# Difference between revisions of "Wetting and Roughness: Part 3"

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− | [[Wetting and Roughness: Part | + | Original entry: Alexander Epstein, APPHY 226, Spring 2009 |

+ | |||

+ | [[Wetting and Roughness: Part 3]] | ||

Authors: David Quere | Authors: David Quere | ||

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microtextures, superhydrophobicity, wicking, slip | microtextures, superhydrophobicity, wicking, slip | ||

− | |||

---- | ---- | ||

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==Soft matters== | ==Soft matters== | ||

+ | |||

+ | ===Superhydrophobicity and Cassie=== | ||

If a hydrophobic solid is rough enough, the liquid will not conform to the solid surface as assumed by the Wenzel model, and instead air pockets will form under the liquid and support it. This is the Cassie state. It is observed if the energy of the liquid-vapor interfaces is lower than the energy of wetting the solid. In the case of our beloved micro/nanoposts, we can assume that the liquid-air interfaces are flat (since the Laplace pressure can be assumed zero at the bottom of the drop) and that the wet surface area <math> \sim\ (r - \phi_s)</math> and liquid-air area <math> \sim\ (1 - \phi_s)</math>. The Cassie state is favored when | If a hydrophobic solid is rough enough, the liquid will not conform to the solid surface as assumed by the Wenzel model, and instead air pockets will form under the liquid and support it. This is the Cassie state. It is observed if the energy of the liquid-vapor interfaces is lower than the energy of wetting the solid. In the case of our beloved micro/nanoposts, we can assume that the liquid-air interfaces are flat (since the Laplace pressure can be assumed zero at the bottom of the drop) and that the wet surface area <math> \sim\ (r - \phi_s)</math> and liquid-air area <math> \sim\ (1 - \phi_s)</math>. The Cassie state is favored when | ||

Line 39: | Line 42: | ||

The Cassie state in the above case can be considered stable. However, lower roughness factors lead to the critical angle criterion not being met, and the Cassie state can then be metastable. As long as the drop does not nucleate a contact point with the bottom of the rough surface, the air will remain trapped. | The Cassie state in the above case can be considered stable. However, lower roughness factors lead to the critical angle criterion not being met, and the Cassie state can then be metastable. As long as the drop does not nucleate a contact point with the bottom of the rough surface, the air will remain trapped. | ||

− | '''An obvious but important fact:''' the more air under the drop in a Cassie state, the closer the apparent angle <math>\scriptstyle{\theta^{\circ}}</math> is to 180°, or no contact. Any deviation from 180° is diagnostic of the fraction of solid surface area in contact with the liquid. From an energy balance, the equilibrium apparent contact angle <math>\scriptstyle{\theta^{*}}</math> is: | + | '''An obvious but important fact:''' the more air under the drop in a Cassie state, the closer the apparent angle <math>\scriptstyle{\theta^{\circ}}</math> is to 180°, or no contact. Any deviation from 180° is diagnostic of the fraction of solid surface area <math>\phi_s</math> in contact with the liquid. From an energy balance, the equilibrium apparent contact angle <math>\scriptstyle{\theta^{*}}</math> is: |

<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math> | <math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math> | ||

− | + | [[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]] | |

− | + | For example, if <math>\theta = 110-120</math>°, and <math>\scriptstyle{\phi_s = 5-10%}</math>, the apparent angle is 160-170°. This condition must be accompanied by the presence of edges on the posts (or more generally of large slopes on the rough surface). A structurally colorful example is the sphere of water on fluorinated silicon microposts in Figure 14. Re-entrant designs make more robust Cassie states and allow even hydrophilic surfaces can trap air under liquid! | |

+ | |||

+ | [[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]] | ||

+ | |||

+ | As just mentioned, the apparent angle is an interesting measure for <math>\phi_s</math> and any properties related to liquid-solid contact, such as electrical conduction, chemical activity, hydrodynamic slip, etc. As <math>\phi_s</math> becomes smaller and smaller, the difference between 180° and <math>\theta^{*}</math> decreases as <math>\scriptstyle{\sqrt{\phi_s}}</math>, so it is difficult to achieve a stric nonwetting situation. Gao and McCarthy [2] used nonwoven assemblies of nanofibers (Figure 7b) to approach angles of 180°. | ||

