Förster Competing-Acceptor Transfer

Worked Example: Competing channels

Consider a donor molecule coupled to two acceptors. The intrinsic lifetime of the donor in the absence of acceptors is tau0 = 1. The first acceptor is at distance r1 = 1 (relative to the Förster radius R0), and the second is at distance r2 = 2. Let g_i = (R0/r_i)^6 be the distance-dependent scaling.

1. For acceptor 1: g1 = (1)^-6 = 1.
2. For acceptor 2: g2 = (2)^-6 = 1/64 = 0.015625.
3. Total scaling: sumg = g1 + g2 = 1.015625.
4. Total transfer efficiency: sumg / (1 + sumg) = 1.015625 / 2.015625 ≈ 0.5038759689922481.
5. Donor lifetime with transfer: tau0 / (1 + sumg) = 1 / 2.015625 ≈ 0.49612403100775193.
6. Acceptor-one probability: g1 / (1 + sumg) = 1 / 2.015625 ≈ 0.49612403100775193.
7. Acceptor-two probability: g2 / (1 + sumg) = 0.015625 / 2.015625 ≈ 0.007751937984496124.

Exploring multiple acceptors

Use the Förster Competing-Acceptor Transfer module to see how competing pathways shift branching ratios.

1. Select Acceptors = 2, Distance one = 1, Distance two = 2, and Donor lifetime = 1. The efficiency will match the 50.4% calculated in the worked example.
2. Change Acceptors to 1. Notice the single acceptor takes over completely, resulting in exactly 50% transfer efficiency and no probability directed to the missing second acceptor.
3. With 2 Acceptors, reduce Distance one to 0.5. Since the transfer rate scales as the inverse sixth power of distance, the first acceptor now dominates almost completely (efficiency > 98%). The relative donor lifetime drops sharply to around 0.015.
4. Change Acceptors to 0. The donor's intrinsic decay path represents 100% of the decay probability, leaving total transfer efficiency at 0 and keeping the relative donor lifetime at 1.0.

Assumptions and Limitations

This formulation relies on the point-dipole idealization, where the distance between the donor and acceptor is assumed to be larger than the size of the chromophores. If r_i approaches 0, the simple r_i^-6 scaling diverges to infinity; physical systems do not exhibit infinite transfer rates because the point-dipole approximation breaks down at extremely short distances.

The model also assumes fixed orientation factors and spectral overlap (embedded within the R0 reference). It treats individual transfer pathways as completely independent and does not account for reverse transfer or acceptor-acceptor energy migration.

References

The foundational description of non-radiative intermolecular energy transfer scaling with the inverse sixth power of distance is found in Förster's original 1948 paper. The derivation of competing independent rates and overall quantum yields builds on this principle.

T. Förster, "Zwischenmolekulare Energiewanderung und Fluoreszenz", Annalen der Physik, 437: 55-75 (1948) details the rate formulation.