Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow

We employ a pairwise force smoothed particle hydrodynamics (PF-SPH) model to simulate sessile and transient droplets on rough hydrophobic and hydrophilic surfaces. PF-SPH allows modeling of free-surface flows without discretizing the air phase, which is achieved by imposing the surface tension and dynamic contact angles with pairwise interaction forces. We use the PF-SPH model to study the effect of surface roughness and microscopic contact angle on the effective contact angle and droplet dynamics. In the first part of this work, we investigate static contact angles of sessile droplets on different types of rough surfaces. We find that the effective static contact angles of Cassie and Wenzel droplets on a rough surface are greater than the corresponding microscale static contact angles. As a result, microscale hydrophobic rough surfaces also show effective hydrophobic behavior. On the other hand, microscale hydrophilic surfaces may be macroscopically hydrophilic or hydrophobic, depending on the type of roughness. We study the dependence of the transition between Cassie and Wenzel states on roughness and droplet size, which can be linked to the critical pressure for the given fluid-substrate combination. We observe good agreement between simulations and theoretical predictions. Finally, we study the impact of the roughness orientation (i.e., an anisotropic roughness) and surface inclination on droplet flow velocities. Simulations show that droplet flow velocities are lower if the surface roughness is oriented perpendicular to the flow direction. If the predominant elements of surface roughness are in alignment with the flow direction, the flow velocities increase compared to smooth surfaces, which can be attributed to the decrease in fluid-solid contact area similar to the lotus effect. We demonstrate that classical linear scaling relationships between Bond and capillary numbers for droplet flow on flat surfaces also hold for flow on rough surfaces.

Figure 1

Pressure for various droplet sizes.

Figure 2

Static contact angle measurements. Here, only the droplet cross section in the $xy$ plane is shown.

Figure 3

Static contact angles for different solid-fluid interaction strengths $s_{\text{sf}}$.

Figure 4

Model verification with respect to Tanner's law: height of the droplet as a function of time. The inset shows the droplet spreading on a horizontal surface.

Figure 5

Cox-Voinov relationship for receding contact angles. The inset shows the simulation results of a plate withdrawal from a pool of liquid ($x=25\mathrm{mm}$, $y=10\mathrm{mm}$).

Figure 6

Different states of droplets depending on wetting conditions (left to right): Cassie state, Wenzel state, and Cassie-Wenzel state.

Figure 7

Surface parameters (four types, from left to right): (1) for rectangular surface—height $H$ and width $l$ of a bar; $d$—distance between bars; (2) for dual-rectangular surface—height $H$ and $l$ width of a block; $d$—distance between blocks; (3) for sinusoidal surface—period $T$ and magnitude $A$ of a sinusoidal function in the $x$ direction; (4) for dual-sinusoidal surface—period $T$ and magnitude $A$ of a sinusoidal function in the $x$ and $z$ directions.

Figure 8

Static contact angles of droplets on hydrophobic rectangular [(a)–(c)] and dual-rectangular [(d)–(f)] surfaces. Surface parameters are $d=0.2\mathrm{mm}$, $l=0.15\mathrm{mm}$ [(a), (d)]; $d=0.25\mathrm{mm}$, $l=0.2\mathrm{mm}$ [(b), (e)]; $d=0.25\mathrm{mm}$, $l=0.25\mathrm{mm}$ [(c), (f)]; $H=0.2\mathrm{mm}$ for all types of surfaces.

Figure 9

Static contact angles of droplets on hydrophobic sinusoidal [(a)–(c)] and dual-sinusoidal [(d)–(f)] surfaces. Surface parameters are $A=0.2\mathrm{mm}$, $T=0.2\mathrm{mm}$ [(a), (d)]; $A=0.2\mathrm{mm}$, $T=0.25\mathrm{mm}$ [(b), (e)]; $A=0.2\mathrm{mm}$, $T=0.3\mathrm{mm}$ [(c), (f)].

Figure 10

Static contact angles of droplets on hydrophilic rectangular [(a)–(c)] and dual-rectangular [(d)–(f)] surfaces. Surface parameters are $d=0.2\mathrm{mm}$, $l=0.15\mathrm{mm}$ [(a), (d)]; $d=0.25\mathrm{mm}$, $l=0.2\mathrm{mm}$ [(b), (e)]; $d=0.25\mathrm{mm}$, $l=0.25\mathrm{mm}$ [(c), (f)]; $H=0.2\mathrm{mm}$ for all types of surfaces.

