5.3 Excited state lifetime and quantum yield

Excited State Dynamics

Excited state dynamics govern how long molecules remain in an excited electronic state and how efficiently they convert absorbed light into useful outcomes. These two measurable quantities, the excited state lifetime and the quantum yield, are central to predicting reaction efficiency, designing fluorescent sensors, and optimizing photocatalysts.

Excited State Lifetime: Significance

The excited state lifetime ($\tau$) is the average time a molecule spends in an excited state before returning to the ground state. It's typically on the order of nanoseconds ($10^{-9}$ s) for fluorescent states or microseconds to seconds for phosphorescent (triplet) states.

Why does $\tau$ matter so much? It sets the window during which any excited-state process can occur. A molecule can only undergo energy transfer, electron transfer, or a photochemical reaction while it's still excited. If the lifetime is too short, the molecule relaxes before anything useful happens.

Several factors shorten or lengthen $\tau$:

• Molecular structure: Rigid, planar molecules (like pyrene) tend to have longer lifetimes because they have fewer vibrational modes that drain energy.
• Solvent environment: Polar solvents can stabilize or destabilize excited states, shifting the balance between decay pathways.
• Temperature: Higher temperatures generally increase non-radiative decay rates, shortening the lifetime.
• Quenchers: Dissolved oxygen is a common triplet-state quencher. Its presence can dramatically reduce $\tau$ by opening additional non-radiative channels.

Calculating Excited State Lifetime

The lifetime depends on the total rate at which the excited state depopulates. If $k_r$ is the radiative decay rate constant and $k_{nr}$ is the sum of all non-radiative decay rate constants, then:

$\tau = \frac{1}{k_r + k_{nr}}$

The key radiative pathways are fluorescence (singlet → singlet) and phosphorescence (triplet → singlet). Non-radiative pathways include internal conversion (same spin multiplicity, energy lost as heat) and intersystem crossing (spin flip from singlet to triplet or vice versa).

A few things to notice about this equation:

• When $k_{nr}$ is small compared to $k_r$, the molecule decays mostly by emitting light, and $\tau$ approaches $1/k_r$. This limiting value is called the natural radiative lifetime ($\tau_0$).
• When $k_{nr}$ dominates, the lifetime shortens and emission becomes weak.
• Adding any new decay channel (a quencher, for instance) increases the denominator and reduces $\tau$.

Experimentally, $\tau$ is measured using time-resolved spectroscopy. A short laser pulse excites the sample, and you monitor the emission intensity as it decays. The decay curve typically follows $I(t) = I_0 \, e^{-t/\tau}$, and fitting this exponential gives $\tau$ directly.

Quantum Yield: Concept

The quantum yield ($\Phi$) quantifies how efficiently absorbed photons produce a specific outcome. The general definition is:

$\Phi = \frac{\text{number of events of interest}}{\text{number of photons absorbed}}$

The "event" depends on what you're measuring. For fluorescence quantum yield, it's photons emitted. For a photochemical reaction quantum yield, it's molecules of product formed (or reactant consumed).

$\Phi$ ranges from 0 to 1 for most processes. A fluorescence quantum yield of 0.90 means 90% of absorbed photons result in fluorescence emission. Values above 1 are possible for chain reactions where one absorbed photon triggers multiple product-forming steps.

The fluorescence quantum yield connects directly to the rate constants:

$\Phi_f = \frac{k_r}{k_r + k_{nr}}$

This tells you something useful: $\Phi_f = k_r \cdot \tau$. So a molecule with a long lifetime and a large radiative rate constant will be a bright emitter.

Factors that affect $\Phi$ mirror those that affect $\tau$:

• Molecular rigidity reduces non-radiative losses (BODIPY dyes are bright partly because of their rigid core).
• Heavy atoms enhance intersystem crossing, lowering $\Phi_f$ but potentially increasing phosphorescence yield.
• Quenchers and energy transfer acceptors (e.g., FRET pairs) open competing pathways that reduce the quantum yield of the initially excited species.

Determining Quantum Yield Experimentally

Two common approaches are used to measure $\Phi$:

Comparative method (for emission quantum yield):

1. Choose a reference fluorophore with a well-known quantum yield (e.g., rhodamine 6G in ethanol, $\Phi_f = 0.95$).
2. Prepare solutions of the reference and your sample with matched absorbance at the excitation wavelength (typically $A < 0.05$ to avoid inner filter effects).
3. Record emission spectra of both under identical instrument conditions.
4. Calculate $\Phi$ using:

$\Phi_{sample} = \Phi_{ref} \times \frac{F_{sample}}{F_{ref}} \times \frac{n_{sample}^2}{n_{ref}^2}$

where $F$ is the integrated emission intensity and $n$ is the solvent refractive index.

Chemical actinometry (for reaction quantum yield):

1. Use a chemical actinometer (e.g., potassium ferrioxalate) to measure the photon flux entering your sample.
2. Irradiate the reaction mixture and measure the amount of product formed (or reactant consumed) over a known time.
3. Apply the formula:

$\Phi_{rxn} = \frac{\text{moles of product formed}}{\text{moles of photons absorbed}}$

For accurate results, ensure the sample absorbs essentially all incident light at the excitation wavelength, correct for any inner filter effects if concentrations are high, and keep conversions low enough that product absorption doesn't distort the measurement.

These measurements have direct practical applications: evaluating photocatalyst performance (e.g., $\text{TiO}_2$ for water splitting), optimizing photopolymerization conditions, and screening fluorescent probes for bioimaging.