# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 3 Data and Discussion
### Surface Tension of Liquids  
Surface tension is the interaction between a liquid phase and air or any other gas. It is thought that, because the cohesion energy is very low in gases, their solubility parameters will be negligible: \(\delta_{D,gas}=\delta_{P,gas}=\delta_{H,gas}=0\). As such, the interaction vector of a liquid phase with a gas phase will, in general, be as follows:  
\[\overset{\rightarrow}{S}_{LG}=\delta_{DL}\hat{t}+\delta_{PL}\hat{j}+\delta_{ HL}\hat{k} \tag{23}\]  
Nonetheless, when a liquid interacts with air (which is non-polar), it can present the following behaviours depending on the molecular structure of the liquid (Figure 2), and these will modify the number of dimensions to be considered in the interaction vector:  
1. The liquid is completely non-polar, and its molecules do not present any special orientation at the interface with air, since its molecules are, like the air molecules, non-polar. The only relevant parameter is \(\delta_{D}\), as \(\delta_{P}=0\) and \(\delta_{H}=0\). This is the case, for instance, for hydrocarbons like pentane. 2. The liquid is polar, but the molecular size is sufficiently large to hide its hydrogen bonds in the inner part of the liquid, preventing them from interacting with air. The electric dipole moment cannot be hidden because it affects the molecule itself. In this case, \(\delta_{H}\) would not be relevant to the air-liquid interaction, while dispersion and polarity would. Figure 2 shows the example of a carboxylic acid like hexanoic acid. The five-atom carbon chain creates a non-polar barrier that would conceal the hydrogen bonds. 3. The liquid is polar, but the molecular size is very small, and non-polar hydrocarbon chains are not able to hide the hydrogen bonding effect versus the air. In this case, the variables of the three solubility parameters, namely dispersion, polarity, and hydrogen bonding, would be relevant. Figure 2 shows the case of formic acid, a small polar molecule with hydrogen bonds that cannot be concealed. According to the suggested hypothesis, the surface tension will be directly proportional to the magnitude of the interaction vector between the liquid and the gas. \[\sigma_{LG}=\tau_{LG}\left|\overset{\rightarrow}{S}_{LG}\right|^{2} \tag{24}\]  
In cases (a) and (b), the vectors of these molecules would operate in the Hansen vector space governed by dimensions \(D\) and \(P\), since hydrogen bonding, \(H\), is non-existent or masked. In this instance, the vector representing these molecules would be two-dimensional:  
\[\overset{\rightarrow}{S}_{LG}=\delta_{DL}\hat{i}+\delta_{PL}\hat{j} \tag{25}\]  
Then, according to (24), the surface tension of most liquids (excluding polar ones, such as water, glycols, glycerine, amines, amides, sulfoxides, and low-molar-mass carboxylic acids) can be calculated using the following expression:  
\[\sigma_{LG}=\tau_{LG}\left(4\delta_{DL}^{2}+\delta_{PL}^{2}\right) \tag{26}\]  
In case (c), vectors of the small polar molecules would operate to the full extent of the Hansen vector field governed by dimensions \(D\), \(P\), and \(H\). In this instance, the vector that represents these molecules would be three-dimensional:  
\[\overset{\rightarrow}{S}_{LG}=\delta_{DL}\hat{i}+\delta_{PL}\hat{j}+\delta_{ HL}\hat{k} \tag{27}\]  
Then, according to (24), the surface tension of polar liquids, including water, glycols, amines, amides, sulfoxides, and low-molar-mass carboxylic acids, can be calculated using the following:  
\[\sigma_{LG}=\tau_{LG}\left(4\delta_{DL}^{2}+\delta_{PL}^{2}+\delta_{HL}^{2}\right) \tag{28}\]  
To prove these hypotheses, the surface tension data of 122 liquids and values measured by the authors using the droplet weight method [12] have been gathered and analyzed. Figure 2: Liquid-air interface for (**A**) pentane, (**B**) hexanoic acid and (**C**) formic acid.