Numerical Study of Vapor Condensation on Patterned Hydrophobic

Jul 21, 2014 - Vapor condensation on solid surfaces plays a crucial role across a wide range of industrial applications. Recent advances of nanotechno...
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Numerical Study of Vapor Condensation on Patterned Hydrophobic Surfaces Using the String Method Yunzhi Li† and Weiqing Ren*,†,‡ †

Department of Mathematics, National University of Singapore, Singapore 119076 Institute of High Performance Computing, Singapore 138632

‡

S Supporting Information *

ABSTRACT: Vapor condensation on solid surfaces plays a crucial role across a wide range of industrial applications. Recent advances of nanotechnology have made possible the manipulation of the condensation process through the control of surface structures. In this work, we study vapor condensation on hydrophobic surfaces patterned with microscale pillars. The critical nuclei, the activation barriers, and the minimum energy paths are computed using the climbing string method. The effects of pillar height, interpillar spacing, the level of supersaturation, and the intrinsic wettability of the solid surface on the nucleation process are investigated. Two nucleation scenarios are obtained from the computation. In the case of high pillar, narrow interpillar spacing, low supersaturation, and/or low surface wettability, the critical nucleus prefers the suspended Cassie state; otherwise, it prefers the impaled Wenzel state. A comparison of the nucleation barrier with that on a flat surface of the same material reveals that vapor condensation is inhibited by the microstructures in the former case, while enhanced in the latter case. The critical values of the pillar height, the interpillar spacing, and the supersaturation at which the critical nucleus changes from the Cassie state to the Wenzel state are identified from the phase diagram of the critical nucleus. It is found that the dependence of the critical interpillar spacing on the supersaturation follows closely the curve of the critical radii in a homogeneous nucleation. The relaxation dynamics of the condensate after the critical nucleus is formed is computed by solving the steepest descent equation. It is observed that when the pillar is low and/or the interpillar spacing is wide, a condensate initially in the Cassie state may evolve into the Wenzel state during the relaxation.

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INTRODUCTION Vapor condensation is ubiquitous in nature and plays an important role in a wide range of applications. It has been the subject of theoretical and experimental studies for a long time. As a first-order phase transition, the condensation occurs via nucleation followed by the growth of the nucleus. The phase transition is a rare event, since it involves the crossing of an energy barrier in order to form the critical nucleus. It may occur via homogeneous nucleation or heterogeneous nucleation, depending on whether the nucleus is formed in the bulk of the vapor phase or in contact with a foreign object.1 In particular, a critical nucleus may form on a solid substrate. The microscopic nucleation process has been investigated using computer simulations, such as biased molecular dynamics, Monte Carlo methods in conjunction with umbrella sampling techniques, and constrained minimization.2−17 In this work, we study the mechanism of vapor condensation on pillared hydrophobic surfaces using the string method.18−20 Superhydrophobic surfaces patterned with microscale structures have attracted much attention in both industry and the scientific community in recent years, due to their unique wetting properties and a wide range of applications, such as selfcleaning, defrosting, and anti-icing.21−24 It is well-known that the hydrophobicity of a solid surface can be greatly enhanced when the surface is patterned with nano- or microscale © 2014 American Chemical Society

structures. On such a surface, water can exhibit either the impaled Wenzel state25 or the suspended Cassie state,26 depending on whether vapor or air is trapped in the microstructures. The mechanism of the wetting transition between the Wenzel and Cassie states has been extensively studied using experimental and theoretical approaches,27−36 as well as novel rare-event type of computer simulations.37−41 Most of these studies assumed the pre-existence of a droplet on the solid surface. In this work, we study the effect of the microstructures on the vapor condensation process. In a supersaturated environment, vapor prefers condensation on a solid surface rather than directly from the bulk of the vapor phase, because of the lowered activation energy of the heterogeneous nucleation in comparison with the homogeneous nucleation. If the condensation occurs inside the microstructures, the surface may lose its superhydrophobic property.42−44 The effect of microstructures on the condensation process has been studied in recent years, mostly by experiments.45−53 In a recent work by Guo et al.,54 the authors studied the effect of the surface structure using a lattice density functional model. The free energy as a function of the volume Received: May 24, 2014 Revised: July 17, 2014 Published: July 21, 2014 9567

