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The thermodynamics of restoring underwater superhydrophobicity Paul R. Jones, Adrian Thomas Kirn, David Ma, Dennis T. Rich, and Neelesh Ashok Patankar Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b04432 • Publication Date (Web): 10 Feb 2017 Downloaded from http://pubs.acs.org on February 11, 2017

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The thermodynamics of restoring underwater superhydrophobicity Paul R. Jones,

†

Adrian T. Kirn,

†

Y. David Ma,

Patankar

†

Dennis T. Rich,

‡

and Neelesh A.

∗,†

†Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road,

Evanston, IL 60208, USA.

‡Illinois Mathematics and Science Academy, 1500 W Sullivan Rd, Aurora, IL 60506 E-mail: [email protected]

Phone: (847) 491-3021. Fax: (847) 491-3915

Abstract Superhydrophobic surfaces submerged in liquids are susceptible to permanently becoming wet. This is especially true when the ambient liquid is pressurized or undersaturated with air. To gain insight into the thermodynamics of restoring underwater superhydrophobicity, nucleation theory is applied to the design of spontaneously dewetting conical pores. It is found that for intrinsically hydrophobic materials, there is a geometric constraint for which reversible superhydrophobic behavior may occur. Molecular dynamics simulations are implemented to support the theory, and steered molecular dynamics simulations are used to investigate the energy landscape of the dewetting process. The results of this work have implications for the ecacy of underwater superhydrophobicity and enhanced nucleation sites for boiling heat transfer.

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INTRODUCTION Underwater superhydrophobicity, manifested in the form of surfaces that remain practically dry in submerged conditions, are desirable for many applications like drag reduction, 13 boiling, 4 reduced biofouling, 5,6 among others. Dry submerged surfaces are unusual; there are only a few naturally occurring instances of air retaining insects 7,8 and plants 9 where wings or leaves remain dry underwater. There too, not all surfaces can remain indenitely dry. 7 Dry submerged surfaces are possible if gases can be sustained in roughness grooves of the submerged surface. 10 These gases are deemed to be a mixture of air and the vapor phase of the submerging liquid. 11,12 If the conditions (surface energy and roughness length scale) are not appropriate, then the vapor in the roughness grooves could condense or the trapped gases, such as air, could dissolve into the liquid. 11 This can lead to wetting transitions, rendering the surface permanently wet. For these reasons, sustaining gases in roughness grooves over long periods has been particularly challenging. 1,1316 The thermodynamics of sustaining vapor 11,1719 or trapped gases like air 12 in roughness grooves of submerged surfaces has been studied. Thermodynamic analyses predict that roughness spacing of less than one micron is necessary to indenitely sustain dry submerged surfaces at room temperature and atmospheric pressure when the liquid is undersaturated with air. 11,12,1719 These predictions are consistent with prior observations where gases were found to deplete after 2-3 days from rough surfaces with tens of micron-scale features, 1,2,11,13,14 whereas submicron-scale roughness was found to sustain gases for more than 120 days. 7,11,13 In spite of the aforementioned progress, one major challenge remains which compromises the robustness of underwater superhydrophobicity to be of practical utility. If a wetting transition does occur in roughness grooves due to, say, a pressure rise in the submerging liquid, then these surfaces remain wet even after the pressure in the submerging liquid reduces back to normal. This is typically because fully wetted roughness grooves are also an energet2


