Article pubs.acs.org/Langmuir
Evaluation of Macroscale Wetting Equations on a Microrough Surface Yang Wang,†,‡ Xiangdong Wang,§ Zhongjie Du,‡ Chen Zhang,*,‡ Ming Tian,† and Jianguo Mi*,† †
State Key Laboratory of Organic-Inorganic Composites and ‡The Key Laboratory of Carbon Fiber and Functional Polymers, Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China § School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing, China ABSTRACT: The wettability of critical droplets on microscale geometric rough surfaces has been investigated using a density functional theory approach. In order to analyze the effect of roughness on nucleation free-energy barriers, the local density fluctuations at liquid−solid interfaces induced by the multi-interactions of a corner substrate are presented to interpret the interfacial free-energy variations, and the vapor− liquid−solid contact line tensions are derived from the contact angles of nuclei to account for the three-phase contact energies. The corresponding wetting diagrams have been constructed in Cassie, Wenzel, and impregnation regions. It is shown that, under the same condition, modest deviations between the microscale and the macroscale models can be observed within the Cassie region, whereas these deviations have been enlarged in the Wenzel and impregnation regions as well as the Cassie−Wenzel transition region. These deviations are also correlated to the roughness of the surface. The reason can be attributed to the cooperative effect of the liquid−solid interfacial free energy and line tension. This study offers a fundamental understanding of wettability of ultrasmall droplets on a microscale geometric rough surface. where θF is the flat surface contact angle. ϕ is the fraction of solid in contact with the liquid. Within the Wenzel regime, the liquid penetrates the interstices of a moderate-interaction substrate, and the corresponding θW is expressed by the Wenzel equation8
1. INTRODUCTION Understanding the wettability of solid surfaces is crucial to both the industrial and research fields, such as separation membranes,1 self-cleaning windows for the automotive and aeronautics industries,2 waterproof textiles,3 and antibacterial adhesion.4 In the past decades, a large number of experimental, theoretical, and computational studies have been performed to investigate the effect of surface characteristics on the wettability. The physics of a liquid drop on flat homogeneous surfaces has been relatively well understood,5,6 while that remains in question on heterogeneous surfaces because of its complexity arising from realistic situations: surface roughness, chemical contamination, or inhomogeneities in the solid surface. To date, even approximately ideal flat surfaces are very difficult to produce and unlikely to maintain their wetting properties over time and use. Real solid surfaces may be rough, have finite rigidity, may be chemically heterogeneous, or may react with the wetting liquid. Thus, it is important to develop theoretical models to describe the wettability of realistic (or designed) surfaces. The general surface wettability can be divided into the Cassie, Wenzel, and impregnation regimes. Within the Cassie regime, the liquid sits above the asperities formed within a weak underlying substrate, and the contact angle θc is expressed by the Cassie equation7 cos θC = ϕ(cos θF + 1) − 1 © XXXX American Chemical Society
cos θ W = r cos θF
(2)
where r is the surface roughness factor, defined as the ratio between the actual area of the rough surface and the geometric projected area, which is always larger than unity. In the impregnation regime, a liquid droplet forms on a relatively strong attractive surface containing fluid-filled grooves, and the impregnation contact angle θI is described by9 cos θI = ϕ(cos θF − 1) + 1
(3)
Using the above equations, a wetting diagram can be obtained to reflect the effect of geometric surface roughness on the contact angle.10,11 On a macroscopic scale, wetting phenomena can be well described in terms of the above three equations with the interface tensions between the different phases to determine the equilibrium contact angle. On a microscopic scale, however, interface phenomena are much richer and provide far more than simply a basis for macroscopic laws. The compressible Received: September 22, 2014 Revised: February 5, 2015
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contact angles are then subsequently extracted from the density profiles of critical nuclei, and thereby the wetting diagrams are constructed. In particular, the microscale Cassie−Wenzel wetting transition at different microrough surfaces is discussed by analyzing these diagrams. Finally, a theoretical evaluation of macroscale wetting equations has been performed when they are extended to describe the wettability of a microrough surface.
liquid in nanoconfinement should be taken into account not only with regard to Laplace’s pressure but also with regard to the oscillatory compression pressure.12 Because of the lack of microscopic information included in this approach, it is unclear to what extent these macroscopic models remain valid as the length scale of the substrate features approaches the characteristic size of fluid molecules. In recent years, increased emphasis has been placed on understanding and modeling the behavior of fluids in the presence of rough substrates with nanoscale or microscale features. For example, Biben and Joly employed the lattice Boltzmann model to measure the various surface tensions and used Young’s law to deduce the wetting angle. As a result, the complete roughness wetting diagrams for a nonpolar crenelated surface are constructed.13,14 These wetting diagrams are quite accurate in the nonwetting regime. In the wetting regime, however, they shift away from a macroscopic wetting curve. This implies that the macroscopic wetting models become unreliable on small length scales. The deviation within the wetting regime is attributed to corner effects:15,16 the interaction potentials between the fluid and wall have particular values at the corners that generate corner energy. These corner effects are mainly visible when a liquid phase fills the corners (wetting situation). In the gas phase, however, the effect resulting from low density can be negligible. This approach is useful in understanding static fluids in contact with nanostructured surfaces decorated with so-called re-entrant features. The validities of macroscopic models on the nanoscale are also examined by Malanoski and his colleagues using lattice density functional theory.17 By studying the wetting of small droplets on grooved surfaces, they found that, for the smallest features, the macroscopic models do not predict the correct contact angles. The well-studied failure of macroscopic models is attributed to the non-negligible line tension contribution which can be neglected for macroscopic droplet. The aforementioned analyses indicate that there are two factors in determining the validity of macroscopic theory at the microscopic roughness surface: the corner effect and line tension. However, the corner effect corresponds to multiinteractions, whereas line tension refers to the excess free energy per unit length at the three-phase line, where three interfaces meet. These explanations may still be too phenomenological to be satisfactory from a more fundamental standing. Difficulty is encountered when trying to understand the relationship between the two factors and their influence on the wettability of a nanoscale or microsale rough surface. How to correctly predict the wetting properties of these systems remains unsolved. Therefore, it is important to clearly define the relationship as it may determine its relevance to evaluating the wetting equations for microfluidic systems. In this article, we report a systematic study of the microscopic wetting on different microrough surfaces using our previously developed three-dimensional density functional theory (3D-DFT) approach.18,19 Unlike previous investigations, the present nucleation free-energy barriers, contact angles, and wetting diagrams are derived from the critical nucleation of droplets. As such, the vapor−liquid, liquid−solid, and vapor− solid interfacial tensions as well as the vapor−liquid−solid three-phase contact line tension are considered simultaneously for a droplet in the critical nucleation state. To study the wetting behavior on a rough surface, we first carry out a series of theoretical calculations of nucleation energy barriers of droplets on different smooth and microrough surfaces. The
