Kinetic roughening of a soft dewetting line under quenched disorder: A numerical study

An elegant simulation method, suitable for investigating the dewetting dynamics of thin and viscous liquid layers, is discussed. The efficiency of the method is exemplified by studying a two-parameter depinninglike model defined on inhomogeneous solid surfaces. The morphology and the statistical properties of the contact line are mapped in the relevant parameter space, and as a result critical behavior in the vicinity of the depinning transition is revealed. The model allows for the tearing of the layer, which leads to a new propagation regime resulting in nontrivial collective behavior. The large deformations observed for the interface are a result of the interplay between the substrate inhomogeneities and the capillary forces.

Figure 1

Discretization of the contact line.

Figure 2

The reconnection mechanism for the tearing of the layer. The label values in braces indicate the succeeding order in the oriented chain. Please note that step 1 and step 2 are made in the same time moment.

Figure 3

Handling the substrate inhomogeneities. In this example, the line segment corresponding to point $i$ crosses four pinning centers, each with its own threshold. The effective threshold experienced by point $i$ is the largest one out of those four. The pinning points are considered pointlike, with no planar extension.

Figure 4

Contact line morphology for equally spaced time moments (plots in the $x\text{−}y$ plane). Evolution of the interface is from top to bottom (from the blue line to the black one). The inset graphs from left to right correspond to increasing ${\tilde{R}}_1$ values (indicated in the horizontal direction), while from bottom to top we consider increasing ${\tilde{L}}_0$ values (indicated in the vertical direction). The roughness and velocity fluctuations increase, and long-range correlation and large deformation develop as the system approaches the depinning transition. In the two bottom-right cases, after sweeping a finite distance, the line is pinned. In order to better visualize every position of the line within the desired interval, different scales on the $y$ axis have been used. The scale in the $x$ direction is always 200 units.

Figure 5

Mean velocity of the interface along the $y$ axis, in the stationary regime, as a function of ${\tilde{L}}_0$ for different ${\tilde{R}}_1$ parameters. The inset shows the mean velocity as a function of ${\tilde{L}}_0/{\tilde{L}}_0^c$. A reasonable collapse is obtained.

Figure 6

Phase diagram of the contact line in the $({\tilde{R}}_1,{\tilde{L}}_0)$ parameter space. Symbols indicate parameter values at which simulations were performed. Blue squares indicate the obtained pinning phase, and green dots indicate the depinning phase. The inset derived from the separation points shows that the two phases are delimited by the curve ${\tilde{L}}_0^c=1/2-{\tilde{R}}_1^{-1}$, the dashed line indicating a slope $-1$. Please note the logarithmic scales for the inset graph.

Figure 7

An example from computer simulations where local backward movement of the interface occurs. The liquid layer sketched in gray contracts by moving in the bottom direction (negative direction of the $y$ axis). Due to the highly deformed shape of the interface, there will be parts of the interface where the velocity vector of the interface will have a positively oriented $y$ component.

Figure 8

Distribution of the $y$ component of velocities along the contact line for ${\tilde{L}}_0=0.5$. Note that when we approach the depinning transition (${\tilde{R}}_1\to {10}^2$) a considerable local backward movement ($v_y>0$) of the interface occurs. Also, far from the transition (${\tilde{R}}_1\ll {10}^2$), the pinned part of the line is quite well separated from the moving part.

Figure 9

Development of the scale-free morphology as the system approaches the critical state, the normalized length $L\left(\Delta \right)$ of the contact line as a function of $\Delta$ (see the text for the definitions), and results for ${\tilde{R}}_1={10}^2$ and different values of ${\tilde{L}}_0$. The dashed line is a guide for the eye, and has a slope $-0.25$. A natural upper cutoff arises due to the finite system size, and a lower cutoff arises from the discretization.

Figure 10

Development of the scale-free morphology as the system approaches the critical state, the structure factor $S_y\left(k_{\Delta }\right)$ as a function of $k_{\Delta }$ (see the text for definitions), and results for ${\tilde{R}}_1=5.0$ and different values of ${\tilde{L}}_0$. The dashed line has a slope $-2.0$ and the range $1\le \Delta <2048$ was used for the Fourier transform.

Figure 11

Average position of the contact line as a function of time, for ${\tilde{R}}_1={10}^2$ and ${\tilde{L}}_0=\{0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$. The arrow indicates increasing values of ${\tilde{L}}_0$. Note how fluctuations increase approaching the transition point and the dynamics becomes intermittent. The inset shows the avalanche size distribution for ${\tilde{R}}_1={10}^2$ and ${\tilde{L}}_0=0.5$, while the solid line has a slope $-2.0$.