modeled. Using the delta function, the contact line force can now be written as a surface force as follows: $$\begin{aligned}
  \int_{L} \vec{F}_{\tau L} {\left( \vec{x}_{L} \right)} d L 
&= 
  \int_{A} \vec{F}_{\tau L} {\left( \vec{x} \right)} \delta_{2D} {\left[ \hat{n}_{2D} {\left( \vec{x}_{L} \right)} \cdot {\left( \vec{x} - \vec{x}_{L} \right)} \right]} d^{2} \vec{x} \nonumber \\
&= 
  \int_{A} \sigma \cos{\theta_{0}} \hat{n}_{2D} {\left( \vec{x} \right)} \delta_{2D} {\left[ \hat{n}_{2D} {\left( \vec{x}_{L} \right)} \cdot {\left( \vec{x} - \vec{x}_{L} \right)} \right]} d^{2} \vec{x}\end{aligned}$$ converting the line integral over the contact line into an integral over the wall surface. Equation \[delta\] can be used as an approximation for the delta function when the interface has a finite thickness. Substitution gives: $$\int_{\Delta L} \vec{F}_{\tau L} {\left( \vec{x}_{L} \right)} d L 
=
  \lim_{h \to 0} \int_{\Delta A_{w}} \sigma \cos{\theta_{0}} \frac{\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}}{{\left[ c \right]}} d^{2} \vec{x}
\label{surfaceForce}$$ and comparing equation \[lineForce\] with \[surfaceForce\], the surface force $\vec{F}_{\tau A}$ can be identified as: $$\vec{F}_{\tau A}
=
  \sigma \cos{\theta_{0}} \frac{\nabla_{2D} \tilde{c} {\left( \vec{x} \right)}}{{\left[ c \right]}}$$ As a last step, the surface integral needs to be converted to a volume integral. Since $\nabla_{2D} \tilde{c}$ is independent of the distance away from the wall, integrating over $\Delta V$ is the same as 