1.3. Non-Equilibrium Excited-State Dynamics
1.3.2. Environmental Relaxation
After optical excitation of a solute, not only does the solute get rid of the excess energy, but also its environment, usually a solvent, will also un-dergo a relaxation process. The most important solvent effect is however often equilibrium in nature: modifying the free energies of the reactants, products, and transition states of a reaction, thereby affecting the activa-tion free energy and sometimes even the course of the chemical process.
As the solvent may affect the energy levels of a solute molecule, the ab-sorption and emission spectra of the latter may depend on the solvent.
This change in the electronic properties — and thus sometimes in the colour of the solution — with the solvent is referred to as )%-8?$%&'*%# -()# and the relationship between solvent properties and spectral parame-ters has been the subject of intense investigation [44].
The solute-solvent interactions have, however, also a dynamic aspect.
The interaction energy actually fluctuates about some average value as a function of time, and the time-dependence of theses fluctuations is closely connected to the frictional forces that mediate a variety of dynamical processes occurring in a solvent environment [37]. With advances in ex-perimental techniques that made it possible to access the femtosecond time domain, the dynamics of solvation itself has become within reach and therefore an interesting and necessary subject of study [37, 45] since other molecular processes that take place on the same time scale as solva-tion, such as charge rearrangement and electron transfer processes, can now also be probed. Solvation should thus be addressed as a both dy-namic and energetic phenomenon. Although its dydy-namics is far from being fully understood on a molecular level, a phenomenological consen-sus view on solvation dynamics involving charged and polar solutes in
dielectric environments (polar solvation dynamics) is given in this sec-tion.6
Be a dipolar solute molecule, embedded, at equilibrium, in a dielec-tric solvent, which undergoes a sudden change of its charge distribution due to photoexcitation. The dipolar solvent molecules find themselves in a high energy configuration and respond to this change in local charge distribution by reorienting and diffusing so as to minimise the interaction energy with the solute. Figure 1.5 describes this process under the as-sumption that the charge redistribution within the solute is fast relative to the time scale of solvent motion.7 Given are snapshots of the system con-figuration just before (a) and just after (b) photoexcitation of the solute, as well as of the equilibrated excited state (c). Under the assumption that the
6 More quantitative physical models describing the dynamics of solvation have been pro-posed. See [37, 39, 45] and references therein.
7 The actual situation is more complicated since the solvent displays not only a slow com-ponent associated with nuclear motion, but also a fast electronic comcom-ponent the character-istic time scale of which is fast or comparable to that of electronic transitions in the solute.
This fast component is however hardly experimentally accessible.
Figure 1.5. Schematic effect of the time-dependent Stokes shift. Right after excita-tion, the solvent is in a non-equilibrium configuration (:(0)). As the solvent starts reorganising, the excited-state population, :($), evolves along the solvation coor-dinate and the fluorescence spectrum shifts to the red until the equilibrium is reached (:(B)).
solvent is slow compared with photoexcitation, its configuration is the same in (a) and (b). The solute is then solvated during the transition from (b) to (c). The energy difference between the configurations (a) and (b) is referred to as 8"*$(&?-, whereas the energy difference between configura-tions (a) and (c) is denoted ?=(?,?$(&.
The dynamics of the solvation process can be followed experimentally by several techniques [46]. One obvious experimental manifestation is an evolving red shift of the emission spectrum of the excited solute. Time-resolved fluorescence spectroscopy appears therefore especially well-suited as it allows the evolution of the excited-state energy to be moni-tored. This type of experiment is often referred to as time-dependent Stokes shift (TDSS) measurement. Alternatively, photon-echo techniques can also address this question.8
In time-resolved fluorescence spectroscopy, the temporal features of solvation are often captured in the normalised solvent response function
# $
the barycentre of the emission spectrum at time $. If the solvation dynam-ics is fast with respect to the excited-state lifetime of the solute, )# $
B can be estimated from the steady-state fluorescence spectrum. Although sys-tematic methods have been proposed [48], it is very difficult to accurately estimate the position of the emission spectrum directly after excitation (at time 0). This suggests that the error on the normalisation factor in the denominator of equation (1.57) is large, as will be the uncertainty of the amplitudes of the components describing 1 $# $
. The absolute time evolu-tion of the spectral parameter, )# $
$ , should therefore be preferred to describe the TDSS:
8 Photon-echo measurements monitor equilibrium solvent fluctuations while TDSS meas-urements follow how a system driven out of equilibrium by optical excitation returns to equilibrium. It has still not yet been demonstrated that photon echo and TDSS measuments lead to the same result on a given system although they should within linear re-sponse theory.
