ANALYSIS FOR WETTING ON ROUGH SURFACES BY A THREE-DIMENSIONAL PHASE FIELD MODEL

. In this paper, we consider the derivation of the modiﬁed Wenzel’s and Cassie’s equations for wetting phenomena on rough surfaces from a three-dimensional phase ﬁeld model. We derive an eﬀective boundary condition by asymptotic two-scale homogenization technique when the size of the roughness is small. The modiﬁed Wenzel’s and Cassie’s equations for the apparent contact angles on the rough surfaces are then derived from the eﬀective boundary condition. The homogenization results are proved rigorously by the Γ-convergence theory.


1.
Introduction. Wetting describes the state and movement of a liquid drop or film on solid surfaces. It has many important applications in various fields such as printing and oil industry [4,14,15,24]. Wetting has been studied for centuries. In the early 19th century, a formula for the contact angle(i.e. the angle between the liquid surface and the solid substrate) on a flat homogeneous surface was derived by T. Young [31]. The Young's equation relates the static contact angle θ s to the solid-liquid interface energy γ SL , the liquid-vapor interface energy γ LV and the solid-vapor interface energy γ SV , γ LV cos θ s = γ SV − γ SL .
Later on, the formulae for the apparent contact angle on geometrically and chemically rough surfaces are proposed by R. N. Wenzel [25] and by A. Cassie and S. Baxter [8], respectively. For geometrically rough surface, the Wenzel's equation shows that the effective contact angle θ e is given in terms of static contact angle θ s by cos θ e = r cos θ s , where r is the roughness factor (ratio of the actual area to the projected area of the surface). For the chemically patterned surface composed by two materials, the Cassie-Baxter equation for the effective contact angle is given by cos θ e = λ cos θ s1 + (1 − λ) cos θ s2 , in terms of the static contact angles θ s1 , θ s2 and area fraction λ, 1 − λ of the component surfaces. 2. The phase field model. As in [28], we consider the phase field model for the equilibrium state of the two phase fluid on solid surface. This is given by the phenomenological Cahn-Landau theory [9,5]. We consider the interfacial free energy in a squared-gradient approximation, with the addition of a surface energy term in order to account for the interaction with the wall: where δ is a small parameter, φ is the composition field, f (φ) is the bulk free energy density in Ω ∈ R 3 and γ f s (φ) is the free energy density at the fluid solid interface ∂Ω. The equilibrium interface structure is obtained by minimizing the total free energy F , which results in the following Cahn-Landau equation In the total free energy functional (1), the double well functionf (φ) is chosen to be with c > 0. In this case, there are two energy minimizing phase φ = 1 and φ = −1.
The equation (2) is reduced to Generally, Young's equation on flat surface can easily be derived from the boundary condition (3), see for example [18,28]. In the following, we use the method in [28] to show such a process. For simplicity, we suppose the liquid-vapor interface is  a surface parallel to the y-axis. Let the solid surface be (x, y)-plane and the fluid region is in the upper half space(see Figure 1). Let us assume that the liquid-vapor interface intersects with the solid surface z = 0 with an angle 0 < θ s < π. When the interface thickness is small, it is reasonable to assume that the phase function φ is a one dimensional function in the direction m normal to the interface and φ does not change in the direction parallel to the interface. We let the diffuse interface meet the solid surface{z = 0}(x − y plane) on the contact line of y-axis({x = 0, z = 0}). Denote m as the unit normal to the liquid-vapor interface and n as the unit normal to the solid surface z = 0. Let m and n be the coordinates along the directions. Therefore we have φ(x) = φ(m) for x = m/ sin θ (see Figure 1). We then have ∂φ ∂n = cos θ s ∂φ ∂m on the solid boundary. Multiplying both sides of (3) by ∂φ ∂x , and integrating across the liquid-vapor interface along the solid boundary, we have Noticing that Here in the second equation, the integral in x is converted to integral in m using the relation that φ(m) = φ(x) for m = x sin θ s . Equation (6) then implies the Young's equation Here γ = ∞ −∞ δ ∂φ ∂m 2 dm denotes the interface tension between the liquid and the vapor [7]. Notice from (8), for partial wetting (i.e. 0 < θ < π), we require |γ 1 − γ 2 | < γ. If |γ 1 − γ 2 | ≥ γ, the surface is either complete wetting with θ s = 0, or complete dry with θ s = π.
As in [28], we can assume γ f s (φ) be an interpolation between γ 1 = γ f s (−1) and where In this paper, we will use the first formula, that will make the analysis below slightly simpler.
Here a, b, d are given constants with d > 0. The roughness of the boundary is modeled by a continuous, piecewise differentiable function h(x, y, y/ ) with microscopic local -periodic oscillations. We assume that h(x, y, Y ) is periodic in variables Y with period 1. We also assume h(·, which represents a rough boundary. Notice that the unit outer normal on the boundary Γ is given by We now concentrate on the behavior of the solution of the Cahn-Landau equation on the rough boundary. Therefore we will consider boundary condition (3) on Γ . On ∂Ω \ Γ , we will prescribe Dirichlet conditions. Specifically, we consider the following system with some given function ϕ. In equation (10), we assume θ s (x, y, Y ) is also a periodic function in Y with period 1. In the following, we study the behavior of the solution on the rough surface when → 0. A boundary layer will develop near the rough We will suppose that φ could be written as φ (x, y, z) =φ (x, y, z)+φ (x, y, y , z ), withφ being the oscillation of φ near the rough boundary. By introducing some fast parameters Y = y and Z = z , we suppose thatφ (x, y, Y, Z) is periodic on the variable Y with period 1, and such thatφ decay exponentally as Z → ∞.
