http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Aepstein&feedformat=atomSoft-Matter - User contributions [en]2020-04-07T14:44:20ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7276Slippery questions about complex fluids flowing past solids2009-05-03T20:41:01Z<p>Aepstein: /* Effect of roughness on slip */</p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
[[Image: G3.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between. <br />
<br />
<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math><br />
<br />
So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.<br />
<br />
Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface. <br />
<br />
<math>\eta v_s \equiv b \sigma_s</math><br />
<br />
where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.<br />
<br />
====Effect of roughness on slip====<br />
<br />
There are two theories to explain why no-slip works in most cases.<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
* Fluid molecules are stuck to solid walls by intermolecular forces to prevent a discontinuity<br />
** BUT, hydro/oleophobic surfaces are nonwetting, so the fluid-solid cohesion is less than fluid-fluid, so this is not always true<br />
* The microscopic roughness (physical and chemical) of real surfaces leads to viscous dissipation at the boundary regardless of surface energies<br />
** Simulations predict slip past perfectly smooth surfaces<br />
** BUT, realistic surfaces possess structur--see Figure 3.<br />
<br />
[[Image: G4.png|thumb|right|500px|'''Fig. 4''' ]]<br />
<br />
Any slip present on a surface decreases as surface roughness increases. Figure 4 shows the influence of wall roughness on flow past partially wetted surfaces. AFM images in a-c shows the r.m.s. roughness, from atomic to 6 nm, The corresponding deviation from no-slip and the slip length are compared in d and e for different flow rates. Controlling these two parameters allows the slip length to be increased to 400 angstroms or more by going to atomic smoothness and high flow rate. At 6 nm r.m.s. roughness (much larger than the size of the fluid molecules), higher flow rate can no longer cause slip. <br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
Figure 5 corroborates the relationship. Here the critical shear rate for onset of slip (left axis) and critical shear stress (right axis) are plotted against r.m.s. surface roughness for flow of DI water (filled circles) and tetradecane (open circles). The datum at the top right with arrows is above experimental resolution and represents only a lower limit.<br />
<br />
====Insoluble gas segregating on smooth surfaces====<br />
<br />
Data for ''smooth'' surfaces at high flow rates turn out to be consistent with a two-layer fluid model in which a layer <1 nm thick and viscosity 10-20 times less than the bulk adjoins the solid surface. An explanation by de Gennes, et al is that shear may nucleate vapor bubbles, which then grow to cover the solid surface such that the liquid now flows on a layer of gas. Importantly, this effect disappears when there is more than atomic surface roughness.<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
Tetradecane, a non-polar liquid, saturated with poorly soluble argon showed slip, whereas when showed no slip when saturated with soluble carbon dioxide. Similarly, on a hydrophobic surface carrying water flow, a vapor phase may thermodynamically form near the wall. Shown in Figure 6, these findings show that gas-mediated slip can even occur on wetted surfaces where the solid surface energy is high, provided that the surface is sufficiently, i.e., atomically, smooth. This control parameter disappears when the surface is rough. In the figure, a slope of unity corresponds to no-slip, while horizontal slope is slip.<br />
<br />
====Riding on air: nanograss====<br />
<br />
A related strategy is to use superhydrophobic micro/nanostructured surfaces (such as in Figure 7) to maximize contact with air, which is exceedingly solvophobic. This is the "lotus effect" that we all know and love. The Cassie state of the fluid traps air underneath, such that the solid-liquid contact is very small. <br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7275Slippery questions about complex fluids flowing past solids2009-05-03T20:39:53Z<p>Aepstein: /* Riding on air: nanograss */</p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
[[Image: G3.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between. <br />
<br />
<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math><br />
<br />
So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.<br />
<br />
Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface. <br />
<br />
<math>\eta v_s \equiv b \sigma_s</math><br />
<br />
where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.<br />
<br />
====Effect of roughness on slip====<br />
<br />
There are two theories to explain why no-slip works in most cases.<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
* Fluid molecules are stuck to solid walls by intermolecular forces to prevent a discontinuity<br />
** BUT, hydro/oleophobic surfaces are nonwetting, so the fluid-solid cohesion is less than fluid-fluid, so this is not always true<br />
* The microscopic roughness (physical and chemical) of real surfaces leads to viscous dissipation at the boundary regardless of surface energies<br />