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ically stable conguration. There is an energy barrier to begin the dewetting transition. To understand this, consider a fully wetted cylindrical pore. Initiating the dewetting of the pore would require the liquid-solid interface to be replaced by a liquid-gas and a gas-solid interface on the bottom of the pore. This transition is always energetically unfavorable leading to an energy barrier that cannot be overcome by the thermal energy, even if the pore diameter is tens of nanometer scale. Active methods, such as electrical tuning, 20 vibrations, 21 and gas generation 2 have been proposed for wet-to-dry or drying-o transitions. Thermodynamics of the wet-to-dry recovery were investigated using theory and molecular dynamics simulations, 22 however, the simulations relied on ambient uids with an undetermined pressure. The eect of pressure and phase change on the dewetting process is critically important for the implementation of underwater superhydrophobicity. This issue is addressed in this work. One approach to induce a spontaneous wet-to-dry transition on submerged surfaces is to design roughness geometries, such that, the wetted state is unstable. Here the energy landscape would be such that the energy monotonously decreases from wet-to-dry states. 22 In other words, the wetted state would be a maximum extrema instead of a minimum extrema like in the case of cylindrical pores. Conical pillars or pores have been identied as potential geometries for this purpose. 23,24 One reason is that a conical pore does not have a base area, unlike a cylindrical pore, that needs to be dewetted to initiate a wet-to-dry transition. There is experimental evidence that surfaces with conical geometries show robust superhydrophobic properties. 25,26 Transmission small-angle x-ray scattering experiments of conical pillars, with feature spacings ∼ 50 nm and macroscopic contact angle ∼ 165◦ , were implemented to assess the role of pressure on the dewetting process. 26 Surfaces that were initially wet demonstrated partial reappearance of the trapped gas within roughness grooves, upon depressurization of the ambient liquid. Two issues are important to understand wet-to-dry transitions. First, would the vapor occupying the dry conical pore condense to cause a wetting transition? Second, would trapped gas in a dry conical pore dissolve into an undersaturated liquid to induce a wetting tran-

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sition? Prior models to explain wet-to-dry transitions considered energy landscapes based on the surface energy and Laplace pressure balance at the liquid-gas interface. 26 Chemical equilibria of phases, as well as dissolved gases are essential to fully resolve the drying-o process during the wet-to-dry transition. 11,12,17 In the following sections, an understanding of the mechanisms underlying drying-o is developed by theoretical analysis of the energy landscape that accounts for phase equilibrium and gas dissolution. It is found that not all cone angles are suitable for dewetting

− appropriate cone angles are identied as a function of other parameters of the problem. Theoretical predictions are veried by way of molecular dynamics (MD) simulations and direct computation of the energy landscape. These ndings are found to be consistent with prior experimental data.

THEORY SECTION To understand how water may dewet a conical pore that is immersed in water, we begin by identifying the four key states that dene the process. These states are illustrated in Fig. 1 and are described below. Note that for the purpose of this discussion it will be assumed that the gas phase is the vapor phase of water. 11,1719 An analogous discussion for the presence of air or a combination of air and vapor is conceptually similar. 12 State 1:

Liquid fully wets the pore immersed in water. This is the reference state. All other

states will be compared to the reference state. State 2:

A vapor nucleus forms at the bottom of the pore. If the energy landscape is such

that State 1 is a maximum extrema, then the vapor nucleus will grow and advance the system to State 3. Else, there will be an energy barrier for dewetting. State 3:

The vapor state reaches the top of the conical pore, with a liquid-vapor interface

hanging from the pore opening. The curvature of this interface is dictated by the 4


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ψ

D (1)

Vapor θ h34 θe (4)

Liquid

hmeniscus re

θ H h123 (2)

z0 (3)

Fig. 1: Schematic for water dewetting a conical pore. Young-Laplace equation. In addition, chemical equilibrium between the liquid and vapor phases is required across the liquid-vapor interface. State 4:

Starting with State 3, another liquid phase may condense within the conical pore.

This second liquid phase may grow from the bottom of the pore until the two liquidvapor interfaces touch. At this point, State 4 transitions back to State 1, the wetted state.

Dewetting of conical pores The process for water dewetting a conical pore of height H has two components. First, the pore starts dewetting, i.e., the liquid starts receding from the bottom of the pore. This would happen if the fully wetted state (State 1) is a maximum extrema. Second, after the liquid starts receding from the pore, it would completely dewet a pore of height H only if there is no stable equilibrium state within the pore at some height less than H . Initiation of dewetting

First we consider the initiation of the dewetting process. To that end, we consider the change in thermodynamic free energy during dewetting. The goal is to identify cone angles for which 5


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spontaneous dewetting would occur. The free energy change from State 1 to State 2 to State 3, is given by 27,28

∆f123 = ∆pVv + Alv σlv + Asv σlv cos θe ,

(1)

where, the area of the liquid-vapor interface is:

Alv = 2πr2 (1 − sin(θ − ψ/2)),

(2)

the area of the solid-vapor interface is:

Asv = π(h123 )2 tan(ψ/2)/ cos(ψ/2),

(3)

the liquid height along the pore walls is:

h123 = −r cos(θ − ψ/2) cot(ψ/2),

(4)

and the volume of vapor within the cone is:

Vv = (1/3)π(h123 )3 tan(ψ/2)2 − (1/3)πr3 (2 + sin(θ − ψ/2))(1 − sin(θ − ψ/2))2 .