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of the nucleus was computed using a constrained minimization, and the critical nucleus was identified from the maximum of the computed free energy. While two nucleation scenarios, in which the critical nucleus takes either the Cassie state or the Wenzel state, were successfully observed, this approach relied on the prescription of the volume of the nucleus as a reaction coordinate, which may lead to an inaccurate estimate of the critical nucleus and the nucleation barrier.55 Furthermore, the sequence of minimizers computed using the constrained minimization, which is parametrized by the volume of nucleus, has no dynamic significance. In this work, we study the nucleation process and accurately compute the critical nuclei, the energy barriers, and the minimum energy path (MEP) using the string method. The string method was designed for the computation of MEPs between metastable states18,19 and the search of saddle points around a given minima.20 For a given potential or free energy, G(ϕ), the MEP, denoted by φ, is a curve in the configuration space which connects critical points of the energy G and along which the force −∇G is parallel to the curve. The MEP satisfies the equation ∇G(φ) − (∇G , τ )̂ τ ̂ = 0

Article

MATHEMATICAL MODEL

The system under study consists of the two-phase fluids and a solid substrate. The solid surface is patterned with a square lattice of rectangular pillars (see Figure 1). The pillars have a square lateral cross section. The width, height, and spacing of the pillars are denoted by w, b, and s, respectively.

Figure 1. Solid substrate patterned with a square lattice of rectangular pillars. The pillars have a square lateral cross section with width w. The height and spacing of the pillars are denoted by b and s, respectively.

The coexistence of the liquid and vapor phases is modeled using a diffuse interface model. In this model, the density of the fluids is represented by the function ϕ(x), and the grand potential is given by58

(1)

where τ̂ is the unit tangent vector to the curve φ, and (·,·) is the Euclidean inner product. Using the large deviation theory,56 it can be shown that, for the overdamped Langevin dynamics and in the small noise limit, the MEP is the most probable transition path, in the sense that the probability that the transition follows any other path is exponentially small. In the string method, the MEP is computed by evolving a string, i.e., a smooth curve with the intrinsic equal arclength parametrization, to the steady state using the steepest descent dynamics in the path space. Recently, an extension of the string method, called the climbing string method, was proposed for the search of saddle points around a given minima.20 The string method has been demonstrated to be an efficient numerical tool for the computation of MEPs and the transition states. It was applied to study phase transitions in various systems, e.g. the magnetization reversal in sub-micrometer-sized ferromagnetic elements,57 the liquid−vapor transition in capillary tubes,58 and, more recently, the Cassie-to-Wenzel wetting transition on pillared surfaces.59 In this work, we use the climbing string method to compute the critical nuclei, the activation barriers, and the MEP for vapor condensation on pillared solid surfaces. We study the problem on the scale of the microstructures. The microstructures are modeled explicitly, and we adopt a continuum description for the fluids using the mean-field theory. Starting from the vapor phase, which is a local minima of the grand potential, we use the climbing string method to compute the saddle point of the grand potential and, at the same time, the MEP connecting the minima and the saddle point. From the saddle point, we obtain the critical nucleus and the activation barrier. The MEP gives the pathway for the formation of the critical nucleus. Once the critical nucleus is identified, the subsequent growth of the nucleus that leads to the liquid phase is computed by solving the steepest descent dynamics with the (perturbed) saddle point as the initial condition. The paper is organized as follows. We first describe the mathematical model and the numerical method. This is followed by a presentation of the numerical results and discussions. In the last section, draw our conclusions.

G(ϕ) =

⎛

⎞

∫Ω ⎝ 12 κ|∇ϕ|2 + f (ϕ) − μϕ⎠ dx ⎜

⎟

(2)

where Ω denotes the physical domain occupied by the fluids. In the above integral, the first term measures the excess free energy density associated with the inhomogeneity of the fluids, the function f(ϕ) is the energy density of the homogeneous phases, and μ is the supersaturation (or the chemical potential measured from the liquid−vapor coexistence). The function f takes the form of a double-well potential: β 2 ϕ (ϕ − 1)2 (3) 2 The two minima of f correspond to the two fluid phases at bulk coexistence: ϕ = 1 for the liquid phase and ϕ = 0 for the vapor phase. The thickness of the liquid−vapor interface is determined by the parameters κ and β: d ∝ (κ/β)1/2. In the paper, we use κ = 10−4 and β = 1. As the system is biased by the supersaturation μ, one of the liquid or vapor phases becomes metastable or even unstable, depending on the value of μ. In particular, the binodal (liquid−vapor coexistence) and spinodal (the boundary of metastability) points for homogeneous bulk phases are given by μ = 0 and μ± = ±√3β/18 ≈ ±0.096β, respectively. A negative value of μ biases the system toward the vapor phase (undersaturated), and a positive value of μ biases the system toward the liquid phase (supersaturated). The boundary condition for the density field at the solid wall determines the wettability of the solid surface. In this work, we use the Dirichlet boundary condition at the solid wall:60 f (ϕ) =