** Simulations predict slip past perfectly smooth surfaces<br />
** BUT, realistic surfaces possess structur--see Figure 3.<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
Any slip present on a surface decreases as surface roughness increases. Figure 4 shows the influence of wall roughness on flow past partially wetted surfaces. AFM images in a-c shows the r.m.s. roughness, from atomic to 6 nm, The corresponding deviation from no-slip and the slip length are compared in d and e for different flow rates. Controlling these two parameters allows the slip length to be increased to 400 angstroms or more by going to atomic smoothness and high flow rate. At 6 nm r.m.s. roughness (much larger than the size of the fluid molecules), higher flow rate can no longer cause slip. <br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
Figure 5 corroborates the relationship. Here the critical shear rate for onset of slip (left axis) and critical shear stress (right axis) are plotted against r.m.s. surface roughness for flow of DI water (filled circles) and tetradecane (open circles). The datum at the top right with arrows is above experimental resolution and represents only a lower limit. <br />
<br />
====Insoluble gas segregating on smooth surfaces====<br />
<br />
Data for ''smooth'' surfaces at high flow rates turn out to be consistent with a two-layer fluid model in which a layer <1 nm thick and viscosity 10-20 times less than the bulk adjoins the solid surface. An explanation by de Gennes, et al is that shear may nucleate vapor bubbles, which then grow to cover the solid surface such that the liquid now flows on a layer of gas. Importantly, this effect disappears when there is more than atomic surface roughness.<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
Tetradecane, a non-polar liquid, saturated with poorly soluble argon showed slip, whereas when showed no slip when saturated with soluble carbon dioxide. Similarly, on a hydrophobic surface carrying water flow, a vapor phase may thermodynamically form near the wall. Shown in Figure 6, these findings show that gas-mediated slip can even occur on wetted surfaces where the solid surface energy is high, provided that the surface is sufficiently, i.e., atomically, smooth. This control parameter disappears when the surface is rough. In the figure, a slope of unity corresponds to no-slip, while horizontal slope is slip.<br />
<br />
====Riding on air: nanograss====<br />
<br />
A related strategy is to use superhydrophobic micro/nanostructured surfaces (such as in Figure 7) to maximize contact with air, which is exceedingly solvophobic. This is the "lotus effect" that we all know and love. The Cassie state of the fluid traps air underneath, such that the solid-liquid contact is very small. <br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7274Slippery questions about complex fluids flowing past solids2009-05-03T20:39:05Z<p>Aepstein: /* Insoluble gas segregating on smooth surfaces */</p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
[[Image: G3.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between. <br />
<br />
<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math><br />
<br />
So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.<br />
<br />
Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface. <br />
<br />
<math>\eta v_s \equiv b \sigma_s</math><br />
<br />
where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.<br />
<br />
====Effect of roughness on slip====<br />
<br />
There are two theories to explain why no-slip works in most cases.<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
* Fluid molecules are stuck to solid walls by intermolecular forces to prevent a discontinuity<br />
** BUT, hydro/oleophobic surfaces are nonwetting, so the fluid-solid cohesion is less than fluid-fluid, so this is not always true<br />
* The microscopic roughness (physical and chemical) of real surfaces leads to viscous dissipation at the boundary regardless of surface energies<br />
** Simulations predict slip past perfectly smooth surfaces<br />
** BUT, realistic surfaces possess structur--see Figure 3.<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
Any slip present on a surface decreases as surface roughness increases. Figure 4 shows the influence of wall roughness on flow past partially wetted surfaces. AFM images in a-c shows the r.m.s. roughness, from atomic to 6 nm, The corresponding deviation from no-slip and the slip length are compared in d and e for different flow rates. Controlling these two parameters allows the slip length to be increased to 400 angstroms or more by going to atomic smoothness and high flow rate. At 6 nm r.m.s. roughness (much larger than the size of the fluid molecules), higher flow rate can no longer cause slip. <br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
Figure 5 corroborates the relationship. Here the critical shear rate for onset of slip (left axis) and critical shear stress (right axis) are plotted against r.m.s. surface roughness for flow of DI water (filled circles) and tetradecane (open circles). The datum at the top right with arrows is above experimental resolution and represents only a lower limit. <br />
<br />
====Insoluble gas segregating on smooth surfaces====<br />
<br />
Data for ''smooth'' surfaces at high flow rates turn out to be consistent with a two-layer fluid model in which a layer <1 nm thick and viscosity 10-20 times less than the bulk adjoins the solid surface. An explanation by de Gennes, et al is that shear may nucleate vapor bubbles, which then grow to cover the solid surface such that the liquid now flows on a layer of gas. Importantly, this effect disappears when there is more than atomic surface roughness.<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
