Mathematical model of cation exchange in nanocrystal quantum dots

Cation exchange is a simple approach for creating doped, alloyed, or entirely new types of nanocrystal quantum dots. We have created a mathematical model describing all three cases, assuming that the central reaction is kickout of a host cation by a guest cation. Our model consists of coupled differential equations for the host and guest cation populations as a function of the kickout reaction rate, the incident flux of guest cations, and the cation diffusion rates. This approach complements atomistic simulation methods like kinetic Monte Carlo, which provide insight in specific scenarios but lack the generalizability of deterministic models. Using our model, we analytically define the time scale of full cation exchange, arrive at numerical solutions for the spatial distribution of guest cations in arrested exchange, and assess how these results change with the guest cation concentration in colloidal solution, the temperature, and nonuniform boundary conditions arising from nanocrystal facets. We also identify 'jammed' and 'starved' regimes, which arise from very large and small flux rates of guest cations, respectively, and can lead to nanocrystals with a core-shell structure. We illustrate our model using Pb-for-Cd exchange in CdSe nanocrystals, but our approach can easily be applied to other materials.