ϕ = ϕs

(4)

Various wettability can be realized by varying ϕs in the range 0 ≤ ϕs ≤ 1, where ϕs = 0 corresponds to a nonwetting surface and ϕs = 1 corresponds to a complete wetting surface. With this boundary condition, there is a transition layer between the solid surface and the bulk fluids (see Figure 2). On a flat surface, the equilibrium contact angle (when μ = 0) is given by60 cos θ = −1 + 6ϕs 2 − 4ϕs 3 9568

(5)

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connecting the minima and the saddle point, which describes the pathway for the formation of the critical nuclei. Once a critical nucleus is located, the subsequent growth of the nucleus that leads to the liquid phase is computed by solving the steepest descent dynamics in eq 7. The computational domain Ω is discretized using a uniform mesh with equal grid size h in the x, y, and z directions. The mesh from a side view is illustrated in Figure 3. The set of grid

Figure 2. Side view of the critical nucleus on a flat solid surface. The supersaturation μ = 0.03, the density field at the solid wall is ϕs = 0.2. The gray scale indicates the density of the fluids, with the black color being close to 1 and the white color being close to 0.

For example, the value ϕs = 0.4 yields a static contact angle of about 125° on a flat surface. We use periodic boundary condition in the directions parallel to the wall (i.e., the x and y directions in Figure 1). Since the main objective of this work is to study vapor condensation on the solid surface, we assume that the fluid is in the vapor phase in the far field; therefore, we fix ϕ = 0 at the upper boundary of the computational domain. The minima and saddle points of the grand potential correspond to the (meta)stable and transition states of the liquid−vapor system. These are critical points of G, which satisfy the Euler−Lagrange equation −

δG = κ ∇2 ϕ − βϕ(ϕ − 1)(2ϕ − 1) + μ = 0 δϕ

Figure 3. Side view of the mesh covering the computational domain. The density field is only computed on the grid points marked by black dots. On the grid points marked with open circles, the density field is given by the boundary conditions.

points {(xi,yj,zk)} in the interior of the computational domain (the black dots) is denoted by Ωh. The phase field function on the grid point (xi,yj,zk) is denoted by ϕi,j,k. In the computation, the integral in the grand potential is approximated using the midpoint numerical quadrature. The spatial derivatives are approximated using the centered finite difference. For example, the derivative ∂ϕ/∂x at the midgrid point (xi+1/2,yj,zk) is approximated by

(6)

in the fluid domain. The minima of the grand potential can be computed using traditional optimization methods, e.g. quasiNewton methods, or simply by solving the steepest descent dynamics ∂ϕ = κ ∇2 ϕ − βϕ(ϕ − 1)(2ϕ − 1) + μ ∂t

∂ϕ 1 ≈ (ϕi + 1, j , k − ϕi , j , k ) ∂x h

(8)

and similarly for the other derivatives. After the discretization, the grand potential G(ϕ) becomes a multivariable function, denoted by Gh(ϕh), where ϕh is the vector consisting of all the unknowns {ϕi,j,k} for (xi,yj,zk) ∈ Ωh. With a slight abuse of notation, we will drop the subscript h and simply use ϕ to denote this long vector and use G to denote the discretized grand potential. For the discretized grand potential G(ϕ), we use the climbing string method to compute the saddle point around the minima corresponding to the vapor phase and the MEP connecting the minima and the saddle point. This is done by evolving a string φ in the d-dimensional configuration space, where d is the length of the vector ϕ. The string is a curve in the configuration space which is parametrized by its normalized arc length α. The string is evolved according to the fictitious dynamics

