The famous factor of 4 - Dr Hansen's view
The measure of "like"
The HSP Distance between two molecules, conventionally called Ra, is the measure of how alike they are. The smaller Ra, the more likely they are to be compatible. The famous formula, used for nearly 50 years is:
Ra² = 4(δD1-δD2)² + (δP1-δP2)² + (δH1-δH2)²
The factor of 4 in front of the δD has aroused controversy for decades but plausible theoretical reasons and overwhelming experimental evidence show that it is valid. Conveniently it makes HSP plots into nice spheres. If from measurements of "good" and "bad" solvents you can define a sphere of radius Ro which contains all the good solvents, then you can define a Relative Energy Difference, RED, which tells you where you are in solubility space:
Why is there a “4” in the Distance equation?
The Hildebrand solubility parameter theory is inherently based on the interaction of pairs of molecules. It was limited to molecules without functional groups, that is, hydrocarbons, or those closely resembling these. Predictions for dipolar and hydrogen bonded molecules were considered as being outside of the realm of the theory. The Hansen solubility parameters encompass all of the energy required to evaporate a liquid thus including the cohesive energy derived from dipolar and hydrogen bonds as well as the dispersive bonds. (The total cohesive energy is found from the latent heat by subtracting a correction for the PV work of expansion gained, this being equal to RT according to the ideal gas law.) With a series of systematic steps, the HSP methodology divides this total cohesive energy into separate energies for these three major effects. It is the square root of these separate cohesive energy densities that gives the three HSP. Considering that this has been done for thousands of liquids and all the practical uses where HSP and in particular the HSPiP have been applied, it would seem that the HSP methodology has a sound thermodynamic foundation. It is not just empirical as many have supposed because of the “4”, for example.
It is important for the present discussion to note that dipoles and hydrogen bonds are directional and involve reasonably stable bonds between two molecules or sections of molecules. These molecules may or may not be of the same kind. The dispersive forces are not directional and change position rapidly, but yield a net cohesive energy when averaged over time. A logical assumption is that the directional nature of the interactions of the molecules having dipolar and hydrogen bonds may effectively prevent about one half of their “expected” interactions with surrounding molecules or pairs of molecules. One “expects” cohesive energy based on the latent heat for evaporation of single molecules, but the majority of the single molecules are not present in the liquid. They have chosen to form pairs. This effectively prevents half of the functional groups from relatively rapid motion and potential interaction with surrounding molecules or pairs of molecules for purposes of forming a solution, for example.
If one accepts that the effective cohesive energy of stable pairs that is available for solubility interactions in the liquid is reduced as discussed above, then the effective cohesive energy of interest in this discussion for the dipolar and hydrogen bonding parameters is reduced by a factor of ½. Since the solubility parameter is the square root of the cohesive energy density, the factor ¼ would appear for the dipolar and hydrogen bonding terms.
The “4” in the Distance equation originally appeared as a practical convenience allowing the dipolar and hydrogen bonding parameters to be plotted with a coefficient “1”. For theoretical purposes the coefficients in the equation should be 1, ¼, ¼, respectively, rather than 4, 1, 1 that practical use dictates. In conclusion the theory discussed above suggests the formation of rather stable pairs where dipolar and hydrogen bonding interactions are present. A logical consequence of this is a reduction of the internal cohesive energy available for solubility for these two types of interaction by a factor of about one-half as a useful average number. This in turn leads to effective Hansen solubility parameters for the dipolar and hydrogen bonding effects that are reduced by a factor close to one-fourth as shown in the equation.
The thoughts here are the result of years of thinking about the mystery of the factor of 4. Dr Hansen would very much welcome feedback and suggestions!