Tetradecane, a non-polar liquid, saturated with poorly soluble argon showed slip, whereas when showed no slip when saturated with soluble carbon dioxide. Similarly, on a hydrophobic surface carrying water flow, a vapor phase may thermodynamically form near the wall. Shown in Figure 6, these findings show that gas-mediated slip can even occur on wetted surfaces where the solid surface energy is high, provided that the surface is sufficiently, i.e., atomically, smooth. This control parameter disappears when the surface is rough. In the figure, a slope of unity corresponds to no-slip, while horizontal slope is slip.<br />
<br />
====Riding on air: nanograss====<br />
<br />
A related strategy is to use our beloved superhydrophobic nanograss (such as in Figure 7) to maximize contact with air, which is exceedingly solvophobic. This is the "lotus effect" that we all know and love. The Cassie state of the fluid traps air underneath, such that the solid-liquid contact is very small. <br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7273Slippery questions about complex fluids flowing past solids2009-05-03T20:38:19Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
[[Image: G3.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between. <br />
<br />
<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math><br />
<br />
So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.<br />
<br />
Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface. <br />
<br />
<math>\eta v_s \equiv b \sigma_s</math><br />
<br />
where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.<br />
<br />
====Effect of roughness on slip====<br />
<br />
There are two theories to explain why no-slip works in most cases.<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
* Fluid molecules are stuck to solid walls by intermolecular forces to prevent a discontinuity<br />
** BUT, hydro/oleophobic surfaces are nonwetting, so the fluid-solid cohesion is less than fluid-fluid, so this is not always true<br />
* The microscopic roughness (physical and chemical) of real surfaces leads to viscous dissipation at the boundary regardless of surface energies<br />
** Simulations predict slip past perfectly smooth surfaces<br />
** BUT, realistic surfaces possess structur--see Figure 3.<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
Any slip present on a surface decreases as surface roughness increases. Figure 4 shows the influence of wall roughness on flow past partially wetted surfaces. AFM images in a-c shows the r.m.s. roughness, from atomic to 6 nm, The corresponding deviation from no-slip and the slip length are compared in d and e for different flow rates. Controlling these two parameters allows the slip length to be increased to 400 angstroms or more by going to atomic smoothness and high flow rate. At 6 nm r.m.s. roughness (much larger than the size of the fluid molecules), higher flow rate can no longer cause slip. <br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
Figure 5 corroborates the relationship. Here the critical shear rate for onset of slip (left axis) and critical shear stress (right axis) are plotted against r.m.s. surface roughness for flow of DI water (filled circles) and tetradecane (open circles). The datum at the top right with arrows is above experimental resolution and represents only a lower limit. <br />
<br />
====Insoluble gas segregating on smooth surfaces====<br />
<br />
Data for ''smooth'' surfaces at high flow rates turn out to be consistent with a two-layer fluid model in which a layer <1 nm thick and viscosity 10-20 times less than the bulk adjoins the solid surface. An explanation by de Gennes, et al is that shear may nucleate vapor bubbles, which then grow to cover the solid surface such that the liquid now flows on a layer of gas. Importantly, this effect disappears when there is more than atomic surface roughness.<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
Tetradecane, a non-polar liquid, saturated with poorly soluble argon showed slip, whereas when showed no slip when saturated with soluble carbon dioxide. Similarly, on a hydrophobic surface carrying water flow, a vapor phase may thermodynamically form near the wall. Shown in Figure 6, these findings show that gas-mediated slip can even occur on wetted surfaces where the solid surface energy is high, provided that the surface is sufficiently, i.e., atomically, smooth. This control parameter disappears when the surface is rough. In the figure, a slope of unity corresponds to no-slip, while horizontal slope is slip.<br />
<br />
A related strategy is to use our beloved superhydrophobic nanograss (such as in Figure 7) to maximize contact with air, which is exceedingly solvophobic. This is the "lotus effect" that we all know and love. The Cassie state of the fluid traps air underneath, such that the solid-liquid contact is very small. <br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7261Slippery questions about complex fluids flowing past solids2009-05-03T19:48:05Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
[[Image: G3.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between. <br />
<br />
<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math><br />
<br />
So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.<br />
<br />
Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface. <br />
<br />
<math>\eta v_s \equiv b \sigma_s</math><br />
<br />