(7)

for the steady state, with the boundary conditions given above and the appropriate initial state. When the level of supersaturation is not too high (specifically, μ < μ+), the vapor phase is a (local) minima of the grand potential. In this state, the density field is close to zero everywhere in the fluid domain, except near the solid wall, where it deviates from zero due to the boundary condition ϕ = ϕs at the wall. The saddle points of the grand potential connected to the vapor phase correspond to the critical nuclei for the vapor condensation. Mathematically, these saddle points are critical points of G and are simply described by the Euler−Lagrange equation (eq 6), but computationally, it is a rather challenging task to locate these saddle points, since they are unstable stationary points of the grand potential. Some numerical methods have been proposed to tackle this difficulty. In this work, we use the climbing string method to search for the saddle points.20

∂φ (α , t ) = −∇G(φ) + λτ ̂, ∂t

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for 0 < α < 1

(9)

with the boundary conditions φ (α , t ) = a ,

NUMERICAL METHODS The computation consists of two steps. First, starting from the vapor phase, a minima of the grand potential, we use the climbing string method to search for the saddle points (i.e. the critical nuclei) around it. This step also gives the MEP

at α = 0

(10)

and ∂φ = −∇G(φ) + 2(∇G , τ )̂ τ ̂, ∂t 9569

at α = 1

(11)

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Figure 4. Snapshots along the MEP for the formation and subsequent growth of the critical nucleus. The four images in each row represent one MEP. Images labeled by a2, b2, c2, d2, and e2 are the critical nuclei. The supersaturation is μ = 0.03. The interpillar spacing and pillar height are (a) s = 0.05, b = 0.12; (b) s = 0.09, b = 0.12; (c) s = 0.10, b = 0.24; (d) s = 0.15, b = 0.24; and (e) s = 0.19, b = 0.24, from top to bottom, respectively. The droplets are represented by the isosurface of ϕ = 0.5. The images in different rows are not on the same scale.

In the above equations, the string φ is parametrized by its normalized arc length α; τ̂ = ∂αφ/|∂αφ| is the unit tangent vector to the string; λ is the Lagrange multiplier associated with the equal arc length parametrization. The potential force −∇G is given by eq 6 after the discretization of the spatial derivatives using the centered finite difference. The evolution of the two end points of the string is governed by eqs 10 and 11. The initial state at α = 0 is fixed at the given minima a (the vapor phase), and the final state of the string at α = 1 is evolved according to the modified force on the righthand side of eq 11. In the modified force, (∇G, τ̂)τ̂ is the projection of ∇G onto the tangent direction of the string; as a result, the potential force acting on the final state is reversed in the direction tangent to the string. This makes the final state climb uphill toward a saddle point on the energy surface. In the subspace perpendicular to the string, the system relaxes following the original steepest descent dynamics. Starting from some initial string, eqs 9−11 are solved until the steady state is reached. In the computation, the string is discretized into N + 1 images: {ϕk0,ϕk1,···,ϕkN}, where ϕkl is the image at αl = l/N for l = 0, 1, ···, N and at the time tk = kΔt with Δt being the time step; ϕk0 and ϕkN are the initial and final states of the string, respectively. After the discretization of the string,

the dynamical equation eq 9 is solved using a time-splitting scheme:18−20 (1) Evolve each image by one time step using the forward Euler method ϕl* = ϕlk − Δt ∇G(ϕlk ),

l = 1, 2, ···, N − 1

(12)

ϕN* = ϕNk − Δt ∇G(ϕNk ) + 2Δt(∇G(ϕNk ), τN̂ )τN̂

(13)

where τ̂N is the unit tangent vector at the final state of the string τN̂ =

ϕNk − ϕNk − 1 |ϕNk − ϕNk − 1|

(14)

(2) Redistribute the images {ϕ*0 ,ϕ*1 ,···,ϕ*N } along the string according to the equal arc length parametrization to obtain the k+1 k+1 string {ϕk+1 0 ,ϕ1 ,···,ϕN } at the new time step. This is done using polynomial interpolation. The second step is to enforce the equal arc length parametrization (i.e. the Lagrange multiplier term in eq 9). This makes the discrete images uniformly distributed along the string. For the algorithmic details of the reparametrization step, we refer to the string method.18−20 The above two steps are repeated until the steady state is attained. At the steady state, the final point ϕN locates a saddle 9570

