Adsorption mechanism investigation of methylthioninium chloride dye onto some metal phosphates using Monte Carlo dynamic simulations and DFT calculations

•Elucidation of the adsorption mechanisms of methylthioninium chloride dye on metal phosphates.•ELUMO and EHOMO energies indicated charge transfer within the molecule.•MCD simulation planar adsorption with a negative adsorption energy and high affinity in an acidic medium.•Higher adsorption energy of dye/Co3(PO4)2 complexes in neutral, acidic, and basic mediums compared to others metal phosphate.