2. THEORY A model system consisting of a monatomic fluid that interacts with itself and solid sites according to the Lennard−Jones potential with hard core repulsion is first constructed20,21 ⎧ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ij ij ⎪ ⎪ 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ , r ≥ σij ⎝ ⎠ ⎝ r⎠⎦ uij(r) = ⎨ ⎣ r ⎪ ⎪∞ , r < σij ⎩
(4)
where εij and σij are energy and size parameters, respectively; r is the distance between any two particles. The overall potential exerted by the solid can be written as22 U (r) = U hom(r) + U pillar(r)
(5)
The first term on the right side of this equation, Uhom, refers to the fluid−solid interaction for the homogeneous substrate and is expressed as a 10−4 potential ⎡ 2 ⎛ σ ⎞10 ⎛ σ ⎞4 ⎤ U hom(r) = 2πεfsσfs 2⎢ ⎜ fs ⎟ − ⎜ fs ⎟ ⎥ ⎝ r ⎠⎦ ⎣5⎝ r ⎠
(6)
where εfs and σfs are the fluid−solid (fs) energy and size parameters. The fluid−fluid (ff) parameters are related as σfs/σff = 1.0. The relative fluid−solid interaction strength ε is defined as ε = εfs/εff. The second term in the equation, Upillar(r), is the potential created by the grooves. A fluid site interacts with grooves through taking a summation of the potential between site i of the fluid and site js of the pillars U pillar(r) =
∑ uij (r) s
js
(7)
Within the framework of the 3D-DFT approach, the grand potential Ω[ρ(r)] of a three-phase system can be represented by the following expression
∫ dr ρ(r)[ln[ρ(r)] − 1] + Fhs[ρ(r)] + Fatt[ρ(r)] + ∫ dr[ρ(r)(U (r) − μ)]
Ω[ρ(r)] = kBT
(8)
where ρ(r) represents the spatial fluid density distribution. Fhs[ρ(r)] indicates the local Helmholtz free energy for the hard-sphere reference, Fatt[ρ(r)] accounts for the local Helmholtz free energy due to the attractive contribution, U(r) means the external field, and μ accounts for the chemical potential of bulk fluid, which can be derived at ρ(r) = ρb. In addition, kB is the Boltzmann constant, and T is the absolute temperature. The contribution of the hard-sphere reference is derived from the fundamental measure theory23 F hs[ρ] = B
∫ dr Φhs[nγ (r)]
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equilibrium contact angle of a macroscopic drop, and τ is the line tension. According to eq 10, the quantities of τ and θ∞ can be determined from a plot of cos θ as a function of droplet size (1/Rd). For theoretical calculations, different rough surfaces are constructed from a homogeneous substrate decorated with an array of rectangular grooves. The grooves are characterized by the width G and height H and are separated by the distance W, as shown in Figure 1. The geometric parameters of rough
and Φhs[nγ(r)] is the Helmholtz free energy density which stems from the modified fundamental measure theory24 including both the scalar and vector contributions ⎡ n1n2 − n V1·n V2 Φhs[nγ (r)] = ⎢− n0 ln(1 − n3) + ⎢⎣ 1 − n3 +
3 n32 ⎞ n2 − 3n2 n V2·n V2 ⎤ 1 ⎛ ⎥ ⎜n3 ln(1 − n3) + ⎟ 2 ⎥⎦ 36π ⎝ (1 − n3) ⎠ n3 3
(10)
where nγ(r) with γ = 0, 1, 2, 3, V1, and V2 are the weighted densities.24 For the attractive interaction, Fatt[ρ(r)] can be expressed with the following weighted density approximation25 Fatt[ρ(r)] =
∫ ρ(r)aatt[ρ̅ (r)] dr
(11)
in which aatt[ρ̅(r)] is the Helmholtz free energy per particle, which can be seen in ref 26. ρ̅(r) is the weighted density, given by the weight function ωatt(r) ρ ̅ (r) =
∫ ρ(r′) ωatt(r − r′) dr′
Figure 1. Side view of a periodically grooved surface characterized by groove width G, groove height H, and separation distance W.
(12)
with ωatt(r ) =
surfaces are described with a fixed width G of 2σff and a separation distance W of 2σff. The rough surface height goes from 1.5σff to 7.5σff. Another rough surface is characterized by a width G of σff, a separation distance W of 2σff, and a height H of 2.5σff. All of the rough surfaces are periodic in the y direction and infinite in the x direction. The friction of a solid ϕ = W/(W + G) and surface roughness factor r = 1 + 2H/(W + G) are calculated in terms of these geometric parameters.15,32 Throughout this work, we employ a calculation box with a size of (25.6σff)3 on a 3D grid of 2563 points which gives a grid resolution of 0.1σff on each axis. Normal to the z direction, there is a nanogrooved substrate at the bottom of the calculation box. The size of the calculation box is chosen so that a certain amount of space near the center of the wall is not affected by edge effects. Nucleation occurs under a supersaturation condition, and the supersaturation ratio is defined as S = p/p0, where p denotes the pressure of the supersaturated vapor and p0 indicates the vapor−liquid coexistence pressure at a fixed temperature. All calculations are performed at temperature T* = 0.7. For the initial configurations of iteration calculations, a hemispherical cap region is placed at the top of the surface. We adopt the Cartesian coordinate with the initial density profile
catt(r )
∫ catt(r ) dr
(13)
where catt(r) is the attractive part of the direct correlation function at the interfacial average density during the nucleation.27 The equation for the fluid density distribution ρ(r) is obtained through the minimization of the grand potential and solving the Euler−Lagrange equation ⎛ ⎞ δ(Fhs[ρ(r)] + Fatt[ρ(r)]) ρ(r) = exp⎜βμ − β − βU (r)⎟ δρ(r) ⎝ ⎠ (14)
The Picard iteration scheme is used to solve the equation. During the iteration process, the grand potential is constrained with ΩC[ρ(r)] = Ω[ρ(r)] + λ(N° − N), where λ is the Lagrange multiplier, N° is the target number of liquid sites (the given volume of the nuclei), and N is the actual value added in eq 8.20 Once the density has been obtained, the constrained free energy for heterogeneous nucleation can be calculated by ΔΩ = +
∫ dr ρ(r)[ln(ρ(r)Λ3) − 1] + Fhs[ρ(r)] + Fatt[ρ(r)]
∫ dr[ρ(r)(U(r) − μ)] + pV
⎧ 0 z ≤ z0 ⎪ ⎪ ρ(x , y , z) = ⎨ ρL* R ≤ R 0 ⎪ ⎪ ρ* R > R ⎩ V 0
(15)
where p is the pressure and V is the volume. It should be noted that the free-energy expression is a complete form, where liquid−solid, vapor−solid, and vapor−liquid interfacial tensions as well as the vapor−liquid−solid three-phase contact line tension are entirely included. The contact angle is determined by following the procedure in refs 28 and 29. Accordingly, the contribution of line tension can be derived from the modified Young equation that accounts for30,31 τ cos θ = cos θ∞ − γvlR d (16)
(17)
where z0 is the position of the solid wall for the boundary effects and R0 is the initial radius of the nucleus with R being the distance to the center of the spherical cap. The initial density of the fluid molecules ρ*L is set to 0.833, and the supersaturated vapor densitiy ρV* is 0.0035 at S = 1.5. The density ρ* is dimensionless and is defined as ρ* = ρσ3.
3. RESULTS AND DISCUSSION We investigate the microscopic wetting behavior under the condition of critical nucleation of droplets on different surfaces. As such, the free-energy cost for droplet formation, including
where θ is the equilibrium contact angle of the microscopic droplet and γvl refers to the vapor−liquid interfacial tensions. Rd is the radius of the droplet in contact with the surface, θ∞ is the C
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Figure 2. Free-energy barrier ΔΩ* as a function of the nucleation radius R of droplets on smooth and rough surfaces. (a) The fluid−solid interaction is ε = 0.20, corresponding to solvophobic surfaces. (b) The fluid−solid interaction is ε = 0.60, corresponding to solvophilic surfaces. The rough surfaces are characterized by the solid friction ϕ = 0.5 and roughness factor r. The black, red, blue, and wine symbols correspond to r = 1.00, 1.75, 2.25, and 3.25, respectively.
Figure 3. Two-dimensional cut of the spatial density profiles of liquid on different surfaces. (a) r = 1.00, ε = 0.20; (b) r = 1.00, ε = 0.60; (c) ϕ = 0.5, r = 2.25, ε = 0.20; and (d) ϕ = 0.5, r = 2.25, ε = 0.60. The profiles are cut at x = 12.8σff.
