# Synergistic Effect and Phase Behavior of SCG-CAPB-H\({}_{2}\)O
## 3 Results and Discussion
### Synergistic Effect of SCG/CAPB Ternary Compound System
#### 3.1.4 The Interaction Parameter (\(\beta\)) for the SCG/CAPB System  
According to conventional solution theory, the interaction strength between the head groups of two surfactant molecules in a solution can be represented by the interaction parameter \(\beta\)[66]. Typically, interaction parameters for mixed micelles (\(\beta^{m}\)) and mixed interfaces (\(\beta^{\prime}\)) can be calculated accordingly. If \(\beta\) < 0, it indicates mutual attraction between the two components; if \(\beta\) > 0, it suggests mutual exclusion between the two components; when \(\beta\) = 0, it signifies no interaction between them. Generally, the interaction parameter \(\beta\) can be determined by measuring the critical micelle concentration (CMC) of the corresponding surfactant mixture. The interaction between surfactants in the SCG/CAPB binary mixture system is evaluated using equations from the Rubingh non-ideal mixing theory [66; 67]. This model is derived from the regular solution theory (RST). According to this theory, the critical micelle concentration (\(C_{12}^{m}\)) of the surfactant mixture can be determined using Equation (8). \[C=X_{1}^{m}\text{ }f_{1}^{m}\text{ }.C_{1}^{m}=X_{1}\text{ }.C_{12}^{m} \tag{8}\]  
In Equation (8), \(C\) represents the concentration of the surfactant SCG in the solution; \(X_{1}\) and \(X_{1}^{m}\) denote the molar fractions of SCG in the solution and in the micelles, respectively; \(C_{1}^{m}\) and \(C_{12}^{m}\) are the critical micelle concentrations (CMC) of SCG and the binary mixed surfactant, respectively; \(f_{1}^{m}\) stands for the activity coefficient of SCG in the mixed micelles. The activity coefficients of SCG (\(f_{1}^{m}\)) and CAPB (\(f_{2}^{m}\)) are associated with the interaction parameters and can be determined by Equations (9) and (10). \[Inf_{1}^{m}=\beta^{m}(1\text{ }-\text{ }X_{1}^{m})^{2} \tag{9}\]  
\[Inf_{2}^{m}=\beta^{m}(X_{1}^{m})^{2} \tag{10}\]  
As \(f_{1}^{m}=X_{1}C_{12}^{m}/\left(X_{1}^{m}C_{1}^{m}\right)\) and \(f_{2}^{m}=C_{12}^{m}(1-X_{1})/\left[C_{2}^{m}\big{(}1-X_{1}^{m}\big{)}\right]\), \(X_{1}^{m}\) could be calculated from Equation (11). In Equation (11), \(C_{2}^{m}\) represents the critical micelle concentration (CMC) of the surfactant CAPB. \[\frac{X_{1}^{m^{2}}\text{ln}\big{[}\left(C_{12}^{m}X_{1}\right)/\left(C_{1}^ {m}X_{1}^{m}\right)\big{]}}{\big{(}1-X_{1}^{m}\big{)}^{2}\text{ln}\big{\{} \left(C_{12}^{m}\left(1-X_{1}\right)\right)/\left[C_{2}^{m}\big{(}1-X_{1}^{m} \big{)}\right]\big{\}}} \tag{11}\]  
The interaction parameter \(\beta^{m}\) can be calculated using Equation (12). \[\beta^{m}=\frac{\text{ln}\big{[}C_{12}^{m}X_{1}/C_{1}^{m}X_{1}^{m}\big{]}}{ \big{(}1-X_{1}^{m}\big{)}^{2}} \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Kraft point (\({}^{\circ}\)C) & 36 \\ \(T_{max}\) (\(\mu\)mol\(\cdot\)m\({}^{-2}\)) & 3.085 \\ \(A_{min}\) (nm\({}^{2}\)) & 0.538 \\ CPP & 0.7935 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical properties of sodium cocoyl glycinate. Similarly, the interaction parameter \(\beta^{\sigma}\) for the mixed monolayer can be calculated using the Rubingh formula [67], as given in Equations (13) and (14). \[\frac{X_{1}^{\sigma^{2}}\text{ln}\big{[}\big{(}C_{12}^{s}X_{1}\big{)}\big{/}\big{(} C_{1}^{s}X_{1}^{\sigma^{\prime}}\big{)}\big{]}}{\big{(}1-X_{1}^{\sigma}\big{)}^{2} \text{ln}\big{\{}\big{(}C_{12}^{s}\;(1-X_{1}\;)\big{)}\big{/}\big{[}C_{2}^{s} \big{(}1-X_{1}^{\sigma}\big{)}\big{]}\big{\}}}=1 \tag{13}\]  
\[\beta^{\sigma}=\frac{\text{ln}\big{[}C_{12}^{s}X_{1}/C_{1}^{s}X_{1}^{\sigma} \big{]}}{\big{(}1-X_{1}^{\sigma}\big{)}^{2}} \tag{14}\]  
In the equations above, \(X_{1}^{\sigma}\) represents the molar fraction of SCG in the mixed monolayer; \(C_{1}^{s}\), \(C_{2}^{s}\), and \(C_{12}^{s}\) are the concentrations of SCG, CAPB, and the binary complex system, respectively, at which the surface tension reaches 40 mN\(\cdot\)m\({}^{-1}\).