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point of the grand potential G, and the string, which is represented by the discrete images {ϕ0,ϕ1,···,ϕN}, converges to the MEP connecting the minima and the saddle point. The MEP gives the optimal pathway for the formation of the critical nucleus from the vapor phase. After passing the saddle point, the nucleus continues to grow and the system relaxes to the liquid phase spontaneously. This relaxation dynamics is computed by solving the steepest descent dynamics in eq 7, with the perturbed saddle point as the initial condition. Specifically, the saddle point is perturbed in the unstable direction given by the MEP: ϕinit = ϕN + ετ̂N, where τ̂N is the unit tangent vector given in eq 14 and ε is a small parameter. In this work, the steepest descent dynamics is solved using the forward Euler method, and the spatial derivatives in eq 7 are discretized using the centered finite difference.

each row are snapshots along the minimum energy path: one before the formation of the critical nucleus, the critical nucleus (the second image), and two for the growth of the nucleus after passing the saddle point (the last two images). We make several observations from the numerical results. First, when the interpillar spacing is narrow, the nucleation process is initiated on the top of the pillars (cases a and b in rows 1 and 2); on the other hand, when the spacing is wide, the nucleation starts from the bottom of the grooves (cases c, d, and e in rows 3, 4, and 5, respectively). In the former cases, the critical nucleus exhibits the Cassie wetting state, in which the droplet sits on the top of the pillars and vapor is trapped in the grooves under the droplet. In the latter cases, the critical nucleus exhibits the Wenzel wetting state and the grooves are filled with liquid. We note that the critical nucleus may also occur on top of a single pillar (not shown), especially when the interpillar spacing is large. However, the energy barrier of this type of nucleation is in general larger than that shown in Figure 4. The second observation we make from Figure 4 is that at some intermediate spacings (case c), the condensation starts from the bottom of the grooves but the nucleus grows all the way to the top of the pillars. The critical nucleus has the features of both Cassie and Wenzel states: On one hand, a droplet is formed on top of the pillars, similar to the Cassie state; on the other hand, the grooves under the droplet are filled with liquid, which is a signature of the Wenzel state. It is interesting to note that when the interpillar spacing is further increased by 0.01, which is only one grid size, the critical nucleus suddenly changes to a much smaller droplet, which falls completely within the grooves (not shown but similar to case d). When the spacing is further increased and becomes larger than the critical nucleus, the condensation starts from one side of a pillar and the critical nucleus becomes asymmetric, which touches one pillar only (case e). A nucleus initially in the Cassie state may evolve into a Wenzel state after passing the saddle point. An example is shown in the second row of Figure 4 (case b). The critical nucleus is a droplet in the Cassie state, but as the droplet continues to grow, liquid gradually fills in the grooves under the droplet and the nucleus evolves into a Wenzel state (the last figure). This Cassie-to-Wenzel transition occurring in the relaxation stage happens when the interpillar spacing is relatively wide (but still narrow so that the critical nucleus is in the Cassie state). This is in contrast to the numerical results of Guo et al.,54 where the Wenzel-to-Cassie transition was observed during the subsequent growth of the droplet after passing the saddle point. In our computation, however, the Wenzel-to-Cassie transition was never observed in the relaxation dynamics. An activation energy is needed in order for a droplet in the Wenzel state to evolve into a Cassie state, since the Wenzel state is energetically preferred due to the contact with the solid wall. The grand potential along the MEPs is shown in Figure 5. The maxima of each curve corresponds to the critical nucleus. It also gives the energy barrier for the formation of the critical nucleus. In each curve, the MEP from the initial state to the saddle point is computed using the climbing string method; the MEP after the saddle point is first computed by solving the steepest descent dynamics and then reparametrized using the arc length of the path. Note that the right end of each curve is not a minima of the grand potential; the grand potential will further decrease as the system continues to evolve toward the liquid phase.