(5)

Here, r is the radius of a nucleate phase, θ is the apparent contact angle, θe is the material contact angle, σlv is the liquid-vapor surface energy, pl , pv are the liquid and vapor pressures, and ψ is the cone angle. Note: pl , pv , σlv are each dependent on the liquid temperature Tl , and at equilibrium, r = re , θ = θe , which leads to the pressure dierence:

2σlv 4σlv ψ ∆p = pl − pv = =− cos θe − , re D 2

6


(6)

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with vapor pressure 27,29

vl pv = psat exp (pl − psat ) . RTl

(7)

Note: D is the diameter of the pore, R is the specic gas constant, vl is the specic volume of the liquid, and psat is the liquid-vapor coexistence pressure. We can non-dimensionalize equation 1 by multiplying by β = 1/ (kB Tl ) to obtain

nd ∆f123 = β∆f123 ,

(8)

where kB is the Boltzmann constant. This non-dimensionalization has been used throughout this work for consistency. This is done so the energy change is compared to the thermal energy in the system. For ambient conditions ( pl = 1 atm, Tl = 300 K) with pores ∼ 759 nm diameter and cone angle ψ = 30◦ , this non-dimensionalization will yield large values O (107 ) for the energy. The dimensional free energy change is typically O (σlv re2 ). Critical cone angle for initiation of dewetting

The threshold for establishing whether a pore is capable of spontaneous dewetting can be observed by setting h123 = 0 in equation 4, and solving for the critical cone angle ψc =

ψ|h123 =0 6= 0. This leaves us with the criterion: ψc = 2θe − 180◦ .

(9)

Equation 9 is plotted in Fig. 2A with a solid black line. This equation partitions the set of material contact angles and cone angles that permit a pore to spontaneously dewet from those that inhibit dewetting. For conical pores with an equilibrium contact angle of θe = 120◦ , the critical cone angle necessary to enable a pore to dewet is ψc = 60◦ . Spontaneous dewetting will begin if ψ < ψc . Equation 9 also implies that hydrophilic pores (θ e ≤ 90◦ ) will never dewet for a non-zero cone angle.

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The criterion established by equation 9 does not determine whether a conical pore of a given height will fully dewet or not, i.e., whether the liquid-vapor interface will reach a height

h = H or not. Instead, it serves as a threshold for assessing whether it is thermodynamically possible for dewetting to begin. This assessment is based on the geometry ψ and chemistry

θe of the surface. Eects due to ambient conditions will manifest in the equilibrium height of equation 4, once ψ < ψc is satised. Degree of dewetting

Next, we consider the degree to which the pore is dewetted. The fraction of the pore lled with liquid is:

χ = Vl /Vcone ,

(10)

1 Vcone = πH 3 tan(ψ/2)2 , 3

(11)

where, the volume of the pore is:

and the volume of liquid within the pore is:

Vl = Vcone − Vv .

(12)

The nal volume of liquid within the pore will then be:

Vl

(state 3)

= Vl |h123 =H .

(13)

Equation 10 is a measure of the degree to which a pore is wet ( χ = 1) or dry (χ = 0). The height of the liquid-vapor interface at the center of the pore is:

hmeniscus 123

ψ ψ ψ cot − r 1 − sin θ − . = −r cos θ − 2 2 2

8


(14)

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(A)

θe (deg)

180 Pore will dewet

150

120

Pore will remain wet 90 0

30

60

90

120

150

ψ (deg)

180

(B)

∆fnd

1

∆fnd 123

0.8

∆fnd 34

0.6

∆fnd (Presumed) 41

0.4

Barrier

0.2 0 0

0.2

0.4

0.6

0.8

1

0.8

1

Degree wet (χ)

(C) 6 5

Barrier

nd

4

∆f

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3 2 1 0 0

Fig. 2:

∆fnd 123 nd ∆f34 ∆fnd (Presumed) 41 0.2

0.4

0.6

Degree wet (χ)

(A) Phase diagram. The boundary for which water will dewet the pore occurs when h123 = 0. This occurs when θ = (π + ψ) /2. When θe < 90◦ there is no spontaneous dewetting. (B) Representative energy cycle for cones that will dewet, ψ = 30◦ , θe = 120◦ . The lowest energy state is dry. (C) Representative energy cycle for cones that will not dewet, ψ = 60◦ , θe = 120◦ . The lowest energy state is wet. The energy barrier for nucleation of liquid condensate is illustrated for both energy cycles Theoretical energy landscape for water dewetting a conical pore.

