Electron Transfer

Besides the above discussed photophysical processes the excited state can lose the energy of excitation through photochemical reactions.

Photoinduced electron transfer represents simplest photochemical reactions, however at the same time it is viewed as one of the most attractive methods to convert light energy or to store it for further applications.¹²

Photoinduced electron transfer (ET) involves the transfer of an electron within an electron donor-acceptor pair, and was described by equations 2.12 and 2.13. Here the processes taking place between two units after excitation, where the excited unit can donate or accept an electron, resulting in the population of a charge separated state (CSS). Afterwards, it usually followed by spontaneous back-electron transfer reactions (electron-hole recombination), and regenerate the starting ground state:

     𝐴^!−𝐵^!→𝐴−𝐵       back oxidative electron transfer (2.25)

     𝐴^!−𝐵^! →𝐴−𝐵       back reductive electron transfer (2.26)

The difference between a photoreaction and a regular thermal reaction that in the last one the system never leaves the potential surface of the ground state, while in a photoreaction, there is a crossover from a excited state potential energy surface to that of the ground state where every reaction eventually ends. The new redistribution of electrons associated with electronic excitation of a molecule effects its physical properties, in particular makes the excited species both stronger oxidants and reductants.

The photoinduced charge separation (CS) occurs when it is thermodynamically favorable, i.e. exergonic. The CSS energy (𝐸!"") as well as the free energy of photoinduced CS (∆𝐺_!") can be estimated from the oxidation potential of the donor 𝐸_!" 𝐴 and the reduction potential of the acceptor 𝐸_!"# 𝐵 using the Weller equation:¹³

     ∆𝐺_!"=𝐸!""−𝐸^∗=𝑒 𝐸_!" 𝐴 −𝐸_!"# 𝐵 +𝐶+𝑆−𝐸^∗       (2.27)  

where 𝐸^∗ is the energy of the excited state from where CS takes place, 𝐶=−𝑒^!/(4𝜋𝜀_!𝜀_!𝑑_!")  is a correction term that represents the Coulombic

interaction between the charged units at a distance 𝑑_!" and screened by a solvent of dielectric constant 𝜀_!. Finally, S is factor accounting for a dielectric constant of the solvent different from that used for the determination of the redox potentials, 𝜀_!^!:

     𝑆=− 𝑒^! 8𝜋𝜀_!

1 𝑟_!+ 1

𝑟_! 1 𝜀_!+ 1

𝜀_!^!       (2.28)   where 𝑟_! and 𝑟_!are the radii of the reacting units.

From a kinetic viewpoint, the photoinduced electron transfer was described by Marcus.^14-15 In this theory, the CS is treated by transition state theory where the reactant state is the excited A^*-B and the product state is the CSS (Figure 2.6). Here the states of the system are represented by identical parabolic functions that are displaced vertically and horizontally along the reaction coordinate. The vertical displacement corresponds to the driving force of the reaction (−∆𝐺_!"), and can be calculated using

2.3 Photoinduced Energy and Electron Transfer Processes

equation 2.27. The horizontal displacement represents the difference between equilibrium geometries of the two states, and described by the nuclear reorganizational energy (𝜆). This energy separated into two independent contributions corresponding to the reorganization of the inner (𝜆_!) shell, which characterizes the changes in bond length and angles, and the outer reorganization energy (𝜆_!), which refers to the reorientation of the surrounding solvent molecules:

     𝜆=𝜆_!+𝜆_!       (2.29)  

Figure 2.6 Schematic representation of free-energy curves for the ground state (AB), the excited state (A^*-B, reactant state), and the charge-separated state (A⁺-B^-, product state) according to the classical Marcus theory.

The outer reorganizational energy is solvent dependent, and can be estimated using the Born-Hush approach:

     𝜆_!= 𝑒^! 4𝜋𝜀_!

