Coarsening dynamics of dewetting nanodroplets on chemically patterned substrates

M. Asgari and A. Moosavi

Phys. Rev. E 86, 016303 – Published 5 July 2012

Abstract

Mesoscopic hydrodynamic equations are solved to investigate coarsening dynamics of two interacting nanodroplets on chemically patterned substrates. The effects of different parameters such as the surface chemical pattern, the slip length, the profile of the disjoining pressure, the size of the droplets, and the contact angles on the coarsening are studied. Our results reveal that the presence of a chemical heterogeneity can enhance or weaken the coarsening dynamics depending on the pattern type and positions of the droplets on the substrate. Also increasing the contact angles to values larger than a critical value may qualitatively change the coarsening process, and the profile of the disjoining pressure and the slippage can appreciably modify the coarsening rate.

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Received 8 April 2012

DOI:https://doi.org/10.1103/PhysRevE.86.016303

Authors & Affiliations

M. Asgari and A. Moosavi^*

Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, P.O. Box 11365-9567, Tehran, Iran

^*moosavi@sharif.edu

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Vol. 86, Iss. 1 — July 2012

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Figure 1
Two droplets placed on a homogeneous surface. The distance between the droplets is $d$ and the sizes of the larger and the smaller droplet are equal to $a_{1}$ and $a_{2}$ , respectively. The initial radius and height of the drops are assumed to be equal to facilitate achieving the equilibrium.Reuse & Permissions
Figure 2
Some explanations about the boundary integral method used for the study. The elements are linear; that is, the nodes are at the ends of the elements and variables change linearly along the element. $p$ can be any point within the system and $q$ can be any point on the boundary $S$ , and $θ$ is the internal angle for boundary points.Reuse & Permissions
Figure 3
The effect of droplet distance on the coarsening dynamics. Droplets are located on a homogeneous surface with $θ_{e q} = 120^{\circ}$ . Collapse of the smaller droplet (a), migration of the smaller droplet (b), growth of the larger droplet (c), and migration of the larger droplet (d) are compared for various distances. By increasing the distance, the collapse time increases. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , and $B = 0$ .Reuse & Permissions
Figure 4
The collapse time vs droplet distance. $t_{c}$ represents the time required for the smaller droplet to vanish and $d$ is the droplet distance. The small circles are the numerical results and, as can be seen, the coarsening time increases linearly with distance. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , and $B = 0$ .Reuse & Permissions
Figure 5
The effect of the contact angle size on the coarsening dynamics. Droplets are located on a homogeneous surface. Collapse of the smaller droplet (a) and migration of the larger droplet (b) are compared for various contact angles. As can be seen, for contact angles larger than $θ_{e q} = 75 . 1^{\circ}$ the direction of migration changes. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 6
The larger droplet overall migration and smaller drop collapse time vs the surface contact angle. The small circles represent the numerical results. It is seen that there is a change in the migration direction at $θ_{e q} = 75 . 1^{\circ}$ . For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ . and $B = 0$ .Reuse & Permissions
Figure 7
The effect of the disjoining pressure parameters $B$ and $C$ on the coarsening dynamics. Droplets are located on a homogeneous surface. Collapse of the smaller droplet (a) and migration of the larger droplet (b) are compared for various parameters of DJP. The values of $B$ and $C$ are chosen such that in all cases $θ_{e q} = 120^{\circ}$ . ( $B, C$ ) values for the minus case are ( $- 1, 1.9$ ), ( $0, 4.0$ ), and ( $1, 11.63$ ) and for the plus case they are ( $- 2.5, 6.38$ ) and ( $- 4, 1.39$ ). For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , and $d = 5$ .Reuse & Permissions
Figure 8
The effect of favorable (I) and unfavorable (II) wettability gradient on the coarsening dynamics. $α$ is the gradient size used to define wettability as $θ (x) = {[110 + α (x - 100)]}^{\circ}$ (I) and $θ (x) = {[110 - α x]}^{\circ}$ (II). Collapse of smaller droplet (a) and migration of larger droplet (b) are compared for various gradient sizes. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 9
Distance between the droplets during the coarsening process for the favorable gradient case. The results for the homogeneous case has also been included in the figure for comparison purpose. The contact angle is $θ (x) = {[110 + α (x - 100)]}^{\circ}$ . For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 10
The collapse time vs favorable (I) and unfavorable (II) wettability gradient size along the surface. $t_{c}$ represents the time required for the smaller droplet to vanish. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 11
