Generalizing solubility parameter theory to apply to one and two-dimensional solutes and to incorporate dipolar interactions

Hildebrand and Hansen solubility parameters are commonly used to identify suitable solvents for the dispersion or dissolution of a range of solutes, from small molecules to graphene. This practice is based on a number of equations which predict the enthalpy of mixing to be minimized when the solubility parameters of solvent and solute match. However, such equations have only been rigorously derived for mixtures of small molecules which interact only via dispersive forces. Here we derive a general expression for the enthalpy of mixing in terms of the dimensionality of the solute where small molecules are considered zero dimensional, materials such as polymers or nanotubes are one-dimensional and platelets such as graphene are two-dimensional. We explicitly include contributions due to dispersive, dipole-dipole and dipole-induced dipole interactions. We find equations very similar to those of Hildebrand and Hansen so long as the solubility parameters of the solute are defined in a manner which reflects their dimensionality. In addition, the equations for 1D and 2D systems are equivalent to known expressions for the enthalpy of mixing of rods and platelets respectively as a function of surface energy. This agreement between our expressions and those commonly used shows that the concept of solubility parameters can be rigorously applied to extended solutes such as polymers, nanotubes and graphene.