1 𝑛^!−1

𝜀_! 1 2𝑟_!+ 1

2𝑟_!− 1

𝑟_!"       (2.30) where e is the electronic charge, 𝑛 is the refractive index of the solvent, 𝜀_! is the permittivity of the solvent, 𝑟_! and 𝑟_! are the radii of the reactants, and 𝑟_!"

is the centre to centre distance between them. An important consequence deriving from equation 2.30 is that 𝜆_! is large when the reaction occurs in a polar solvent between reactants separated by a long distance.

The inner reorganization energy can be estimated within the harmonic-oscillator approximation, and is given in terms of its force constants 𝑓_! and the change in equilibrium positions Δ𝑥_! between the reactants and the products:

     𝜆_!= 𝑓_!(𝑅)∙𝑓_!(𝑃)

𝑓_! 𝑅 +𝑓_!(𝑃)  (Δ𝑥_!)^!       (2.31)

where 𝑓_! 𝑅  is the force constant for the reactants and 𝑓_!(𝑃) is that for the products.

Within this theory, knowing the above discussed parameters for

∆𝐺_!" and 𝜆, the activation barrier (∆𝐺_!"^! ) for the thermal electron transfer

can be estimated via quadratic relationship:

     ∆𝐺_!"^! =(∆𝐺_!"+𝜆)^!

4𝜆       (2.32) When ET is a thermally activated process with known barrier and weak electronic coupling between reactants, the rate of charge separation

(𝑘_!") can be expressed through the Arrhenius equation:¹⁶

     𝑘_!"=𝐴𝑒𝑥𝑝 −∆𝐺_!"^!

𝑘_!𝑇 =𝐴𝑒𝑥𝑝 − ∆𝐺_!"+𝜆 ^!

4𝜆𝑘_!𝑇       (2.33)

where 𝑘_! is the Boltzmann constant and 𝑇 is the temperature.

Equation 2.33 predicts that the dependence of 𝑘_!" on ∆𝐺_!" is Gaussian for constant 𝜆, and involves three regimes (Figure 2.7):

• an activationless (or optimal) regime (−𝜆≈∆𝐺_!"), when the electron transfer rate reaches its maximum value;

• a normal regime (−𝜆<∆𝐺_!"), when the electron transfer rate increases with larger driving forces;

• an ‘‘inverted’’ regime (−𝜆>∆𝐺_!"), when the electron transfer rate decreases with increasing driving force

2.3 Photoinduced Energy and Electron Transfer Processes

Figure 2.7 (top) The three regimes of electron transfer according to the Marcus theory; (bottom) the corresponding reaction rate dependence as a function of the.

The classical Marcus model works well in the normal region but the electron transfer rates in the inverted region were found to be higher than predicted. The situation in the inverted region is better described by semi-classical Marcus theories based on a quantum-mechanical description of electron transfer, a detailed description of which can be found, for example, in ref 16. In a few words, from a quantum mechanical viewpoint, the electron transfer processes can be viewed as radiationless transitions between two electronic states, and thus its rate constant can be expressed through Fermi's golden rule (equation 2.15) considering the density of states at the intersection point of the reactant and the product energy surface (Figure 2.8, left). Taking higher-energetic molecular vibrations into account opens additional electron transfer channels that can occur in parallel, thus resulting in a greater rate of charge separation in the inverted region. Consequently, the quantum mechanical model predicts a more linear decrease of the rate constant with increasing driving force in the inverted region then the parabolic one found with the classical Marcus theory (Figure 2.8, right).

Figure 2.8 (left) Free energy surfaces including the high-frequency intramolecular vibrational modes of the product state (A⁺-B^-); (right) schematic representation of the Marcus according to the classical and semi-classical Marcus theories.

Chapter 3

Experimental Part

3.1 Nonlinear Optics

The aim of this chapter is to introduce the most used spectroscopic techniques during this thesis for investigating excited-state dynamic processes. Firstly, lets briefly discuss some basic concepts of nonlinear optics in order to better understand the working principles of the setups.

