# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 3 Data and Discussion
### Physical Meaning of \(\tau\) Constant
The polymer-solvent interaction parameter, \(\chi_{12}\), is defined as follows:  
\[\chi_{12}=\frac{z\Delta w}{kT} \tag{59}\]  
where \(z\) is the coordination number (number of neighbor units occupied by either a polymer segment or a solvent molecule), \(\Delta w\) is the energy increment because of the interaction between the substance and solvent, \(k\) is the Boltzmann constant, and \(T\) is the absolute temperature. The interaction parameter can be estimated using the following expression [5]:  
\[\chi_{12}=\frac{v(\delta_{1}-\delta_{1})^{2}}{RT}+\beta \tag{60}\]  
where \(v\) is the molar volume of the solvent, \(\delta_{1}\) is the Hildebrand solubility parameter of the solvent, \(\delta_{2}\) is the Hildebrand solubility parameter of the polymer, \(R\) is the gas constant and \(T\) is the absolute temperature. The empirical constant, \(\beta\), is used as a correction for Florycombinatorial entropy. If the Hansen solubility parameters are considered, \(\chi_{12}\) can also be estimated using the following expression:  
\[\chi_{12}=\frac{vA_{12}}{RT} \tag{61}\]  
where \(A_{12}\) is as follows:  
\[A_{12}=(\delta_{D2}-\delta_{D1})^{2}+0.25(\delta_{P2}-\delta_{P1})^{2}+0.25( \delta_{H2}-\delta_{H1})^{2} \tag{62}\]  
It can be observed that for an interaction between polymer 2 with solvent 1, \(A_{12}\) is equal to quarter of the square of the interaction vector magnitude \(\overset{\rightarrow}{S}_{12}\) according to the vector field model:  
\[A_{12}=\frac{1}{4}\left|\overset{\rightarrow}{S}_{12}\right|^{2} \tag{63}\]  
For this reason, \(\left|\overset{\rightarrow}{S}_{12}\right|^{2}\) is proportional to \(\chi_{12}\):  
\[\left|\overset{\rightarrow}{S}_{12}\right|^{2}=\frac{4 RT}{v}\chi_{12} \tag{64}\]  
By solving (53), the following is obtained:  
\[\tau=\frac{\sigma_{12}}{\left|\overset{\rightarrow}{S}_{12}\right|^{2}} \tag{65}\]  
If we compare this equation with the pressure coefficient in (58), it can be observed that it is also a free surface energy divided by an equivalent pressure magnitude, so the units will be of length. Pressure by itself, as suggested by [PERSON] [18; 19], is not enough to exactly represent the thickness of the interface because, in some way or another, it must contain the parameter \(\chi_{12}\) in its formulation. Actually, \(\left|\overset{\rightarrow}{S}_{12}\right|^{2}\) is a magnitude with pressure as units (a mandatory condition) that is also related to the \(\chi_{12}\) parameter. Therefore, it can be concluded that the expression in (65) represents the thickness of the interface. When fitting the data by regression, it has been found that \(\tau\) is very close to 0.025 nm in all instances. In other words, it is less than 0.1 nm, just like [PERSON] and collaborators predicted [17]. The thickness is smaller than an atom's diameter, and it must be so. Otherwise, the interface would be occupied by the atoms and molecules of one of the phases. The interface can also be visualized as a zone in which the atomic and molecular orbitals of both phases get close but do not belong to either of them, acting a transition between them. The obtained value \(\tau\) is independent of whether it represents surface or interfacial tension, the chemical family of the substances involved, or the state of matter, whether solid, liquid, or gas. This leads to the conclusion that \(\tau\) is a dimensional characteristic of the interface by itself, rather than being dependent on the nature of the phases. It could be due to chance or an unknown causal relationship, but the interface thickness, \(\tau\), is approximately (and with great accuracy) half of the Bohr radius, \(a_{0}=0.05292\) nm. \[\tau\approx\frac{1}{2}a_{0} \tag{66}\]  
The Bohr radius is the radius of the orbit of the electron of the hydrogen atom in the atomic model developed by [PERSON] in 1913 for the quantum number \(n=1\). According to the Schrodinger model, the Bohr radius is actually the distance to the nucleus at which the probability of finding the electron is maximum for the 1s atomic orbital of the hydrogen atom.