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.