## 3 Results  
### London Dispersive Surface Energy of Modified Copolymers  
Two principal methods were used in the literature to determine the London dispersive surface energy, both based on the Fowkes equation [18]. The first one was proposed by [PERSON] and [PERSON] [49], correlating the work of adhesion \(W_{a}\) to the free energy of adsorption by the help of the surface area \(a\) of the adsorbed molecule and the geometric mean of the dispersive components of the surface energy of the liquid solvent \(\gamma_{l}^{d}\) and the solid \(\gamma_{s}^{d}\). \[\Delta G_{a}^{0}=\mathcal{N}a\;W_{a}=2\mathcal{N}a\;\sqrt{\gamma_{l}^{d}\gamma _{s}^{d}} \tag{13}\]where \(\mathcal{N}\) is Avogadro's number. [PERSON] and [PERSON] [49] then determined the \(\gamma_{s}^{d}\) of solid surfaces using the following relation:  
\[\gamma_{s}^{d}=\frac{\left[RTln\left[\frac{V_{n}\left(C_{n+1}H_{2(n+2)}\right)} {V_{n}\left(C_{n}H_{2(n+1)}\right)}\right]\right]^{2}}{4\mathcal{N}^{2}\;a_{- CH2-}^{2}7-CH2-} \tag{14}\]  
where \(V_{n}\) is the net retention volume of the adsorbed n-alkane, \(C_{n}H_{2(n+1)}\) and \(C_{n}H_{2(n+1)}\) represent the general formula of two consecutive n-alkanes, \(a_{-CH2-}\) is the surface area of the methylene group equal to 6 A\({}^{2}\), and \(\gamma_{-CH2-}\) is the surface energy of the methylene group, which is given by the following:  
\[\gamma_{-CH2-}\left(\text{in}\;\text{m}\;\text{m}/\text{m}^{2}\right)=52.603-0.058\;T\;(\text{in}\;\text{K}) \tag{15}\]  
The critiques to this method were formulated by [PERSON]'s works [26; 27; 28; 29; 30; 31; 32; 33], which proved that the surface area of organic molecules varies as a function of temperature, and showed an important dependence of the surface area of methylene group \(a_{-CH2}\). versus the temperature. [PERSON] [26] gave an expression of \(a_{-CH2-}(T)\) (in A\({}^{2}\) with \(T\) in K):  
\[a_{-CH2-}(T)=\frac{69.939}{\left(563.02-T\right)^{1/2}} \tag{16}\]  
This proved the non-validity of results obtained by several scientists [50; 51; 52; 53; 54; 55; 56; 57] using the [PERSON] and Gray method. The second method was that of [PERSON] et al. [58], which was almost identical to the method of [PERSON] and [PERSON] [39]; both were based on the Fowkes relation [18], and used the following relation:  
\[RTln\;Vn=\left(\gamma_{s}^{d}\right)^{1/2}\left[2\mathcal{N}a\left(\gamma_{l} ^{d}\right)^{1/2}\right]+C \tag{17}\]  
where \(C\) is constant. These authors supposed that the surface area \(a\) and the dispersive surface energy of molecules are constant at any temperature and obtained a n-alkanes straight line whose slope led to the value of \(\gamma_{s}^{d}\) of the solid. This method was widely applied by many researchers [54; 55; 56; 57; 59; 60; 61; 62; 63; 64; 65], despite the inaccuracy due to the dependence of the surface areas of molecules on the temperature. The Hamieh thermal model [26] gave the expressions of the dependence of the surface areas and \(\gamma_{l}^{d}\) of n-alkanes polar molecules against the temperature. It was then concluded that the methods of [PERSON] et al. and [PERSON], supposing the surface area of solvent molecules are constant, are inaccurate and cannot be used to determine the London dispersive surface energy, nor to evaluate the specific interactions. The expressions of the surface areas and the dispersive surface energy of n-alkanes and polar molecules obtained by the Hamieh thermal model against the temperature were used to determine the correct London dispersive surface energy of the different modified copolymers. Based on the Hamieh thermal model [26; 27; 28; 29; 30; 31; 32; 33; 34] and the Fowkes equation [18], the representation of the variations in \(RTlnV_{n}(T)\) of n-alkanes adsorbed on the modified copolymers as a function of \(2\mathcal{N}a(T)\Big{(}\gamma_{l}^{d}(T)\Big{)}^{1/2}\) led to the \(\gamma_{s}^{d}(T)\) of the various copolymers versus the temperatures [45; 46; 47]. The results are plotted in Figure 1, where a decrease in \(\gamma_{s}^{d}(T)\) was observed as the temperature increased.