+ | |||

+ | ===Nonsticking Water?=== | ||

+ | |||

+ | Besides a near nonwetting surface with apparent contact angle approaching 180°, the other requirement for water not sticking to surfaces is small contact angle hysteresis. The familiar sight of raindrops sticking to the outside of window panes would change if the drops' hysteresis were decreased and they were more mobile. In fact, the drops could then bounce off the window and not stick at all. The Leidenfrost quality of drops on many superhydrophobic surfaces is limited by a residual hysteresis (and thus adhesion), whose value is unclear. However, the mechanism is as shown in Figure 15. | ||

+ | |||

+ | In a Cassie state, a drop is likely to pin on the top edges of the defects as the contact line moves. The drop becomes distorted, and the energy stored in this deformation fixes the amplitude of the hysteresis. Quere goes through some force analysis, but the key results are twofold. | ||

+ | |||

+ | [[Image: Q15.png|thumb|right|200px|'''Fig. 15''' ]] | ||

+ | |||

+ | <math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math> | ||

+ | |||

+ | '''First:''' the hysteresis vanishes with an infinitesimal density of posts, but the log term means the decrease itself slows down and residual hysteresis is experimentally inevitable. | ||

+ | |||

+ | '''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when: | ||

+ | |||

+ | <math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math> | ||

+ | |||

+ | This gives us the scaling between the density of defects, e.g., posts, and the degree of adhesion of the drop on a solid. Small densities are clearly required for low adhesion; but the tradeoff is increasing loss of Cassie stability. | ||

+ | |||

+ | ===Metastable Cassie=== | ||

+ | |||

+ | Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16. | ||

+ | |||

+ | [[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]] | ||

+ | |||

+ | The substrate has a low post density and low roughness. The millimetric drop on the left was gently planted and remained in Cassie state; the one on the right was dropped from a large height and took on the Wenzel state. The impacted drop had sufficient energy to break through the activation barrier from Cassie to the ground Wenzel state. More generally, any pertubation of a Cassie drop, such as vibration, pressure, or impact, can drive the otherwise unfavorable wetting of the post walls and impalement of the drop. This energy per unit area for the hydrophobic case is (reverse sign for hydrophilic): | ||

+ | |||

+ | <math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math> | ||

+ | |||

+ | and the energy barrier in terms of the surface geometry is: | ||

+ | |||

+ | <math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math> | ||

+ | |||

+ | The conclusions we draw from this relation are | ||

+ | |||

+ | 1. Energy barrier for micron scale posts cannot be overcome by thermal energy | ||

+ | 2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter | ||

+ | |||

+ | The Cassie-Wenzel transition occurs in a zipping fashion, as rows of cavities get filled in sequentially at a speed of 10 <math>\mu</math>s per 100 <math>\mu</math>m cavity. Progression is desribed by the same Washburn law that applies for capillary invasion of porous materials. Figure 17 illustrates the liquid-air interface curvature that precedes the transition. The depth of penetration <math>\delta</math> scales as <math>p^2/R</math>, where ''R'' is the drop radius; so, the smaller the drop, the greater the interface penetration into the cavity, until liquid-solid contact is made, and the Wenzel regime takes over. This implies a critical radius for a Cassie drop scaling as: | ||

+ | |||

+ | <math>R^{*} \sim\ p^2/h</math> | ||

+ | |||

+ | '''Note:''' the critical drop radius can be much larger than ''p'' if ''h'' < ''p'', meaning the Cassie state is weak. A small critical radius means a strong Cassie state, and is achievable by making posts tall or by reducing both post height and pitch. | ||

