# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 3 Data and Discussion
### Surface Tension of Liquids
2.1 Surface Tension of Non-Polar Liquids or Polar with Non-Small Molecular Size as Function of the Interaction Vector  
Table 11, in Appendix A, shows the surface tension and Hansen solubility parameter data for liquids of very diverse chemical categories such as aliphatic hydrocarbons, cyclic hydrocarbons, aromatic compounds, chlorinated derivatives, alcohols, ketones, carboxylic acids, esters, ethers, glycol ethers, nitrogen compounds, etc. They all are liquids included in the above-mentioned cases (a) and (b) and represent most liquids. In Figure 3, the surface tension versus the square of the interaction vector magnitude and the linear regression for these compounds are shown. The correlation is excellent as a fitting coefficient \(R^{2}=0.990\). The standard deviation of the absolute errors of the surface tension is 3.2 mN/m, and the mean is 2.9 mN/m. It is proven that in this case, the surface tension is directly proportional to the square of the magnitude of the interaction vector, with the following expression:  
\[\sigma_{LG}=0.0243\Big{(}4\delta_{DL}^{2}+\delta_{PL}^{2}\Big{)} \tag{29}\]  
Generalizing:  
\[\sigma_{LG}=0.0243\Big{|}\overrightarrow{S}_{LG}\Big{|}^{2} \tag{30}\]  
where the surface tension, \(\sigma_{LG}\), is expressed in mN/m, the vector magnitude is expressed in MPa and \(\tau_{LG}=0.0243\) nm. It is worth noting that \(\tau_{LG}\) does not depend on the molar volume and its units are those of length. The equation of a circle with its origin as the center of a plane is as follows:  
\[x^{2}+y^{2}=R^{2} \tag{31}\]  
where \(x\) and \(y\) are abscissa and ordinate axis coordinates, respectively, and \(R\) is the radius of the circle. If (29) is reorganized, it is found that it also fits the equation of a circle with its center as the origin of coordinates within the Hansen \(DP\) plane:  
\[\sigma_{LG}=\tau_{LG}\left|\overset{\rightarrow}{S}_{LG}\right|^{2} \tag{24}\]0  
\[\sigma_{LG}=\tau_{LG}\left|\overset{\rightarrow}{S}_{LG}\right|^{2} \tag{24}\]1 Correlation of surface tension, \(\sigma_{LG}\), and \(4\delta_{D}{}^{2}+\delta_{P}{}^{2}\), for every molecule type except for low-molecular-mass polar molecules. The abscissa, being in a Hansen vectorial field, is multiplied by 2 (4 when squared), and the radius circle, \(R_{s}\), is as follows:  
\[\sigma_{LG}=\tau_{LG}\left|\overset{\rightarrow}{S}_{LG}\right|^{2} \tag{24}\]2  
This expression allows us to draw lines of equal tension (iso-tension lines) in a \(DP\) diagram. Knowing the solubility parameters \(\delta_{D}\) and \(\delta_{P}\) of any given liquid (except for low molar mass, polar ones), its surface tension can be graphically determined with ease (Figure 4). 2.2 Surface Tension of Polar or Non-Polar Liquids with Not-Small Molecular Size as Function of the Interaction Vector  
Table 15, in Appendix A, shows the surface tension data of small polar molecules including water, glycols, glycerine, nitrogen compounds and dimethyl sulfoxide. Their molecules are small and cannot hide their hydrogen bonds from air, which is why the interaction vector operates in the entire Hansen space volume. The case of 1,5-Pentanediol is not an anomaly: although its molecule is not small, it has a double bond that prevents rotation around the molecule's axis, making the hydroxyl groups difficult to hide. Similarly, a similar situation occurs with 3-hydroxymethylpyridine, a flat, rigid molecule that also does not allow the reorientation of polar substituents. The square of the magnitude of the interaction vector has been calculated by including the three solubility parameters: \(4\delta_{D}{}^{2}+\delta_{P}{}^{2}+\delta_{H}{}^{2}\). Figure 5 shows surface tension versus the square of the magnitude of the interaction vector, along with the linear regression.