The effect of slip boundary condition on the coarsening. $β$ is the slip length. The chemical gradient considered along the substrate is $θ (x) = {[110 + 0.1 (x - 100)]}^{\circ}$ . Collapse of the smaller droplet (a) and migration of the larger droplet (b) are compared for a range of slip lengths. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 12
The collapse time as a function of the slip length. $t_{c}$ represents the time required for smaller droplet to vanish. $t_{c}$ can be well described in terms of a power-law function of the slip length. The chemical gradient considered along the substrate is $θ (x) = {[110 + 0.1 (x - 100)]}^{\circ}$ . For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 13
Two series of droplets considered on a substrate with a favorable chemical gradient ( $α = 0.021$ , $θ_{\max} = 110^{\circ}$ , and $x_{m} = 100$ ) to investigate the effect of droplet size on the coarsening while keeping the size ratio of the droplets fixed. In the first series $a_{1} = 10$ and $a_{2} = 5$ and in the second series $a_{1} = 15$ and $a_{2} = 7.5$ . For case (I) of the first series the distance between the center of mass of the droplets is equal to that of the second series while for case (II) the distance between the edges, $d$ , is equal to that of the second series.Reuse & Permissions
Figure 14
The effect of droplet size on the coarsening rate. Cases (I) and (II) in the figures refer to the first series of droplets ( $a_{1} = 10$ and $a_{2} = 5$ ). In case (I) the distance between the center of mass of the droplets is the same as that of the larger droplets ( $a_{1} = 15$ and $a_{2} = 7.5$ ) but in case (II) the droplets have an edge-to-edge distance equal to that of the larger droplets. Collapse of smaller droplet (a) and migration of larger droplet (b) are compared to show the size effect. Increasing the size of the droplets does not change the velocity of the larger droplet during the coarsening. However, the coarsening rate has been decreased considerably. For all the considered cases $B = 0$ .Reuse & Permissions
Figure 15
Two droplets placed on a chemical substrate to investigate the effect of this heterogeneity on the coarsening process. In case (I) the heterogeneous part is below the larger droplet and in case (II) the chemical part is below the smaller droplet.Reuse & Permissions
Figure 16
The effect of a more wettable part, which is placed below the larger droplet (upper cases) or the smaller drop (lower cases), on the coarsening dynamics. $Φ = θ_{1} - θ_{2}$ is the contact angle difference between the less ( $θ_{1}$ ) and more ( $θ_{2}$ ) wettable parts of the substrate. Collapse of the smaller droplet (a) and migration of the larger droplet (b) are compared for various values of the wettability change. For all the considered cases the size of the heterogeneity is equal to 14 for (I) and 6 for (II), which is smaller than the base width of the droplet, and $θ_{1} = 120^{\circ}$ , $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 17
The collapse time vs the contact angle difference $Φ = θ_{1} - θ_{2}$ for the case of a more wettable chemical heterogeneity (I) below the larger droplet and (II) below the smaller droplet. The behavior of the system can be well described by a power-law-type distribution. For all the considered cases the size of the heterogeneity is equal to (I) 14 or (II) 6 and $θ_{1} = 120^{\circ}$ , $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 18
The effect of a less wettable stripe on the coarsening dynamics. The contact angle of the stripe is equal to $130^{\circ}$ and the rest parts of the substrate have contact angles equal to $120^{\circ}$ . The less wettable stripe is located below (a) the larger drop and (b) the smaller one, respectively. For all the considered cases $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , $B = 0$ , and the stripe size is equal to 14 and 6 for the cases (a) and (b), respectively. In the figure the times during the transient evolution of the system are also shown.Reuse & Permissions
Figure 19
Two droplets placed on a chemical substrate to investigate the effect of the heterogeneity on the coarsening process. In case (I) the heterogeneity is located between two droplets and in (II) the heterogeneity is in the form of a chemical step.Reuse & Permissions
Figure 20
The effect of the chemical step on the coarsening dynamics. $Φ = θ_{1} - θ_{2}$ represents the contact angle difference between the less ( $θ_{1}$ ) and more ( $θ_{2}$ ) wettable parts of the substrate. Collapse of the smaller droplet (a) and migration of the larger droplet (b) are compared for various values of the wettability change. For all the considered cases $θ_{2} = 120^{\circ}$ , $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions
Figure 21
The collapse time vs the contact angle difference $Φ = θ_{1} - θ_{2}$ for a chemical step. $t_{c}$ represents the time required for the smaller droplet to vanish. The small circles are the numerical results. A power-law distribution is used to model the behavior of the system. For all the considered cases $θ_{2} = 120^{\circ}$ , $a_{1} = 10$ , $a_{2} = 5$ , $d = 5$ , and $B = 0$ .Reuse & Permissions

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