General Formulations for Predicting Longevity of Submerged

Article pubs.acs.org/Langmuir

General Formulations for Predicting Longevity of Submerged Superhydrophobic Surfaces Composed of Pores or Posts A. A. Hemeda and H. Vahedi Tafreshi* Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284-3015, United States ABSTRACT: Superhydrophobicity can arise from the ability of a submerged rough hydrophobic surface to trap air in its surface pores, and thereby reduce the contact area between the water and the frictional solid walls. A submerged surface can only remain superhydrophobic (SHP) as long as it retains the air in its pores. SHP surfaces have a short underwater life, and their longevity depends strongly on the hydrostatic pressure at which they operate. In this work, a comprehensive mathematical framework is developed to predict the mechanical stability and the longevity of submerged SHP surfaces with arbitrary pore or post geometries. We start by deriving an integro-partial differential equation for the 3-D shape of the air− water interface, and use this information to predict the rate of dissolution of the entrapped air into the ambient water under different hydrostatic pressures. For the special case of circular pores, the above integro-partial differential equation is reduced to easy-to-solve ordinary differential equations. In addition, approximate nonlinear algebraic solutions are also obtained for surfaces with circular pores or posts. The effects of geometrical parameters and hydrostatic conditions on surface stability and longevity are discussed in detail. Moreover, a simple equivalent pore diameter method is developed for SHP surfaces composed of posts with ordered or random configurationan otherwise complicated task requiring the solution of an integro-partial differential equation.

1. INTRODUCTION Superhydrophobic (SHP) surfaces are characterized by water droplets beading on them with contact angles exceeding 150°. This is thanks to the peculiar ability of such surfaces in trapping air in their pores, thereby reducing the contact between water and the solid surface (the Cassie state).1 The reduced solid− water contact area can result in a reduction in the skin-friction drag, if the surface is exposed to a moving body of water.2 For such applications, the stability of the air−water interface (AWI) under pressure is critically important. An elevated pressure can force the water into the air-filled pores of the surface and lead to a partial or complete wetting of the surface (the Wenzel state). A similar phenomenon also takes place during the evaporation of a droplet on an SHP surface: pressure inside the droplet increases due to evaporation and forces the AWI into the air-filled pores beneath the droplet.3−6 The method of balance of forces has been used to predict the pressure at which the surface starts transitioning from the Cassie state to the Wenzel state, the critical pressure, for surfaces made up of grooves, posts, and particles.7−14 Critical pressure has also been measured experimentally in many pioneering studies.15−19 While critical pressure can be used to judge if the Cassie state is mechanically stable under elevated pressures, it is the surface longevity that matters the most for a submerged SHP surface. Longevity is the time that it takes for an SHP surface to transition to the Wenzel state. Optical techniques have been used to experimentally estimate the longevity of SHP surfaces with different microstructures under elevated pressures.15,20−24 © XXXX American Chemical Society

These studies have shown that longevity decreases with increasing hydrostatic pressure. No theory, however, has yet been developed to establish a quantitative relationship between the longevity of an SHP surface and its microstructural parameters (e.g., diameter and height of posts or pores). Our group was the first to propose a mathematical framework to quantify the longevity of an SHP surface.25 Our previous study, however, was only applicable to surfaces made of parallel grooves as the equations were derived in a 2-D space (see also ref 26). In the current work, new formulations are derived in a generalized 3-D form such that predicting the critical pressure and longevity of surfaces composed of dissimilar pores and posts with arbitrary round cross sections (not necessarily circular) with ordered or random arrangements is made possible. The formulation presented in this work can directly be used to design and optimize the microstructure of an SHP surface for different applications ranging from self-cleaning to underwater drag reduction. Our generalized formulations for critical pressure and longevity predictions are presented in section 2 for surfaces made of arbitrary pores and posts. Specific solution methods and closed-form analytical expressions are derived for surfaces with circular pores and circular posts in sections 3 and 4, respectively. Proposing new equivalent pore diameter defiReceived: May 25, 2014 Revised: August 6, 2014

A

dx.doi.org/10.1021/la501894u | Langmuir XXXX, XXX, XXX−XXX

Langmuir

Article

the YLCA θ is the critical pressure.27−29 Consider capillary pressure defined as Pcap = σ∇·n̂ where σ is the surface tension and n̂ is the unit vector normal to the interface. Assuming z = F(x,y,t) to be the profile of the interface, the unit normal becomes n̂F = (Fx′ Fy′ − 1)/(1 + F2x + F2y )1/2 where the subscript F is for the interface F and it could be G or f for the critical and initial AWI, respectively. The capillary pressure or the divergence term therefore become,

nitions for SHP surfaces made of ordered and/or random posts, we have developed easy-to-use analytical expressions for predicting critical pressure and longevity of such surfaces in section 5. This is followed by our conclusions in section 6.

2. GENERAL FORMULATIONS The AWI in a pore (or between vertical posts) is considered to be at equilibrium when the hydrostatic, Pg, and ambient pressures, P∞, are balanced by the capillary pressure, Pcap, and the pressure caused by the air entrapped in the pore, Pbub (see Figure 1). The angle between the solid walls and the AWI

Pcap = σ ∇·nF̂ = σ[(1 + Fy2)Fxx + (1 + Fx2)Fyy − 2FxFyFxy] × (1 + Fx2 + Fy2)−3/2

(1)

Let the critical profile at t = 0 be z = F(x,y,0) = G(x,y). From the balance of forces, Pcap − Pg − P∞ + Pbub ≈ 0

(2)

In order to calculate the pressure of the entrapped bubble, an isothermal compression Pbubvbub = P∞v0 is assumed, with the bubble volume represented as vbub = v0 + ∫ ∫ s F dS. Rearranging these two equations yields, Pbub = P∞v0/v bub = P∞v0/(v0 +

∬ G dS )

(3)

where v0 is the pore volume (v0 = S·h). Equation 2 then becomes

Figure 1. Balance of forces acting on the AWI, z = F(x,y,t) formed inside a pore (figure to the left) or between the posts (figure to the right) on a submerged superhydrophobic surface. The pores or posts can have any arbitrary round cross sections.