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When liquid completely dewets, the liquid-vapor interface will rest from the top of the pore with height:

z0 = H − r (1 − sin (θ − ψ/2)) ,

(15)

relative to the bottom of the pore.

Condensation within conical pores Once liquid has fully dewet the pore (State 3), is it possible for vapor to condense inside the pore and rewet the surface by a pathway from State 3 to State 4 to State 1? 11 The free energy change in this process is described in the Supporting Information. The free energy change from State 1 to State 4 is obtained by adding the changes in free energy: nd nd nd ∆f1234 = ∆f123 + ∆f34 .

Energy landscape The energy landscape for water dewetting a conical pore is shown in Fig. 2. Representative energy cycles for cone angles of ψ = [30◦ , 60◦ ] are shown in Fig. 2B and Fig. 2C. There,

T = 300 K , σlv = 71.686 mN/m, vl = 1.0035 × 10−3 m3 , pl = 1.0132 × 105 P a, pv = 3539.3 P a, psat = 3536.8 P a, 30 θe = 120◦ , re = 1.4662 µm, and H = 1.419 µm. Fig. 2B and Fig. 2C provide the theoretical change in free energy as a function of the degree of wetting (χ). When ψ = 30◦ (Fig. 2B) the lowest energy corresponded to the dry state, whereas, for ψ = 60◦ (Fig. 2C) the lowest energy corresponded to the wet state. Thus, for water to begin in the fully wetted state (State 1), it may proceed to State 2 and then to State 3 by spontaneous dewetting, as long as the parameters ( ψ, θe ) are suitable (Fig. 2A). However, the condensation pathway from State 3 to State 4 to State 1 requires an energy barrier to be overcome. This energy barrier is signicantly greater than the thermal energy when pores have dimensions greater than tens of nanometers. Hence, it is unlikely that a spontaneously dewetted pore will rell by way of condensation, unless the pores are small

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(few nanometers). nd |h=H associated with the fully dewetted conguration (State The free energy change ∆f123 nd |h=H < 0 3) is plotted in Fig. 3 as a function of the cone angle ψ . It is observed that for ∆f123 nd |h=H > 0 spontaneous dewetting of the entire conical pore will occur, whereas for ∆f123

the pore will remain at least partially wet. This is veried in later sections by performing molecular dynamics simulations. 8

1

x 10

123 h=H

0.5

∆fnd |

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0

−0.5

−1 0

30

60

90

120

150

180

ψ (deg)

Fig. 3: Non-dimensional free energy for a fully dewetted pore under atmospheric conditions, as determined by classical nucleation theory.

Air in conical pores The theory presented, thus far, has relied on the sustenance of vapor within the pore. If instead, there was air or some other gas dissolved in water, would it eervesce into the conical pore and dewet it? This question can be answered by following the same calculations as those for the liquid-vapor case discussed, above, but with one exception. The vapor pressure pv in equation 6 should be replaced by the air pressure pair in the conical pore. An equation for pair can be obtained by imposing chemical equilibrium between the air dissolved in the liquid and air that has eervesced into the conical pore. 12 Henry's Law, which relates the degree of saturation α of air dissolved in the liquid to the pressure of air in the pore, has been applied to the design of superhydrophobic surfaces, yielding the 11


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approximation: 12

pair ≈ αpl ,

(16)

By denition α ∈ [0, 1]. This leaves the pressure dierence as:

ψ 4σl,air cos θe − . ∆p = pl (1 − α) = − D 2

(17)

where σl,air is the liquid-air interfacial energy. In general, both vapor and eervesced gases will occupy a pore that spontaneously dewets. In this case, the calculations above can be modied in terms of the partial pressures of vapor and gases. 12 Note that the criterion for initiation of dewetting, given by equation 9, remains unchanged.