D
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Figure 4. Droplets on the rough surface characterized by the solid friction ϕ = 0.5 and roughness factor r = 2.25 with different fluid−solid interactions. (a) ε = 0.15, (b) ε = 0.25, (c) ε = 0.40, (d) ε = 0.50, (e) ε = 0.60, and (f) ε = 0.70. The red region denotes the liquid phase, the blue area represents the vapor phase, and the thick white line indicates the liquid/vapor interface. The thin white line represents the solid surface.
the two-step nucleation process, including critical nuclei for groove filling, the intermediate state, and those out of the groove is shown in the inset of Figure 2b. The variation of nucleation free energy barriers can be further illustrated by the liquid−solid interfacial free energy, which is represented by the local density distributions of nuclei. The results are shown in Figure 3. According to our previous investigations,34,35 high-density peaks correspond to low local free energy. In the case of a strong solvophobic state, it is shown that the nucleated droplets on the smooth (Figure 3a) and microrough (Figure 3b) surfaces display similar morphology outside the liquid−solid interface. Near the smooth surface, the density distribution displays a complete layered structure, and the density peaks are higher than the liquid platform. However, such layered structure has been disturbed in the vicinity of the rough surface. Because of the effect of surface roughness, the high-density regions have been largely reduced, leading to the increase in interfacial free energy. As a consequence, the nucleation free energy barrier increases. If the substrate is transferred to a solvophilic interaction, then the layered profile displayed at the smooth surface (Figure 3c) becomes more obvious, whereas the density distribution has complex structure at the rough surface (Figure 3d). More importantly, the density peaks of the droplet near the smooth surface are much shorter than that near the rough surface. It can be attributed to the strong attraction at the inner corner of roughness, which causes liquid molecules to compress and attain higher densities than those in the bulk phase at very small narrow grooves.12,13,15 Therefore, the roughness plays an important role in extensively improving the interfacial density packing, leading to the extreme drop in interfacial free energy and the nucleation energy barrier. According to the droplet morphology at critical nucleation, the Cassie−Wenzel−impregnation wetting transition process is clearly displayed in Figure 4 to evaluate the wettability of surfaces. On the supersolvophobic surface (ε = 0.10), the
the vapor−liquid, liquid−solid, and vapor−solid interfacial tensions and the vapor−liquid−solid three-phase contact line tension, can be systematically considered. When the free energy of droplet formation reaches a maximum, the size of the droplet is defined as the critical radius Rc, and the energy is the nucleation barrier, which correlates to a state of critical nucleation in the system. Thus, if a formed droplet on a solid surface has a radius smaller than Rc, then the droplet will evaporate, but if its radius is larger than Rc, then the drop will grow larger. Figure 2 shows the constrained free-energy curves for droplet formation on smooth and microrough surfaces with solvophobic (Figure 2a) or solvophilic (Figure 2b) characteristics. The roughness is characterized by the solid friction and roughness factor. Here we alter the roughness factor by changing the height H of grooves. In the solvophobic case, the nucleation free-energy barrier on the smooth surface is the lowest. As the roughness increases, the free-energy barrier increases, and the nucleation size also slightly increases, with droplet nucleation becoming comparably difficult. This is because the surface interaction of liquid−groove is weaker than that of liquid−smooth. Instead, if the surface is solvophilic, droplet nucleation with a lower energy barrier with increasing surface roughness is convenient. Compared to droplet nucleation on a smooth surface, the two-step nucleation on a microrough surface can be observed: there exist two local maxima, corresponding to two nucleation barriers. The results confirm the suggestion of Page and Sear.33 After the droplet crosses the first nucleation barrier, the nucleus exceed the first critical size and grows spontaneously until it fills the grooves to reach an intermediate state having a local minimum in free energy. The sudden decrease in free energy near the first critical nucleus is caused by the fact that the nucleus touches the opposite side of the groove. Then the intermediate state is required to overcome the second nucleation barrier to grow up, inducing the droplet nucleation out of the groove. A typical drawing of critical nuclei on a microrough surface to illustrate E
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Figure 5. Two-dimensional cut of the spatial density profiles of liquid on rough surfaces characterized by the solid friction ϕ = 0.5 and roughness factor r = 2.25 with different fluid−solid interactions. (a) ε = 0.15, (b) ε = 0.25, (c) ε = 0.40, and (d) ε = 0.70. The profiles are cut at x = 12.8σff.
correlates with the depletion effect, which is induced by the weak fluid−solid force. Figure 5b shows that, on the solvophobic rough surface, the density packing increases in the first and second layers. Part of the space is occupied by a low-density vaporlike fluid between the grooves and droplets. There are strong density oscillations due to the ordering of fluid molecules that form several liquid layers of various densities.36 Inside the droplets, such a fluctuation has been depressed. If the fluid−solid interaction is improved from 0.25 to 0.40, as shown in Figure 3c, then the density fluctuation in the corners has been remarkably enriched due to the multiple interactions and becomes more obvious, and this excess attraction at the internal corners allows these sites to be natural nucleation sites: liquid droplets form at the corners and progressively fill the cavity. If the fluid−solid interaction is further improved to 0.7, then liquid with a high packing density fully occupies the grooves, leading to the strong solvophilic attribute (Figure 5d). Apart from the liquid−solid interfacial density fluctuation, the nucleation free energy barrier is also influenced by the vapor−liquid−solid three-phase contact line tension. According to eq 16, the determination of the line tension is achieved by determining the slope of the data plotted as cos θ versus (1/
droplet is in the Cassie state (Figure 4a). As the interaction increases to ε = 0.25, a few liquid molecules penetrate the grooves, leading to partial wetting and the reduction in solvophobicity (Figure 4b). The droplet is located in the Cassie−Wenzel transition region. A further increase in the interaction to ε = 0.40 or 0.5 can lead to an excessive number of liquid molecules condensed at the substrate surface, and the wetting of the rough surface transforms to the Wenzel state (Figure 4c,d). By maintaining the increase in the interaction at 0.60, one sees that the surface becomes solvophilic and the droplet is located in the Wenzel−impregnation transition region (Figure 4e). Finally, the surface becomes strongly solvophilic upon increasing the interaction to 0.7, and the droplet is in the impregnation state (Figure 4f). The results show that the wetting transition is fulfilled via the following patch: Cassie state → Cassie−Wenzel transition → Wenzel state → Wenzel−impregnation transition → impregnation state. Figure 5 presents the two-dimensional cuts of density distributions of critical nuclei on different surfaces to illustrate the above wetting transition. On the supersolvophobic rough surface, as shown in Figure 5a, one can easily see that the density fluctuation is unobvious and the layered structure F
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Figure 6. (a) Cosine of the contact angle vs inverse of the base radius of the droplets at fluid−solid interaction ε = 0.20. The black, red, blue, olive, and wine symbols correspond to r = 1.00, 1.75, 2.25, 2.75, and 3.25, respectively. (b) Line tension of droplets on smooth surfaces and rough surfaces characterized by friction of solid ϕ = 0.5 as a function of the fluid−solid interaction strength. The symbols are defined in the same manner as in panel a.
Rd), as shown in Figure 6a. After scaling these dimensionless results with a characteristic force (argon-like system), the calculated line tension values is about −10−12 N. The magnitude of the line tension is similar to the results in the literature.37−40 Figure 6b shows that as the fluid−solid interaction strength increases, the value declines gradually. It reaches a minimum value at the fluid−solid interaction strength of ε = 0.5. Subsequently, the value rise slowly. Although the tendency of line tension variation on rough surfaces is similar to that on smooth surfaces, sudden drops emerge at rough surfaces, corresponding to the transition from Cassie to Wenzel wetting. The significantly reduced line tension is the result of the local density fluctuation at three-phase contact lines. If the interaction strength is further improved, then the adsorption of the supersaturated vapor also increases extremely, and the difference in the two densities becomes smaller, leading to the declining line tension contribution. As a consequence, the line tension plays an important role in its wetting behavior. Figure 7 surmises the wetting diagram for droplets on different microrough surfaces. The abscissa provides the cosine of the contact angle that the liquid droplet forms on the smooth surface. The ordinate is defined by the cosine of the contact angle that liquid droplets form on a rough surface. An asymmetric wetting property can be seen from the nonwetting region to the wetting one. As the roughness increases, the Cassie region has been enlarged, and the Cassie−Wenzel transition region has been delayed. The result suggests that the design of the supersolvophobic surface that relies upon increasing the height of grooves to improve the roughness is likely to become feasible. In the Cassie−Wenzel and Wenzel− impregnation transition regions, the data given by the DFT approach deviate from the curve denoted by cos θR = cos θF, and the deviation has been enlarged as the roughness increases. Because a rough substrate has a higher surface area than a smooth one, the surface roughness increases the hydrophobicity of an intrinsically hydrophobic substrate and similarly increases the solvophilicity of a hydrophilic substrate. The effect of roughness on the Cassie−Wenzel transition is shown in Figure 8. If the fluid−solid interaction is smaller than ε = 0.10, then the nucleated droplets are always in the Cassie
Figure 7. Wetting diagram for critical nuclei on rough surfaces characterized by friction of solid ϕ = 0.5. The symbols are defined in the same manner as in Figure 6a.