Pcr = σ ∇·nĜ − P∞[1 − v0/(v0 +

∬ G dS)]

(4)

Equation 4 is solved using an iterative interpolation method. The integro-partial differential equation (IPDE) given in eq 4 is solved in each iteration using the finite element engine of a commercial PDE solver (PDE Solutions Inc.) controlled by an in-house MATLAB code. 2.2. Failure Time in Regime I. In Regime I, the hydrostatic pressure is less than the critical pressure. Here the transition from the Cassie state to the Wenzel state takes place in two steps. The interface first deforms (sags) while being pinned (t ≤ tcr) and then detaches from the edges and moves deeper into the pore (tcr < t < tf1). For a pinned interface, one should first calculate the initial interface profile, i.e., the shape of the interface at t = 0 by applying the balance of forces across the interface (eq 2). Let z = F(x,y,0) = f(x,y) be the interface shape at t = 0. By following a procedure similar to the one discussed in the previous section, eq 2 can be written as

increases with increasing hydrostatic pressure. At a certain hydrostatic pressure, i.e., the critical pressure, Pcr, this angle reaches the Young−Laplace contact angle (YLCA) θ. Our study on the time evolution of the AWI is divided into two major categories in terms of the hydrostatic pressure. We define Regime I and Regime II for systems operating under hydrostatic pressures smaller or greater than the critical pressure of the surface, respectively. At a given hydrostatic pressure in Regime I, the AWI first deforms (sags) over time, as the pressure of the entrapped air Pbub decreases due to air dissolution in water, and then translates downward with further air dissolution. In the deforming stage, the AWI remains pinned to the sharp edges of the wall, but the angle between the interface and the wall increases with time until it reaches the YLCA. This interface profile is referred to as the critical profile, and the time required for the interface to reach this profile is called the critical time tcr in this work. Once the AWI reaches the critical profile, it translates downward at a constant speed (due to the constant rate of air dissolution from the bubble). The time needed for the interface to touch the bottom of the pore in Regime I is referred to as the failure time tf1 here. Unlike Regime I, the AWI in Regime II conforms to its critical profile instantly (at time t = 0) but at a depth h0 inside the pore, depending on the difference between the hydrostatic pressure and critical pressure. This interface then moves down further into the pore (similar to the case of Regime I) until it touches the bottom of the pore in a time period denoted here as tf2. In the next sections, we first derive our general formulations for the calculation of the critical pressure, and its corresponding AWI profile (section2.1), and then present a method for calculating the failure times (section 2.2). 2.1. Critical Pressure and Critical Air−Water Interface. For a given pore (or a post), the hydrostatic pressure that makes the angle between the AWI and the solid walls equal to

σ∇·n f̂ − P∞(1 − v0/(v0 +

∬ f dS)) − Pg = 0

(5)

The above IPDE is subjected to the “pinned interface” boundary condition at the wall f = 0. The shape of the initial interface can then be calculated by numerically integrating eq 5 using an IPDE solver. The transient interface is calculated using eq 2, where the pressure of the entrapped air changes as air dissolves in water Pbub = Pbub (t). The rate of dissolution of a gas from a bubble into water is proportional to the difference between the partial pressure of the gas in the bubble and its mole fraction in water,30 vi· = ξiA(kici − pbub, i )

(6)

where vi, ki, ci, and ξi are volume flow rate, Henry’s constant, mole fraction in water, and the invasion coefficient of gas i, respectively. The surface area A can be obtained as A = ∫ ∫ s (1 + F2x + F2y )1/2 dS (dS = dx × dy). Following ref 30, the bubble pressure can be obtained as B


Langmuir

Article

Pbub = Pv − v·/(ξ ̅ A) where ξ̅ = ∑ni ξi, Pv =

The initial condition for this equation is τ(t = tcr) = 0. Therefore, τ can easily be calculated as shown in eq 18 in the interval tcr < t < tf1. In summary, the transient shape of the AWI in Regime I can be obtained as

(7) 1 ξ̅

∑ki ci ξi = P∞ because p∞,i = kici. As

mentioned in sec 2.1, the volume of the bubble is v(t) = v0 + ∫ ∫ s F dS where v0 = (h)(S) is a constant, and therefore, v· = ∫ ∫ sFt dS. Substituting eq 7 into eq 2 and integrating over the area S we obtain, −σ

∬ ∇·nF̂ dS − ΠS + ξS̅A ∬ Ft dS ≈ 0

−βt/ α ⎧ − γ /β ]f (x , y) 0 < t < tcr ⎪ [(1 + γ / β ) e F(x , y , t ) = ⎨ ⎪ ⎩G(x , y) + (κ + Π)ξ ̅A(t − tcr)/S tcr < t < tf1 (18)

(8)

The failure time tf1 is reached when F(x*,y*,tf1) = −h, where (x*,y*) is the location of the minimum height of the critical profile G(x,y). From eq 18, the failure time can be calculated as

where Π = Pv − Pg −P∞. Equation 8 can directly be solved with a generic IPDE solver. Alternatively, one can assume that the AWI profile at each moment during its evolution is similar the initial interface profile but scaled with a factor that is a function of time, i.e., F (x , y , t ) = T (t ) f (x , y )

tf1 = tcr − S[h + G(x*, y*)]/([κ + Π]ξ ̅ A)

where (x*,y*) is the coordinates of the minimum height of the critical interface and it is the center point of the critical profile in the case of axi-symmetric (or symmetric) cross sections. 2.3. Failure Time in Regime II. In Regime II, the hydrostatic pressure is greater than the critical pressure Pg >Pcr and the interface at t = 0 is detached from the edges. With an appropriate coordinate system, the initial interface can be described by F = G(x, y), which can be calculated from eq 4 (the balance of forces over the interface at a distance h0 from the edges at t = 0). Equation 2 indicates that the pressure of the entrapped air remains constant during air dissolution in Regime II, Pbub = Pbub |t=0. Assuming the entrapped air (ideal-gas) to be in thermal equilibrium with water,

(9)

where f(x,y) is the initial profile obtained from eq 5. The time evolution of the interface can be obtained by calculating the scale function T(t). Using eqs 8 and 9, one obtains αTt + βT + γ = 0

(10)

where γ = −ΠS and α=

S ξ ̅ A (t )

β = −σ

∬ f dS

(11a)