During a long time all optical media could be explained as linear, up to 1960 when the first invented laser¹⁷ allowed to examine the light behavior in optical materials at high intensities. After the first laser, optical media were shown not to follow linear behavior at high intensities and to exhibit changes of refractive index and frequency of light by going through nonlinear optical material, breakdown of the superposition principle, and ability to interact and impact of one beam on another.¹⁸

The interaction between matter and light induces a macroscopic polarization, P.¹⁹ Thus, as long as the applied electric field is weak, the dependence between the polarization and the electric field, E, is linear:

     𝑃=𝜀_!𝜒𝐸       (3.1) where 𝜒 is the proportionality constant, called the electrical susceptibility of the medium, 𝜀_! is the permittivity of free space. However, when the electric field is very intense, this dependence is not liner anymore and the polarization is expressed as a sum of linear, PL, and nonlinear, PNL, part of it:

     𝑃=𝑃_!+𝑃_!"=𝜀_! 𝜒 ^! ∙𝐸+𝜒^! ∙𝐸^!+𝜒 ^!𝐸^!+. . .       (3.2)

where 𝜒 ^! ,  𝜒^! ,… are the second-, third- and higher-order nonlinear electric susceptibilities, respectively, that account for nonlinear effects.

3.1.1 Second-order Nonlinear Optics

Let us now consider only the second-order term in the equation 3.2 as the response to two electric fields oscillating at angular frequency ω1 and ω2:

     𝑃^! 𝑡 =𝜀_!𝜒^! 𝐸^! 𝑡       (3.3)

The electric field can be described as:

     𝐸 𝑡 =𝐸_!𝑒^!!!!!+𝐸_!𝑒^!!!!!+𝑐.𝑐       (3.4)

Then the nonlinear polarization is given by:²⁰

𝑃^! 𝑡 =𝜀_!𝜒^! 𝐸_!^!𝑒^!!!!!!+𝐸_!^!𝑒^!!!!!!+2𝐸_!𝐸_!𝑒^!!(!!!!!)!+

     2𝐸_!𝐸_!^∗𝑒^!!(!!!!!)!+𝑐.𝑐 +𝜀_!𝜒 ^! 𝐸_!𝐸_!^∗+𝐸_!𝐸_!^∗      (3.5)

The resulting polarization can oscillate at the frequency, ω3, equal to the double frequencies, 2ω1 or 2ω2, as well as oscillating at the sum frequency, ω1+ω2, or the difference frequency |ω1-ω2| (Figure 3.1).¹⁹ In practice, all these cases are not present at the same time. In order to create one or another of these new frequencies, the linear momentum conservation conditions called phase matching conditions have to be fulfilled. Due to it, the new field intensity is the greatest when:²¹

     ∆𝑘=𝑘_!"#− 𝑘_!" =0      (3.6) where ∆k is the wavevector (or phase) mismatch, where k_in and k_out are the

3.1 Nonlinear Optics

wavevectors of the incoming and of the new outgoing field, respectively. In fact, phase matching conditions is the representation of momentum conservation (p=ħk).^{19, 21}

Figure 3.1 Energy level representation of second-harmonic generation (SHG), sum-frequency generation (SFG) and difference-frequency generation (DFG).

The effect when ω3 = 2ω1 (or 2ω2) is called second-harmonic generation (SHG). It is valid when the k_ω3=  2kω₁ phase matching condition is fulfilled. SHG is commonly used to generate light at the double frequency of the applied one, and is widely used in the experiments presented below to convert the output laser light, which is usually at 800 nm, to 400 nm.

The phenomenon with ω3 = ω1 + ω2 is called sum frequency generation (SFG), and is used during the thesis to convert a weak spontaneous fluorescence signal by mixing it with the laser beam in the up-conversion technique (section 3.3.2). While the one at ω3 = ω1 - ω2 is called difference frequency generation (DFG), and underlies two other important effects, as optical parametric generation (OPG) and amplification (OPA). In OPG an incident laser pulse, called pump pulse at ω1, is divided into two lower-frequency pulses that are called idler, ω2, and signal, ω3. OPA is used to amplify the signal. In this process, a high-intensity beam transfers its energy to a low-intensity signal beam and generates a third idler beam. The non-collinear geometry of OPA (NOPA) is used in this work to generate pulses between 380 and 700 nm. The required phase match condition for SFG is defined as k_ω3=  kω₁+  kω₂, and for DFG as k_ω3=  kω₁−k_ω2.