METHODS SECTION Numerical simulation Molecular dynamics and steered molecular dynamics (SMD) simulations were implemented to verify the spontaneous dewetting of conical pores and the corresponding energy landscapes. The open source molecular dynamics package, LAMMPS 31 (version June 28, 2014) software was used for all simulations. The SPC/E 32 water model was used with SHAKE 33 to permit a 2.0 fs time step. A cut-o distance of 12 Å was used for non-bonded interactions with a switch distance of 10 Å. A particle-particle-particle-mesh solver 34 was used to calculate long-range electrostatic interactions. The specied relative error in forces was 10−4 . Pressure was applied to the liquid water using a rigid graphene sheet piston. The piston was constrained to move in the z-direction only, while the surface remained at a constant location. Temperature was controlled using a Nosé-Hoover thermostat 35 with a relaxation time of 200 fs. The water oxygen (O), top surface (S), pore/bottom surface (pore), and piston (PST) 12


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Lennard-Jones interactions were: σO,S = σO,pore = σO,P ST = 3.19004 Å, and εO,S = εO,pore =

εO,P ST = 0.0449 kcal/mol. The well-depth εO,pore was chosen to yield a hydrophobic contact angle. This approach of increasing the contact angle by decreasing the interaction energy is discussed further in Ref. 36 During equilibration a ctitious surface was used to block water from entering the pore. The Lennard-Jones parameters for the ctitious surface matched those for the top surface. After equilibration, the Lennard-Jones parameters were set to zero to allow water to enter the pore.

Molecular dynamics simulations of hydrophobic conical pores Dewetting simulations of conical pores with diameter D = 10 nm were initiated with water fully wetting the pore. See Fig. 4A for model. This was achieved by signicantly pressurizing the water up to pl = 1500 bar. Pores with cone angles of ψ = [30◦ , 60◦ , 90◦ ] were simulated at constant temperatures of Tl = 300 K and Tl = 500 K . The simulations consisted of 179,097, 163,481, and 158,518 atoms, respectively, with a constant number of 39,086 water molecules adjacent to each pore. The applied liquid pressure pl was decremented, then held constant to allow the system to adjust to the new conguration. This de-pressurization process was repeated until water either fully dewet the pore, or the applied pressure reached zero.

Steered molecular dynamics to determine energetic landscape The free energy landscape governs the wetting process for hydrophobic surfaces. 22,37,38 The calculation of free energies can be divided into two general groups: (1) equilibrium methods, such as thermodynamic integration or free energy pertubation, and (2) non-equilibrium methods which often employ Jarzynski's equality. 39,40 Jarzynski's equality relates non-equilibrium

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work to the equilibrium free energy of a system. The relation is written as

1 ∆f = − ln (e−βW ), β

(18)

were β = 1/ (kB Tl ), Tl is the temperature, and kB is the Boltzmann constant. Jarzynski's equality has been veried experimentally for a single molecule of RNA. 41 To determine the energy landscape for the dewetting of a conical pore, we implemented the non-equilibrium technique of steered molecular dynamics (SMD) 4244 to investigate the potential of mean force along the z-direction of the pore. The potential of mean force Φ is the free energy landscape as a function of a single parameter. 42,43 In this case, the parameter is the z-component of the center of mass of water zcm . As a rst-order approximation, the potential of mean force is ∆Φ ≈ ∆f . See Supporting Information for details. The steered molecular dynamics simulations used pores of diameter D = 3 nm that were signicantly smaller than those used in molecular dynamics simulations. For cone angles of ψ = [30◦ , 60◦ , 90◦ ], 12,778, 11,085, and 10,662 atoms were used. Each system consisted of 1,664 water molecules. This was done for computational eciency. Water was subject to a constant temperature of Tl = 300 K and pressure of pl = 200 bar. The water initially resided outside the pore in the dry state. From there, water was pulled into the pore at a rate of v = 1 × 10−6 Å/f s using a harmonic potential with spring constant −1

k = 1 × 107 kcal · mol−1 Å . The spring was bound to the z-component of the center of mass of the water to impose a body force.

