Wetting theory for small droplets on textured solid surfaces

Conventional wetting theories on rough surfaces with Wenzel, Cassie-Baxter, and Penetrate modes suggest the possibility of tuning the contact angle by adjusting the surface texture. Despite decades of intensive study, there are still many experimental results that are not well understood because conventional wetting theory, which assumes an infinite droplet size, has been used to explain measurements of finite-sized droplets. Here, we suggest a wetting theory applicable to a wide range of droplet size for the three wetting modes by analyzing the free energy landscape with many local minima originated from the finite size. We find that the conventional theory predicts the contact angle at the global minimum if the droplet size is about 40 times or larger than the characteristic scale of the surface roughness, regardless of wetting modes. Furthermore, we obtain the energy barrier of pinning which can induce the contact angle hysteresis as a function of geometric factors. We validate our theory against experimental results on an anisotropic rough surface. In addition, we discuss the wetting on non-uniformly rough surfaces. Our findings clarify the extent to which the conventional wetting theory is valid and expand the physical understanding of wetting phenomena of small liquid drops on rough surfaces.

for each wetting mode. are set to satisfy the roughness factors (r, f) in the legend. The contact angle difference between the proposed theory and the conventional theory ( ) converges to ~ ∘ when .

Supplementary Note 1: Free energy comparison between circular boundary and vertical boundary
It was reported that the liquid droplet forms the part of the circle when the effect of the gravity is neglected 1 . However, it is mathematically very hard to model the liquid droplet on periodic rectangular protrusion with circular boundary with the constant droplet volume constraint. Therefore, we used vertical liquid boundary (path B in Fig. S1a) to model the droplet in Wenzel mode (CB mode and P mode do not need the assumption) rather than the circular liquid boundary (path A in Fig. S1a). To check the validity of our assumption, we compare the free energy of two boundary for a simple case where the three-phase contact line touches the bottom of the groove.
The free energy of each liquid boundary can be formulated as below.

sin sin
Here, and refer to the free energy of the circular and the vertical boundary. The free energy can be formulated by E σ . α refers to the angle between circular boundary within the groove as depicted in Fig. S1a. The relative error between two energy barriers can be defined by . The relative error between two boundaries are calculated for the surface with r=1.5, when ̅ 0.5 (small droplet) and ̅ 0.005 (large droplet). The region with high contact angle (θ 140 ∘ ) when the droplet volume is small is not investigated because α cannot be decided in the case because the three-phase contact line cannot reach the floor of the groove. As shown in Fig. S1b and Fig. S1c, the relative error is smaller than 15% unless the surface is highly hydrophilic (θ 60 ∘ ) or highly hydrophobic (θ 120 ∘ ). Because the droplet on the highly hydrophilic (hydrophobic) surface prefers P (CB) mode, we can conclude that our assumption is reasonable within 15% of errors.

1-1. Cassie-Baxter & Penetrate mode
From Young's relation 2 , one can say the free energy from same area of liquid -solid boundary is always larger than that from the liquid -vapor boundary for any substrate when θ 0 ∘ . Because Case II always contains larger n than in Case I, regardless of L (Fig. 2b), Case I contains less area of the liquid-vapor boundary. Consequently, Case I become more stable if the same θ is assumed. In the case of Penetrate mode, on the contrary, Case II become more stable because the liquid-vapor boundary is substituted by a liquid-liquid boundary, which contains 0 surface energy.

1-2. Wenzel mode
If the same amount of n is assumed, Case I always has a larger L or smaller θ than Case II. As we noted in the text, the E θ graph contains convex contours of conserved n, where the minimum is at θ (red dotted in Figs. S1a, S1b). Because Case I has a larger θ for conserved n, one can notice the convex contour of Case II is always on the right side of Case I for conserved n. Recalling that the Wenzel mode has a larger equilibrium contact angle than θ , when the substrate is hydrophilic, the free energy of the liquid decreases as θ increases.
Therefore, the local free energy minimum near the equilibrium contact angle is on the right side of the contour, Case II. If a hydrophobic substrate is assumed, by the opposite logic, the local free energy minimum is on Case I.

Supplementary Note 3: Convergence to conventional theory with respect to for each wetting modes
As depicted in Figs. 2a-c, the predicted contact angle from the proposed theory (θ ) converges to the contact angle from the conventional theory (θ when ̅ 0.025. We are not able to find a closed form expression for the difference between the two contact angles (Δθ θ θ ), but find that the envelope of the oscillating Δθ curve decreases with ̅ for all wetting modes (Fig. S2). The contact angle difference converges with vibration and becomes 1~2 ∘ range when ̅ 0.025.

Supplementary Note 4: Free energy and curvature radius calculation about
The curvature radius of the liquid droplet of each wetting mode (W,CB,P) can be formulated with a circular trace of the boundary of the liquid and the constant volume constraint as Shahraz et al. 1 reported. In Wenzel mode, the sum of the area of the circular part and of the groove should be constrained. Therefore, the curvature radius of the Wenzel mode R can be formulated as follows.

R π nG H θ sin θcosθ
After the curvature radius is formulated, the free energy of the droplet can be calculated by summing up the free energy from the liquid-vapor boundary and the liquid-solid boundary as follows. Because the n θ relation is known if the liquid tip condition (Case I or Case II) is suggested, a set of (R , n, E ) can be shown with respect to a specific θ. In the case of CB mode and P mode, because R , , E , , and n are explicitly expressed by θ, a pair of (R , , n, E , ) can be easily obtained for a specific θ. In W mode, R , E , and n are in an implicit relation; therefore we adopted the bisection method to numerically calculate the set of (R , n, E ). With this process, the free energy of the droplet can be numerically obtained from a specific θ and can be used to find the contact angle of the minimum free energy.