It is important to mention here that second-order (or even order) optical properties occur only in non-centrosymmetric materials, as the

second order nonlinear susceptibility of centrosymmetric materials is zero.¹⁹ Centrosymmetric materials in turn exhibit the third-order nonlinear optical susceptibility as the lowest nonlinear order response.

3.1.2 Third-order Nonlinear Optics

Third-order nonlinear processes can occur in all materials. In general, these effects are much weaker than second-order analogues and thus require high intensity electric fields. Such increase of the intensity effect the refractive index by making it intensity-dependent:²²

     𝑛=𝑛_!+𝑛_!𝐼 𝑡       (3.7)

where 𝐼 𝑡 is the intensity of laser radiation, n0 is the linear refractive index, and 𝑛_! is the nonlinear refractive index related to the third-order nonlinear susceptibility, 𝜒^! :²²

     𝑛_!= 2𝜋

𝑛_!

!

𝜒^!       (3.8)

Such a change of the refractive index in response to an applied electric field is known as optical Kerr effect (OKE). It is the underlying principle of such important effects such as self-phase and cross-phase modulation.

Self-phase modulation causes a time-dependent phase shift in the pulse that leads to spectral broadening. It comes from the time dependence of the intensity of electric field and of the refractive index (equation 3.8). As a consequence, it causes a nonlinear or time-dependent phase, 𝜙 𝑡 , in the material:²²

     𝜙 𝑡 =𝜔

𝑐 𝑛_!𝐼 𝑡 𝐿      (3.9) where 𝐿 is the pathlength, and so it gives rise to a time dependent frequency of the pulse:²²

3.1 Nonlinear Optics

     ∆𝜔 𝑡 =𝜕𝜙 𝑡

𝜕𝑡 =𝜔 𝑐 𝑛_!𝐿𝐼_!

𝜏      (3.10) where ∆𝜔 the resulting spectral broadening of the pulse, 𝐼_! is the peak intensity of the optical pulse, and τ is the duration of the pulse. We used self-phase modulation in this work to generate a white light continuum in the transient absorption experiments by focusing the output of a standard Ti:sapphire amplified laser system into a CaF2 window, or a sapphire plate in NOPA.

Another similar effect is cross-phase modulation. The difference is that in this case the phase shift caused by interaction of two laser pulses, where the phase of a weak pulses is affected by another intense pulse due to the change in the refractive index of the medium. As a result, it gives rise to a phase shift of the weak pulse.

3.1.3 Optical Pulses

In order to monitor ultrafast events, an experiment needs time resolution, i.e. the optical pulses should be short enough to follow the processes under investigation. Thus time resolution in ultrafast spectroscopy comes from such short optical laser pulses. They are usually characterized by the duration, 𝜏, and its spectral width, ∆𝜈, and for a Gaussian pulse is expressed as:²³

     ∆𝜈𝜏≥2ln2/𝜋      (3.11)

where ∆𝜈 and 𝜏 are the full width at half maximum of the laser-pulse spectrum and time, respectively. Consequently, equation 3.11 shows inverse proportionality between time and frequency. This is an important characteristic that allows to establish the right spectral width required to get a desired pulse duration. For example a short pulse that has a pulse duration of 100 fs has a frequency bandwidth of at least 4 × 41¹² Hz (133 cm^-1).

3.2 Steady-State Spectroscopy

Absorption spectra were measured on a Cary 50 (Varian) spectrometer, whereas fluorescence emission and excitation spectra were recorded on a Cary Eclipse (Varian) or a FluoroMax-4 (Horiba Scientific) fluorometer. The latter setup has the advantages of a higher sensitivity, a larger dynamic range and an extended detection into the near-infrared region.