state, regardless of the roughness factor. Once the interaction is larger than ε = 0.40, the droplets are always in the Wenzel state. The Cassie−Wenzel transition region appears in the middle area of the phase diagram. For a relatively small rough factor (r = 1.75), the Cassie−Wenzel transition can be realized at a weak interaction (ε = 0.20). As the roughness factor increases (r = 2.75), a relatively strong interaction (ε = 0.25) is necessary for the transition. If the roughness factor increases to a certain degree (r = 3.75), then the interaction for the transition region does not change again, indicating that the Cassie−Wenzel transition region dependence of the interaction strength is no longer sensitive. Figure 9 compares the wetting diagrams given by the present microscopic model with those calculated by the three ordinary wetting equations. Accordingly, the calculated data given by the microscopic model are fitted with three branches corresponding to the Cassie, Wenzel, and impregnation states, respectively. In Figure 9a, the regressed solid fractions are 0.498 and 0.939 in the Cassie and impregnation domains, and the regressed G
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models can be achieved within the Cassie regime. Within the Wenzel regime, however, the difference between the microscopic and macroscopic models increases with increasing roughness factor. Figure 9 also shows that the regressed parameters in the Wenzel and impregnation regions are similar; thus, we can expect that the Wenzel−impregnation transition is approximately continuous for relatively narrow grooves, which is consistent with the result given by Kumar and Errington.15 The result shows clearly that the physical meaning of the geometric parameters in wetting equations should be unreliable for the narrow-grooved rough surfaces.
4. CONCLUSIONS The microscale wetting behavior of a droplet on a microrough surface has been investigated under the condition of metestable equilibria. This indicates that droplet formation on a solvophobic surface becomes more difficult as the roughness increases. At the solvophilic surface, two-step nucleation on a microrough surface can be observed: nucleation inside the groove and nucleation outside the groove. According to the morphology of nucleated droplets, the liquid density fluctuations at a rough surface have been presented to analyze the multiple interactions in the corner of grooves, and the line tensions have been derived from the contact angles. The wetting diagrams have been constructed by contact angles traced from the Cassie to Wenzel to impregnation regime with increasing fluid−solid strength. Accordingly, the deviations between the microscale and the macroscale models have been evaluated. It has been shown that the macroscale equations cannot correctly describe the wetting diagrams in the Wenzel and impregnation regions as well as in the Cassie−Wenzel transition region. These deviations are different for different microrough surfaces. In other words, realistic microrough characteristics should be taken into account in describing its wettability. The investigation of microscale droplets on solid surfaces offers a wide range of research opportunities on both a fundamental and an applied level.
Figure 8. Wenzel−Cassie transition diagram for droplets on different rough surfaces. Here the roughness factor variation is realized via varying the height of the grooves. The red squares correspond to the wetting transition from the Wenzel to Cassie state. The filled black and open black squares represent the Wenzel and Cassie wetting states.
roughness factor r is 1.284. In contrast, the corresponding parameters for wetting equations are 0.5, 2.25, and 0.5, respectively. In the Cassie region, the two models show a modest slope difference, although the region given by the microscopic model has been depressed. In the Wenzel and impregnation domains, however, the slopes given by the microscopic model are different from the macroscopic slopes. As the fluid−solid interaction increases, more and more fluid molecules are adsorbed and occupy the grooves, leading to the liquid−solid interfacial density fluctuation. Because the strong multiple interactions of the corner substrate on microscale droplets have been overlooked, the prediction of the macroscopic model is inaccurate. In Figure 9b, the microscale wetting parameters are 0.724 and 0.859 in the Cassie and impregnation domains, and the fitted roughness factor r is 1.530, whereas the corresponding macroscale wetting parameters are 0.67, 2.67, and 0.67, respectively. Comparing the two subfigures, one sees that the agreement between microscopic and macroscopic
Figure 9. Comparison of wetting diagrams given by the present theory and the wetting equations. The symbols are the calculated results. The solid lines are the linear regression of the symbols. The dashed lines correspond to the wetting equations. (a) Microrough surface with H = 2.5σff, G = 2.0σff, and W = 2.0σff. (b) Microrough surface with H = 2.5σff, G = 1.0σff, and W = 2.0σff. H
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(20) Berim, G. O.; Ruckenstein, E. Nanodrop on a nanorough solid surface: Density functional theory considerations. J. Chem. Phys. 2008, 129, 014708. (21) Berim, G. O.; Ruckenstein, E. Nanodrop on a nanorough hydrophilic solid surface: Contact angle dependence on the size, arrangement, and composition of the pillars. J. Colloid Interface Sci. 2011, 359, 304−310. (22) Porcheron, F.; Monson, P. A. Mean-Field Theory of Liquid Droplets on Roughened Solid Surfaces: Application to Superhydrophobicity. Langmuir 2006, 22, 1595−1601. (23) Rosenfeld, Y. Free-energy model for the inhomogeneous hardsphere fluid mixture and density-functional theory of freezing. Phys. Rev. Lett. 1989, 63, 980−983. (24) Yu, Y.-X.; Wu, J. Structures of hard-sphere fluids from a modified fundamental-measure theory. J. Chem. Phys. 2002, 117, 10156−10164. (25) Soon-Chul, K.; Song Hi, L. A density functional perturbative approach for simple fluids: the structure of a nonuniform LennardJones fluid at interfaces. J. Phys.: Condens. Matt. 2004, 16, 6365−6374. (26) Tang, Y.; Tong, Z.; Lu, B. C. Y. Analytical equation of state based on the Ornstein-Zernike equation. Fluid Phase Equilib. 1997, 134, 21−42. (27) Tang, Y.; Wu, J. Modeling inhomogeneous van der Waals fluids using an analytical direct correlation function. Phys. Rev. E 2004, 70, 011201. (28) Werder, T.; Walther, J. H.; Jaffe, R. L.; Halicioglu, T.; Koumoutsakos, P. On the Water−Carbon Interaction for Use in Molecular Dynamics Simulations of Graphite and Carbon Nanotubes. J. Phys. Chem. B 2003, 107, 1345−1352. (29) Wei, N.; Lv, C.; Xu, Z. Wetting of Graphene Oxide: A Molecular Dynamics Study. Langmuir 2014, 30, 3572−3578. (30) Marmur, A.; Krasovitski, B. Line Tension on Curved Surfaces: Liquid Drops on Solid Micro- and Nanospheres. Langmuir 2002, 18, 8919−8923. (31) Boruvka, L.; Neumann, A. W. Generalization of the classical theory of capillarity. J. Chem. Phys. 1977, 66, 5464−5476. (32) Shahraz, A.; Borhan, A.; Fichthorn, K. A. Wetting on Physically Patterned Solid Surfaces: The Relevance of Molecular Dynamics Simulations to Macroscopic Systems. Langmuir 2013, 29, 11632− 11639. (33) Page, A.; Sear, R. Heterogeneous Nucleation in and out of Pores. Phys. Rev. Lett. 2006, 97, 065701. (34) Wang, Y.; Sang, D. K.; Du, Z.; Zhang, C.; Tian, M.; Mi, J. Interfacial Structures, Surface Tensions, and Contact Angles of Diiodomethane on Fluorinated Polymers. J. Phys. Chem. C 2014, 118, 10143−10152. (35) Xu, M.; Zhang, C.; Du, Z.; Mi, J. Role of Interfacial Structure of Water in Polymer Surface Wetting. Macromolecules 2013, 46, 927− 934. (36) Alts, T.; Nielaba, P.; D’Aguanno, B.; Forstmann, F. A local density functional approximation for the ion distribution near a charged electrode in the restricted primitive model electrolyte. Chem. Phys. 1987, 111, 223−240. (37) Errington, J. R.; Wilbert, D. W. Prewetting Boundary Tensions from Monte Carlo Simulation. Phys. Rev. Lett. 2005, 95, 226107. (38) Heim, L.-O.; Bonaccurso, E. Measurement of Line Tension on Droplets in the Submicrometer Range. Langmuir 2013, 29, 14147− 14153. (39) Quere, D. Surface wetting: Model droplets. Nat. Mater. 2004, 3, 79−80. (40) Baumgart, T.; Hess, S. T.; Webb, W. W. Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 2003, 425, 821−824.
AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This work is supported by the National Natural Science Foundation of China (nos. 21276010 and 51373019) and by Chemcloudcomputing of Beijing University of Chemical Technology.
