# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 3 Data and Discussion
### Physical Meaning of \(\tau\) Constant  
Classical approaches to interface thermodynamics, according to [PERSON], involve conceiving interfaces as per their mathematical definition; in other words, these are structures with no volume, only area [15]. Each extensive property of the system, such as free energy, entropy, or the quantity of matter, is divided into three contributions. The first contribution is to a bulk phase in which it is assumed that said extensive property is homogeneous until reaching the maximum mathematical surface. The second is a similar contribution in  
Figure 9: Correlation between free surface energy and the square of the magnitude of the interaction vector. the other homogeneous bulk phase, and the third is formed by residues of the extensive property that get assigned to the mathematical surface. That is to say, extensive properties like free energy or entropy get assigned to the mathematical surface but do not get assigned to volume. From a mathematical point of view, this assumption could be valid; however, from a physical perspective, it does not make sense that something without volume could store free energy or entropy. Another obstacle that these approximations face is that the no-volume mathematical surface can be spatially situated at will, allowing selected extensive properties to have an arbitrarily chosen value, including zero. It must be accepted that the mathematical surface, which by definition does not have volume, does not correspond to the physical representation of an interfacial layer. These objections were manifested by [PERSON], who proposes a new approach in which interfaces are not a mathematical construct, but a real physical layer with volume and thickness [16]. [PERSON] uses theoretical reasoning to develop the thermodynamics of interfaces in systems formed by various compounds; this comes from the physical existence of the interfacial layer with a thickness estimated to be less than 10 nm, and he does not propose any experimental value. [PERSON] et al. have theoretically studied adsorption in binary systems of indium-tin, gallium-bismuth, and thallium-bismuth. Their aim was to determine the influence of the concentrations of the components on the distance between different positions of an interfacial surface defined according to [PERSON] [17]. The obtained equations predict that the said surface must be in a range that would correspond to a physical layer with a thickness smaller than the size of an atom or, in other words, less than 0.1 nm. It has been proposed that the pressure coefficient, which is a derivative of the interfacial tension with respect to pressure, can be used as a measurement of the interfacial layer thickness, \(\tau\):  
\[\tau=\frac{\partial\sigma}{\partial P} \tag{58}\]  
The pressure coefficient, defined as the interfacial tension divided by pressure, has units of distance and is thus a suitable candidate for determining the thickness of an interfacial layer. As such, it has been considered for a long time. However, recently, [PERSON] et al. studied compressible polymer mixtures and found that, while this coefficient presents values compatible with a potential interfacial layer, it can exhibit negative values in some regions, rendering it incompatible with thickness [18]. They suggest that pressure by itself cannot predict thickness because the interfacial tension is also strongly dependent on the Flory-Huggins theory of the polymer solution interaction parameter, \(\chi_{12}\). In addition, there is also a dependence on the density of polymer packing, \(\eta\)[19]. In a Monte Carlo simulation about capillary waves in homopolymer interfaces, proportionality between interfacial tension and \(\chi_{12}\) parameter has been found [20].