RESULTS AND DISCUSSION Molecular dynamics simulations of hydrophobic conical pores Molecular trajectories of conical pores that dewetted, once the applied pressure subsided, are shown in Fig. 4B. For a temperature of Tl = 300 K , water fully dewet the pore with

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ψ = 30◦ and applied pressure of pl = 50 bar. This pressure was larger than the coexistence pressure psat = 1.017 × 10−2 bar 45 for the SPC/E water model. This case demonstrates a pore becoming dry underwater (State 3), in-spite of beginning with a wetted conguration (State 1). The pore with ψ = 60◦ exhibited an ability to partially dewet, in agreement with our criterion (equation 9). However, this pore was not capable of fully dewetting. With the lowest applied pressure pl = 0 bar, which is below the vapor pressure, vapor is the thermodynamically favorable state. This means the pooled liquid above the pore is metastable. When ψ = 90◦ , the pore never dewetted, as predicted by equation 9. At an elevated temperature of Tl = 500 K , similar results were achieved but at dierent applied pressures. Dewetting was obtained for ψ = 30◦ at pl = 50 bar, and ψ = 60◦ at

pl = 25 bar. In both cases, liquid water ( pl > psat = 16.51 bar 45 ) fully dewet the pore and was replaced with vapor within the pore. The thermodynamic state of the vapor is such that it would be metastable in the bulk. For ψ = 90◦ , water also dewet the pore, but only once the applied pressure decreased below the vapor pressure. The results for hydrophobic pores dewetting at temperatures of Tl = 300 K and Tl =

500 K are organized in Fig. 5. The expected threshold for dewetting (equation 6) is plotted separately for each temperature. Dewetting pressures can be made dimensionless using equation 6, pnd l = (pl − pv ) D/ (4σlv ). This is shown in Fig. 6A as a function of the angle φ =

θe −ψ/2. Contact angles were calculated by tting a second-order multivariate polynomial to the liquid-vapor interface. 46 This process was repeated for 500 trajectory frames to establish the material contact angles reported in Table 1. A subset of the non-dimensional plot (Fig. 6A) is shown in Fig. 6B. Note: This method of controlling the pore geometry to inuence the behavior of water may be extended to promote wetting as well (Fig. S1.). The ability of a conical pore to dewet may not require perfect geometry. Fig. S2-S3 show simulations of truncated hydrophobic pores that either remained wet or dewetted at

T = 300 K once ambient pressure subsided. This is signicant, as defects may occur when fabricating surfaces for experiments.

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(A)

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Molecular dynamics model Piston

Applied pressure

ψ=90˚ Pore

D

ψ=60˚ Liquid H2O

ψ=30˚ (B)

Dewetting simulations T = 300 K

ψ = 30◦ 1500 bar

100 bar

50 bar

800 bar

50 bar

0 bar

300 bar

25 bar

0 bar

1500 bar

100 bar

50 bar

800 bar

50 bar

25 bar

300 bar

10 bar

2 bar

ψ = 60◦

ψ = 90◦ T = 500 K

ψ = 30◦

ψ = 60◦

ψ = 90◦

Fig. 4:

Molecular dynamics simulations of hydrophobic conical pores. (A) Molecular dynamics model with cone angles ψ = [30◦ , 60◦ , 90◦ ]. (B) Dewetting simulations at temperatures of 300 K and 500 K. Simulations began with water fully wetting the pore. The applied pressure was decremented until water dewet the pore.

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Pressure (bar)

(A) 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

Liquid fully wets surface Liquid partially dewets surface Liquid fully dewets surface Expected threshold for full dewetting

30

60 Cone angle ψ (deg)

90

30

60 Cone angle ψ (deg)

90

(B)

Pressure (bar)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

Fig. 5: Phase diagram for dewetting simulations of conical pores with cone angles ψ = [30◦ , 60◦ , 90◦ ]. (A) Temperature = 300 K. (B) Temperature = 500 K

Table 1: Material contact angles calculated from dewetting molecular dynamics simulations of fully dry conical pores Temperature Tl (K) 300 300 300 500 500 500

Cone angle ψ (deg) 30 60 90 30 60 90

Pressure pl (bar) 50 − − 50 25 02

17

Material contact angle θe (T ) ± ∆θ (deg) 121.34 ± 2.9 − − 137.05 ± 4.78 132.36 ± 6.36 138.81 ± 7.67


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(A) 15

Liquid fully wets surface Liquid partially dewets surface

plnd=(pl−pv)D/(4σlv)

Liquid fully dewets surface Expected threshold for full dewetting

10

Threshold for stable liquid (pl−pv = 0)

5

0 80

90

100 110 φ = θ −ψ/2 (deg)

120

90 100 110 φ = θe−ψ/2 (deg)

120

e

(B) 1 0.8 plnd=(pl−pv)D/(4σlv)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.6 0.4 0.2 0 −0.2 −0.4

80

Fig. 6: Non-dimensional phase diagram for dewetting simulations of conical pores.

