20.5: Frenkel exciton transport

The Frenkel exciton model provides a mechanism for describing excitation energy transport in chains of identical electronic chromophores. If an excitation is initialized at site \(|n_i\rangle\), the subsequent time evolution is determined by the superposition of the chain's eigenstates at that location

\[\begin{gathered}
\left| n_{i} \right\rangle = \mathop \sum \limits_{m = 1}^{N} b_{m,i} \left| k_{m} \right\rangle \\
b_{m,i} = \sqrt {\frac{2}{N + 1}} \sin \left( \frac{\pi mn_{i}}{N + 1} \right) \\
\end{gathered} \label{eq20.5.01}\]

We can then calculate the probability of the excitation being at a final site \(n_{f}\), \(P_{n_{f}}(t) = |c_{n_{f}}(t)|^{2}\), from the amplitude at the final site at time \(t\):

\[c_{n_f}(t)=\left\langle n_f\right| U(t)\left|n_i\right\rangle=\frac{2}{N+1} \sum_{m=1}^N \sin \left(\frac{\pi m n_i}{N+1}\right) \sin \left(\frac{\pi m n_f}{N+1}\right) e^{-\mathrm{i} E_k t / \hbar} \label{eq20.5.02}\]

Here \(U=e^{-\mathrm{i}H_0t/\hbar}\) is the usual time-propagator. We see that this oscillates in a complex manner with time, reflecting the initial exciton superposition state.

However, this expression isn’t particularly useful, because it doesn’t describe traditional motion of a particle along a chain, and the question that we posed and answered wouldn’t occur in reality. Why? Because even if the initial and final states are designated at the ends of the chain (\(n_{i} = 1,n_{f} = N\)), the exciton states are delocalized over all sites. That is, since the exciton states in eq. (\ref{eq20.5.01}) involve the whole chain, even the act of preparing the initial state at one end of the chain involves creating a superposition of eigenstates that carry well-defined phase relationships between all sites. In this wave—rather than particle—formulation, there isn’t a real meaning to the concept of being at a site. Nonetheless, there is a lot of interest in realizing this type of coherent energy transport. For instance, a more realistic design of directional transport, one can imagine coupling energetically distinct donor and acceptor molecules to the ends of the chain.

Without digging into this topic in detail, we can make some predictions about how fast the excitation can move along a chain by calculating the group velocity of an exciton wavepacket using:

\[v_{\text{gr}}(k) = \frac{1}{\hbar }\frac{dE}{dk} \label{eq20.5.03}\]

For the Frenkel Hamiltonian eq. (20.2.23) we obtain

\[v_{\text{gr}}(k) = - \frac{2J\alpha }{\hbar }\sin \left( k\alpha \right) \label{eq20.5.04}\]

This indicates that the speed at which an electronic excitation can move in the lattice is governed by the coupling between sites. The group velocity depends strongly on \(k\), and the maximum group velocity occurs at \(k = \pi /2\):

\[v_{\text{max}} = \frac{2Ja}{\hbar } \label{eq20.5.05}\]

This maximal velocity sets the fastest possible ballistic propagation down the chain. We can estimate the minimum time it takes for an exciton to cross a chain of \(N\) chromophores as

\[t_{\text{min}} = \frac{L}{v_{\text{max}}} = \frac{N\hbar }{2J} \label{eq20.5.06}\]

Let’s evaluate these variables using some characteristic parameter values. If the lattice spacing is \(\alpha=0.5 \, \text{nm}\) and the coupling is \(J=100 \, \text{cm}^{-1}\), then the maximum phase velocity \(v_\text{max}=1.9 \times 10^4 \, \text{m}/\text{s}\), and the time it takes to travel down a \(N=12\) or \(6 \, \text{nm}\) chain is \(320 \, \text{fs}\). Note that the scaling in eq. (\ref{eq20.5.06}) is linear with chain length, reflecting ballistic quantum transport reminiscent of a particle or wave traveling at fixed speed: \(x = vt\). Excitation transfer in this manner is known as the coherent transport regime, which relies on maintaining well-defined phase relationships between the wavefunctions on each site.

In realistic settings, environmental fluctuations such as vibrations or solvent motions induce dephasing, which dynamically alters the phase relationships between chain sites. This has the effect of localizing the excitation to one or two sites within the chain and disrupts coherent transport down the chain. With increasing dynamic or static disorder, the transport process transitions from ballistic (wave-like) to diffusive, with excitations hopping from one chromophore to an adjacent site. In this “strong dephasing” or Forster/Dexter regime, energy transfer is governed by uncorrelated transfer of excitations between sites with no memory (phase-relationship) of where the excitation was previously. If each site is assigned a dephasing rate \(\Gamma\) that describes the loss of coherence, one can derive an effective hopping rate

\[k_{\text{eff}} = \frac{J^{2}}{\hbar ^{2}\Gamma} \label{eq20.5.07}\]

by solving the Liouville equation for the population difference between adjacent sites. Then one can describe energy flow through the chain in terms of rate equations for the energy transport between adjacent sites:

\[\frac{dp_{n}}{dt} = k_{\text{eff }}\left[ p_{n - 1} - 2p_{n} + p_{n + 1} \right] \label{eq20.5.08}\]

In this limit, excitation propagates diffusively through the chain, and the it takes for the excitation to reach the end of the chain is

\[t_{\text{diff }} \approx \frac{N^{2}}{2k_{\text{eff}}} \label{eq20.5.09}\]

Note that the transport time scale is now quadratic in the chain length, as expected for diffusive (random walk) transport, where \(\left\langle x^{2} \right\rangle = \mathscr{D}t \) and \(\mathscr{D}\) is a diffusion constant. The scaling behavior for transport time with distance is the essential discriminator of the transport mechanism. In coherent (ballistic) transport the transport time \(t\) is proportional to distance, \(t \sim N\), whereas diffusive transport scales with the square of distance, \(t \sim N^{2}\).

