Asymptotic analysis of the dewetting rim

Consider a film of viscous liquid covering a solid surface, which it does not wet. If there is an initial hole in the film, the film will retract further, forming a rim of fluid at the receding front. We calculate the shape of the rim as well as the speed of the front using lubrication theory. We employ asymptotic matching between the contact line region, the rim, and the film. Our results are consistent with simple ideas involving dynamic contact angles and permit us to calculate all free parameters of this description, previously unknown.

Figure 1

Sketch of the dewetting rim profile $h\left(x\right)$. A liquid film of thickness $h_f$ is invaded by a moving rim of height $h_r$ and width $w$. The apparent contact angle ${\theta }_{ap}$ is defined as the intersection of the solid surface and a parabolic fit of the rim (dashed line).

Figure 2

Schematic representation of the matching procedure. The contact line and rim are described by the solid line, from Eq. (7). The dashed line represents the profile in the film region, from Eq. (8). The profiles are matched at the advancing side of rim around $x^{\ast }$.

Figure 3

The Airy function $\text{Ai}\left(s\right)$ and the values of $s$ that are relevant in the analysis. The solution runs between $s=\infty$ (the contact line $\xi =0$) and $s=s_1$ (infinitely far from the contact line $\xi =\infty$). In particular, we define $s_0=-2.338\dots$ as the rightmost zero of Ai, and $s_1=s_0+ϵ$. The rightmost maximum of Ai is defined as $s_n=-1.088\dots$, which corresponds to the minimum at the neck behind the rim $\left({\xi }_{min}\right)$. The value $s_r\sim {\left(-\text{ln}ϵ\right)}^{2/3}$ corresponds to the maximum thickness of the rim $\left({\xi }_{max}\right)$.

Figure 4

The similarity solution $G\left(\zeta \right)$ near the flat film that has to be matched to the rim. This curve was also computed in [23].

Figure 5

A numerical simulation of Eq. (1), showing the growing rim. The frame of reference is chosen such that the tip of the receding front remains at the origin. The slip length is $\lambda ={10}^{-4}{h}_f$ and ${\theta }_{eq}=0.3$.

Figure 6

A numerical test of our theoretical prediction (46) for the speed of retraction, using the simulation shown in Fig. 5. As the rim width $w$ increases, the speed decreases. The full line comes from the numerical simulation of Eqs. (48, 49), while the dashed line represents Eq. (46). The inset shows the difference between theory and simulation.

Figure 7

Apparent contact angle ${\theta }_{ap}$ for the receding and advancing fronts (solid and dashed lines, respectively) for ${\theta }_{eq}=0.3$. Length scales were chosen as $\lambda ={10}^{-9}\text{m}$, $h_f={10}^{-6}\text{m}$, and $w={10}^{-4}\text{m}$. The intersection of the curves provides the selection of angle and speed.