Topotactic cation exchange. In real II–VI and IV–VI nanocrystal chemistry, the Se²⁻ anion sublattice acts as a rigid scaffold: soaking a CdSe nanocrystal in a solution of a more favorable guest cation (Pb²⁺ or Ag⁺) lets those cations diffuse in and displace the smaller, more mobile Cd²⁺ ions while the anion framework barely moves — the particle keeps its size and shape but its composition (and electronic structure) completely changes.
CdSe + Pb²⁺ → PbSe + Cd²⁺(aq) (1:1, both divalent)
CdSe + 2Ag⁺ → Ag₂Se + Cd²⁺(aq) (charge-balanced 2:1)
Shrinking-core kinetics. Exchange starts at the surface and the reacted shell grows inward as a sharp front at radius r(t), shrinking from the full particle radius R at constant, temperature-activated velocity:
dr/dt = −v(T), v(T) = A·exp(−Eₐ / R_gas·T) (Arrhenius)
Converted fraction: X(t) = 1 − (r(t)/R)³
Ag⁺ is given a lower activation energy than Pb²⁺ in this model — a real, well-documented effect: Ag⁺ is an exceptionally mobile interstitial cation in chalcogenide lattices (the same property behind superionic Ag₂S/Ag₂Se), so exchange to silver selenide is characteristically faster than exchange to lead selenide at the same temperature.
Size-dependent bandgap. Because the particle radius R stays fixed through the reaction, only the composition changes the confinement term. The effective gap uses the Brus effective-mass approximation:
E_g(R) = E_g,bulk + (ħ²π²)/(2R²)·(1/mₑ* + 1/m_h*) − 1.8e²/(4πεε₀R)
The first correction is quantum confinement (grows as 1/R²), the second is the attenuated electron–hole Coulomb attraction. PbSe has a far smaller effective mass than CdSe, so it shows dramatically stronger confinement at the same radius — the bandgap swatch color updates live from the current mix of unconverted CdSe core and converted shell.
Simplification: each cation lattice site is recolored in place rather than literally rearranged, and Ag₂Se's real 2:1 stoichiometry / lattice contraction is not modeled atomistically — the shrinking-core front and Arrhenius rate law are the physically accurate parts being demonstrated.