∬ ∇·nF̂ dS

(11b)

h0 = (1 − P∞/(Pg + P∞ − κ ))h +

Equation 10 is a nonlinear PDE in time as α = α(t), and β = β(t). In order to find an analytical solution for this equation, we assume that α = α(t = 0), and β = β(t = 0) with an initial condition of T(t = 0) = 1, T = (1 + γ /β) e

−βt/ α

− γ /β

1 ξ ̅A

(12)

F(x , y , t ) = G(x , y) − h0 + (κ + Π)ξ ̅ At /S

(21)

(22)

(23)

The failure time tf2 is reached when F(x*, y*, tf2) = −h and can be calculated using tf2 = S[h0 − h − G(x*, y*)]/([κ + Π]ξ ̅ A)

(24)

With the critical pressure (and its corresponding profile G(x, y)) numerically obtained from eq 4, one can use eq 24 to predict the lifetime of the entrapped air. The general formulations derived here in section 2 can predict the critical pressure and longevity of SHP surfaces comprised of pores or posts with arbitrary round cross sections (not necessarily circular). For the special case of circular pores, we obtain easyto-use analytical expressions for critical pressure and longevity calculations, as will be discussed in the next section.

(15)

where G(x,y) is the critical profile as defined in eq 4. After substituting in eq 14 and integrating the result over the crosssectional area, we obtain

∬ ∇·nĜ dS

(20)

Substituting into eq 21, with ; (t = 0) = 0, the AWI equation is (13)

Since the shape of a detached interface remains invariant, we assume a solution for eq 14 in the form of

where κ is σ κ= S

∬ Ft dS − σ ∇·nF̂ − Π ≈ 0

F (x , y , t ) = G (x , y ) − h 0 + ;( t )

∬

Sτt /(ξ ̅ A) ≈ κ + Π

∬ G dS

With the procedure described in section 2.2, the transient interface profile becomes

For times greater than the critical time (tcr ≤ t), the AWI is detached from the edges of the pore and slowly moves down as the entrapped air continues to dissolve. Substituting eqs 1 and 7 in eq 2 and integrating over the cross-section of the AWI, one obtains 1 Ft dS − σ ∇·nF̂ − Π ≈ 0 ξ ̅A (14)

F (x , y , t ) = G (x , y ) + τ ( t )

1 S

Substituting eqs 1 and 7 into eq 2, we obtain

This equation is valid until the YLCA is reached at the boundaries, i.e., |∇F|max = |cot θ|. Finally, the critical time t = tcr, can be analytically calculated using α ⎛ γ |∇f |max − β cot θ ⎞ tcr = − ln⎜ ⎟ β ⎝ (β + γ )|∇f |max ⎠

(19)

3. SUPERHYDROPHOBIC SURFACES MADE OF CIRCULAR PORES For the special case of circular pores, the AWI conforms to the shape of a spherical cap. Therefore, the general PDE given for critical pressure calculation in section 2, eq 4, can significantly be simplified to yield an analytical solution (section 3.1). Also

(16)

(17) C


Langmuir

Article

numerically by solving eq 4. This figure also presents the effects of pore depth on the critical pressure. It can be seen that shallow pores exhibit better resistance against elevated pressures. This is because the ratio of the volume of the air displaced by the sagging interface to the volume of the pore is greater when the pores are shallow causing higher compression pressures. We have also calculated the minimum pore depth hminthe depth below which failure occurs before the interface reaches the critical profilefor circular pores with different diameters (the inset in Figure 2). It is interesting to note that for longer pores critical pressure becomes almost independent of the pore diameter (determined mostly from the balance between the capillary pressure and hydrostatic pressure). This can easily be seen in eq 27 where the contribution of the second term becomes almost negligible for long pores. Note in Figure 2 that hmin is independent of the pore depth as it only depends on the shape of the critical profile. 3.2. Failure Time. For longevity calculations, the AWI for both regimes can generally be written as

the PDE presented in eq 8, for the transient AWI (needed for longevity calculation), can be reduced to an easy-to-solve ordinary differential equation (ODE) (section 3.2). Unless otherwise specified, default parameters considered in this paper are h = 30 μm, θ = 115°, and D = 112.8 μm. Hydrostatic pressures corresponding to 0.4 and 3.0 m of water are considered for the calculations conducted in Regime I and Regime II, respectively. Note that these parameters are chosen arbitrarily. 3.1. Critical Pressure and Critical Air−Water Interface for Circular Pores. The AWI takes the shape of a spherical cap inside a circular pore. Therefore, the critical interface can be described as G=

R cr2 − D2 /4 −

R cr2 − x 2 − y 2

(25)

where Rcr is the critical radius of curvature of the interface. For a circular pore, capillary pressure can be defined as Pcap = −2σ cos θ/D, and so the critical radius can be obtained from eq 1 as R cr = −D/(2 cos θ )

(26)

F (x , y , t ) =

Using eq 25 and 26, the critical AWI, and its corresponding pressure, can explicitly be determined using eq 4. Pcr =

R2 − x 2 − y 2 + h0

(28)

where R =R(t) and h0 is the depth at which the initial interface reaches an equilibrium with the hydrostatic pressure at t = 0. This depth can be calculated using eq 20, and is zero in Regime I. Equation 28 can be used to convert the PDE given in eq 5 for the initial AWI ( f at t = 0), to a simple algebraic equation,

2σ − P∞ R cr

⎛ ⎡ πD 2 2π × ⎜1 − v0/⎢v0 + R cr2 − D2 /4 + 4 3 ⎣ ⎝ ⎤⎞ [(R cr2 − D2 /4)3/2 − R cr3]⎥⎟ ⎦⎠

R2 − D2 /4 −

⎛ ⎡ 2σ D2 2π − Pg − P∞⎜⎜1 − v0/⎢v0 + S R 02 − + ⎢ R0 4 3 ⎣ ⎝ 3/2 ⎤⎤⎞ ⎡⎛ D2 ⎞ 3 ⎥⎥⎟ 2 ⎢ × ⎜R 0 − ⎟ − R0 ⎟ = 0 ⎥⎦⎥ ⎢⎣⎝ 4 ⎠ ⎦⎠

(27)

Figure 2 shows the effects of pore diameter D on the critical pressure of an SHP surface calculated using the above analytical equation. For comparison, critical pressure is also calculated