Pressures were made dimensionless using equation 6. (A) Non-dimensional phase diagram for cone angles ψ = [30◦ , 60◦ , 90◦ ] using temperatures of 300 K and 500 K. (B) Subset of the non-dimensional phase diagram for pressures pnd l ∈ [−0.4, 1.0]. Note: For the two cases with ◦ ◦ ψ = [60 , 90 ] at Tl = 300 K, pl = 0 bar, liquid never fully dewetted the pore. Thus, the contact angles were obtained from the ψ = 30◦ case at the same temperature and pressure.

Steered molecular dynamics to determine energetic landscape The potential of mean force (PMF) was calculated for cone angles of ψ = [30◦ , 60◦ , 90◦ ]. This is shown in Fig. 7 and Fig. S4. Corresponding images of the cones are superimposed on the image. The potential of mean force was binned according to the height of the meniscus z , rather than the center of mass of the water zcm , as the meniscus is the relevant parameter for 18


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wetting. A non-uniform bin size was used to partition the energy landscape with roughly the same number of measurements per bin. See Tables S1-S3 for data of the bin sizes, bin counts, and uncertainty estimates. 40,4749 This achieved probability distributions that were roughly normal (Fig. S5). A calibration curve between zcm and z was established using a third-order polynomial (Fig. S6). The coecients of determination for the calibration were satisfactory (Fig. S7). In Fig. 7B, the percent change in meniscus position ∆z = 0 corresponds to the dry state, and ∆z = 1 the nal wet state. For cone angle ψ = 30◦ , the minimum of the potential of mean force occurred when water remained outside the pore in the dry conguration (State 3). For ψ = 60◦ and ψ = 90◦ , the minimum of the potential of mean force occurred when the liquid wet the pore (State 1). This means the dry state (State 3) was energetically unfavorable, as it required additional energy to keep water there. The increase in the potential of mean force after the wetted state corresponds to liquid compressing against the pore walls, in response to pulling the system beyond its physical equilibrium state. While Jarzynski's equality (equation 18) does not require reversible trajectories, reducing the work dissipated during forward pulling can reduce the number of trajectories needed to estimate the free energy. The number of trajectories required was roughly approximated by 40

N F ∼ exp βWdR ,

(19)

where WdR is the work dissipated during reverse pulling. To estimate the work dissipated, we pulled water into the pore (forward pulling) and then subsequently pulled water out of the pore (reverse pulling) for a single trajectory at each cone angle. The work associated with this pulling process was computed and shown in Fig. S8. The pulling rate v was chosen to reduce the work dissipated. The maximum dierence between the initial state of the forward pulling and the nal state of the reverse pulling was ∼ 6 kB T . If we assume the work dissipated during the forward and reverse processes to be the same, then WdR ∼ 3 kB T and N F ∼ 20 trajectories. We used 72 trajectories to estimate the potential of mean force 19


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for each cone angle. Convergence remains an issue in free energy calculations, often requiring large numbers of simulations to achieve accurate estimates. In our system, the potential of mean force is plotted for each bin as a function of the number of trajectories averaged (Fig. S9). The estimates appear to reach a steady-state value for each bin. Low estimates for work inuenced the PMF more than larger estimates due to the exponential average in calculating the PMF (equations 18, S12). These rare events, along with using a nite number of trajectories will incur some amount of sampling error in our estimate. (A) Applied pressure

Water center of mass

z cm

Spring Pull direction

λ

Tether point

(B) 50 40

ψ=30 deg ψ=60 deg ψ=90 deg

30

PMF (kBT)

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20 10 0 −10 −20 0

0.2

0.4

0.6

0.8

1

∆z (%)

Fig. 7: Energetic landscape of water within a conical pore.