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RESULTS AND DISCUSSION In the computation, the grid size of the mesh covering the computational domain is h = 0.01, the pillar width is fixed at w = 0.04, and the parameters in the grand potential are κ = 10−4 and β = 1. The string is discretized into N + 1 = 11 images, including the two end points. Unless otherwise specified, the density field at the wall is ϕs = 0.3. Other parameters, such as the pillar height b, the interpillar spacing s, and the supersaturation μ, are specified later when we discuss the numerical results. To construct the initial string, we first make a small perturbation to the minima (the vapor phase) and then join the minima and this perturbed state using a straight line in the configuration space. Specifically, we perturb the minima by increasing the density by 0.01 at one grid point, while keeping the density unchanged at all other grid points. The grid point where the density is increased plays the role of a nucleation seed. It is chosen either near the top of a pillar or near the bottom of the groove. The former choice makes the condensate preferentially converge to a Cassie state, whereas the later choice makes it more likely converge to a Wenzel state. We note that for a complex energy landscape, many saddle points may coexist. The saddle point found by the climbing string method depends on the choice of the initial string. It is always possible that some saddle points (especially those with high energy) exist but are not found by this procedure due to the lack of knowledge of those saddle points. To illustrate the features of the diffuse interface model, we first present an example of the heterogeneous nucleation on a flat solid surface. The critical nucleus computed using the climbing string method is shown in Figure 2. Note that there exists a boundary layer between the solid wall and the bulk fluids due to the Dirichlet boundary condition at the wall; this is similar to the transition layer between the vapor phase and the liquid phase. The width of the transition/boundary layer is determined by the values of κ and β [d ∼ (κ/β)1/2)]. The energy barrier at the critical nucleus is 7.66 × 10−5. We also computed the critical nucleus in a homogeneous nucleation from the bulk of the vapor phase using the climbing string method. The critical nucleus has a spherical shape (not shown), and the energy barrier is 8.21 × 10−5. Comparing the two energy barriers, we see that the heterogeneous nucleation is preferred and the presence of the solid wall enhances vapor condensation. Typical Nucleation Scenarios. Next we turn to vapor nucleation on pillared surfaces. Typical scenarios of the nucleation process are shown in Figure 4. The images in 9571

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the Cassie state and the Wenzel state, respectively, while at intermediate spacings in region II, the critical nuclei exhibit the features of both Cassie and Wenzel states. Typical configurations of the critical nuclei in these three regions are shown in the lower panel of the figure. From the phase diagram, we also see that the wetting state of the critical nucleus is more sensitive to the interpillar spacing rather than the pillar height. In particular, the critical nuclei are always in the Cassie state at small spacings (e.g., s = 0.03, 0.04), and in the Wenzel state at large spacings (e.g., s = 0.11, 0.12), in the range of pillar height shown in Figure 6. The nucleation barriers are shown in Figure 7. The two figures in the upper panel show the energy barrier as a function of the interpillar spacing s at fixed pillar height b = 0.07 (left) and 0.12 (right). In the regions indicated by circles (Cassie state) and squares (Wenzel state), the energy barrier increases with s; this is mainly due to the decrease of the contact area between the critical nucleus and the solid wall. When the spacing is so large that the Wenzel-type of critical nucleus only touches one pillar, as shown in case e of Figure 4, the nucleation barrier reaches a plateau (not shown); it will remain unchanged if the spacing is further increased. In the intermediate region indicated by triangles (region II), the energy barrier decreases with w; this is due to the relaxation of the confinement of the critical nucleus by the pillars. We also observe that at certain values of interpillar spacing, different types of critical nuclei may coexist. For example, at b = 0.12 and s = 0.07, we obtained two critical nuclei, one in the Wenzel state and the other in the Cassie state. The former was obtained from an initial perturbation near the bottom of the groove, and the latter was obtained from an initial perturbation near the top of a pillar. In the case that two or more critical nuclei coexist, the system will adopt the critical nucleus with lower energy barrier. In the phase diagram, only the critical nucleus with lower barrier is shown. Also shown in the upper panel of Figure 7 is the nucleation barrier on a flat solid surface of the same material (the dotted line). It is seen that the use of the microstructures increases the energy barrier and thus inhibits vapor nucleation when the interpillar spacing is small; in contrast, the use of the microstructures with large interpillar spacing lowers the energy barrier and enhances vapor condensation on the solid surface. The nucleation barrier also depends on the pillar height. Two typical profiles are shown in the lower panel of Figure 7, where the energy barrier is plotted as a function of the pillar height b with the interpillar spacing s being fixed at 0.07 (left) and 0.12 (right), respectively. In the case of s = 0.07, the critical nucleus is in the Wenzel state (region II) when b is small and the nucleation barrier increases with b; when b is large (b = 0.11), the critical nucleus becomes the Cassie type and further increase of the pillar height has no effect on the energy barrier. In the case of s = 0.12, the critical nucleus is always in the Wenzel state (region III), regardless of the pillar height. The energy barrier decreases with the pillar height, due to the increasing contact area of the nucleus with the pillars. When the pillars are tall enough so that the grooves can completely accommodate the critical nucleus, the energy barrier reaches a plateau, and further increase of the pillar height has no effect on the energy barrier. Phase Diagram on the μ−s Plane. From the phase diagram in Figure 6, we see that there exists a critical value for the interpillar spacing at which the wetting state of the critical nucleus changes from the Cassie state to the Wenzel state.