In his original work, Forster derived an expression for the diffusion constant of the electronic energy transfer (\(\mathscr{D}_\text{EET}\)) on a lattice of chromophores through FRET. By summing the single-pair rate \(k_{0}(r) = R_0^{6}/r^{6}\tau _{0}\) over the distribution of sites on the lattice, he found

\[\mathscr{D}_\text{EET} = \frac{\eta }{\tau _{0}}R_0^{6}C^{4/3} \label{eq20.5.10}\]

where \(\tau _{0}\) is the radiative lifetime, \(C\) is the number density of chromophores, and \(\eta \) is a dimensionless constant between \(2\) and \(3\) that depends on the type of lattice. This expression overpredicts the actual diffusion constant in systems with disorder, and it generally over-estimates the effective energy transfer rate, likely due to other nonradiative relaxation processes that ensue. However, even small variations in the assumptions of how chromophores are distributed can have qualitative impacts on the scaling with \(C\) and \(R_{0}\). Over the decades since, the theory of exciton diffusion has been extended greatly to include different descriptions of the Coulombic coupling, multiple competing transfer and relaxation processes, and many other experimental observables, such as the excitation survival probability.

Example: Exciton Diffusion in Solution

Calculate the average rate of energy transfer and the exciton diffusion constant in a homogeneous solution of chromophores with a concentration (number density) \(C\), using the single donor-acceptor rate \(k_{0}(r) = R_0^{6}/r^{6}\tau _{0}\).

We will assume that an orientationally averaged value of \(\kappa ^{2} = 2/3\) has been included in \(R_{0}\), and the positions of acceptors have a uniform angular distribution. Then, for acceptors characterized by a radial distribution function \(g(r)\), the effective rate of transfer in solution is

\[k_{\text{eff}} = \int dr \, 4\pi r^{2} C g(r) k_{0}(r) \label{eq20.5.11}\]

For a uniform distribution of acceptors, \(g(r) = 1\) once one is beyond the radius of the molecule, \(a\). In practice, a lower limit of \(0\) would diverge in the integral, but acceptors can’t be in the same position as donors anyway. We deal with this by setting \(g(r) = 0\) for \(r \lt a\)—the hard sphere radial distribution function. This gives us

\[k_{\text{eff}} = \int_{a}^{\infty } \kern-.5em dr\, 4\pi r^{2} C k_{0}(r) = \left( \frac{4\pi }{3} \right)\frac{C R_0^{6}}{\tau _{0} a^{3}} \label{eq20.5.12}\]

For the diffusion constant, we want to calculate rate of change in the mean square displacement of the exciton. In three-dimensions, \(dr^{2}/dt = 6\mathscr{D}_\text{EET}\). We can obtain this by averaging the rate of energy transfer over the distribution of the acceptor distances: \(6\mathscr{D} = \mathop \sum \nolimits_{j} k\left( r_{j} \right)r_j^{2}\) or as an integral

\[\mathscr{D}_\text{EET} = \frac{1}{6}\int_{0}^{\infty } \kern-.5em 4\pi \rho g(r) r^{2} k(r) r^{2} \, dr = \left( \frac{2\pi }{3} \right)\frac{C R_0^{6}}{\tau _{0} a} \label{eq20.5.13}\]

This describes how the electronic excitation will radially diffuse from one site, however, this process doesn’t continue indefinitely. Ultimately, the exciton will relax due to radiative and non-radiative processes. From the diffusion constant, we can estimate the exciton diffusion length, \(L_\text{EET}\), the root-mean-squared distance over which the exciton diffuses over the fluorescence lifetime, \(\tau _{0}\):

\[L_\text{EET} = \sqrt {6\mathscr{D}_\text{EET}\tau _{0}} = \sqrt {\frac{4\pi C}{a}} R_0^{3} \label{eq20.5.14}\]

Now let’s make some predictions for excitation energy transport in a dilute solution of a bright fluorescent chromophore. We choose a critical transfer distance of \(R_0=5 \, \text{nm}\), which we would obtain by calculating the Forster overlap integral between the chromophores absorption and emission spectrum. Let’s also choose a radius of \(a=0.5\text{nm}\) from the approximate size of a bright dye molecules, and a fluorescence lifetime of \(\tau_0=5\, \text{ns}\). Then let’s look at the energy diffusion parameters for a solution with \(C=1 \text{mM}\) concentration. We calculate the effective rate of energy transfer as \(k_{ \text{eff}}=69 \, \text{ns}^{-1}\), the exciton diffusion constant as \(\mathscr{D}=8.7\times{10}^{-5} \, \text{cm}^2 \text{s}^{-1}\), and the exciton diffusion length as \(L_\text{EET}=16 \, \text{nm}\). In practice, these numbers far exceed what one observes in experiment, reflecting limitations to the assumptions of the model, particularly by neglecting the role of disorder and competing relaxation processes.

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