Fluorescence Decay of Acridine Orange

• The paper demonstrates that Acridine Orange exhibits a bi-exponential decay, identifying two distinct lifetimes that reflect different emissive states.
• A time-correlated single photon counting method combined with nonlinear least-squares fitting yields lifetimes of approximately 1.73 ns and 5.95 ns with high precision.
• The study highlights a quantum mechanical framework that explains the decay dynamics and shows that multi-state mixing can suppress expected late-time power-law decay signatures.

Fluorescence decay of Acridine Orange refers to the time-dependent decrease in fluorescence intensity following pulsed excitation, arising from the relaxation dynamics of its excited electronic states. The decay profile serves as a sensitive probe of molecular environment, electronic structure, and quantum mechanical processes governing state lifetimes and transitions. Acridine Orange, a widely studied heterocyclic fluorophore, displays a decay pattern best captured by the sum of two exponential functions over nanosecond time scales, reflecting the presence and admixture of two distinct emissive states in aqueous solution.

1. Quantum Mechanical Framework and Survival Probability

The formal description of fluorescence decay in molecular systems is governed by the time-evolution of the survival probability of an excited electronic state. Within quantum mechanics, the survival probability $P(t)$ is defined as the squared modulus of the Fourier transform of the energy distribution $\rho(E)$:

$$P(t) = \left| \int_{E_\mathrm{th}}^{\infty} \rho(E) e^{-iEt/\hbar}\, dE \right|^2.$$

In this framework, the observed fluorescence intensity $I(t)$ is related to the time derivative of the population of excited states:

$I(t) = -\frac{dN}{dt} = - N_0 P'(t),$

where $N(t) = N_0 P(t)$ represents the number of excited molecules at time $t$. A purely exponential decay law emerges only for idealized cases (e.g., Breit–Wigner distributions extending over the real axis), whereas realistic molecules exhibit deviations due to physical thresholds and multiple decay channels.

2. Phenomenological Description and Bi-Exponential Decay

Empirically, the fluorescence decay of many dyes, and particularly Acridine Orange, is dominated by one or more exponential components. For Acridine Orange, the best fit is a bi-exponential form:

$I(t) = A_1 e^{-t/\tau_1} + A_2 e^{-t/\tau_2},$

where $\tau_1$ and $\tau_2$ are the lifetimes associated with two distinct excited-state populations, and $A_1$, $A_2$ are their normalized amplitudes. This form reflects two nonoverlapping resonances in $\rho(E)$, each corresponding to a separate relaxation pathway or molecular configuration. The explicit bi-exponential decay law incorporating instrument response offset $t_0$ and background $b$ is:

$$I(t) = A_1 \exp{\left[-(t - t_0) / \tau_1 \right]} + A_2 \exp{\left[-(t - t_0) / \tau_2 \right]} + b,$$

with $A_1 = 0.9776$, $A_2 = 0.0224$ as determined from fitting coefficients, and $t_0 = 2.24\,\mathrm{ns}$.

3. Experimental Methods: Time-Correlated Single Photon Counting

Precise measurement of Acridine Orange fluorescence decay was accomplished via time-correlated single photon counting (TCSPC). The experimental protocol included:

• Solution: Acridine Orange in deionized water at $10^{-5}\ \mathrm{mol/dm^3}$.
• Excitation: PicoQuant LDH-D-C-485 laser diode ($485\,\mathrm{nm}$, $10\,\mathrm{MHz}$ repetition, pulse width $<120\,\mathrm{ps}$).
• Detection: Nikon Eclipse Ti-E confocal microscopy with two PicoQuant PMA-Hybrid 40 single-photon detectors, spatially separated and spectrally filtered (Channel 1: $485-555\,\mathrm{nm}$, Channel 2: $550-650\,\mathrm{nm}$).
• Timing Electronics: PicoHarp 300 TCSPC module for photon arrival timestamping and histogram generation.

Multiple runs (three per channel, $10\,\mathrm{min}$ each) were aggregated to construct high-statistics decay curves. The time offset $t_0$ excluded the instrument response region, ensuring analysis focused on true fluorescence lifetime dynamics.

4. Data Fitting and Model Selection

Decay curves $I_n \equiv I(t_n)$ were fitted by minimizing the Poisson-based chi-square:

$$\chi^2 = \sum_{n=1}^{N_t} \frac{[I_n - I(t_n)]^2}{I(t_n)}.$$

Two candidate models were assessed:

Model Type	Functional Form	Parameters
Bi-exponential	$$I(t) = C_1 e^{-(t-t_0)/\tau_1} + C_2 e^{-(t-t_0)/\tau_2} + b$$	$C_1, \tau_1, C_2, \tau_2, b$
Exponential + power-law	$$I(t) = C e^{-(t-t_0)/\tau} + C_p (t-t_0)^{-\beta} + b$$	$C, \tau, C_p, \beta, b$

Parameters were optimized via nonlinear least-squares, with uncertainties taken from the covariance matrix (inverse Hessian). Goodness-of-fit was assessed with the reduced chi-square ($\chi_\nu^2$). The bi-exponential model achieved $\chi_\nu^2 \approx 1.04$ while the non-exponential model gave $\chi_\nu^2 > 11$, excluding significant power-law contributions under experimental conditions.

5. Results and Quantitative Determinations

Inverse-variance weighted analysis of both spectral channels produced definitive lifetime values:

• $\tau_1 = 1.7331 \pm 0.001$ ns
• $\tau_2 = 5.948 \pm 0.012$ ns

For Channel 1, fitted coefficient examples are:

• $C_1 = 2.78967 \times 10^5$ (primary amplitude)
• $C_2 = 6.3713 \times 10^3$ (secondary amplitude)
• $b = 21.99$ counts (background offset)

The dominant component ($\tau_1$) accounts for $97.76\%$ of the decay amplitude, with the secondary component ($\tau_2$) comprising $2.24\%$. The measured lifetimes are consistent with literature and support the assignment of two emissive species, likely including micelle-like aggregates.

6. Absence of Late-Time Power-Law Deviations

Quantum mechanical theory predicts very late-time power-law decay tails ($I(t) \sim t^{-\beta}$), observable only after many lifetimes and at low amplitudes. In Acridine Orange, no significant power-law tail was detected up to $t \approx 10\,\tau_1$. This finding contrasts with analogous studies of erythrosine B under the same setup, where $t^{-\beta}$ behaviour emerged beyond $\sim10\,\tau$; the difference is plausibly attributed to the longer secondary lifetime and multi-state admixture in Acridine Orange, which suppresses potential tails.

7. Implications for Molecular Photophysics and Future Investigations

Precise quantification of the dual lifetimes in Acridine Orange validates the TCSPC apparatus for detailed late-time decay studies and confirms the persistence of two emissive species in aqueous solution. This methodology highlights crucial experimental variables for future searches for nonexponential quantum mechanical decay, notably selecting dyes with primary lifetimes below $1$ ns and minimal multi-state mixing. A plausible implication is that such species provide an optimal setting for detecting late-time power-law behaviour predicted by quantum mechanics but not resolved in the present work.