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Depinning of Evaporating Sessile Droplet



Schematic of drying sessile droplet of comprising particles

In this article, the derivation for the depinning time associated with the sessile drop evaporating on a substrate and maintained at different temperatures is illustrated. The derivation goes below - 
  

The droplet evaporating at dissimilar rates are observed to depin nearly at the same scaled time $t/t_f$ (where $t_f$ is the time for complete evaporation). The argument can be supported by a rudimentary mathematical framework illustrated below:
\begin{equation}
   \frac{dV}{dt}|_{\Delta T} = -V_E|_{\Delta T}
\end{equation}
where, $V$ is the instantaneous volume of water in the droplet, $V_E$ is the evaporation rate which is a constant at the particular $\Delta T $ (difference of substrate and environment temperature), the negative sign at r.h.s indicates the reduction in volume with time.

Now by integrating Eq(1),
\begin{equation}
  V(t)|_{\Delta T} = -t\,V_E|_{\Delta T} + k|_{\Delta T}
\end{equation}
Since, the initial volume ($V_\circ$) of an evaporating droplet is same ($\sim 2\mu$L), which implies at t = 0, $V(t=0) = V_\circ (say)$ = constant. This condition simplifies  equation(2),\\
\begin{equation}
  V(t)|_{\Delta T} = -t\,V_E|_{\Delta T} + V_\circ
\end{equation}
Now substituting another constraint that at $t=t_f$ the droplet completely evaporates and V(t) = 0, we obtain,
\begin{equation}
V'_E t_f = V''_E t_{f1} = V_\circ (constant)
\end{equation}
where, $V'_E$, $V''_E$ are the evaporation rates at different  $\Delta T $'s with respective $t_f$, $t_{f1}$ being the total evaporation time. Now by further using  Eq(2) and Eq(4) and assuming that at an instant of depinning the instantaneous volume of the droplet is $V_1$ we can write,

\begin{equation}
    V_1 = V_\circ\left( 1 - \frac{t}{t_f}\right) = V_\circ\left( 1 - \frac{t}{t_{f1}}\right)
\end{equation}
\[
\therefore \frac{t}{t_f} = \frac{t}{t_{f1}}
\]
Therefore, for droplets with the same initial volume and the similar substrate wettability the scaled time depicting an evaporation kinetics will be independent of $\Delta T$.




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