(29)

where R(t = 0) = R0. Moreover, using eq 28, one can reduce the PDE in eq 8 to a simpler equation in which the space and time dependence are decoupled, ⎛ ⎞ RS 2σ 1 ⎜ ⎟ dR + Π ≈ 0 − − A ⎟ dt ξ ̅ A ⎜⎝ R 2 − D2 /4 R ⎠ cr

(30)

Equation 30 is an ODE which can easily be solved via a simple Runge−Kutta method for the radius of curvature of the interface from R(0) = R0 until the critical time R(tcr) = Rcr. For times greater than the critical time, one can simplify eq 14 with the critical interface profile known from eqs 25−26, where κ is 2σ/Rcr.Therefore, the interface profile and longevity can easily be calculated in Regime I using eq 18 and eqs 25−26. In Regime II, the critical interface is the same as that represented by eqs 25−26. The equation for h0, (eq 20) can now be simplified to h0 = (1 − P∞/(Pg + P∞ − κ ))h + +

2π [(R cr2 − D2 /4)3/2 − R cr3] 3S

R cr2 − D2 /4 (31)

With κ = 2σ/Rcr the interface and longevity can be obtained using the method of section 2.3. The AWI inside a circular pore having a diameter of D = 112.8 μm, a depth of h = 30 μm, and a contact angle of θ = 115° is studied in Figure 3. For these calculations, our interface tracking process started by solving eq 29 for the initial profile

Figure 2. Compression between the critical pressure predictions obtained from a numerical solution of the PDE in eq 4 and the analytical calculation using eq 27 for different diameter pore diameters with θ = 115°. The minimum pore height hmin is shown in the inset. D


Langmuir

Article

Figure 3. A compression between the longevity predictions obtained from a numerical solution of the ODE in eq 30 and the analytical calculation using eq 32 for SHP surfaces made up of pores with different diameters is given in (a) and under different hydrostatic pressures in (b) in Regime I (tf1) and Regime II (tf2). The AWI under a hydrostatic pressure of 0.4 m of water is shown in the inset of (b) for t = 0, t = 3.9 min, t = 10 min, and t = 19.4 min from top-left to bottom-right, respectively. The instantaneous pressure of the air entrapped in the pore Pbub is shown for two different depths of water in (c) and two different pore diameters in (d).

the pore diameter. In this figure, the longevity predictions obtained from analytical solution using eq 32 are compared with those obtained from the ODE of eq 30, showing excellent agreement. Figure 3b shows that longevity decreases with increasing the hydrostatic pressure. This is in agreement with the experimental observation reported for square pores.20 It again can be seen that longevity predictions obtained from analytical solution in eq 32 are in excellent agreement with those obtained from the simple ODE given in eq 30. For this reason, we use eq 30 later in section 5 when presenting our equivalent pore diameter method for an SHP surface made of ordered or random posts. The inset in this figure shows the AWI at four different moments of time under a hydrostatic pressure of 0.4 m of water. The effects of hydrostatic pressure and pore diameter on longevity (a and b of Figure 3), can be explained by monitoring the pressure inside the entrapped bubble Pbub over time. We therefore considered the air bubbles inside two pores with diameters of 112.8 and 56.4 μm at a depth of 0.4 m in Figure 3c, and inside a pore with a diameter of 112.8 μm but under two different hydrostatic pressures of 0.4 and 3.0 m in Figure 3d. It can be seen that Pbub decreases with time until t = tcr (e.g., 2 min for the pore with D = 11.8 μm at H = 0.4 m). Beyond the critical time, the bubble pressure

R(t = 0) = R0. Then, the transient air−water interface was obtained by solving eq 30 until the interface slope at the wall reached the Young−Laplace Slope (YLS). Note that in deriving the formulations presented in this section, we did not use the method of separation of variables as described in eq 9. Now eq 12 can be used with α=

⎛ ⎡⎛ ⎤⎞ 2 2 ⎞3/2 S ⎜S R 2 − D + 2π ⎢⎜R 2 − D ⎟ − R 3⎥⎟ 0 0 0 ⎥⎦⎟ ξ ̅A(t = 0) ⎜⎝ 4 3 ⎢⎣⎝ 4 ⎠ ⎠ (32a)

β=−

2Sσ R0

(32b)

With this information, the critical and failure times in Regime I can be calculated using eqs 13 and 19, with the maximum slope at the boundary |▽f |max = D/(2(R20 − D2/4)1/2). Longevity of the entrapped air in Regime I tf1 and Regime II tf2 are shown for circular pores with different diameters in Figure 3a under two hydrostatic pressures of H = 0.4 m and H = 3.0 m, respectively (the longevity calculations in Regime II follow the same analytical procedure as described earlier using eq 32). It can be seen that longevity decreases with increasing E


Langmuir

Article

when posts are placed in ordered configurations. The solution domain is only a unit cell with symmetric boundary conditions not shown for the sake of brevity. Equation 4 is numerically solved to obtain critical pressure values for SHP surfaces comprised of ordered posts with different SAFs (careful attention has been paid to ensure grid-independence). Here we considered posts with h = 9.5 μm, diameter, d = 6.0 μm, and two different contact angles of θ = 103.5° and 122.1°. These particular dimensions are chosen so that our results can be compared with the experimental data of ref 16. These authors have also reported theoretical predictions for the critical pressure of their surfaces using the equations of ref 11. Perfect agreement between our numerical results and those reported in ref 11 can be seen in Figure 4a. When compared to the experimental data, our theoretical predictions (and obviously those of ref 11) show good agreement for SAFs between 0.2