Figure 5. (Shifted) Grand potential along the MEPs in Figure 4. The points indicated by crosses correspond to the images in the first column (a1−e1) of Figure 4, and the maxima indicated by circles correspond to the critical nuclei in the second column (a2−e2) of Figure 4

Phase Diagram on the s−b Plane and Energy Barriers. To systematically investigate the influence of the microstructures on the vapor condensation process, we computed the phase diagram for the critical nuclei on the plane of the interpillar spacing s and the pillar height b. The numerical result is shown in Figure 6. The phase diagram is divided into three regions according to the monotonicity of the activation energy: At fixed pillar height, the activation energy increases with the spacing s in regions I (circles) and III (squares), but decreases with s in region II (triangles); see Figure 7 (upper panel) for the energy barriers. In regions I and III, the critical nuclei are in

Figure 6. Phase diagram of the critical nucleus on the plane of the interpillar spacing s and the pillar height b. The phase diagram is divided into three regions according to the monotonicity of the activation energy: For fixed b, the activation energy increases with s in regions I (open circles) and III (squares) and decreases in region II (triangles). Typical configurations of the critical nuclei in the three regions are shown in the lower panel for b = 0.12 and s = 0.06, 0.08, and 0.10 (from left to right). The supersaturation is μ = 0.04. 9572

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Figure 7. Energy barrier versus the interpillar spacing s (upper panel) and the pillar height b (lower panel). The symbols have the same meaning as in Figure 6: The Cassie state is indicated by circles (region I), and the Wenzel state is indicated by triangles (region II) and squares (region III). The dotted line in the upper panel indicates the activation energy (ΔG = 3.33 × 10−5) on a flat surface of the same material. The supersaturation is μ = 0.04.

Intuitively, this critical interpillar spacing depends on the size of the critical nucleus, which in turn depends on the level of supersaturation. To quantify the relation between the critical spacing and the size of the critical nucleus, we computed the phase diagram of the critical nucleus on the plane of the supersaturation μ and the interpillar spacing s. The numerical result is shown in Figure 8. In this computation, the pillar height is fixed at b = 0.14. It is seen that the critical interpillar spacing (the region of triangles) decreases as the level of supersaturation increases. We also computed the critical nucleus for vapor condensation in the bulk (i.e. homogeneous nucleation) using the climbing string method. The radius of the critical nucleus, denoted by Rc, is plotted against the supersaturation in the phase diagram (the dashed line). It is seen that the critical interpillar spacing follows closely the curve of the radius of the critical nucleus. The radius of the critical nucleus is inversely proportional to μ: Rc ∼ 1/μ. The profile of the curve agrees well with theoretical values computed from the classical nucleation theory using the notion of surface tension, except for large μ (μ ≳ 0.06) where the critical nucleus is so small that the interface cannot be well described using the notion of surface tension and deviation occurs. The nucleation barrier on pillared surfaces depends on the level of supersaturation. An example is shown in Figure 9, where the pillar height and interpillar spacing are fixed at b = 0.14 and s = 0.06, respectively. It is seen that the nucleation

Figure 8. Phase diagram of the critical nucleus on the plane of the supersaturation μ and the interpillar spacing s. The phase diagram is divided into three regions according to the monotonicity of the activation energy: For fixed μ, the activation energy increases with s in regions I (open circles) and III (squares) and decreases in region II (triangles). Typical configurations of the critical nuclei in the three regions are shown in the lower panel of Figure 6. The dashed line shows the radius of the critical nucleus in a homogeneous nucleation. The pillar height is b = 0.14.

barrier decreases with the supersaturation, regardless of the wetting state of the critical nucleus. However, as shown in the 9573

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Figure 9. Energy barrier versus the supersaturation at fixed pillar height b = 0.14 and interpillar spacing s = 0.06. The inset shows the log−log plot of the data. The symbols have the same meaning as in Figure 8: The Cassie state is indicated by circles (region I), and the Wenzel state is indicated by triangles (region II) and squares (region III).