(A) Model of a hydrophobic pore with cone angle ψ = 30 . (B) Potential of mean force versus meniscus height for water held at a constant temperature of 300 K, and pressurized with 200 bar. The simulation began with ∆z = 0, and completed when ∆z = 1. Note: ∆z was normalized independently for each cone angle ψ . ◦

20


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The steered molecular dynamics calculations of the potential of mean force were initiated from the pooled liquid state to assess the dewetting capability of each pore. For completeness, molecular dynamics simulations of the same conical pores were implemented to show the equilibrium state may be achieved by either dewetting a surface (Fig. S10) or by condensing an ambient vapor near the surface (Fig. S11).

CONCLUSIONS Superhydrophobic surfaces immersed in liquids are prone to wetting when the ambient liquid is pressurized. Once wetted, these surfaces typically do not regain their dry state when the pressure subsides. Superhydrophobic surfaces possessing a conical geometry oer a passive means for pressurized water to dewet by minimizing the contact area of the bottom/base surface. Conical geometries are not the only types of geometries that exhibit dewetting behavior, 22 but they are a candidate that does not rely on density uctuations and thus they permit the dewetting process to be scaled to larger length-scales. In fact, theory presented prior predicts micron-scale geometries are obtainable under atmospheric conditions. Experiments have demonstrated this for hundreds of nanometer-scale conical pillars. 26 The nucleation theory describes this process with account for phase change. This is novel, and useful as it allows surfaces to be systematically designed based on the critical cone angle ψc , contact angle θe , and environmental conditions Tl , pl , pv . Not accounting for the vapor, which is typically done, does not permit the use of these surfaces in applications, such as enhanced nucleation sites or pressurized underwater surfaces. The thermodynamics of conical pores that de-wet, cannot be fully understood by simply looking at the Laplace equation. Cylindrical pores cannot de-wet, but still wet according to the Laplace equation. 11 The results presented in this work use a single contact angle for the material, instead of advancing and receding contact angles. The relevant contact angle during dewetting would be the receding contact angle, which could be used in the theoretical analysis. There can

21


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be contact angle hysteresis due to surface heterogeneity, surface roughness, and surface impurities/contaminates. 29 In our computational study, the conical pores were composed of uniform material, no surface roughness within the pore, and the adjacent uid was pure water. Because of this, we did not observe noticeable contact angle hysteresis in computations. The nucleation theory is supported by molecular dynamics simulations demonstrating a pore remaining dry underwater. These simulations were organized into a phase diagram and illuminated conditions when water will dewet. Non-equilibrium calculations of the potential of mean force identied the eect of the cone angle on pressurized liquids. The results of this work provide rationale for the design of dewetting surfaces immersed in pressurized liquids.

Acknowledgement The authors thank Professor Sinan Keten for benecial discussions. Support from the Institute for Sustainability and Energy at Northwestern (ISEN) is gratefully acknowledged. This research was supported in part through the computational resources and sta contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Oce of the Provost, the Oce for Research, and Northwestern University Information Technology. Support from the McCormick Catalyst Award at Northwestern University is also acknowledged.

Author Contributions N.A.P. and P.R.J. conceived and planned the research. P.R.J. and A.T.K. performed simulations. P.R.J. and D.T.R. implemented the free energy simulations. N.A.P., Y.D.M., and P.R.J. did theoretical analysis. P.R.J. and N.A.P. wrote and edited the manuscript.

22


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Author Information Correspondence and requests for materials should be addressed to N.A.P. ([email protected]).

Competing Financial Interests The authors have no competing nancial interests or other interests that might be perceived to inuence the results and/or discussion reported in this paper.

Supporting Information Available The following les are available free of charge.

• conical_langmuir-SI.pdf : Supporting text and gures • tableS1_cone30Uncertainty.csv : Supporting table used to create the potential of mean force plot for cone angle ψ = 30◦ .

• tableS2_cone60Uncertainty.csv : Supporting table used to create the potential of mean force plot for cone angle ψ = 60◦ .

• tableS3_cone90Uncertainty.csv : Supporting table used to create the potential of mean force plot for cone angle ψ = 90◦ . This material is available free of charge via the Internet at http://pubs.acs.org/.

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Graphical TOC Entry WATER DEWETTING A HYDROPHOBIC PORE

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