All emission spectra were corrected for the wavelength-dependent sensitivity of the detector. The steady-state measurements were performed in 1 cm thick quartz cells.

The fluorescence quantum yields, Ф_!", were calculated as:⁸

     Ф_!"= 𝐹(𝜆) 𝐹_!"#(𝜆)

𝑛^! 𝑛_!"#^!

1−10^!!"#

1−10^! Ф!",!"#      (3.12)

where Ф!",!"#is the fluorescence quantum yield of a reference compound;

𝐹(𝜆) and 𝐹_!"#(𝜆) are the integrated intensities of sample and reference spectra, respectively; 𝑛 and 𝑛_!"# the refractive index of the sample and reference solution, respectively; 𝐴 the absorbance at the excitation wavelength.

3.3 Time-Resolved Fluorescence

One of the ways for an excited molecule to dissipate its energy is fluorescence. Many processes, like radiative and non-radiative relaxation, intersystem crossing, excitation energy and charge transfer, etc. can influence the excited-state dynamic of the molecule, hence its fluorescence intensity and dynamics. In time-resolved fluorescence measurements, the molecules are promoted to the excited state by using a laser pulse and the resulting fluorescence spectrum is monitored as a function of time. In this work, fluorescence dynamics were measured on sub-nanosecond and femtosecond timescales using the time correlated single photon counting (TCSPC) and fluorescence up-conversion techniques, respectively.

3.3 Time-Resolved Fluorescence

3.3.1 Time-Correlated Single Photon Counting

The TCSPC technique is one of the most widely used methods to measure fluorescence dynamics on the nanosecond timescale.^24-25 In this technique, the time between the excitation pulse and the detected photon is measured electronically. Sometimes it is compared to a fast stopwatch, where the clock is started by the Start signal pulse and stopped by the Stop signal pulse. In a typical TCSPC set-up, the Start pulse is generated from a small part of the excitation light and forwarded to a photodiode to produce the trigger pulse, which starts to charge a capacitor. An emitted photon from the sample is directed to the detector (photomultiplier or micro channel plate), and stops the charging of the capacitor. The time measured for a Start-Stop cycle represents the arrival time of the photon. The measurement is repeated many times and stored in a histogram, where x-axis represents the time difference and y-axis the number of the detected photons over the measured time difference. TCSPC is a statistical method and is based on the detection of single photons to represent the fluorescence decay of the sample.

Setup. The TCSPC setup has been described in detail previously.^26-27 The sample was excited at 395 nm (PicoQuant model LDH-P-C-400B) or 470 nm (PicoQuant model LDH-P-C-470) with ~60 ps excitation pulses. The average power was about 0.5 mW at the repetition rate in the range of 10-40 MHz. Fluorescence was collected at 90^o, and passed through an analyzer set at the magic angle with respect to the excitation polarization, and a narrow bandpass filter located in front of a photomultiplier tube (Hamamatsu, H5783- P-01). The detector output was connected to the input of a TCSPC computer board module (Becker and Hickl, SPC-300- 12). The full width at half maximum (FWHM) of the instrument response function (IRF) was ~200 ps. The IRF was determined from the signal scattered from a solution of LUDOX at the sample position.

All measurements were performed in 1 cm quartz cells. The accuracy on the lifetimes is of ca. 0.1 ns. The obtained fluorescence time profiles were analyzed with the convolution of the measured IRF and mono- or multiexponential functions with MATLAB (The Mathworks Inc.) using a script written by Dr. Arnulf Rosspeintner.

