The values of the solvent reorganization energy, Erm in eV, are shown near the lines: 0.5 (solid and dashed red lines), 0.53 (black line), 0.4 (solid and dashed blue lines). D
DOI: 10.1021/acs.jpcb.7b02537 J. Phys. Chem. B XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry B value of Vel is out of the stochastic model region of applicability (Vel < kBT).55 However, the reorganization of high-frequency vibrational modes alters the estimation and the limits of the MCSPT model applicability can be estimated as,55 Veff Erm.8 The simulation results are also in agreement with the experimental data on the CR dynamics in excited DACs composed of PMDA as acceptor and of methylbenzene derivatives (MeBs) or larger polyaromatic hydrocarbons (Ph) as donors in several solvents investigated by Mataga and coworkers.26 The groups of DACs explored in ref 26 (MeBs− PMDA, Ph−PMDA) and ref 17 (MoBs−PMDA) cover approximately the same FEG region. However, the CR kinetics in excited DACs composed of MeBs−PMDA and Ph−PMDA are substantially slower than that of MoBs−PMDA complexes and demonstrate weak influence of the solvent relaxation time scale on CR rate constant.26 CR in MeBs−PMDA and Ph− PMDA complexes occurs on time scale noticeably exceeding the solvent relaxation time that means a larger part of the CR to proceed after solvent relaxation, that is, in the equilibrium mode. This result is in full accord with the simulations that predict the DSE for thermal charge transfer in the region of large exergonicity, −ΔGCR > Erm, to be small.23 Thus, the CR in excited DACs can proceed in both the thermal and nonequilibrium modes. The nonequilibrium mechanism is realized if the CR time scale is shorter than the solvent relaxation time.
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CONCLUSIONS In the article both solvent and free energy experimental dependencies of the CR rate constant were well reproduced within the MCSPT model with nonequilibrium initial state of the nuclear subsystem produced by a laser pump pulse. Although, both trends are indicative for nonequilibrium E
DOI: 10.1021/acs.jpcb.7b02537 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
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In this article the term ultrafast CR refers to that reaction. In the opposite case CR mainly proceeds in the thermal regime.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Anatoly I. Ivanov: 0000-0002-4420-5863 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The study was performed by a grant from the Russian Science Foundation (Grant No. 16-13-10122). REFERENCES
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