Ising model for nanocrystal ligand exchange

To account for inter-ligand interactions during ligand exchanges on colloidal quantum dots, we have developed an Ising model that uses a Monte-Carlo simulation where a two dimensional grid with periodic boundary conditions represents the nanocrystal surface. During the simulation, a random grid site populated with a ligand is selected along with a random free ligand in solution, and if the identities of those ligands are different, an exchange is attempted based on the Gibbs free energy of exchange. This process is repeated until the system reaches thermal equilibrium, and then the enthalpy and ligand coverage are calculated for a given number of starting exchange ligands. The simulation results are then compared to the raw experimental data, and the parameters are adjusted to fit isothermal titration calorimetry data and the 1H NMR data. This simulation was developed in Python, and the code for the simulation is deposited herein.