(1) Cheng, Z.; Du, M.; Fu, K.; Zhang, N.; Sun, K. pH-Controllable Water Permeation through a Nanostructured Copper Mesh Film. ACS Appl. Mater. Interfaces 2012, 4, 5826−5832. (2) Xiu, Y.; Hess, D. W.; Wong, C. P. UV-Resistant and Superhydrophobic Self-Cleaning Surfaces Using Sol−Gel Processes. J. Adhes. Sci. Technol. 2008, 22, 1907−1917. (3) Zhou, H.; Wang, H.; Niu, H.; Gestos, A.; Wang, X.; Lin, T. Fluoroalkyl Silane Modified Silicone Rubber/Nanoparticle Composite: A Super Durable, Robust Superhydrophobic Fabric Coating. Adv. Mater. 2012, 24, 2409−2412. (4) Wang, L.; Wang, H.; Yuan, L.; Yang, W.; Wu, Z.; Chen, H. Stepwise control of protein adsorption and bacterial attachment on a nanowire array surface: tuning surface wettability by salt concentration. J. Mater. Chem. 2011, 21, 13920−13925. (5) Kang, H. C.; Jacobi, A. M. Equilibrium Contact Angles of Liquid Droplets on Ideal Rough Solids. Langmuir 2011, 27, 14910−14918. (6) Borcia, R.; Borcia, I. D.; Bestehorn, M. Drops on an arbitrarily wetting substrate: A phase field description. Phys. Rev. E 2008, 78, 066307. (7) Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Trans. Faraday Soc. 1944, 40, 546−550. (8) Wenzel, R. N. Resistance of solid surfaces to wetting by water. Ind. Eng. Chem. 1936, 28, 988−994. (9) Bico, J.; Tordeux, C.; Quéré, D. Rough wetting. Europhys. Lett. 2001, 55, 214−220. (10) Tuteja, A.; Choi, W.; Ma, M.; Mabry, J. M.; Mazzella, S. A.; Rutledge, G. C.; McKinley, G. H.; Cohen, R. E. Designing Superoleophobic Surfaces. Science 2007, 318, 1618−1622. (11) Kumar, V.; Sridhar, S.; Errington, J. R. Monte Carlo simulation strategies for computing the wetting properties of fluids at geometrically rough surfaces. J. Chem. Phys. 2011, 135, 184702. (12) Chen, Y.; Wetzel, T.; Aranovich, G. L.; Donohue, M. D. Generalization of Kelvin’s equation for compressible liquids in nanoconfinement. J. Colloid Interface Sci. 2006, 300, 45−51. (13) Joly, L.; Biben, T. Wetting and friction on superoleophobic surfaces. Soft Matter 2009, 5, 2549−2557. (14) Biben, T.; Joly, L. Wetting on Nanorough Surfaces. Phys. Rev. Lett. 2008, 100, 186103. (15) Kumar, V.; Errington, J. R. Impact of Small-Scale Geometric Roughness on Wetting Behavior. Langmuir 2013, 29, 11815−11820. (16) Leroy, F.; Müller-Plathe, F. Can Continuum Thermodynamics Characterize Wenzel Wetting States of Water at the Nanometer Scale? J. Chem. Theory Comput. 2012, 8, 3724−3732. (17) Malanoski, A. P.; Johnson, B. J.; Erickson, J. S. Contact angles on surfaces using mean field theory: nanodroplets vs. nanoroughness. Nanoscale 2014, 6, 5260−5269. (18) Zhou, D.; Mi, J.; Zhong, C. Three-Dimensional Density Functional Study of Heterogeneous Nucleation of Droplets on Solid Surfaces. J. Phys. Chem. B 2012, 116, 14100−14106. (19) Zhou, D.; Zhang, F.; Mi, J.; Zhong, C. Line tension and contact angle of heterogeneous nucleation of binary fluids. AIChE J. 2013, 59, 4390−4398. I
DOI: 10.1021/la505035k Langmuir XXXX, XXX, XXX−XXX
Evaluation of Macroscale Wetting Equations on a Microrough Surface Yang Wang,†,‡ Xiangdong Wang,§ Zhongjie Du,‡ Chen Zhang,*,‡ Ming Tian,† and Jianguo Mi*,† †
State Key Laboratory of Organic-Inorganic Composites and ‡The Key Laboratory of Carbon Fiber and Functional Polymers, Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China § School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing, China ABSTRACT: The wettability of critical droplets on microscale geometric rough surfaces has been investigated using a density functional theory approach. In order to analyze the effect of roughness on nucleation free-energy barriers, the local density fluctuations at liquid−solid interfaces induced by the multi-interactions of a corner substrate are presented to interpret the interfacial free-energy variations, and the vapor− liquid−solid contact line tensions are derived from the contact angles of nuclei to account for the three-phase contact energies. The corresponding wetting diagrams have been constructed in Cassie, Wenzel, and impregnation regions. It is shown that, under the same condition, modest deviations between the microscale and the macroscale models can be observed within the Cassie region, whereas these deviations have been enlarged in the Wenzel and impregnation regions as well as the Cassie−Wenzel transition region. These deviations are also correlated to the roughness of the surface. The reason can be attributed to the cooperative effect of the liquid−solid interfacial free energy and line tension. This study offers a fundamental understanding of wettability of ultrasmall droplets on a microscale geometric rough surface. where θF is the flat surface contact angle. ϕ is the fraction of solid in contact with the liquid. Within the Wenzel regime, the liquid penetrates the interstices of a moderate-interaction substrate, and the corresponding θW is expressed by the Wenzel equation8
1. INTRODUCTION Understanding the wettability of solid surfaces is crucial to both the industrial and research fields, such as separation membranes,1 self-cleaning windows for the automotive and aeronautics industries,2 waterproof textiles,3 and antibacterial adhesion.4 In the past decades, a large number of experimental, theoretical, and computational studies have been performed to investigate the effect of surface characteristics on the wettability. The physics of a liquid drop on flat homogeneous surfaces has been relatively well understood,5,6 while that remains in question on heterogeneous surfaces because of its complexity arising from realistic situations: surface roughness, chemical contamination, or inhomogeneities in the solid surface. To date, even approximately ideal flat surfaces are very difficult to produce and unlikely to maintain their wetting properties over time and use. Real solid surfaces may be rough, have finite rigidity, may be chemically heterogeneous, or may react with the wetting liquid. Thus, it is important to develop theoretical models to describe the wettability of realistic (or designed) surfaces. The general surface wettability can be divided into the Cassie, Wenzel, and impregnation regimes. Within the Cassie regime, the liquid sits above the asperities formed within a weak underlying substrate, and the contact angle θc is expressed by the Cassie equation7 cos θC = ϕ(cos θF + 1) − 1 © XXXX American Chemical Society
cos θ W = r cos θF
(2)
where r is the surface roughness factor, defined as the ratio between the actual area of the rough surface and the geometric projected area, which is always larger than unity. In the impregnation regime, a liquid droplet forms on a relatively strong attractive surface containing fluid-filled grooves, and the impregnation contact angle θI is described by9 cos θI = ϕ(cos θF − 1) + 1
(3)
Using the above equations, a wetting diagram can be obtained to reflect the effect of geometric surface roughness on the contact angle.10,11 On a macroscopic scale, wetting phenomena can be well described in terms of the above three equations with the interface tensions between the different phases to determine the equilibrium contact angle. On a microscopic scale, however, interface phenomena are much richer and provide far more than simply a basis for macroscopic laws. The compressible Received: September 22, 2014 Revised: February 5, 2015
(1) A
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contact angles are then subsequently extracted from the density profiles of critical nuclei, and thereby the wetting diagrams are constructed. In particular, the microscale Cassie−Wenzel wetting transition at different microrough surfaces is discussed by analyzing these diagrams. Finally, a theoretical evaluation of macroscale wetting equations has been performed when they are extended to describe the wettability of a microrough surface.
