# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 1 Introduction
[PERSON] considered that the Hildebrand Solubility Parameter is valid when it comes to the global energy balance, but it does not include structural considerations of molecules [5]. That would explain why substances with a similar heat of vaporization and different molecular structures cannot be dissolved between them. To overcome this obstacle, [PERSON] considers the solubility parameter to be the combination of three solubility parameters:  
\[\delta^{2}=\delta_{D}^{2}+\delta_{P}^{2}+\delta_{H}^{2} \tag{2}\]  
where \(\delta_{D}\), \(\delta_{P}\) and \(\delta_{H}\) are the dispersion, polarity, and hydrogen bond solubility parameters, respectively. The \"distance\", \(Ra\), between two substances, called 1 and 2, is determined by the following equation:  
\[Ra=\sqrt{4(\delta_{D2}-\delta_{D1})^{2}+(\delta_{P2}-\delta_{P1})^{2}+(\delta _{H2}-\delta_{H1})^{2}} \tag{3}\]  
The coordinates of each substance are the centre of a sphere with radius \(R_{0}\), termed the solubility sphere. Within the solubility sphere, the 'good solvents' for the substance are located, while outside the solubility sphere, the 'bad solvents' are found. The parameter called _RED_ (Relative Energy Difference) is defined as follows:  
\[RED=\frac{Ra}{R_{0}} \tag{4}\]  
When \(RED>1\), the substance is not soluble in the considered solvent, while if \(RED\leq 1\), the opposite will be true. The Hansen solubility parameters have been related not only to the phenomenon of solubility but to a multitude of phenomena in which surface free energy is involved. Various correlations have been found between surface and interfacial tension, \(\sigma\), with solubility parameters that respond to the general expressions [4; 5; 6; 7; 8; 9; 10]:  
\[\sigma=f\Big{(}a\delta_{D}^{\kappa},bo\delta_{P}^{\beta},c\delta_{H}^{\gamma}, v^{d}\Big{)} \tag{5}\]  
where \(v\) is the molar volume and the constants \(a\), \(b\), \(c\), \(d\), \(\alpha\), \(\beta\) and \(\gamma\) are empirical adjustment constants without physical meaning. Various correlations between surface and interfacial tension versus the Hansen solubility parameters are shown in Table 1.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline \multicolumn{1}{c}{**Equation**} & \multicolumn{1}{c}{**Comments**} & \multicolumn{1}{c}{**References**} \\ \hline \(\delta=v^{-2/3}\sigma^{0.5}\) & Surface free energy, \(\sigma\); Hildebrand solubility parameter, \(\delta\); molar volume, \(v\). & [PERSON] and [PERSON] (1950) [4] \\ \hline \(\sigma=0.0688v^{\frac{1}{3}}\big{[}\delta_{D}^{2}+k\big{(}\delta_{P}^{2}+ \delta_{H}^{2}\big{)}\big{]}\) &
\begin{tabular}{p{113.8 pt} p{113. The present work aims to establish, both theoretically and experimentally, a vector field model for the Hansen solubility parameters. This model will allow, in the first place, the characterisation of substances as components of a vector. The coordinates of this vector will be the dispersion (\(\delta_{D}\)), polarity (\(\delta_{P}\)), and hydrogen bonding (\(\delta_{H}\)) parameters of the substances. In the second place, it will enable the evaluation of physical processes and phenomena in which different substances interact as a function of the vectors that characterise these processes, referred to as interaction vectors. The immediate practical applications of the Hansen vector field and interaction vectors are evident in surface processes, where free surface energy is a critical factor, manifesting as surface tension, interfacial tension, and the contact angle. The hypothesis assumes that the free surface energy is proportional to the square of the magnitude of the interaction vector between the phases involved, where the proportionality constant will have an identifiable physical meaning. This interaction vector, which will be defined later, is closely related to the cohesive energy density.