3.3.2 Fluorescence Up-conversion

Faster dynamics up to 2 ns were investigated by fluorescence up-conversion. Similarly to the previous setup, the initial excitation light is split in two parts, where the first one is used to excite the sample and another one acts as the probe pulses, called gate. The excitation beam is focused on the sample, and the spontaneous fluorescence emitted by the sample is mixed with an intense gate pulse in a nonlinear crystal to generate the sum (up-conversion) frequency (section 3.1.1). The result light intensity is measured by a detection system. The gate pulse is passed to the optical delay line before reaching the nonlinear crystal. Thus, the time evolution of the fluorescence can be measured by varying the delay of the gate pulse. The up-conversion process is efficient only when the phase matching condition is satisfied, which requires a different angle between the crystal optical axis and the laser pulses at each fluorescence wavelength. Therefore time traces are recorded at each emission wavelengths.

Setup. In the setup used in this work, excitation was performed between 350 and 520 nm produced by frequency doubling the output of a Ti:sapphire oscillator (Spectra-Physics, Mai Tai). The pulse had a duration of

~100 ps and a repetition rate of 82 MHz. The polarization of the pump pulses was at the magic angle relative to that of the gate pulses. The UV signal produced as a result of sum-frequency mixing of the fluorescence and the gate was detected by a photomultiplier tube operating in the photon counting mode. The pump intensity on the sample was around 5 µJ/cm², and the FWHM of the IRF was ca. 200 fs. The sample solutions were located in a 0.4 mm rotating cell and at an absorbance between 0.1 and 0.4 at the excitation wavelength.

The experimental data were recorded several times and averaged to improve the signal-to-noise ratio. Subsequently, the fluorescence time profiles were analyzed with a trial function representing the convolution of a Gaussian function (𝐺), i.e. the IRF, with an exponential function (𝐸):

3.4 Transient Absorption

𝐼   𝜆,𝑡 = 𝐼_!⨂!,! 𝜆,𝑡

!

!!!

=𝐴_! 𝜆 + 𝑎_!(𝜆)

2 𝑒𝑥𝑝 𝑡−𝑡_!(𝜆) 𝜏_!

!

!!!

∙

     ∙𝑒𝑥𝑝 𝜔(𝜆)^!

4𝜏_!^! 1+𝑒𝑟𝑓 𝑡−𝑡_!(𝜆)−𝜔(𝜆)^!/2𝜏_!

𝜔(𝜆)      (3.13)

where 𝐼 is the measured fluorescence intensity, 𝜏_! and 𝑎_! are time constants and their amplitude, respectively, 𝑡_! is the position and 𝜔 the width of the Gaussian, 𝐴_! is the offset accounting for a constant background. The data analysis was performed with MATLAB (The Mathworks Inc.) using a script written by Dr. Romain Letrun.

3.4 Transient Absorption

The other pump-probe technique, which has been mainly used during this thesis, is transient absorption (TA).^28-29 Contrary to the time-resolved fluorescence techniques, it is based on monitoring changes in the absorbance in the sample that allows monitoring non-emissive and dark states. The main idea is that different intermediate states of the photoinduced processes exhibit different absorption bands.³⁰

3.4.1 Principles

In TA spectroscopy, a first laser pulse, called pump pulse, perturbs the sample promoting the molecules to an electronically excited state (Figure 3.2). A second weaker pulse, called probe pulse, is sent through the sample to measure an effect of the pump pulse. A low intensity probe pulse is used to avoid any influence on the sample being investigated. A difference absorption (∆𝐴) spectrum is then calculated as the comparison of the intensities transmitted through the excited (𝐼^∗) and non-excited (𝐼_!) sample:

In TA spectroscopy, a first laser pulse, called pump pulse, perturbs the sample promoting the molecules to an electronically excited state (Figure 3.2). A second weaker pulse, called probe pulse, is sent through the sample to measure an effect of the pump pulse. A low intensity probe pulse is used to avoid any influence on the sample being investigated. A difference absorption (∆𝐴) spectrum is then calculated as the comparison of the intensities transmitted through the excited (𝐼^∗) and non-excited (𝐼_!) sample:

Dans le document Excited-state dynamics in electron-donor-acceptor systems of increasing complexity (Page 26-0)