+ | |||

+ | The common mosquito uses this latter strategy to great effect. Figure 18 shows the "face" of the ''Culex pipiens'' after exposure to water aerosol. Droplets condense on the antennae, but the eyes remain dry--a necessary condition to preserve sight for navigation. The texture on the surface of the eye (Fig. 6d) features both pitch and height on the order of 100 nm, so the critical Cassie radius <math>\scriptstyle{R^{*} \sim\ 100\ nm}</math>, a size of droplet that normally evaporates in an instant. | ||

+ | |||

+ | As already alluded to, oils with contact angles of about 40° on a flat surface can bead up to 160° on superoleophobic surfaces that have re-entrant overhanging micro/nanostructures, such as mushroom caps or nail heads. | ||

+ | |||

+ | ===Anisotropy=== | ||

+ | |||

+ | Strategically patterning a surface with wetting and nonwetting defects can generate anisotropy and directional wetting. For example, parallel grooves or microwrinkles will pin contact lines in the perpendicular direction far more than in the parallel direction. Axial flow of liquid is preferred along such a "smart surface." We can imagine guiding liquid along a complex network of these axial paths on a surface. | ||

+ | |||

+ | [[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]] | ||

+ | |||

+ | I am not sure if this principle has been widely exploited in synthetic surfaces. However, there are certainly examples of anisotropic wetting in nature. One is the butterly ''Papilio ulysses'', whose wings have a directional microtexture (Fig. 19). The other is the water strider already mentioned. The water strider really ought to be called the "water skater": it strikes the surface perpendicular to grooves between its hairs, generating a large contact force, before swinging the legs by 90° to align them | ||

+ | in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. | ||

+ | ===Wettability switches=== | ||

+ | |||

+ | Since roughness amplifies chemical hydrophobicity and hydrophilicity, there has been much interest in using this "transistor" quality to make surfaces that switch from completely wetting to completely nonwetting. Light on photocatalytic textures or heat for thermal coatings are two possible triggers. | ||

+ | |||

+ | The stumbling block with this idea is that, generally speaking, the Cassie-Wenzel transition is irreversible. The Wenzel state is normally the ground state for the system. And the liquid gets pinned in the superhydrophilic state, making it difficult to expel back into Cassie. The one reported approach for transitioning back to Cassie state is the use of a short, intense pulse of current through the Wenzel state drop, vaporizing a film underneath, and rocketing the drop upwards from the surface. However, this is an extreme technique. | ||

+ | |||

+ | No materials that can condense dew directly into Cassie state, such that the dew drops are mobile and roll off, have been achieved. This is a ripe area for research! | ||

+ | |||

+ | ===Giant slip=== | ||

+ | |||

+ | [[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]] | ||

+ | |||

+ | [[Image: Q21.png|thumb|right|300px|'''Fig. 21''' ]] | ||

+ | |||

+ | One other superb application that awaits the benefits of superhydrophobicity is hydrodynamic slip. As seen in Figure 20, the slip length <math>\lambda</math> is the extrapolated distance inside the solid at which the velocity profile of a flowing liquid vanishes. On a classical flat surface, the slip length is molecular scale. On a flat hydrophobic surface it can be on the order of 10 nm. And on a rough superhydrophobic surface, it is reported to be tens of <math>\mu</math>ms. | ||

+ | |||

+ | For a Poiseuille flow, the flux varies as <math>\scriptstyle{W^4 \nabla p/\eta}</math>, but for a large slip, it varies as <math>\scriptstyle{\lambda W^3 \nabla p/\eta}</math>, where ''W'' is channel width and depth, <math>\scriptstyle{\nabla}</math> is the pressure gradient, and <math>\eta</math> is viscosity. Slip at the wall reduces the pressure gradient needed to drive a given flow by a factor <math>\lambda/W</math>, and an experimental result of 40% has been reported by Ou, et al, corresponding to <math>\lambda = 10 \mu m</math>. [4] | ||