We suppose thatφ andφ has the expansions: withφ i (x, y, Y, Z) are periodic in Y and such that lim Z→∞φi = 0 decaying exponentally. First, we consider the expansion far away from the rough boundary. Substituting the above expansion (12) into equation (10), noticing the decay ofφ , we obtain, for the leading order, the following equation Next we consider the inner expansions near the rough surface. Notice that h(x, y, y ) = h(x, y, Y ) and θ s (x, y, y ) = θ(x, y, Y ). Then Equation (10) is rewritten as Substituting the expansions (11)- (12) into (14), we have, for the leading order From standard analysis for Laplace equation, we know that Then the next order of equation (14) could be written as When → 0, the leading order outer solutionφ 0 is defined in domain Ω with a flat boundary Γ := {(x, y, x) : z = 0, a < x < b, a < y < b}(See Figure 2 b.). The solvability condition of Equation (16) gives the effective boundary condition forφ 0 on the boundary z = 0 as following.
on the homogenized surface Γ.
In summary, when → 0, we have that the leading order solution,φ 0 (x, y, z) satisfies the following equation with an effective boundary condition modified by the roughness of the surface: 4. Γ-convergence theorem for the homogenization problem. In this section, we are going to prove rigorously the convergence of the problems (10) to the problem (20) as → 0 by Γ-Convergence theory for variational minimizing problems. It is known that the elliptic equation (20) is equivalent to the following energy minimizing problem: with B(x, y) Similarly, the equation (10) is equivalent to the following energy minimizing problem: min with Here we define F (φ ) on V , not on V . This is customary in dealing with minimizing problems and is useful when considering the Γ-convergence [6,11].
The existence of minimizers to the problems (21) and (22) could be established from the standard method [16]. In this section, we are concerned mainly with the limit of the minimizers of the problems (22) as → 0. The convergence result is the following, Theorem 4.1. Let F and F be functionals defined in (21) and (23), then we have i). F are uniformly coercive in the weak topology of H 1 (Ω), i.e., for every t > 0, there exist a K t ⊂ H 1 (Ω), which is precompact in the weak topology of H 1 (Ω) and such that {φ : F (φ) < t} ⊂ K t for all > 0. ii). As → 0, the functionals F Γ-convergence to F in the weak sense of H 1 (Ω). iii). Let φ be the minimizers of F in V for all > 0, then, up to a subsequence, φ weakly convergence to some φ in H 1 (Ω) and φ is a minimizer of F . Remark 1. The statement iii) of the theorem also implies that the solutions of Equation (10) converge weakly to that of Equation (20). The theorem is a generalization of Theorem 5.1 in [28] in R 3 .
Proof of the theorem. i) The uniformly coercivity is easy to prove. We use the following inequality. For fixed δ > 0, there exists a constant C 0 > 0, such that where C 1 is a -independent constant and For any t > 0 and F (φ) < t, we have Thus and K t is precompact in weak topology in H 1 (Ω). We have proved the uniformly coercivity. ii) We first prove the lower-bound inequality. That is, for any given φ and for any sequence φ ∈ V such that φ φ in H 1 (Ω), we have If lim inf →0 F (φ ) = +∞, the inequality is obvious. Otherwise, we know that φ ∈ V and for some constant C 3 > 0. It is easy to prove the weak lower continuity for the first two terms of F . From the convexity of the energy density on ∇φ and the continuity of f (φ) on φ, we have, [16] We now consider the third term in F , Γ δγ 2 cos θ s (x, y, y ) sin πφ 2 dS = Γ δγ 2 cos θ s (x, y, y ) sin πφ (x, y, h(x, y, y )) 2 For I 1 , from the Rellich-Kondrachov theorem [2], we have, up to a subsequence, It is easily to know that, in L 2 (Γ), Thus, we know that Now we need to show that lim →0 I 2 = 0. This is easily seen from the following where C 4 is a positive constant. Combining (29)-(31), we have proved that which together with (28) imply the lower-bound inequality (26). Now we will prove the upper bound inequality. That is, for any φ ∈ V , there exists a consequenceφ φ in H 1 (Ω), and For any φ ∈ V , we defineφ in Ω as an expansion of φ, as following For simplicity, we assume that h = 0 on the boundary of Γ, so thatφ defined above belong to V . Then, we only need to prove that and Γ δγ 2 cos θ s sin Equation (34) is obvious from the definition of φ and φ ∈ H 1 (Ω), and Equation (35) could be proved similarly as Equation (32).
From the lower-bound and upper-bound inequalities, we have proved the Γconvergence of F to F . iii). By the basic theorem of Γ-convergence [6], the third conclusion is achieved immediately from i) and ii). As in the derivation of the Young's formula, we assume that the liquid-vapor interface intersects the homogenized surface Γ near the line {x = x 0 , y = 0} with an effective contact angle 0 < θ e < π. Multiplying both sides of the second equation in (20) by ∂φ ∂x , which is generally nonzero across the interface, and integrating across the liquid-vapor interface, we have cos(θ s (x 0 , Y )) 1 + (∂ X h(x 0 , Y )) 2 dY, and (from equation (7)) int∩{z=0} δ ∂φ ∂n where θ e is the apparent contact angle, Equation (36) implies that cos θ e = 1 0 cos(θ s (x 0 , Y )) 1 + (∂ Y h(x 0 , Y )) 2 dY.
(40) The factor λ(x 0 ) represents the length faction of material 1 on the contact line x = x 0 . It is easy to see that the apparent angle 0 < θ e < π, if 0 < λ < 1 and θ s1 and θ s2 do not equal to 0 and π at the same time. Equation (40) is the so-called modified Cassie's equation [29].