# An AMR-Based Liquid Film Simulation with Surfactant Transport Using PLIC-HF Method
## 2 Governing Equations
### Surfactant Transport Equations and the Marangoni Effect  
In the current research, surfactant transport was taken into consideration. The surface tension coefficient \(\sigma\) changes according to the concentration of the interface surfactant \(f\). Based on work by [PERSON] and [PERSON] [34], a model that has the ability to calculate surfactant transport on a deformed interface was proposed. The transportation equations for the bulk concentration \(F\) in solution and the interface concentration \(f\) defined on the interface can be expressed by the following equations that model the advection-diffusion and adsorption-desorption effects:  
\[\frac{\partial F}{\partial t}+\mathbf{u}\cdot\
abla F=D_{F}\
abla^{2}F+j\delta(\psi) \tag{20}\]  
\[\frac{Df}{Dt}-\mathbf{u}\cdot\
abla_{s}f(\mathbf{u}\cdot\mathbf{n})(\
abla_{s}\cdot\mathbf{n}) =-\
abla_{s}\cdot(f\mathbf{u})+D_{f}\
abla_{f}^{2}f+j \tag{21}\]  
In Equations (20) and (21), \(\
abla_{s}=(\mathbf{I}-\mathbf{n}\otimes\mathbf{n})\
abla\) represents the gradient on the surface. \(D_{F}\) is the bulk diffusive coefficient, \(D_{f}\) is the interface diffusive coefficient, and \(\delta(\psi)\) is a delta function of the level set. \(j\) is the source term that models the adsorption and desorption between the bulk surfactant and interface surfactant. In the present research, \(j\) is expressed by the following equation using the Langmuir model:  
\[j=k_{\rm ad}F(f_{\rm lim}-f)-k_{\rm de}f \tag{22}\]  
Here, \(k_{\rm ad}\) is the adsorption rate coefficient, \(f_{\rm lim}\) is the saturation concentration, and \(k_{\rm de}\) is the desorption rate coefficient. Equation (21) can be expressed as the derivative on the Cartesian coordinate:  
\[\frac{\partial f}{\partial t}+\mathbf{u}\cdot f-f(\mathbf{n}\cdot\
abla\mathbf{u}\cdot\bm {n})=D_{f}\Big{(}\
abla^{2}f-\mathbf{n}\cdot\
abla\
abla f\cdot\mathbf{n}-\kappa\mathbf{n }\cdot\
abla f\Big{)}+j \tag{23}\]  
Since the present method implicitly expresses the interface position by using the VOF method, the following equation [32] is solved to create a field of \(f\) that makes \(\mathbf{n}\cdot\
abla f=0\) in each phase:  
\[\frac{\partial f}{\partial\tau}+S(\psi)\mathbf{n}\cdot\
abla f=0 \tag{24}\]For the bulk concentration \(F\), the transportation equation is solved only in the liquid phase. The values of \(F\) on the gas phase side should not affect the values of \(F\) on the liquid phase side. Therefore, the values on the liquid phase side are extrapolated [31] to the gas phase side:  
\[\frac{\partial F}{\partial\tau}-S(\psi)\mathbf{n}\cdot\
abla F=0 \tag{25}\]  
Equation (25) is only solved in gas phase side. In Equations (24) and (25), \(S(\psi)\) is defined by Equation (19) as a smoothed sign function. The Marangoni effect refers to the free surface movement or flow along the interface caused by the surface tension gradient due to the temperature difference and the surfactant concentration difference. In the present research, an isothermal state was assumed, so the Marangoni effect due to the temperature difference was not considered. In the present research, the Marangoni effect was only induced by the difference in the interface concentration \(f\) of the surfactant. The change in the surface tension coefficient \(\sigma\) can be represented by the following Langmuir state equation:  
\[\sigma(f)=\sigma_{0}\left[1+\frac{RTf_{\text{lim}}}{\sigma_{0}}\ln\left(1- \frac{f}{f_{\text{lim}}}\right)\right] \tag{26}\]  
Here, \(R\) is the gas constant of the ideal gas, \(T\) is the absolute temperature, and \(\sigma_{0}\) is the surface tension coefficient when the surfactant concentration is 0. As shown in Figure 2, since Equation (26) is a non-linear equation that includes logarithms, when the interface concentration \(f\) approaches \(f_{\text{lim}}\), the surface tension coefficient asymptotically approaches \(-\infty\), resulting in a very steep gradient. A linearized equation can be used to stabilize the calculation.