2.14: Solvent Effect of Fluorescence
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- 366322
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The concept of solvation can be understood from interactions between a fluorophore (the solute), and the surrounding solvent molecules. The dominating solute-solvent interactions arise from electrostatic dipole-dipole interactions, which lead to lowering the potential energies of all energy levels involved in absorption and fluorescence processes.
![clipboard_ee2186080aec285be1d60184cafe12988.png](https://chem.libretexts.org/@api/deki/files/367268/clipboard_ee2186080aec285be1d60184cafe12988.png?revision=1)
This effect can be explained by Onsager's model of solvation. According to this model, the dipole moment of the fluorophore in the ground state, \(μ_g\), interacts with the dipole moments of the surrounding solvent molecules, rearranging them in a way that minimizes the potential energy of the whole system. If we would "freeze" the molecules for a while and remove the fluorophore, the special arrangement of the solvent dipole moments would result in a non-balanced electric field \(R_g\), called the "reaction field" or "vacuum field"
Solvation Energies
In Onsager's model, the solute-solvent interaction is identified as an interaction of the fluorophore dipole moment, μg, with the reaction field Rg, namely:
\[U_{g}^{\text {rel }}=-\vec{\mu}_{g} \cdot \vec{R}_{g} \nonumber \]
The energy level of the ground state is therefore lowered by this value. The symbol 'rel' indicates that the solvent is in a state of thermodynamic equilibrium (relaxed).
Electronic excitation of the fluorophore causes a rapid (~10-15 s) change of its dipole moment to μe. This time is much too short for the solvent molecules to rearrange their orientations. Thus, immediately after excitation the interaction energy will be:
\[U_{e}^{F C}=-\vec{\mu}_{e} \cdot \vec{R}_{g} \nonumber \]
indicating that the reaction field will be still the same as it was before excitation. The symbol 'FC' indicates a non-equilibrated, Franck-Condon state. The solvent molecules need usually picoseconds (10-12-10-10 s) to perform solvent relaxation achieving finally the solute-solvent interaction energy (Figure 2):
\[U_{e}^{\text {rel }}=-\vec{\mu}_{e} \cdot \vec{R}_{e} \nonumber \]
![clipboard_e43b6ef57c9ad3e5a09a68db49d712bd2.png](https://chem.libretexts.org/@api/deki/files/400293/clipboard_e43b6ef57c9ad3e5a09a68db49d712bd2.png?revision=1)
The process of fluorescence brings the fluorophore dipole moment back to its ground-state value \(μ_g\), so just after fluorescence:
\[U_{e}^{F C}=-\vec{\mu}_{e} \cdot \vec{R}_{e} \nonumber \]
which finally evolves during ground-state solvent relaxation to \(U_{g}^{\text {rel }}\). Direct consequences of the different solute-solvent interaction energies at different stages of absorption and fluorescence events are the spectral shifts in absorption (\(ΔU_{abs}\) and fluorescence (\(ΔU_{flu}\)) spectra (Figure 3):
![clipboard_e8c48e5f90407b63693fc8277bf522229.png](https://chem.libretexts.org/@api/deki/files/400294/clipboard_e8c48e5f90407b63693fc8277bf522229.png?revision=1)
which finally evolves during ground-state solvent relaxation to \(U_{g}^{\text {rel }}\). Direct consequences of the different solute-solvent interaction energies at different stages of absorption and fluorescence events are the spectral shifts in absorption (\(ΔU_{abs}\)) and fluorescence (\(ΔU_{flu}\)) spectra:
\[\begin{aligned}
&\Delta U_{abs}=U_{e}^{F C}-U_{g}^{r e l} \\
&\Delta U_{flu}=U_{g}^{F C}-U_{e}^{r e l}
\end{aligned} \nonumber \]
Reorganization Energy
If we assume the stabilization energies in the excited and ground state are identical, we can assign them to the reorganization energy (\(\lambda\)) of the system:
\[U_{g}^{rel} = U_{e}^{r e l} = \lambda \nonumber \]
Reorganization energies are critical parameters in Marcus theory for charge transfer.