Normally, the dynamic surface excess from fresh surface in aqueous surfactant solution is given by Ward-Tordai equation
$$ \begin{align*} \Gamma(t) &= \sqrt{\frac{D}{\pi}} (2c_b \sqrt{t}-\int_0^t \frac{c_s(t)}{\sqrt{t - \tau}} d\tau) \end{align*} $$ The equation was derived from Fick's second law of diffusion $$ \begin{align*} \frac{{\partial c}}{{\partial t}} = D \left( \frac{{\partial c}}{{\partial x}} \right)_{x=0} \end{align*} $$ with the initial condition that $ c_s(0)=0 $, where $c_s, c_b$ are the surface and bulk concentration, respectively.
However, if the surface concentration at the beginning is a non-zero value $c_s(0)=c_e < c_b$, corresponding to the surface excess at the beginning $\Gamma_e$.
How does this change in the initial surface concentration influent the dynamic surface excess behavior?
Thanks!