where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7260Slippery questions about complex fluids flowing past solids2009-05-03T19:28:09Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging. <br />
<br />
[[Image: G3.png|thumb|left|400px|'''Fig. 2''' ]]<br />
<br />
There are actually two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.<br />
<br />
====Deviation from no-slip, quantified====<br />
<br />
<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 3''' ]]<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7259Slippery questions about complex fluids flowing past solids2009-05-03T19:10:02Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
====Lack of slip in everyday life====<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
====Exceptions to the "no-slip dogma"====<br />
<br />
The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate. <br />
<br />
The exceptions fall into these categories:<br />
* Flow of multicomponent fluids with different viscosity components<br />
** Suspensions , foodstuffs, and emulsions<br />
** Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)<br />
* Viscous polymers (where there is a range of molecular weights)<br />
* "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition<br />
* Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)<br />
* Superfluid helium<br />
<br />
<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
[[Image: G3.png|thumb|left|400px|'''Fig. 3''' ]]<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7257Slippery questions about complex fluids flowing past solids2009-05-03T18:52:33Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.<br />
<br />
===Lack of slip in everyday life===<br />
<br />
This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water. <br />
<br />
Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.<br />
<br />
===<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
[[Image: G3.png|thumb|left|400px|'''Fig. 3''' ]]<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7196Slippery questions about complex fluids flowing past solids2009-05-02T20:09:34Z<p>Aepstein: /* Soft matters */</p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
***I'm finishing reading the paper, which is fascinating. This wiki will be done by Sunday.***<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
[[Image: G2.png|thumb|right|400px|'''Fig. 2''' ]]<br />
<br />
[[Image: G3.png|thumb|left|400px|'''Fig. 3''' ]]<br />
<br />
[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
<br />
[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
<br />
[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
<br />
[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7195Slippery questions about complex fluids flowing past solids2009-05-02T20:09:12Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
***I'm finishing reading the paper, which is fascinating. This wiki will be done end of Sunday.***<br />
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[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
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[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
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===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7194Slippery questions about complex fluids flowing past solids2009-05-02T19:52:24Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
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[[Image: G2.png|thumb|right|400px|'''Fig. 2''' ]]<br />
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[[Image: G3.png|thumb|left|400px|'''Fig. 3''' ]]<br />
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[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
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[[Image: G5.png|thumb|left|300px|'''Fig. 5''' ]]<br />
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[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
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[[Image: G7.png|thumb|left|300px|'''Fig. 7''' ]]<br />
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===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G7.png&diff=7193File:G7.png2009-05-02T19:51:44Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G6.png&diff=7192File:G6.png2009-05-02T19:51:34Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G5.png&diff=7191File:G5.png2009-05-02T19:51:25Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G4.png&diff=7190File:G4.png2009-05-02T19:51:15Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G3.png&diff=7189File:G3.png2009-05-02T19:51:07Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G2.png&diff=7188File:G2.png2009-05-02T19:50:57Z<p>Aepstein: </p>
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<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7187Slippery questions about complex fluids flowing past solids2009-05-02T19:50:50Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
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[[Image: G2.png|thumb|right|300px|'''Fig. 2''' ]]<br />
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[[Image: G3.png|thumb|right|300px|'''Fig. 3''' ]]<br />
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[[Image: G4.png|thumb|right|300px|'''Fig. 4''' ]]<br />
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[[Image: G5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
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[[Image: G6.png|thumb|right|300px|'''Fig. 6''' ]]<br />