remains constant equal to the sum of the ambient and hydrostatic pressures minus the capillary pressure at the critical condition, if the surface is in Regime I. If the surface is in Regime II (e.g., at H = 3.0 m), the bubble pressure remains constant over time (see Figure 3d when H = 3.0 m). Note that the rate of air dissolution in water (absolute value) is proportional to the bubble pressure. Therefore, one can expect that increasing the hydrostatic pressure from 0.4 to 3.0 m reduces the longevity of the surface. Similarly, increasing the pore diameter increases the bubble pressure and thus reduces the surface longevity. In a recent study, a set of formulations similar to those proposed for failure in Regime II in ref 25 were used for predicting the longevity of SHP surfaces made of circular pores.31 The authors in ref 31 used an expression for calculating the rate of air dissolution involving an empirical value, referred to as the characteristic length l, which directly controls the rate of air dissolution from the bubble. Using an ad-hoc value of l = 0.45 mm, these authors were able to show that their numerical predictions match their experimental data obtained via confocal microscopy. Our longevity formulations show perfect agreement with the experimental data reported in ref 31 only after we alter our original invasion coefficient definition (taken from ref 30) to read as ξ̅mod = (R̅ Tair/kN2) (1/Pbub) (DN2/l), where R̅ is the universal gas constant, Tair is the air temperature, and DN2 is the diffusion coefficient of nitrogen in water. Note that the formulation of ref 31 assumes that the entrapped air is 2.3% vapor and 97.7% nitrogen. In the absence of clear information as to how this expression was obtained and how a characteristic length was chosen, we continue this paper with the original formulation given in section 2. It is worth mentioning that, given the limited number of studies in the literature reporting on underwater longevity of SHP surfaces, there is still no perfect set of experimental data available that can be used to benchmark a computational model. This seems, in part, to be due to the lack of perfect control over the saturation level of the water used for such experiments.20,31 For instance, our formulation can produce perfect agreement with the experimental data of ref 20 but only if the air saturation was 1% less than the assumed value (the formulations of ref 31 cannot be used for these experiments as the failure mechanism falls under the Regime I category). Note also that the experiment of ref 20 was conducted with square pores, leading to critical pressures different from those of circular pores with identical crosssectional area. Any attempts in trying to predict the longevity values reported for square pores using formulations derived for circular pores will only be an approximation.

4. SUPERHYDROPHOBIC SURFACES MADE OF CIRCULAR POSTS The formulations developed in section 2 are used here to predict the critical pressure and longevity of SHP surfaces composed of circular microposts arranged in ordered and random configurations. Note that the work presented in this work is the first to include the effects of air compression in calculating the critical pressure of an SHP surface made of randomly distributed dissimilar posts. More importantly, the current work is the first to present a mathematical framework for predicting the longevity of such surfaces. 4.1. Surfaces with Ordered Posts. We characterize surfaces made of microposts with their solid area fraction (SAF) defined as φ = S/L2 where L is the length of the unit cell

Figure 4. Our critical pressure predictions obtained from a numerical solution of eq 4 for SHP surfaces composed of circular posts with ordered arrangements are compared with those of Lobaton and Salamon11 and the experimental data of Forsberg et al. 16 in (a). The minimum post height is shown in (b) for different SAFs. F


Langmuir

Article

and 0.3. Outside this range, however, the theoretical predictions and experimental data deviate from one another by up to about 35%. To better understand the reason for the mismatch, one should note that our computational method is expected to slightly under-predict the true critical pressure. This is because critical pressure in our work is defined as the pressure at which the local slope of the AWI at some point along the contact-line reaches the YLS (the pressure needed for a local interface detachment). It is not impossible for a partially-detached interface to withstand hydrostatic pressures somewhat greater than this pressure until all the other points along the contactline reach the critical slope. However, in the absence of more detailed quantitative information regarding partially-detached interfaces in the literature, we proceed with the conservative assumption that the entire interface detaches from the post once a local detachment takes place. We conjecture that the mismatch between the theoretical predictions and experimental data at SAFs less than 0.2 (see Figure 4a) is probably due to experimental errors. Nevertheless, given the complexity of the problem from both the computational and experimental viewpoints, the general agreement shown in Figure 4a is appreciable. It is also interesting to note the interplay between the contribution of air compression (dominant at low SAFs) and capillary pressure (dominant at high SAFs) in balancing the hydrostatic pressure in Figure 4a (see also Figure 2). The minimum post height required to avoid premature hmin is calculated and shown in Figure 4b for two different YLCAs. As expected, hmin is greater for more hydrophobic posts. In other words, when the posts are more hydrophobic, they should be taller to take full advantage of the hydrophobicity of the surface. It is also interesting to note that posts which are closely packed can be shorter than those farther away from each other. The evolution of the AWI between posts is studied in Figure 5. The inset in this figure shows the AWI for a when φ = 0.08 at t = 0, t = 0.24 min (critical time), t = 1.30 min, and t = 2.66 min (failure time) under a hydrostatic pressure of 0.6 m of water. It is interesting to note the similarity between the instantaneous shape of the AWI shown in this figure and those imaged during the evaporation of a droplet placed on an SHP surface made of ordered posts (the sagging and impalement of the AWI underneath an evaporating droplet follows the same physics the Laplace pressure increases as the droplet evaporates and forces the interface into the open space between the posts).3 Effects of posts’ packing fraction and the hydrostatic pressure on longevity are shown in Figure 5b. It can be seen that longevity increases with φ but decreases as hydrostatic pressure increases. The reason for this is that bubble pressure increases with hydrostatic pressure or when the posts are placed far from one another (due to a decrease in the total capillary force per unit area). Note that surfaces with φ = 0.08 and φ = 0.3 shown here have almost identical critical pressure (about 16 kPa, from Figure 4a) although their longevity values are different. In a recent study, experimental critical pressure values were reported for submerged SHP surfaces composed of circular nanoposts (pillars) with diameter, spacing, height, and YLCA of 35 nm, 51.8 nm, 180 nm, and 112°, respectively, in a hexagonal packing.32 The authors in ref 32 measured a critical pressure of about 27.5 atm for their nanotextured surfaces. Using the formulation given in section 2, we obtain a critical pressure of Pcr = 25.2 atm, which is in close agreement with the measurement of ref 32. 4.2. Surfaces with Randomly Distributed Posts. With the random arrangement not having a unit cell, our

Figure 5. Effects of hydrostatic pressure and SAF on longevity of the entrapped air on a SHP surface made of ordered posts are shown in (a) and (b), respectively. The inset in (a) shows the AWI in the space between circular posts with φ = 0.08 under a hydrostatic pressure of 0.6 m of water at t = 0, t = tcr = 0.24 min (critical time tcr), t = 1.3 min, and t = tf = 2.66 min.