+ | |||

+ | |||

+ | Figure 21 sensibly shows that the slip length increases on a superhydrophobic nanotube surface for increasing post pitch in the Cassie state, as there is less solid-liquid contact area. The large viscosity ratio between water and air (~100) means that viscous drag is dominated by the solid-liquid interface until we get to large pitches and very thin posts (<math>\eta V b/p^2 > \eta_{air} V /h</math>) In the Wenzel state, there is no slip whatsoever on such a rough surface. | ||

+ | |||

+ | Quere deduces that the effective slip length scales in terms of the surface geometry as: | ||

+ | |||

+ | <math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math> | ||

+ | |||

+ | So we can increase the slip length by increasing post pitch and reducing post area density, i.e., radius. The inevitable problem that limits arbitrarily large slip is that the liquid will eventually sink inside the texture, nucleate contact, and come to a screeching Wenzel halt. | ||

===References=== | ===References=== | ||

− | 1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 | + | 1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 |

+ | |||

+ | 2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53 | ||

− | + | 3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function. | |

+ | Adv. ''Insect Physiol.'' '''34''':117–92 | ||

− | + | 4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic | |

+ | surfaces. Phys. Fluids 16:4635–43 |

## Latest revision as of 01:35, 24 August 2009

Original entry: Alexander Epstein, APPHY 226, Spring 2009

**Wetting and Roughness: Part 3**

Authors: David Quere

Annu. Rev. Mater. Res. 2008. 38:71–99

## Contents

#### Soft matter keywords

microtextures, superhydrophobicity, wicking, slip

### Abstract from the original paper

We discuss in this review how the roughness of a solid impacts its wettability. We see in particular that both the apparent contact angle and the contact angle hysteresis can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well-controlled micro- or nanotextures on a solid, these effects are being exploited to induce novel wetting properties, such as spontaneous filmification, superhydrophobicity, superoleophobicity, and interfacial slip, that could not be achieved without roughness.

In Part 3, we examine the sectionsSuperhydrophobicityandSpecial Properties

## Soft matters

### Superhydrophobicity and Cassie

If a hydrophobic solid is rough enough, the liquid will not conform to the solid surface as assumed by the Wenzel model, and instead air pockets will form under the liquid and support it. This is the Cassie state. It is observed if the energy of the liquid-vapor interfaces is lower than the energy of wetting the solid. In the case of our beloved micro/nanoposts, we can assume that the liquid-air interfaces are flat (since the Laplace pressure can be assumed zero at the bottom of the drop) and that the wet surface area <math> \sim\ (r - \phi_s)</math> and liquid-air area <math> \sim\ (1 - \phi_s)</math>. The Cassie state is favored when

<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math>

and the corresponding critical Young angle relation is

<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math>

For very rough solids (<math>\scriptstyle{r \gg 1}</math>), <math>\scriptstyle{cos\ \theta_c\ \to\ 90^{\circ}}</math>, and the criterion for air trapping is satisfied since we already assume chemical hydrophobicity (<math>\scriptstyle{\theta > 90^{\circ}}</math>). Materials with long hairs can have a roughness of 5 to 10, and a beautiful example of this is the *Microvelia* water strider pictured in Figure 12. Its legs have high aspect ratio hydrophobic hairs that trap air in a Cassie state, allowing the insect to skate on water.

The Cassie state in the above case can be considered stable. However, lower roughness factors lead to the critical angle criterion not being met, and the Cassie state can then be metastable. As long as the drop does not nucleate a contact point with the bottom of the rough surface, the air will remain trapped.

**An obvious but important fact:** the more air under the drop in a Cassie state, the closer the apparent angle <math>\scriptstyle{\theta^{\circ}}</math> is to 180°, or no contact. Any deviation from 180° is diagnostic of the fraction of solid surface area <math>\phi_s</math> in contact with the liquid. From an energy balance, the equilibrium apparent contact angle <math>\scriptstyle{\theta^{*}}</math> is:

<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math>

For example, if <math>\theta = 110-120</math>°, and <math>\scriptstyle{\phi_s = 5-10%}</math>, the apparent angle is 160-170°. This condition must be accompanied by the presence of edges on the posts (or more generally of large slopes on the rough surface). A structurally colorful example is the sphere of water on fluorinated silicon microposts in Figure 14. Re-entrant designs make more robust Cassie states and allow even hydrophilic surfaces can trap air under liquid!