liquid in nanoconfinement should be taken into account not only with regard to Laplace’s pressure but also with regard to the oscillatory compression pressure.12 Because of the lack of microscopic information included in this approach, it is unclear to what extent these macroscopic models remain valid as the length scale of the substrate features approaches the characteristic size of fluid molecules. In recent years, increased emphasis has been placed on understanding and modeling the behavior of fluids in the presence of rough substrates with nanoscale or microscale features. For example, Biben and Joly employed the lattice Boltzmann model to measure the various surface tensions and used Young’s law to deduce the wetting angle. As a result, the complete roughness wetting diagrams for a nonpolar crenelated surface are constructed.13,14 These wetting diagrams are quite accurate in the nonwetting regime. In the wetting regime, however, they shift away from a macroscopic wetting curve. This implies that the macroscopic wetting models become unreliable on small length scales. The deviation within the wetting regime is attributed to corner effects:15,16 the interaction potentials between the fluid and wall have particular values at the corners that generate corner energy. These corner effects are mainly visible when a liquid phase fills the corners (wetting situation). In the gas phase, however, the effect resulting from low density can be negligible. This approach is useful in understanding static fluids in contact with nanostructured surfaces decorated with so-called re-entrant features. The validities of macroscopic models on the nanoscale are also examined by Malanoski and his colleagues using lattice density functional theory.17 By studying the wetting of small droplets on grooved surfaces, they found that, for the smallest features, the macroscopic models do not predict the correct contact angles. The well-studied failure of macroscopic models is attributed to the non-negligible line tension contribution which can be neglected for macroscopic droplet. The aforementioned analyses indicate that there are two factors in determining the validity of macroscopic theory at the microscopic roughness surface: the corner effect and line tension. However, the corner effect corresponds to multiinteractions, whereas line tension refers to the excess free energy per unit length at the three-phase line, where three interfaces meet. These explanations may still be too phenomenological to be satisfactory from a more fundamental standing. Difficulty is encountered when trying to understand the relationship between the two factors and their influence on the wettability of a nanoscale or microsale rough surface. How to correctly predict the wetting properties of these systems remains unsolved. Therefore, it is important to clearly define the relationship as it may determine its relevance to evaluating the wetting equations for microfluidic systems. In this article, we report a systematic study of the microscopic wetting on different microrough surfaces using our previously developed three-dimensional density functional theory (3D-DFT) approach.18,19 Unlike previous investigations, the present nucleation free-energy barriers, contact angles, and wetting diagrams are derived from the critical nucleation of droplets. As such, the vapor−liquid, liquid−solid, and vapor− solid interfacial tensions as well as the vapor−liquid−solid three-phase contact line tension are considered simultaneously for a droplet in the critical nucleation state. To study the wetting behavior on a rough surface, we first carry out a series of theoretical calculations of nucleation energy barriers of droplets on different smooth and microrough surfaces. The
2. THEORY A model system consisting of a monatomic fluid that interacts with itself and solid sites according to the Lennard−Jones potential with hard core repulsion is first constructed20,21 ⎧ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ij ij ⎪ ⎪ 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ , r ≥ σij ⎝ ⎠ ⎝ r⎠⎦ uij(r) = ⎨ ⎣ r ⎪ ⎪∞ , r < σij ⎩
(4)
where εij and σij are energy and size parameters, respectively; r is the distance between any two particles. The overall potential exerted by the solid can be written as22 U (r) = U hom(r) + U pillar(r)
(5)
The first term on the right side of this equation, Uhom, refers to the fluid−solid interaction for the homogeneous substrate and is expressed as a 10−4 potential ⎡ 2 ⎛ σ ⎞10 ⎛ σ ⎞4 ⎤ U hom(r) = 2πεfsσfs 2⎢ ⎜ fs ⎟ − ⎜ fs ⎟ ⎥ ⎝ r ⎠⎦ ⎣5⎝ r ⎠
(6)
where εfs and σfs are the fluid−solid (fs) energy and size parameters. The fluid−fluid (ff) parameters are related as σfs/σff = 1.0. The relative fluid−solid interaction strength ε is defined as ε = εfs/εff. The second term in the equation, Upillar(r), is the potential created by the grooves. A fluid site interacts with grooves through taking a summation of the potential between site i of the fluid and site js of the pillars U pillar(r) =
∑ uij (r) s
js
(7)
Within the framework of the 3D-DFT approach, the grand potential Ω[ρ(r)] of a three-phase system can be represented by the following expression
∫ dr ρ(r)[ln[ρ(r)] − 1] + Fhs[ρ(r)] + Fatt[ρ(r)] + ∫ dr[ρ(r)(U (r) − μ)]
Ω[ρ(r)] = kBT
(8)
where ρ(r) represents the spatial fluid density distribution. Fhs[ρ(r)] indicates the local Helmholtz free energy for the hard-sphere reference, Fatt[ρ(r)] accounts for the local Helmholtz free energy due to the attractive contribution, U(r) means the external field, and μ accounts for the chemical potential of bulk fluid, which can be derived at ρ(r) = ρb. In addition, kB is the Boltzmann constant, and T is the absolute temperature. The contribution of the hard-sphere reference is derived from the fundamental measure theory23 F hs[ρ] = B
∫ dr Φhs[nγ (r)]
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equilibrium contact angle of a macroscopic drop, and τ is the line tension. According to eq 10, the quantities of τ and θ∞ can be determined from a plot of cos θ as a function of droplet size (1/Rd). For theoretical calculations, different rough surfaces are constructed from a homogeneous substrate decorated with an array of rectangular grooves. The grooves are characterized by the width G and height H and are separated by the distance W, as shown in Figure 1. The geometric parameters of rough
and Φhs[nγ(r)] is the Helmholtz free energy density which stems from the modified fundamental measure theory24 including both the scalar and vector contributions ⎡ n1n2 − n V1·n V2 Φhs[nγ (r)] = ⎢− n0 ln(1 − n3) + ⎢⎣ 1 − n3 +
3 n32 ⎞ n2 − 3n2 n V2·n V2 ⎤ 1 ⎛ ⎥ ⎜n3 ln(1 − n3) + ⎟ 2 ⎥⎦ 36π ⎝ (1 − n3) ⎠ n3 3
(10)
where nγ(r) with γ = 0, 1, 2, 3, V1, and V2 are the weighted densities.24 For the attractive interaction, Fatt[ρ(r)] can be expressed with the following weighted density approximation25 Fatt[ρ(r)] =
∫ ρ(r)aatt[ρ̅ (r)] dr
(11)
in which aatt[ρ̅(r)] is the Helmholtz free energy per particle, which can be seen in ref 26. ρ̅(r) is the weighted density, given by the weight function ωatt(r) ρ ̅ (r) =
∫ ρ(r′) ωatt(r − r′) dr′
Figure 1. Side view of a periodically grooved surface characterized by groove width G, groove height H, and separation distance W.
(12)
with ωatt(r ) =
surfaces are described with a fixed width G of 2σff and a separation distance W of 2σff. The rough surface height goes from 1.5σff to 7.5σff. Another rough surface is characterized by a width G of σff, a separation distance W of 2σff, and a height H of 2.5σff. All of the rough surfaces are periodic in the y direction and infinite in the x direction. The friction of a solid ϕ = W/(W + G) and surface roughness factor r = 1 + 2H/(W + G) are calculated in terms of these geometric parameters.15,32 Throughout this work, we employ a calculation box with a size of (25.6σff)3 on a 3D grid of 2563 points which gives a grid resolution of 0.1σff on each axis. Normal to the z direction, there is a nanogrooved substrate at the bottom of the calculation box. The size of the calculation box is chosen so that a certain amount of space near the center of the wall is not affected by edge effects. Nucleation occurs under a supersaturation condition, and the supersaturation ratio is defined as S = p/p0, where p denotes the pressure of the supersaturated vapor and p0 indicates the vapor−liquid coexistence pressure at a fixed temperature. All calculations are performed at temperature T* = 0.7. For the initial configurations of iteration calculations, a hemispherical cap region is placed at the top of the surface. We adopt the Cartesian coordinate with the initial density profile
catt(r )
∫ catt(r ) dr
(13)
where catt(r) is the attractive part of the direct correlation function at the interfacial average density during the nucleation.27 The equation for the fluid density distribution ρ(r) is obtained through the minimization of the grand potential and solving the Euler−Lagrange equation ⎛ ⎞ δ(Fhs[ρ(r)] + Fatt[ρ(r)]) ρ(r) = exp⎜βμ − β − βU (r)⎟ δρ(r) ⎝ ⎠ (14)
The Picard iteration scheme is used to solve the equation. During the iteration process, the grand potential is constrained with ΩC[ρ(r)] = Ω[ρ(r)] + λ(N° − N), where λ is the Lagrange multiplier, N° is the target number of liquid sites (the given volume of the nuclei), and N is the actual value added in eq 8.20 Once the density has been obtained, the constrained free energy for heterogeneous nucleation can be calculated by ΔΩ = +
∫ dr ρ(r)[ln(ρ(r)Λ3) − 1] + Fhs[ρ(r)] + Fatt[ρ(r)]
∫ dr[ρ(r)(U(r) − μ)] + pV
⎧ 0 z ≤ z0 ⎪ ⎪ ρ(x , y , z) = ⎨ ρL* R ≤ R 0 ⎪ ⎪ ρ* R > R ⎩ V 0
(15)
where p is the pressure and V is the volume. It should be noted that the free-energy expression is a complete form, where liquid−solid, vapor−solid, and vapor−liquid interfacial tensions as well as the vapor−liquid−solid three-phase contact line tension are entirely included. The contact angle is determined by following the procedure in refs 28 and 29. Accordingly, the contribution of line tension can be derived from the modified Young equation that accounts for30,31 τ cos θ = cos θ∞ − γvlR d (16)
(17)
where z0 is the position of the solid wall for the boundary effects and R0 is the initial radius of the nucleus with R being the distance to the center of the spherical cap. The initial density of the fluid molecules ρ*L is set to 0.833, and the supersaturated vapor densitiy ρV* is 0.0035 at S = 1.5. The density ρ* is dimensionless and is defined as ρ* = ρσ3.