# $ # $
1abs $ ") $ . (1.58)
The time dependence of this function should be the same as that of its normalised analogue, only the amplitudes of the decay components dif-fer.
The solvent response function is often equated to the solvation time correlation function E $
# $
within linear response theory [37]:9# $ # $ # $ # $ # $ # $
Equation (1.59) states that the response of the system when driven by the solute perturbation mirrors the dynamics of the energy gap I "2 ') ex-perienced by an unperturbed solute in equilibrium with its solvent envi-ronment [37]. It allows the connection of the solvent response to the sol-vent fluctuations L2
# $
0 L2 $# $
which cause the solvation dynamics.A wide range of different intermolecular interactions contribute to sol-vation. They encompass dispersion and other induced-dipole interac-tions, dipole-dipole or higher-order multipolar interacinterac-tions, and specific interactions such as hydrogen bonding [37]. Despite this, it seems that the solvation dynamics in pure solvents depends only to a minor extent on the nature of the probe and mostly on the solvent, although no large-scale systematic study has been conducted so far. Because of its biological rele-vance, many simulations and experiments have been performed with organic fluorophores in water. Figure 1.6 represents the decay of 1 $
# $
observed by Maroncelli and co-workers with coumarin 343 (C343) in water [50]. In agreement with simulations [51], it shows that the solvation dynamics in water consists mainly of two parts. The first part is ultrafast, with time constants in the 50–150 fs range, and represents more than half
9 The reason why solvation experiments have mostly been discussed in the framework of linear response theory is actually not that linear response has been proven to be valid for solute-solvent systems, but that most results could be described within this framework so far. Recent transient 2D IR experiments provide evidence for a coupling between the commonly observed fast and slow solvation processes and suggest that in some cases one might need to go beyond the linear response [49].
of the total amplitude of the decay.10 Simulations indicate that it is created by small rotational deviations of only a few water molecules nearby the solute [52]. This part is most of the time not fully resolved experimentally and arises from librational motions (hindered and damped rotations) and intermolecular vibrations in the water network around the probe. This response is sometimes considered as polarisation of the bulk induced by the change in the charge distribution of the solute molecule [53]. The second part is much slower and most experimental studies find that it is associated with a time constant of 1.0 ± 0.3 ps [50, 53, 54]. This process involves rotational and translational diffusive motions mainly in the first solvation shell of the solute. It can thus be regarded as the polarisation of the first shell in response to the new charge distribution.
This two-step solvation picture seems actually to be transposable to many solvents, although the time constants and amplitudes used to re-produce the solvation dynamics change with the nature of the solvent.
10 The large amplitude associated with this ultrafast solvent response is expected to have a profound importance on aqueous reaction dynamics as it suggests that solvents are able to respond to changes in reactant configuration on time scales relevant to the crossing of a reaction barrier [50].
Figure 1.6. Simulation of the experimental solvation response function as meas-ured by Maroncelli and co-workers with coumarin 343 in water [50]. The trace was reconstructed by using a Gaussian component (frequency of 38.5 fs!1, total amplitude of 45 %) and two exponential components: 126 fs (20 %) and 880 fs (35 %).
Maroncelli and co-workers have thoroughly studied the solvation dynam-ics of coumarin 153 (C153) in a large range of solvents, including many ionic liquids, and determined the time constants associated with each particular solvent [55, 56]. Each solvent seems to be characterised by a dynamics which, so far, is assumed to depend only to a minor extent on the nature of the probe, at least with polar probes in polar solvents. In its simplest picture, the overall decay of the solvent correlation function is related to the dipole density and the dielectric constant of the solvent by a power law [37].
As the solvation dynamics with a given solute molecule depends on the solvent and therefore on the nature of its direct surroundings, this process can be used to follow changes in the close environment of a probe molecule. The dynamics of the local environment is for example expected to be different in the bulk or at the interface between a solvent and another solvent or a large biomolecule. It has indeed been shown that solvation dynamics of a probe close to a biomolecule is slower than in the bulk [53] and some of the experimental findings in this regard are reviewed in Chapter 2.