As just mentioned, the apparent angle is an interesting measure for <math>\phi_s</math> and any properties related to liquid-solid contact, such as electrical conduction, chemical activity, hydrodynamic slip, etc. As <math>\phi_s</math> becomes smaller and smaller, the difference between 180° and <math>\theta^{*}</math> decreases as <math>\scriptstyle{\sqrt{\phi_s}}</math>, so it is difficult to achieve a stric nonwetting situation. Gao and McCarthy [2] used nonwoven assemblies of nanofibers (Figure 7b) to approach angles of 180°.

### Nonsticking Water?

Besides a near nonwetting surface with apparent contact angle approaching 180°, the other requirement for water not sticking to surfaces is small contact angle hysteresis. The familiar sight of raindrops sticking to the outside of window panes would change if the drops' hysteresis were decreased and they were more mobile. In fact, the drops could then bounce off the window and not stick at all. The Leidenfrost quality of drops on many superhydrophobic surfaces is limited by a residual hysteresis (and thus adhesion), whose value is unclear. However, the mechanism is as shown in Figure 15.

In a Cassie state, a drop is likely to pin on the top edges of the defects as the contact line moves. The drop becomes distorted, and the energy stored in this deformation fixes the amplitude of the hysteresis. Quere goes through some force analysis, but the key results are twofold.

<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math>

**First:** the hysteresis vanishes with an infinitesimal density of posts, but the log term means the decrease itself slows down and residual hysteresis is experimentally inevitable.

**Second:** a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:

<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math>

This gives us the scaling between the density of defects, e.g., posts, and the degree of adhesion of the drop on a solid. Small densities are clearly required for low adhesion; but the tradeoff is increasing loss of Cassie stability.

### Metastable Cassie

Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.

The substrate has a low post density and low roughness. The millimetric drop on the left was gently planted and remained in Cassie state; the one on the right was dropped from a large height and took on the Wenzel state. The impacted drop had sufficient energy to break through the activation barrier from Cassie to the ground Wenzel state. More generally, any pertubation of a Cassie drop, such as vibration, pressure, or impact, can drive the otherwise unfavorable wetting of the post walls and impalement of the drop. This energy per unit area for the hydrophobic case is (reverse sign for hydrophilic):

<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math>

and the energy barrier in terms of the surface geometry is:

<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math>

The conclusions we draw from this relation are

1. Energy barrier for micron scale posts cannot be overcome by thermal energy
2. Higher posts (*h*) increase the Cassie-Wenzel barrier, and *h* is a good tuning parameter

The Cassie-Wenzel transition occurs in a zipping fashion, as rows of cavities get filled in sequentially at a speed of 10 <math>\mu</math>s per 100 <math>\mu</math>m cavity. Progression is desribed by the same Washburn law that applies for capillary invasion of porous materials. Figure 17 illustrates the liquid-air interface curvature that precedes the transition. The depth of penetration <math>\delta</math> scales as <math>p^2/R</math>, where *R* is the drop radius; so, the smaller the drop, the greater the interface penetration into the cavity, until liquid-solid contact is made, and the Wenzel regime takes over. This implies a critical radius for a Cassie drop scaling as:

<math>R^{*} \sim\ p^2/h</math>

**Note:** the critical drop radius can be much larger than *p* if *h* < *p*, meaning the Cassie state is weak. A small critical radius means a strong Cassie state, and is achievable by making posts tall or by reducing both post height and pitch.

The common mosquito uses this latter strategy to great effect. Figure 18 shows the "face" of the *Culex pipiens* after exposure to water aerosol. Droplets condense on the antennae, but the eyes remain dry--a necessary condition to preserve sight for navigation. The texture on the surface of the eye (Fig. 6d) features both pitch and height on the order of 100 nm, so the critical Cassie radius <math>\scriptstyle{R^{*} \sim\ 100\ nm}</math>, a size of droplet that normally evaporates in an instant.