3. RESULTS AND DISCUSSION We investigate the microscopic wetting behavior under the condition of critical nucleation of droplets on different surfaces. As such, the free-energy cost for droplet formation, including
where θ is the equilibrium contact angle of the microscopic droplet and γvl refers to the vapor−liquid interfacial tensions. Rd is the radius of the droplet in contact with the surface, θ∞ is the C
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Figure 2. Free-energy barrier ΔΩ* as a function of the nucleation radius R of droplets on smooth and rough surfaces. (a) The fluid−solid interaction is ε = 0.20, corresponding to solvophobic surfaces. (b) The fluid−solid interaction is ε = 0.60, corresponding to solvophilic surfaces. The rough surfaces are characterized by the solid friction ϕ = 0.5 and roughness factor r. The black, red, blue, and wine symbols correspond to r = 1.00, 1.75, 2.25, and 3.25, respectively.
Figure 3. Two-dimensional cut of the spatial density profiles of liquid on different surfaces. (a) r = 1.00, ε = 0.20; (b) r = 1.00, ε = 0.60; (c) ϕ = 0.5, r = 2.25, ε = 0.20; and (d) ϕ = 0.5, r = 2.25, ε = 0.60. The profiles are cut at x = 12.8σff.
D
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Figure 4. Droplets on the rough surface characterized by the solid friction ϕ = 0.5 and roughness factor r = 2.25 with different fluid−solid interactions. (a) ε = 0.15, (b) ε = 0.25, (c) ε = 0.40, (d) ε = 0.50, (e) ε = 0.60, and (f) ε = 0.70. The red region denotes the liquid phase, the blue area represents the vapor phase, and the thick white line indicates the liquid/vapor interface. The thin white line represents the solid surface.
the two-step nucleation process, including critical nuclei for groove filling, the intermediate state, and those out of the groove is shown in the inset of Figure 2b. The variation of nucleation free energy barriers can be further illustrated by the liquid−solid interfacial free energy, which is represented by the local density distributions of nuclei. The results are shown in Figure 3. According to our previous investigations,34,35 high-density peaks correspond to low local free energy. In the case of a strong solvophobic state, it is shown that the nucleated droplets on the smooth (Figure 3a) and microrough (Figure 3b) surfaces display similar morphology outside the liquid−solid interface. Near the smooth surface, the density distribution displays a complete layered structure, and the density peaks are higher than the liquid platform. However, such layered structure has been disturbed in the vicinity of the rough surface. Because of the effect of surface roughness, the high-density regions have been largely reduced, leading to the increase in interfacial free energy. As a consequence, the nucleation free energy barrier increases. If the substrate is transferred to a solvophilic interaction, then the layered profile displayed at the smooth surface (Figure 3c) becomes more obvious, whereas the density distribution has complex structure at the rough surface (Figure 3d). More importantly, the density peaks of the droplet near the smooth surface are much shorter than that near the rough surface. It can be attributed to the strong attraction at the inner corner of roughness, which causes liquid molecules to compress and attain higher densities than those in the bulk phase at very small narrow grooves.12,13,15 Therefore, the roughness plays an important role in extensively improving the interfacial density packing, leading to the extreme drop in interfacial free energy and the nucleation energy barrier. According to the droplet morphology at critical nucleation, the Cassie−Wenzel−impregnation wetting transition process is clearly displayed in Figure 4 to evaluate the wettability of surfaces. On the supersolvophobic surface (ε = 0.10), the
the vapor−liquid, liquid−solid, and vapor−solid interfacial tensions and the vapor−liquid−solid three-phase contact line tension, can be systematically considered. When the free energy of droplet formation reaches a maximum, the size of the droplet is defined as the critical radius Rc, and the energy is the nucleation barrier, which correlates to a state of critical nucleation in the system. Thus, if a formed droplet on a solid surface has a radius smaller than Rc, then the droplet will evaporate, but if its radius is larger than Rc, then the drop will grow larger. Figure 2 shows the constrained free-energy curves for droplet formation on smooth and microrough surfaces with solvophobic (Figure 2a) or solvophilic (Figure 2b) characteristics. The roughness is characterized by the solid friction and roughness factor. Here we alter the roughness factor by changing the height H of grooves. In the solvophobic case, the nucleation free-energy barrier on the smooth surface is the lowest. As the roughness increases, the free-energy barrier increases, and the nucleation size also slightly increases, with droplet nucleation becoming comparably difficult. This is because the surface interaction of liquid−groove is weaker than that of liquid−smooth. Instead, if the surface is solvophilic, droplet nucleation with a lower energy barrier with increasing surface roughness is convenient. Compared to droplet nucleation on a smooth surface, the two-step nucleation on a microrough surface can be observed: there exist two local maxima, corresponding to two nucleation barriers. The results confirm the suggestion of Page and Sear.33 After the droplet crosses the first nucleation barrier, the nucleus exceed the first critical size and grows spontaneously until it fills the grooves to reach an intermediate state having a local minimum in free energy. The sudden decrease in free energy near the first critical nucleus is caused by the fact that the nucleus touches the opposite side of the groove. Then the intermediate state is required to overcome the second nucleation barrier to grow up, inducing the droplet nucleation out of the groove. A typical drawing of critical nuclei on a microrough surface to illustrate E
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Figure 5. Two-dimensional cut of the spatial density profiles of liquid on rough surfaces characterized by the solid friction ϕ = 0.5 and roughness factor r = 2.25 with different fluid−solid interactions. (a) ε = 0.15, (b) ε = 0.25, (c) ε = 0.40, and (d) ε = 0.70. The profiles are cut at x = 12.8σff.
