Area per Ligand as a Function of Nanoparticle Radius: A Theoretical and Computer Simulation Approach

Inorganic nanoparticles (NPs) display unique size-dependent properties and have applications in many different areas such as medicine and the semiconductor industry. In order to take advantage of these properties, the organization of the NPs must be controlled, either to promote crystallization or to prevent agglomeration. This control is typically acheived by using covalently bound amphiphilic ligands. While the properties of the NPs themselves have been well-characterized, much less is known about the organic ligand coating. Here, we present a theoretical and computer simulation approach to compute the surface area occupied per ligand molecule as a function of the NP radius and of the ligand hydrophilic to lipophilic balance. We employ a self-consistent method which takes into account the full free energy of the NP/ligand/solvent system, which for this study is composed of hydrophobic NPs, alkyl poly(oxyethylene) ligands, and water. We find an order of magnitude higher ligand coverage on NPs compared to flat surfaces, in agreement with some experimental reports. Our approach is fundamentally different from existing computational methods in the literature and builds a foundation for studies of the organization of colloidal NPs in solvents or at interfaces.