As already alluded to, oils with contact angles of about 40° on a flat surface can bead up to 160° on superoleophobic surfaces that have re-entrant overhanging micro/nanostructures, such as mushroom caps or nail heads.

### Anisotropy

Strategically patterning a surface with wetting and nonwetting defects can generate anisotropy and directional wetting. For example, parallel grooves or microwrinkles will pin contact lines in the perpendicular direction far more than in the parallel direction. Axial flow of liquid is preferred along such a "smart surface." We can imagine guiding liquid along a complex network of these axial paths on a surface.

I am not sure if this principle has been widely exploited in synthetic surfaces. However, there are certainly examples of anisotropic wetting in nature. One is the butterly *Papilio ulysses*, whose wings have a directional microtexture (Fig. 19). The other is the water strider already mentioned. The water strider really ought to be called the "water skater": it strikes the surface perpendicular to grooves between its hairs, generating a large contact force, before swinging the legs by 90° to align them
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events.

### Wettability switches

Since roughness amplifies chemical hydrophobicity and hydrophilicity, there has been much interest in using this "transistor" quality to make surfaces that switch from completely wetting to completely nonwetting. Light on photocatalytic textures or heat for thermal coatings are two possible triggers.

The stumbling block with this idea is that, generally speaking, the Cassie-Wenzel transition is irreversible. The Wenzel state is normally the ground state for the system. And the liquid gets pinned in the superhydrophilic state, making it difficult to expel back into Cassie. The one reported approach for transitioning back to Cassie state is the use of a short, intense pulse of current through the Wenzel state drop, vaporizing a film underneath, and rocketing the drop upwards from the surface. However, this is an extreme technique.

No materials that can condense dew directly into Cassie state, such that the dew drops are mobile and roll off, have been achieved. This is a ripe area for research!

### Giant slip

One other superb application that awaits the benefits of superhydrophobicity is hydrodynamic slip. As seen in Figure 20, the slip length <math>\lambda</math> is the extrapolated distance inside the solid at which the velocity profile of a flowing liquid vanishes. On a classical flat surface, the slip length is molecular scale. On a flat hydrophobic surface it can be on the order of 10 nm. And on a rough superhydrophobic surface, it is reported to be tens of <math>\mu</math>ms.

For a Poiseuille flow, the flux varies as <math>\scriptstyle{W^4 \nabla p/\eta}</math>, but for a large slip, it varies as <math>\scriptstyle{\lambda W^3 \nabla p/\eta}</math>, where *W* is channel width and depth, <math>\scriptstyle{\nabla}</math> is the pressure gradient, and <math>\eta</math> is viscosity. Slip at the wall reduces the pressure gradient needed to drive a given flow by a factor <math>\lambda/W</math>, and an experimental result of 40% has been reported by Ou, et al, corresponding to <math>\lambda = 10 \mu m</math>. [4]

Figure 21 sensibly shows that the slip length increases on a superhydrophobic nanotube surface for increasing post pitch in the Cassie state, as there is less solid-liquid contact area. The large viscosity ratio between water and air (~100) means that viscous drag is dominated by the solid-liquid interface until we get to large pitches and very thin posts (<math>\eta V b/p^2 > \eta_{air} V /h</math>) In the Wenzel state, there is no slip whatsoever on such a rough surface.

Quere deduces that the effective slip length scales in terms of the surface geometry as:

<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math>

So we can increase the slip length by increasing post pitch and reducing post area density, i.e., radius. The inevitable problem that limits arbitrarily large slip is that the liquid will eventually sink inside the texture, nucleate contact, and come to a screeching Wenzel halt.

### References

1. Quere, David. Wetting and Roughness. *Annu. Rev. Mater. Res.* 2008. **38**:71-99

2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. *JACS* **128**:9052–53

3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.
Adv. *Insect Physiol.* **34**:117–92

4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16:4635–43