computational domain should now encompass a group of posts. These posts are randomly distributed in a square domain with periodic boundary conditions and dimensions much larger than the average spacing between the posts. To reduce statistical uncertainty, the results were averaged over an ensemble of five statistically identical surfaces. As mentioned earlier, in the absence of more detailed information on the stability of a partially detached interface, we have assumed that the entire AWI detaches from all the posts once a local detachment on one of the posts has occurred. The detached interface will then move downward as the entrapped air dissolves in water (Regime I) or if more penetration is needed to balance the hydrostatic pressure (Regime II). Gridindependence for the critical pressure calculations was guaranteed by using 30 grid-points per post perimeter (the analysis is eliminated for brevity). Note that grid-independence for the longevity predictions is automatically guaranteed once the critical pressure predictions are grid-independent. This is because the accuracy of the longevity calculations depends only on the accuracy of the interface profile calculations (see eq 10), which is obtained from solving eq 4 for critical pressure. Figure 6 compares the critical pressure values obtained for SHP G


Langmuir

Article

Figure 6. Critical pressure vs SAF obtained for SHP surfaces with ordered and random post distributions.

surfaces with random and ordered post arrangements at different SAFs ranging from φ = 0.15 to φ = 0.4. As expected, critical pressure is lower when the posts are arranged randomly. Note again that the critical pressures in this work are obtained by including the compression forces generated by the entrapped air in the balance of forces acting on the meniscus and is therefore different from those presented in ref 12 in which the hydrostatic pressure was solely balanced by the capillary pressure under the assumption that the posts are so tall that air compression is negligible. When capillary forces are the only forces balancing the hydrostatic pressure, critical pressure monotonically increases with increasing SAF. Parts a and b of Figure 7 show the effects of SAF and pressure on the longevity of an SHP surface composed of randomly distributed posts, respectively. The inset in Figure 7a shows the AWI between the posts with an SAF of φ = 0.15 at t = 0, t = 0.04 min (critical time), t = 0.60 min, and t = 1.19 min (failure time) under a hydrostatic pressure of 1.0 m of water. As expected, longevity increases with SAF but decreases with increasing the hydrostatic pressure. A comparison between longevity values obtained for SHP surfaces with ordered and random posts at different hydrostatic pressures is given in Figure 8 for an SAF of φ = 0.15. It can be seen that random post arrangement can affect the longevity of an SHP surface. Note that the surface with random posts operates in the Regime II for the entire range of hydrostatic pressures considered in this figure (i.e., from 1 to 2 m of water). However, the surface with ordered posts operates in Regime II only when the hydrostatic pressure is above 1.5 m of water.

Figure 7. Effects of hydrostatic pressure and SAF on longevity of the entrapped air on an SHP surface composed of random posts are given in (a) and (b), respectively. The inset in (a) shows the AWI in the space between randomly distributed circular posts with φ = 0.15 under a hydrostatic pressure of 1.0 m of water at t = 0, t = tcr = 0.02 min (critical time tcr), t = 0.6 min, and t = tf = 1.19 min.

5. EQUIVALENT PORE DIAMETERS FOR SURFACE MADE OF POSTS As discussed earlier, predicting the critical pressure and longevity of an SHP surface requires a complicated IPDE to be solved numerically. This equation, however, can significantly be simplified if the surface is made of circular pores (see analytical formulations in section 3). Therefore, it is highly desirable to define an equivalent circular pore that can resemble the porous structure of an SHP surface made of posts with arbitrary arrangements. The first study to define equivalent pore diameters for estimating the critical pressure of SHP surfaces composed of ordered posts is reported in ref 33. In the

Figure 8. Comparison between the longevity values obtained for SHP surfaces with ordered and random post distributions.

H


Langmuir

Article

The definition for Deq g was then modified in ref 33 to read as

current work, we define equivalent pore diameters for both the critical pressure and longevity predictions and for both the ordered and random post arrangements. As mentioned before, capillary pressure and air compression are the two forces balancing the hydrostatic pressure acting on an AWI. A geometry-based equivalent pore diameter representing the contribution of gas compression in the balance of forces can be defined using the maximum distance between the posts in a unit cell (see Figure 9a), Dgeq = ( π /(2ϕ) − 1)d

Dgeq *

(35)

The authors in ref 33 neither discussed how they arrived at eq 35 nor presented a reason as to why such modifications were eq justified. We therefore use only Dgeq and Dcap in the formulations given here. With the above equivalent diameters representing the capillary and air compression, the critical pressure (eq 27) can be written as

(33)

eq Pcr = −4σ cos θ /Dcap − P∞

The concept of hydraulic diameter was used in ref 33 to define a second pore diameter to represent the capillary forces acting on the AWI, eq Dcap = (1 − ϕ)d /ϕ

eq 3 eq ⎛ eq ⎞3 Dg Dcap ϕ (Dg ) ⎜⎜ ⎟⎟ = = 2 1 − ϕ 2d 2 ⎝ d ⎠

× (1 − 1/(1 + Dgeq [2 − 3 sin θ + sin 3 θ ])/(6h cos3 θ))

(36)

Note in this equation that the volume of the entrapped air is represented in terms of Deq g and YLCA, which is the same as eq that in eq 27 for a circular pore (Deq g = Dcap = D). Figure 9a shows a comparison between the critical pressures obtained by numerically solving eq 4 for an SHP surface with ordered posts (from Figure 4a) and the analytical predictions obtained from the equivalent circular pore equation (eq 36) using both definitions given for geometric equivalent pore Deq g (current study) and Deq g * (ref 33). Note that both equivalent pore diameter formulations provide reasonable predictions. For longevity calculations, we have developed an approximate method that does not require solving a PDE. In this method, we treat the initial AWI between the posts in a unit cell as the initial interface in a circular pore. We then calculate the angle θ0 between the interface and the walls of the pore (instead of calculating the initial radius of curvature R0 as in sections 2 and 3). Equation 29 can therefore be modified as

(34)

eq − 4σ cos θ0/Dcap − Pg − P∞

× [1 − 1/(1 + Dgeq [2 − 3 sin θ0 + sin 3 θ0])/(6h cos3 θ0)] = 0

(37)

From section 2, parameters β = −4σ cos eq eq cos θ/Deq cap, and Rcr = −Dcap/2 cos θ) and α written as eq

