This is the same Starks' extraction mechanism as the 3D interfacial-shuttle model, but represented as a genuinely 2D reaction-diffusion-advection field instead of discrete particles orbiting a 3D box. Two stacked concentration fields on a grid — organic (top) carrying [R-Br]/[R-X], aqueous (bottom) carrying [X⁻]/[Br⁻] — are coupled through a catalyst field [QX]+[QBr] that diffuses freely across the whole grid and is advected across the interface by a constant "extraction" drift, exactly like the 3D model's per-particle bias, just solved as a continuum flux instead of individual trajectories:
∂[RBr]/∂t = D∇²[RBr] − k2[QX][RBr] (organic domain only)
∂[RX]/∂t = D∇²[RX] + k2[QX][RBr]
∂[Xaq]/∂t = D∇²[Xaq] − k1[QBr][Xaq] (aqueous domain only)
∂[Braq]/∂t= D∇²[Braq]+ k1[QBr][Xaq]
∂[QX]/∂t = D∇²[QX] − v·∂[QX]/∂y (+k1[QBr][Xaq] − k2[QX][RBr]) (full grid, v = extraction bias)
∂[QBr]/∂t = D∇²[QBr] + v·∂[QBr]/∂y (+k2[QX][RBr] − k1[QBr][Xaq])
k2(T) = 0.15·exp((T−20)/45) v(lipophilicity) = 0.35 + lipoNorm·1.3
The finite-volume diffusion and advection terms are solved with a no-flux boundary (nothing leaves the vessel), so total substrate mass, total aqueous-ion mass and total catalyst mass are each conserved by construction — verified numerically to machine precision. Turnover number here is the textbook definition: moles of product formed per mole of catalyst present.
- Catalyst loading — sets the initial catalyst field concentration relative to the substrate field (mol%), exactly as in the 3D model.
- Stirring rate — raises the diffusion coefficient (turbulent eddy diffusivity) for every field, the same real-world lever for interfacial contact area.
- Anion lipophilicity — sets the constant vertical drift velocity that carries the loaded catalyst (QX) up into the organic field and the discharged catalyst (QBr) back down, reusing the 3D model's own extraction-bias formula.
- Temperature — scales the organic-phase SN2 rate constant k2 through the same Arrhenius-like factor as the 3D model.