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[[Image: G7.png|thumb|right|300px|'''Fig. 7''' ]]<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7160Slippery questions about complex fluids flowing past solids2009-05-02T00:45:02Z<p>Aepstein: </p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
Rough surfaces, slip control, hydrophobicity<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:G1.png&diff=7159File:G1.png2009-05-02T00:44:10Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Slippery_questions_about_complex_fluids_flowing_past_solids&diff=7158Slippery questions about complex fluids flowing past solids2009-05-02T00:40:27Z<p>Aepstein: New page: Slippery questions about complex fluids flowing past solids Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee ====Soft matter keywords==== TBA By Alex Epstein ---- ===Abstract f...</p>
<hr />
<div>[[Slippery questions about complex fluids flowing past solids]]<br />
<br />
Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee<br />
<br />
====Soft matter keywords====<br />
TBA<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent<br />
experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly<br />
when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that<br />
can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases.<br />
To discover what boundary condition is appropriate for solving continuum equations requires investigation of<br />
microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to<br />
capitalize on ‘slip at the wall’ for purposes of materials engineering.<br />
<br />
==Soft matters==<br />
<br />
<br />
<br />
[[Image: G1.png|thumb|right|300px|'''Fig. 1''' ]]<br />
<br />
<br />
<br />
===References=== <br />
1.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=User:Aepstein&diff=7157User:Aepstein2009-05-02T00:35:59Z<p>Aepstein: </p>
<hr />
<div>== Weekly Wiki Entries ==<br />
<br />
[[Skeleton of Euplectella sp.: Structural Hierarchy from the Nanoscale to the Macroscale]]<br />
<br />
[[Self-Assembly of Hexagonal Rods Based on Capillary Forces]]<br />
<br />
[[Photoreactive coating for high-contrast spatial patterning of microfluidic device wettability]]<br />
<br />
[[Biomimetic self-assembly of helical electrical circuits using orthogonal capillary interactions]]<br />
<br />
[[Planarization of Substrate Topography by Spin Coating]]<br />
<br />
[[Smart responsive surfaces switching reversibly between super-hydrophobicity and super-hydrophilicity]]<br />
<br />
[[Wetting and Roughness: Part 1]]<br />
<br />
[[Wetting and Roughness: Part 2]]<br />
<br />
[[Wetting and Roughness: Part 3]]<br />
<br />
[[Slippery questions about complex fluids flowing past solids]]</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Morrisonisms&diff=6957Morrisonisms2009-04-23T13:40:35Z<p>Aepstein: </p>
<hr />
<div>"People will look at you and say you are either wasting your time, dangerous or both" ~ About Research<br />
<br />
"There is a way to get physicists all hot and bothered. Tell them that there is a new force."<br />
<br />
"If you're not doing something wrong. You're not doing hard enough stuff."<br />
<br />
"These big pieces of chalk are frightening."<br />
<br />
"Fat thumb, small pointer." ~ Skipping Slides<br />
<br />
"If you were a capillary tube, you would need to know about these things." ~ Capillary Rise etc<br />
<br />
"We'll take it easy at first and go prancing along merrily." ~ Survey Course Versus Fluids <br />
<br />
"In the magical world of de Gennes and his friends..."<br />
<br />
"It was the 60s. We were wide awake... I think." ~ Saturday Morning Classes<br />
<br />
"When you take the cheap easy way... like de Gennes... AND win the Nobel Prize." ~ Choosing what information is extraneous<br />
<br />
"Who knows what he had been drinking or smoking at the time?" ~ de Gennes and AFM tips<br />
<br />
"I have a hammer. Now what can I hit with it?" ~ Creating an equation and then applying it.<br />
<br />
"There is only one group of people you can legitimately make fun of...Canadians." ~ Russians at fault for kappa^-1 capillary length<br />
<br />
"The keys to the kingdom are in this little spot." ~ A singularity on a film energy-thickness diagram<br />
<br />
"Think about that when you are going to sleep. It will put you to sleep." ~ high energy = low contact angles, low energy = high contact angles<br />
<br />
"That's a nasty description of a mathematician... Someone who doesn't try to understand much."<br />
<br />
"In grad school if you see a math course with algebra...think again."<br />
<br />
"When you are 6 feet tall it makes a difference..."~When you are talking about nothing<br />
<br />
"For 22 years we tell them, 'Don't think about it.' Then when they get married we ask, when will they? When will there be little Morrisons?" ~ Sex<br />
<br />
"You'll tell your children about the first time you learned the screening length."<br />
<br />
"It's hard to do derivations while the Red Sox are losing." ~ Another great reason for them to win more often. Higher scientific productivity in Boston.<br />
<br />
"If you ever see log(n) you know someone in statistical mechanics has been there." ~ Derivation of repulsion in a polymer chain<br />
<br />
"This equation is older than sin...." ~ Hydrostatics <br />
<br />
Drunks = Random walk and Blob theory</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6909Wetting and Roughness: Part 32009-04-22T02:17:14Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 3]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|200px|'''Fig. 15''' ]]<br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