θ0/Deq cap, eq

κ = −4σ can now be eq

αeq = S2 × (h + Dgeq [2 − 3 sin θ0 + sin 3 θ0]/(6 cos3 θ0))/(ξ ̅ A 0) (38)

With the maximum initial slope of the interface at the wall given as |▽f |max = |cot θ0|, in this method, we do not obtain an explicit equation for the shape of the AWI. Therefore, we used the projected area of interface surface (i.e., A0, or Acr) in our calculations as the deflection of the initial AWI is not generally significant, A 0 = (1/ϕ − 1)πd 2/4

(39a)

To predict the surface area of the critical AWI, we assumed that the critical-to-initial surface area ratio for the interface between posts is the same as the critical-to-initial surface area ratio for the interface inside its equivalent pore (with a diameter Deq g ). Hence,

Figure 9. Comparison between the critical pressures obtained from a numerical solution of the PDE in eq 4 and the predictions of the equivalent pore diameter method for an SHP surface made of ordered posts is shown in (a). Two definitions for the equivalent pore diameter are considered, that of ref 33 and that of the current study. The inset in (a) shows the geometric equivalent pore diameter Deq g . Our equivalent pore diameter method (eq 40) is used to predict the longevity of SHP surfaces with different SAFs in (b), and the results are compared with the predictions from the numerical solution of eqs 5 and 10

Acr = 8A 0R creq[R creq −

[R creq ]2 − [Dgeq /2]2 ]/[Dgeq ]2 (39b)

Req cr

Note that the critical radius is also used to estimate the coordinate of the lowest point on the interface (the point that I


Langmuir

Article

makes the first contact with the bottom). The critical and failure times can now be calculated as tcr = −

eq αeq ⎛ γ |∇f |max − β cot θ ⎞ ln ⎜ ⎟ β eq ⎝ (β eq + γ )|∇f |max ⎠

⎡ ⎢h + tf1 = tcr − S⎢ ⎣

⎤ (R creq)2 − [Dgeq /2]2 − R creq ⎥ ⎥ (κ eq + Π)ξ ̅ Acr ⎦

(40a)

(40b)

Figure 9b shows a comparison between the longevity values obtained for SHP surfaces made of ordered posts with different SAFs and under different hydrostatic pressures using the equivalent pore diameter method (eq 40) and the numerical solution of the PDE given in eq 10 (see Figure 5). It can be seen that the equivalent pore diameter method produces reasonable predictions while being significantly less complicated mathematically. Note that the predictions of the equivalent pore diameter method tend to become less accurate when the SAF of the surface is high especially at low hydrostatic pressures. This is due perhaps to the increasing mismatch between the shape of an AWI inside a pore and that formed between four closely packed posts. Nonetheless, with the equivalent pore diameter method one can easily estimate the longevity of an SHP surface without actually solving a PDE. The equivalent pore diameters discussed here can also be utilized for critical pressure and longevity estimation when the posts are distributed randomly. For such calculations, we produced the so-called Voronoi diagram for each surface (see the inset in Figure 10a). In a Voronoi diagram, each post is placed in a cell with its boundaries defined as the locations of the points on the surface that are equidistant from the two nearest posts.34 Voronoi diagram is used here to determine the local minimum SAF, i.e., φmin, for calculating the capillary pore diameter, i.e., eq Dcap = (1 − ϕmin)d /ϕmin

Figure 10. Critical pressure predictions obtained from our equivalent pore diameter method (eq 41) are compared in (a) with those obtained from solving the PDE in eq 4. An example Voronoi diagram with φ = 0.25 is given in the inset in (a). Longevity values obtained from solving PDEs in eq 5 and eq 10 for an SHP surface with φ = 0.15 are compared with those of our equivalent diameter method (eq 41) in (b).

(41)

For the geometric pore diameter Deq g we used eq 33. The remainder of the procedure is similar to that described earlier for surfaces with ordered posts. Parts a and b of Figure 10 show a comparison between critical pressure and longevity values obtained from solving the PDEs given in section 4 (eq 4 for critical pressure and eq 10 for longevity) and those from the simple equivalent pore diameter calculations of this section. As mentioned before, the equivalent pore diameter method becomes less accurate in the case of large SAFs under low pressures.

hydrostatic pressure. It is shown that the air compression forces are more significant when the pores are shallow and wide whereas the capillary forces are dominant for long and narrow pores (or posts). This leads to a U-shaped behavior for the mechanical stability of an SHP surface in terms of pore diameter (for surfaces with pores) or SAF (for surfaces with posts). Surface longevity is found to drastically decrease with increasing the hydrostatic pressure. Increasing the pore diameter or decreasing the SAF is found to also decrease the longevity of the surface. Comparing surfaces with ordered and random posts, it was found that randomness in the posts’ spatial positions decreases both the mechanical stability and the longevity of the surface.

6. CONCLUSIONS The work presented here is a generalized theory that allows one to predict the mechanical stability and the service life (longevity) of SHP surfaces when used for underwater applications. The formulations presented are general enough to handle surfaces composed of pores or posts of any arbitrary cross sections, as long as they are round (i.e., do not have sharp corners). Moreover, our formulations can significantly be simplified to algebraic equations or ODEs if the surface is comprised of circular pores. On this basis, an equivalent pore diameter method is developed to predict the stability and longevity of surfaces with ordered or random posts. Our work quantifies the contributions of the capillary forces and the forces generated by the entrapped air in balancing the