correlates with the depletion effect, which is induced by the weak fluid−solid force. Figure 5b shows that, on the solvophobic rough surface, the density packing increases in the first and second layers. Part of the space is occupied by a low-density vaporlike fluid between the grooves and droplets. There are strong density oscillations due to the ordering of fluid molecules that form several liquid layers of various densities.36 Inside the droplets, such a fluctuation has been depressed. If the fluid−solid interaction is improved from 0.25 to 0.40, as shown in Figure 3c, then the density fluctuation in the corners has been remarkably enriched due to the multiple interactions and becomes more obvious, and this excess attraction at the internal corners allows these sites to be natural nucleation sites: liquid droplets form at the corners and progressively fill the cavity. If the fluid−solid interaction is further improved to 0.7, then liquid with a high packing density fully occupies the grooves, leading to the strong solvophilic attribute (Figure 5d). Apart from the liquid−solid interfacial density fluctuation, the nucleation free energy barrier is also influenced by the vapor−liquid−solid three-phase contact line tension. According to eq 16, the determination of the line tension is achieved by determining the slope of the data plotted as cos θ versus (1/
droplet is in the Cassie state (Figure 4a). As the interaction increases to ε = 0.25, a few liquid molecules penetrate the grooves, leading to partial wetting and the reduction in solvophobicity (Figure 4b). The droplet is located in the Cassie−Wenzel transition region. A further increase in the interaction to ε = 0.40 or 0.5 can lead to an excessive number of liquid molecules condensed at the substrate surface, and the wetting of the rough surface transforms to the Wenzel state (Figure 4c,d). By maintaining the increase in the interaction at 0.60, one sees that the surface becomes solvophilic and the droplet is located in the Wenzel−impregnation transition region (Figure 4e). Finally, the surface becomes strongly solvophilic upon increasing the interaction to 0.7, and the droplet is in the impregnation state (Figure 4f). The results show that the wetting transition is fulfilled via the following patch: Cassie state → Cassie−Wenzel transition → Wenzel state → Wenzel−impregnation transition → impregnation state. Figure 5 presents the two-dimensional cuts of density distributions of critical nuclei on different surfaces to illustrate the above wetting transition. On the supersolvophobic rough surface, as shown in Figure 5a, one can easily see that the density fluctuation is unobvious and the layered structure F
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Figure 6. (a) Cosine of the contact angle vs inverse of the base radius of the droplets at fluid−solid interaction ε = 0.20. The black, red, blue, olive, and wine symbols correspond to r = 1.00, 1.75, 2.25, 2.75, and 3.25, respectively. (b) Line tension of droplets on smooth surfaces and rough surfaces characterized by friction of solid ϕ = 0.5 as a function of the fluid−solid interaction strength. The symbols are defined in the same manner as in panel a.
Rd), as shown in Figure 6a. After scaling these dimensionless results with a characteristic force (argon-like system), the calculated line tension values is about −10−12 N. The magnitude of the line tension is similar to the results in the literature.37−40 Figure 6b shows that as the fluid−solid interaction strength increases, the value declines gradually. It reaches a minimum value at the fluid−solid interaction strength of ε = 0.5. Subsequently, the value rise slowly. Although the tendency of line tension variation on rough surfaces is similar to that on smooth surfaces, sudden drops emerge at rough surfaces, corresponding to the transition from Cassie to Wenzel wetting. The significantly reduced line tension is the result of the local density fluctuation at three-phase contact lines. If the interaction strength is further improved, then the adsorption of the supersaturated vapor also increases extremely, and the difference in the two densities becomes smaller, leading to the declining line tension contribution. As a consequence, the line tension plays an important role in its wetting behavior. Figure 7 surmises the wetting diagram for droplets on different microrough surfaces. The abscissa provides the cosine of the contact angle that the liquid droplet forms on the smooth surface. The ordinate is defined by the cosine of the contact angle that liquid droplets form on a rough surface. An asymmetric wetting property can be seen from the nonwetting region to the wetting one. As the roughness increases, the Cassie region has been enlarged, and the Cassie−Wenzel transition region has been delayed. The result suggests that the design of the supersolvophobic surface that relies upon increasing the height of grooves to improve the roughness is likely to become feasible. In the Cassie−Wenzel and Wenzel− impregnation transition regions, the data given by the DFT approach deviate from the curve denoted by cos θR = cos θF, and the deviation has been enlarged as the roughness increases. Because a rough substrate has a higher surface area than a smooth one, the surface roughness increases the hydrophobicity of an intrinsically hydrophobic substrate and similarly increases the solvophilicity of a hydrophilic substrate. The effect of roughness on the Cassie−Wenzel transition is shown in Figure 8. If the fluid−solid interaction is smaller than ε = 0.10, then the nucleated droplets are always in the Cassie
Figure 7. Wetting diagram for critical nuclei on rough surfaces characterized by friction of solid ϕ = 0.5. The symbols are defined in the same manner as in Figure 6a.
state, regardless of the roughness factor. Once the interaction is larger than ε = 0.40, the droplets are always in the Wenzel state. The Cassie−Wenzel transition region appears in the middle area of the phase diagram. For a relatively small rough factor (r = 1.75), the Cassie−Wenzel transition can be realized at a weak interaction (ε = 0.20). As the roughness factor increases (r = 2.75), a relatively strong interaction (ε = 0.25) is necessary for the transition. If the roughness factor increases to a certain degree (r = 3.75), then the interaction for the transition region does not change again, indicating that the Cassie−Wenzel transition region dependence of the interaction strength is no longer sensitive. Figure 9 compares the wetting diagrams given by the present microscopic model with those calculated by the three ordinary wetting equations. Accordingly, the calculated data given by the microscopic model are fitted with three branches corresponding to the Cassie, Wenzel, and impregnation states, respectively. In Figure 9a, the regressed solid fractions are 0.498 and 0.939 in the Cassie and impregnation domains, and the regressed G
DOI: 10.1021/la505035k Langmuir XXXX, XXX, XXX−XXX
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Langmuir
models can be achieved within the Cassie regime. Within the Wenzel regime, however, the difference between the microscopic and macroscopic models increases with increasing roughness factor. Figure 9 also shows that the regressed parameters in the Wenzel and impregnation regions are similar; thus, we can expect that the Wenzel−impregnation transition is approximately continuous for relatively narrow grooves, which is consistent with the result given by Kumar and Errington.15 The result shows clearly that the physical meaning of the geometric parameters in wetting equations should be unreliable for the narrow-grooved rough surfaces.
4. CONCLUSIONS The microscale wetting behavior of a droplet on a microrough surface has been investigated under the condition of metestable equilibria. This indicates that droplet formation on a solvophobic surface becomes more difficult as the roughness increases. At the solvophilic surface, two-step nucleation on a microrough surface can be observed: nucleation inside the groove and nucleation outside the groove. According to the morphology of nucleated droplets, the liquid density fluctuations at a rough surface have been presented to analyze the multiple interactions in the corner of grooves, and the line tensions have been derived from the contact angles. The wetting diagrams have been constructed by contact angles traced from the Cassie to Wenzel to impregnation regime with increasing fluid−solid strength. Accordingly, the deviations between the microscale and the macroscale models have been evaluated. It has been shown that the macroscale equations cannot correctly describe the wetting diagrams in the Wenzel and impregnation regions as well as in the Cassie−Wenzel transition region. These deviations are different for different microrough surfaces. In other words, realistic microrough characteristics should be taken into account in describing its wettability. The investigation of microscale droplets on solid surfaces offers a wide range of research opportunities on both a fundamental and an applied level.
Figure 8. Wenzel−Cassie transition diagram for droplets on different rough surfaces. Here the roughness factor variation is realized via varying the height of the grooves. The red squares correspond to the wetting transition from the Wenzel to Cassie state. The filled black and open black squares represent the Wenzel and Cassie wetting states.
roughness factor r is 1.284. In contrast, the corresponding parameters for wetting equations are 0.5, 2.25, and 0.5, respectively. In the Cassie region, the two models show a modest slope difference, although the region given by the microscopic model has been depressed. In the Wenzel and impregnation domains, however, the slopes given by the microscopic model are different from the macroscopic slopes. As the fluid−solid interaction increases, more and more fluid molecules are adsorbed and occupy the grooves, leading to the liquid−solid interfacial density fluctuation. Because the strong multiple interactions of the corner substrate on microscale droplets have been overlooked, the prediction of the macroscopic model is inaccurate. In Figure 9b, the microscale wetting parameters are 0.724 and 0.859 in the Cassie and impregnation domains, and the fitted roughness factor r is 1.530, whereas the corresponding macroscale wetting parameters are 0.67, 2.67, and 0.67, respectively. Comparing the two subfigures, one sees that the agreement between microscopic and macroscopic
Figure 9. Comparison of wetting diagrams given by the present theory and the wetting equations. The symbols are the calculated results. The solid lines are the linear regression of the symbols. The dashed lines correspond to the wetting equations. (a) Microrough surface with H = 2.5σff, G = 2.0σff, and W = 2.0σff. (b) Microrough surface with H = 2.5σff, G = 1.0σff, and W = 2.0σff. H
DOI: 10.1021/la505035k Langmuir XXXX, XXX, XXX−XXX
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
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REFERENCES
This work is supported by the National Natural Science Foundation of China (nos. 21276010 and 51373019) and by Chemcloudcomputing of Beijing University of Chemical Technology.
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