inset of the figure, the rate of decrease is different in the two regions: ΔG ∼ μ−2.8 for Cassie-type of critical nuclei, while ΔG ∼ μ−7 for Wenzel-type of critical nuclei. Effect of Surface Wettability. Finally, we study the influence of the intrinsic wettability of the solid surface on the nucleation process. In the phase-field model used in this work, the surface wettability is determined by the boundary condition ϕs of the density field at the wall. Figure 10 shows the wetting state of the critical nucleus and the associated energy barrier versus the interpillar spacing for different values of ϕs. The overall trends of the energy barrier and the wetting state of the critical nucleus are similar in all the three cases. It is also seen that the surface wettability plays a similar role as the supersaturation: The critical interpillar spacing at which the Cassie-to-Wenzel transition occurs decreases with ϕs. The transition occurs at s ≈ 0.17, when ϕs = 0.1, which decreases to s ≈ 0.11 when ϕs = 0.3.

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CONCLUSIONS We studied the vapor condensation process on pillared hydrophobic surfaces using a diffuse-interface model and the climbing string method. Unlike previous studies, the string method does not require the prescription of a reaction coordinate; thus, it allows us to accurately compute the critical nuclei, the activation energies, and the minimum energy paths for the vapor condensation. We studied in detail the effects of pillar height, the interpillar spacing, the supersaturation, and the intrinsic wettability of the solid surface on the condensation process. This study offers insights into the role of the surface structure in the condensation process and provides a quantitative basis for designing surfaces that is optimized to inhibit or enhance condensation in engineered systems. The main results of this study can be summarized as follows. First of all, we obtained two nucleation scenarios from the computation. In cases of high pillar, narrow interpillar spacing, low level of supersaturation, and/or low surface wettability, the condensation occurs on the top of the pillars and the critical nuclei exhibit the Cassie wetting state; otherwise, the condensation preferentially occurs inside the grooves and the critical nuclei exhibit the Wenzel wetting state. At certain

Figure 10. Energy barrier versus interpillar spacing s on surfaces with different wettability: ϕs = 0.1, 0.2, and 0.3 (from top to bottom). The dashed line in each panel shows the nucleation barrier on a flat surface of the same material (ΔG = 8.13 × 10−5, 7.66 × 10−5, 6.58 × 10−5, from top to bottom). Different symbols are used according to the monotonicity of the energy barrier as a function of s. Typical configurations of the critical nuclei in the regions indicated by circles, triangles, and squares are shown in the lower panel of Figure 6. The pillar height and the supersaturation are fixed at b = 0.24 and μ = 0.03, respectively.

intermediate spacings, the critical nuclei exhibit the features of both the Wenzel and the Cassie states. A comparison of the nucleation barrier with that on a flat surface of the same 9574

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material shows that the nucleation is inhibited when the critical nucleus is in the Cassie state while enhanced when the critical nucleus is in the Wenzel state. Second, we computed the phase diagram for the critical nuclei, from which we identified the critical values of the pillar height, the interpillar spacing, and the supersaturation at which the wetting state of the critical nucleus changes from the Cassie state to the Wenzel state. It was found that the critical interpillar spacing depends on the level of supersaturation and the intrinsic wettability of the solid surface. In particular, the dependence on the supersaturation follows closely the curve of the critical radii in a homogeneous nucleation. Besides the critical nuclei and the activation energy, the climbing string method also gives the MEP connecting the vapor phase and the critical nucleus. The MEP provides the optimal pathway for the formation of the critical nucleus. After the critical nucleus is formed, the condensate grows and the system relaxes to the liquid phase spontaneously. We studied this relaxation process by solving the steepest descent dynamics. In this work, we used a diffuse interface model in which the fluids were described by a phase field function. However, the numerical method is rather general and is not restricted to the specific model. Other models, such as the model used in the lattice density functional methods, can be used as well. The numerical method can also be applied to solid surfaces with other patterns without difficulty.

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ASSOCIATED CONTENT

S Supporting Information *

Convergence study using a refined mesh and a comparison of the numerical results computed using different pillar width and interfacial width. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported in part by Singapore A*STAR SERC PSF grant R-146-000-173-305 (Project No. 1321202071) and A*STAR SERC “Complex Systems Programme” R-146-000171-305 (Project No. 1224504056).

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