This is the 2D counterpart to the 3D collision-counting Stern-Volmer simulator — not that box flattened onto a plane, but an independent measurement technique: a real 2D reaction-diffusion field, imaged the way exciton-diffusion-length microscopy actually works in OLED and photovoltaic research. A focused excitation spot continuously creates excited-state population n*(x,y,t), which spreads by spatial diffusion while decaying both radiatively and by collisional quenching:
∂n*/∂t = D·∇²n* + P(x,y)·(1−n*) − n*·(1/τ₀ + kq·[Q])
kq = diffusion-controlled rate constant ∝ D_Q (Smoluchowski)
Away from the excitation spot this is a screened diffusion equation — mathematically identical to a 2D Yukawa/Helmholtz problem — whose steady-state solution decays as n*(r) ∝ K₀(r/ℓ_D), a modified Bessel function whose asymptotic tail is a straight line in ln n vs r with slope −1/ℓ_D:
ℓ_D = √(D · τ_eff), τ_eff = 1 / (1/τ₀ + kq·[Q])
- Imaging method (ℓ_D fitted) — the panel measures the actual spatial glow radius by linear-fitting ln n(r) from the live field, exactly as a real PL-quenching-imaging measurement extracts an exciton diffusion length from a micrograph. It should track the analytic prediction √(Dτ_eff) as [Q] rises and shrinks the glow spot.
- Intensity method (I₀/I) — same cuvette, read out the classic way: total radiative output vs. total excitation input gives the textbook Stern-Volmer ratio 1 + Ksv[Q], Ksv = kq·τ₀. Both numbers are produced by the same field and should agree.
- [Q] slider shortens τ_eff, which shrinks both the fitted glow radius and boosts I₀/I in lockstep — the same physical cause, two independent observables.
- D_Q slider sets the diffusion-controlled encounter rate kq (Smoluchowski: quenching is diffusion-limited, so kq scales with how fast the quencher itself moves).
- D slider is the exciton's own spatial diffusion coefficient — it does not change Ksv or I₀/I at all, only how far the glow spreads before decaying, which is exactly what makes the imaging method a genuinely separate measurement of the same τ_eff.
Real-world relevance: PL-quenching-imaging (measuring how a quencher shrinks the diffusion-limited glow radius around a point excitation) is a real technique for extracting exciton diffusion lengths in organic photovoltaics and OLED films — complementary to, and independently verifiable against, the classic bulk-intensity Stern-Volmer titration.