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: 804-828-9936. Notes

The authors declare no competing financial interest. J


Langmuir

■

Article

(23) Poetes, R.; Holtzmann, K.; Franze, K.; Steiner, U. Metastable underwater superhydrophobicity. Phys. Rev. Lett. 2010, 105, 1661041− 1661044. (24) Samaha, M. A.; Ochanda, F. O.; Tafreshi, H. V.; Tepper, G. C.; Gad-el-Hak, M. In situ, noninvasive characterization of superhydrophobic coatings. Rev. Sci. Instrum. 2011, 82, 0451091−0451097. (25) Emami, B.; Hemeda, A. A.; Amrei, M. M.; Luzar, A.; Gad-elHak, M.; Tafreshi, H. V. Predicting longevity of submerged superhydrophobic surfaces: Surfaces with parallel grooves. Phys. Fluids. 2013, 25, 062108. (26) Hemeda, A. A.; Gad-el-Hak, M.; Tafreshi, H. V. Effect of hierarchical features on longevity of submerged surfaces with parallel grooves. Phys. Fluids. 2014, 26, 082103. (27) Patankar, N. A. Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach. Langmuir 2010, 26, 8941−8945. (28) Whyman, G.; Bormashenko, E. How to make the Cassie wetting state stable? Langmuir 2011, 27 (13), 8171−8176. (29) Lee, C.; Kim, C.-J. Maximizing the giant liquid slip on superhydrophobic microstructures by nanostructuring their sidewalls. Langmuir 2009, 25 (21), 12812−12818. (30) Flynn, M. R.; Bush, W. M. Underwater breathing: The mechanics of plastron respiration. J. Fluid Mech. 2008, 608, 275−296. (31) Lv, P.; Xue, Y.; Shi, Y.; Lin, H.; Duan, H. Metastable states and wetting transition of submerged superhydrophobic structures. Phys. Rev. Lett. 2014, 112, 196101. (32) Checco, A.; Ocko, B. M.; Rahman, A.; Black, C. T.; Tasinkevych, M.; Giacomello, A.; Dietrich, S. Collapse and reversibility of the superhydrophobic state on nanotextured surfaces. Phys. Rev. Lett. 2014, 112, 216101. (33) Xue, Y.; Chu, S.; Lv, P.; Duan, H. Importance of hierarchical structures in wetting stability on submersed superhydrophobic surfaces. Langmuir 2012, 28 (25), 9440−9450. (34) Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S. N. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams; Wiley: Chichester, 2000.

ACKNOWLEDGMENTS H.V.T. acknowledges NSF CMMI Grant # 1029924 for financial support.

■

REFERENCES

(1) Quere, D. Wetting and roughness. Annu. Rev. Mater. Res. 2008, 38, 71−99. (2) Rothstein, J. P. Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 2010, 42, 89−109. (3) Moulinet, S.; Bartolo, D. Life and death of a fakir droplet: Impalement transitions on superhydrophobic surfaces. Eur. Phys. J. 2007, E 24, 251−260. (4) Kusumaatmaja, H.; Blow, M. L.; Dupuis, A.; Yeomans, J. M. The collapse transition on superhydrophobic surfaces. Europhys. Lett. 2008, 81, 36003. (5) Reyssat, M.; Yeomans, J. M.; Quere, D. Impalement of fakir drops. Europhys. Lett. 2008, 81, 26006. (6) Papadopoulos, P.; Mammen, L.; Deng, X.; Vollmer, D.; Butt, H. J. How superhydrophobicity breaks down. Proc. Natl. Acad. Sci. U.S.A. 2013, 110 (9), 3254−3258. (7) Extrand, C. W. Model for contact angles and hysteresis on rough and ultraphobic surfaces. Langmuir 2002, 18, 7991−7999. (8) Extrand, C. W. Criteria for ultralyophobic surface. Langmuir 2004, 20, 5013−5018. (9) Extrand, C. W. Repellency of the lotus leaf: Resistance to water intrusion under hydrostatic pressure. Langmuir 2011, 27, 6920−6925. (10) Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Effects of hydraulic pressure on the stability and transition of wetting modes of superhydrophobic surfaces. Langmuir 2005, 21 (26), 12207−12212. (11) Lobaton, E. J.; Salamon, T. R. Computation of constant mean curvature surfaces: Application to the gas-liquid interface of a pressurized fluid on a superhydrophobic surface. J. Colloid Interface Sci. 2007, 314 (1), 184−198. (12) Emami, B.; Tafreshi, H. V.; Gad-el-Hak, M.; Tepper, G. C. Predicting shape and stability of air−water interface on superhydrophobic surfaces with randomly distributed, dissimilar posts. Appl. Phys. Lett. 2011, 98, 2031061−2031063. (13) Emami, B.; Tafreshi, H. V.; Gad-el-Hak, M.; Tepper, G. C. Simulation of meniscus stability in superhydrophobic granular surfaces under hydrostatic pressures. Colloids Surf., A 2011, 385, 95−103. (14) Emami, B.; Tafreshi, H. V.; Gad-el-Hak, M.; Tepper, G.C. Predicting shape and stability of air−water interface on superhydrophobic surfaces comprised of pores with arbitrary shapes and depths. Appl. Phys. Lett. 2012, 100, 0131041−01310414. (15) Samaha, M. A.; Tafreshi, H. V.; Gad-el-Hak, M. Sustainability of Superhydrophobicity under Pressure. Phys. Fluids. 2012, 24, 112103. (16) Forsberg, P.; Nikolajeff, F.; Karlsson, M. Cassie−Wenzel and Wenzel−Cassie transitions on immersed superhydrophobic surfaces under hydrostatic pressure. Soft Matter. 2011, 7, 104−109. (17) Rathgen, H.; Mugele, F. Microscopic shape and contact angle measurement at a superhydrophobic surface. Faraday Discuss. 2010, 146, 49−56. (18) Sheng, X.; Zhang, J. Air layer on superhydrophobic surface underwater. Colloids Surf., A 2011, 377, 374−378. (19) Samaha, M. A.; Tafreshi, H. V.; Gad-el-Hak, M. Effects of hydrostatic pressures on drag-reduction performance of submerged aerogel particle coatings. Colloids Interfaces A: Physicochem. Eng. Aspects 2012, 399, 62−70. (20) Bobji, M. S.; Kumar, S. V.; Asthana, A.; Govardhan, R. N. Underwater sustainability of the Cassie state of wetting. Langmuir 2009, 25, 12120−12126. (21) Sakai, M.; Yanagisawa, T.; Nakajima, A.; Kameshima, Y.; Okada, K. Effect of surface structure on the sustainability of an air layer on superhydrophobic coatings in a water−ethanol mixture. Langmuir 2009, 25, 13−16. (22) Lei, L.; Li, H.; Shi, J.; Chen, Y. Diffraction patterns of a watersubmerged superhydrophobic grating under pressure. Langmuir 2010, 26 (5), 3666−3669. K


General Formulations for Predicting Longevity of Submerged

Recommend Documents