Figure 11

Static contact angles of droplets on hydrophilic sinusoidal [(a)–(c)] and dual-sinusoidal [(d)–(f)] surfaces. Surface parameters are $A=0.2\mathrm{mm}$, $T=0.2\mathrm{mm}$ [(a), (d)]; $A=0.2\mathrm{mm}$, $T=0.25\mathrm{mm}$ [(b), (e)]; $A=0.2\mathrm{mm}$, $T=0.3\mathrm{mm}$ [(c), (f)].

Figure 12

Effective static contact angles ${\theta }_{\text{eff}}^x$ and ${\theta }_{\text{eff}}^z$ for scaling ratio $\lambda$ between $3.5\times {10}^{-3}$ and $8.5\times {10}^{-3}$ for hydrophobic rectangular and dual-rectangular (a), hydrophilic rectangular and dual-rectangular (b), hydrophobic sinusoidal and dual-sinusoidal (c), and hydrophilic sinusoidal and dual-sinusoidal (d) surfaces. Solid symbols represent droplets in a Cassie state, and open symbols represent droplets in a Wenzel state. Symbols “plus” and “cross” represent a droplet in a Cassie-Wenzel state.

Figure 13

The effective static contact angle difference of droplets on rough hydrophobic and hydrophilic surfaces. Red: hydrophilic; blue: hydrophobic; square: rectangular surface; circle: sinusoidal surface; solid: dual surface; empty: nondual surface.

Figure 14

Comparison of effective contact angles on a fine-roughness dual-rectangular surface obtained from a high-resolution (particle spacing $2.5\times {10}^{-5}\mathrm{mm}$; ${\theta }_{\text{eff}}^x=149.{36}^{\circ }$; ${\theta }_{\text{eff}}^z=150.{28}^{\circ }$) and a low-resolution (particle spacing $5\times {10}^{-5}\mathrm{mm}$; ${\theta }_{\text{eff}}^x=151.{52}^{\circ }$; ${\theta }_{\text{eff}}^z=150.{84}^{\circ }$) simulation. Green particles: solid surface; red particles: low resolution; blue particles: high resolution.

Figure 15

Droplets on a fine-roughness dual-rectangular surface. Equilibrated radii of droplets are 1.5, 1.3, 1.11, 0.92, 0.54, and $0.44\mathrm{mm}$.

Figure 16

Pinning effect of droplets on a fine-roughness dual-rectangular surface. Equilibrated radii of droplets are 1.11, 1.3, and $1.5\mathrm{mm}$.

Figure 17

Cassie-to-Wenzel transition based on critical capillary pressure and internal pressures of droplets with $R_{\text{eq}}$ ranging from $0.44$ to $1.5\mathrm{mm}$ for a fine-roughness dual-rectangular surface.

Figure 18

Cassie-to-Wenzel transition based on critical capillary pressures and internal pressures of droplets ranging from $97.00$ to $330.68\text{Pa}$ (corresponding $R_{\text{eq}}$ are ranging from $1.5$ to $0.44\mathrm{mm}$) for fine-, medium-, and coarse-roughness dual-rectangular surfaces. Open symbols: Wenzel regime; filled symbols: Cassie regime.

Figure 19

Droplet states depending on initial conditions: (a) droplets with $R_{\text{eq}}=0.64\mathrm{mm}$ are brought into contact with a rough surface; (b) droplets with $R_{\text{eq}}=0.64\mathrm{mm}$ are dropped from $1.3\mathrm{mm}$ height; (c) droplets with $R_{\text{eq}}=1.3\mathrm{mm}$ are brought into contact with a rough surface; (d) droplets with $R_{\text{eq}}=1.3\mathrm{mm}$ are dropped from $1.75\mathrm{mm}$ height.

Figure 20

Hydrophobic and hydrophilic droplets flowing on a rough rectangular surface with rectangular bars oriented parallel to the flow direction (a) and rectangular bars oriented perpendicular to the flow direction (b) at the time step $t=50000$ ($46.296\mathrm{ms}$). Surface inclination angle is $\alpha ={90}^{\circ }$.

Figure 21

Dimensionless scaling for smooth and rough hydrophobic and hydrophilic surfaces with different orientations of roughness relative to the droplet flow direction. (par.): flow parallel to the orientation of the bars; (perp.): flow perpendicular to the orientation of bars. The surface inclination angles $\alpha$ are ${10}^{\circ }$, ${20}^{\circ }$, ${30}^{\circ }$, ${40}^{\circ }$, ${50}^{\circ }$, ${60}^{\circ }$, ${70}^{\circ },{80}^{\circ }$, and ${90}^{\circ }$.