# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 1 Introduction  
Determining the free surface energy between phases, typically referred to as surface tension in liquid-gas systems and interfacial tension in liquid-liquid systems, is of crucial importance in surface physical chemistry. Many phenomena and applications where free surface energy plays a key role can be cited: the size and shape of liquid droplets or gas bubbles in sprays and foam formation; the formulation of emulsions in cosmetics, food, and other uses; the shape and growth of crystals in industrial crystallizers; capillarity, as in plants where sap rises through xylem vessels; the dispersion and cleaning of surfaces; the formation and properties of nanoparticles influencing catalysis and medicine; hydrophilic and hydrophobic materials for various technical applications like textiles, glass, or paints; oil recovery by surfactant injection; the formulation of high-strength adhesives for industrial and domestic applications; the coating of biomaterials for medical implants; the improved absorption of pesticides and fertilizers in agriculture for greater efficiency and reduced environmental impact; and many more. The prediction of free surface energy (surface/interfacial tension) and associated properties such as the contact angle or adhesion energy can be achieved using classical methods based on Gibbs thermodynamics formulations. A recent equation in this context is the Shardt-Elliott-Connors-Wright equation [1]. In Gibbs thermodynamics, the free surface energy is accumulated at the interface, which is considered to have zero thickness, and its relative position can be arbitrarily chosen. In contrast, the Kirkwood-Buff equation for surface tensions uses a Gibbsian surface with thickness \(\tau\), where \(\tau\) is a macroscopic length that is large relative to the range of intermolecular forces [2]. According to these authors, surface tension is proportional to the product of the normal pressure on the interface, P, and the thickness of the surface, \(\tau\). For determining pressure, statistical mechanics and molecular density considerations are used, allowing the calculation of surface tension without needing the exact value of \(\tau\). Another approach is the classical density functional theory [3], where the solid-liquid interfacial tension is obtained as the ratio of the difference in grand potentials of the fluid confined in a slit-like pore and the bulk fluid, and the interface area. These methods generally require molecular parameters, which are often difficult to determine accurately. Furthermore, in Gibbs thermodynamics, the interface is assumed to be a surface with no thickness, which is physically counterintuitive, or if thickness is considered, its size is unknown. Another alternative extensively researched, although not yet fully satisfactory, is correlating free surface energy with Hansen solubility parameters. This approach will be applied in the present work to establish a general law for evaluating free surface energy (surface/interfacial tension) and contact angles, independent of the chemical nature and physical state of the phases involved. Additionally, the thickness of the interface will be determined. To achieve this, the Hansen solubility parameter space will be reformulated as a vector field. Prior to this, a brief review of the fundamentals of solubility parameters and an examination of previous attempts to correlate them with free surface energy is necessary. [PERSON] and [PERSON] suggest that the process of solubilisation is similar to the process of evaporation, as both processes involve the separation of molecules by breaking their intermolecular bonds [4]. To energetically quantify the solubilisation of a substance, [PERSON] defined the solubility parameter, \(\delta\), as the square root of the cohesive energy density of said substance:  
\[\delta=\sqrt{\frac{E}{v}} \tag{1}\]  
where \(v\) is the molar volume of the pure substance (m\({}^{3}\)/mol), \(E\) is the energy of vaporization (J/mol) and \(E/v\) is the cohesive energy (J/m\({}^{3}\)).