Contact angle hysteresis on doubly periodic smooth rough surfaces in Wenzel's regime: The role of the contact line depinning mechanism

We report here on the contact angle hysteresis, appearing when a liquid meniscus is in contact with doubly sinusoidal wavelike patterned surfaces in Wenzel's wetting regime. Using the full capillary model we obtain numerically the contact angle hysteresis as a function of the surface roughness factor and the equilibrium contact angle for a block case and a kink case contact line depinning mechanism. We find that the dependencies of the contact angle hysteresis on the surface roughness factor are different for the different contact line depinning mechanisms. These dependencies are different also for the two types of rough surfaces we studied. The relations between advancing, receding, and equilibrium contact angles are investigated. A comparison with the existing asymptotical, numerical, and experimental results is carried out.

Figure 1

Schematic image of a $3\mathrm{D}$ liquid meniscus, attached to a vertical rough solid plate.

Figure 2

Surface roughness factor $r$ as a function of the ratio $A/\lambda$ (all variables are in dimensionless units).

Figure 3

Schematic drawing of different CL depinning mechanisms from a CL $L_1$ toward a CL $L_2$, when a rough solid plate, described by $\left|f_1(y,z)\right|$, is withdrawing from a pool of liquid: (a) block case and (b) kink case. The sequences of lines in (a) $L_1,L_2$, and in (b) $L_1,L_{1a},...,L_2$, suggests the kinematics of depinning for each mechanism.

Figure 4

Schematic illustration of the “corresponding” equilibrium states of the liquid meniscus in the “vertical rough solid plate dipped in a tank of liquid' geometry: (a) meniscus state forming with the plate CA ${\theta }_{\text{eq}}\le {90}^{\circ }$; (b) “corresponding” meniscus state forming with the plate CA ${180}^{\circ }-{\theta }_{\text{eq}}$. (${\mathrm{\Sigma }}_s^{+}$, ${\mathrm{\Sigma }}_{lg}^{+}$, ${\mathrm{\Sigma }}_s^{-}$, and ${\mathrm{\Sigma }}_{lg}^{-}$ are defined in the text).

Figure 5

(a) 3D meniscus solution ${\mathrm{\Sigma }}_{lg}^{\left(1\right)}$(i.e., when $x\le l_d$) in the kink depinning regime is shown over one period $l=20\lambda$ along the $y$ axis. [Part of the solution enclosed in the rectangle is displayed in Fig. 6.] The plate's rough surface is described by the function $f_1(y,z)$ and ${\theta }_{\text{eq}}={60}^{\circ }$, $A/\lambda =4/30$. Among all solutions, this solution is the one with the highest averaged CL height, just preceding the CL depinning. (b) CL height $h\left(y\right)$ of the meniscus solution [displayed in (a)] on half of the interval along the $y$ axes ($l/2$)—thick solid line; CL height of the meniscus solution when $l=30\lambda$—dashed line; height $h\left(y\right)$ of an intermediate CL, which appears in the CL depinning process—thin solid (red) line. All variables are in dimensionless units.

Figure 6

Parts of the numerical solutions for the equilibrium meniscus of the highest averaged CL height in the neighborhood of the rough wall for $f_1(y,z)$ for ${\theta }_{\text{eq}}={60}^{\circ }$ in (a) the block depinning regime ($A/\lambda =11/60$) and (b) the kink depinning regime ($A/\lambda =4/30$) [this is the solution shown in Fig. 5, enclosed in the rectangle]. All variables are in dimensionless units.

Figure 7

Parts of the numerical solutions for the equilibrium meniscus of the lowest averaged CL height in the neighborhood of the rough wall for $\left|f_1(y,z)\right|$ for ${\theta }_{\text{eq}}={60}^{\circ }$ in (a) the block depinning regime ($A/\lambda =7/60$) and (b) the kink depinning regime ($A/\lambda =4/30$). All variables are in dimensionless units.

Figure 8

${\theta }_{\text{adv}}$ (empty symbols and lines) and ${\theta }_{\text{rec}}$ (solid symbols and lines) as functions of the ratio $A/\lambda$ (in dimensionless units) in the block depinning regime (shown by circles) and in the kink depinning regime (shown by stars). The plate roughness is described by $f_1(y,z)$ and ${\theta }_{\text{eq}}$ is (a) ${30}^{\circ }$: (blue) solid lines; ${60}^{\circ }$: (black) dashed lines; (b) ${45}^{\circ }$: (blue) solid lines; ${75}^{\circ }$: (black) dashed lines. The analytical results [Eqs. (7) and (8)] are displayed by the dot-dashed lines. The dotted lines show the results for Wenzel's angle.

Figure 9

${\theta }_{\text{adv}}$ and ${\theta }_{\text{rec}}$ as functions of the ratio $A/\lambda$ (in dimensionless units) when the plate roughness is described by $\left|f_1(y,z)\right|$ and when ${\theta }_{\text{eq}}$ is (a) ${30}^{\circ }$, ${60}^{\circ }$; (b) ${45}^{\circ }$, ${75}^{\circ }$. The notations are the same as in Fig. 8.