# Free Surface Energy and Hansen Solubility Parameter Vector Field. Interface Thickness
## 3 Data and Discussion
### Interfacial Tension between a Solid and a Liquid, or Any Given Two Substances  
The diagram of Figure 7 displays a ternary system comprising a solid (S), a liquid (\(L\)) and a gas (\(G\)). The following is true:  
\[\overset{\rightarrow}{S}_{SG}+\overset{\rightarrow}{S}_{SL}=\overset{ \rightarrow}{S}_{LG} \tag{42}\]  
The interaction vector between the solid and liquid, \(\overset{\rightarrow}{S}_{SL}\), will then be as follows:  
\[\overset{\rightarrow}{S}_{SL}=\overset{\rightarrow}{S}_{LG}-\overset{ \rightarrow}{S}_{SG} \tag{43}\]Substituting (26) and (36) into (43), the following is obtained:  
\[\overset{\rightarrow}{S}_{SL}=\left(\delta_{DL}\overset{\rightarrow}{i}+\delta_{ PL}\overset{\rightarrow}{j}+\delta_{HL}\overset{\rightarrow}{k}\right)-\left( \delta_{DS}\overset{\rightarrow}{i}+\delta_{PS}\overset{\rightarrow}{j}+ \delta_{HS}\overset{\rightarrow}{k}\right) \tag{44}\]  
Operating leads to the following:  
\[\overset{\rightarrow}{S}_{SL}=(\delta_{DL}-\delta_{DS})\overset{\rightarrow} {i}+(\delta_{PL}-\delta_{PS})\overset{\rightarrow}{j}+(\delta_{HL}-\delta_{ HS})\overset{\rightarrow}{k} \tag{45}\]  
The magnitude of the interaction vector between the liquid and solid will be as follows:  
\[\left|\overset{\rightarrow}{S}_{SL}\right|=\sqrt{4(\delta_{DL}-\delta_{DS})^{2 }+(\delta_{PL}-\delta_{PS})^{2}+(\delta_{HL}-\delta_{HS})^{2}} \tag{46}\]  
And this matches with the Hansen distance, \(Ra\), between \(S\) and \(L\):  
\[\left|\overset{\rightarrow}{S}_{SL}\right|=Ra \tag{47}\]  
For this reason:  
\[\sigma_{SL}=\tau_{SL}\left|\overset{\rightarrow}{S}_{SL}\right|^{2}=\tau_{SL }Ra^{2} \tag{48}\]  
\[\sigma_{SL}=\tau_{SL}\left[4(\delta_{DL}-\delta_{DS})^{2}+(\delta_{PL}-\delta_ {PS})^{2}+(\delta_{HL}-\delta_{HS})^{2}\right] \tag{49}\]  
Figure 7: Interaction vectors of a three-phase system: solid (\(S\)), liquid (\(L\)), and gas (\(G\)).  
Table 14 shows the experimental interfacial tension data of several liquids with a few solids: polystyrene polymers (PS), polymethylmethacrylate (PMMA), Polyamide 6,6 (PA66), polytetrafluoroethylene (PTFE), and polyvinylbutyl (PVB). The solubility parameters of polymers and various liquids, the square of the module of the interaction vector \(\overset{\rightarrow}{S}_{SL}\) and the interfacial tension calculated according to (49) are also shown.  
The interfacial tensions between solid polymers and liquids and the linear regression are displayed in Figure 8:  
\[\sigma_{SL}=0.0268\Big{[}4(\delta_{DL}-\delta_{DS})^{2}+(\delta_{PL}-\delta_{ PS})^{2}+(\delta_{HL}-\delta_{HS})^{2}\Big{]} \tag{50}\]  
The value \(\tau_{SL}=0.0268\) nm is also very similar to previous values found for liquid-gas and solid-gas interactions.  
The previous solid-liquid interaction equations can also be extended to the interaction between two liquids. Therefore, for a liquid L1 and a liquid L2, the following can be expressed:  
\[\sigma_{L1,L2}=t_{L1,L2}\Big{|}\overset{\rightarrow}{S}_{L1,L2}\Big{|}^{2}= \tau_{L1,L2}Ra^{2} \tag{51}\]  
\[\sigma_{L1,L2}=\tau_{L1,L2}\Big{[}4(\delta_{DL2}-\delta_{DL1})^{2}+(\delta_{ PL2}-\delta_{P1})^{2}+(\delta_{HL2}-\delta_{HL1})^{2}\Big{]} \tag{52}\]