[[Image: Q21.png|thumb|right|300px|'''Fig. 21''' ]]<br />
<br />
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. <br />
<br />
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] <br />
<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6908Wetting and Roughness: Part 32009-04-22T02:16:34Z<p>Aepstein: /* Giant slip */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|200px|'''Fig. 15''' ]]<br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
[[Image: Q21.png|thumb|right|300px|'''Fig. 21''' ]]<br />
<br />
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. <br />
<br />
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] <br />
<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q21.png&diff=6907File:Q21.png2009-04-22T02:15:49Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q20.png&diff=6906File:Q20.png2009-04-22T02:15:40Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q19.png&diff=6905File:Q19.png2009-04-22T02:15:28Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q16.png&diff=6904File:Q16.png2009-04-22T02:15:13Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6903Wetting and Roughness: Part 32009-04-22T02:14:55Z<p>Aepstein: /* Nonsticking Water? */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|200px|'''Fig. 15''' ]]<br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6902Wetting and Roughness: Part 32009-04-22T02:14:43Z<p>Aepstein: /* Nonsticking Water? */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|300px|'''Fig. 15''' ]]<br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6901Wetting and Roughness: Part 32009-04-22T02:13:52Z<p>Aepstein: /* Nonsticking Water? */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|300px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q15.png&diff=6900File:Q15.png2009-04-22T02:13:04Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q7b.png&diff=6899File:Q7b.png2009-04-22T02:12:55Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q14.png&diff=6898File:Q14.png2009-04-22T02:12:44Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6897Wetting and Roughness: Part 32009-04-22T02:12:38Z<p>Aepstein: /* Superhydrophobicity and Cassie */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q14.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°.<br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|400px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q12.png&diff=6896File:Q12.png2009-04-22T02:11:46Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6895Wetting and Roughness: Part 32009-04-22T02:11:34Z<p>Aepstein: /* Giant slip */</p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q12.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°. <br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|400px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6894Wetting and Roughness: Part 32009-04-22T02:09:48Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q12.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°. <br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|400px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
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. <br />
<br />
[[Image: Q19.png|thumb|right|400px|'''Fig. 19''' ]]<br />
<br />
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<br />
in the direction of the motion for skating. Motion arises from alternating pinning and gliding events. <br />
===Wettability switches===<br />
<br />
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. <br />
<br />
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. <br />
<br />
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!<br />
<br />
===Giant slip===<br />
<br />
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. <br />
<br />
[[Image: Q20.png|thumb|right|400px|'''Fig. 20''' ]]<br />
<br />
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] <br />
<br />
[[Image: Q21.png|thumb|right|400px|'''Fig. 21''' ]]<br />
<br />
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.<br />
<br />
Quere deduces that the effective slip length scales in terms of the surface geometry as:<br />
<br />
<math>\lambda \sim\ p^2/b \sim\ p/\phi_s^{1/2}</math><br />
<br />
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. <br />
<br />
<br />
<br />
<br />
<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3. Bush JWM, Hu D, Prakash M. 2008. The integument of waterwalking arthropods: form and function.<br />
Adv. ''Insect Physiol.'' '''34''':117–92<br />
<br />
4. Ou J, Perot B, Rothstein JP. 2004. Laminar drag reduction in microchannels using ultrahydrophobic<br />
surfaces. Phys. Fluids 16:4635–43</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6884Wetting and Roughness: Part 32009-04-21T22:39:56Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q12.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°. <br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|400px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
===Anisotropy===<br />
<br />
<br />
===Wettability switches===<br />
<br />
Generally speaking, the Cassie-Wenzel transition is irreversible. <br />
<br />
===Giant slip===<br />
<br />
<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6877Wetting and Roughness: Part 32009-04-21T21:03:48Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
===Superhydrophobicity and Cassie===<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
[[Image: Q12.png|thumb|right|200px|'''Fig. 14''' ]]<br />
<br />
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!<br />
<br />
[[Image: Q7b.png|thumb|right|200px|'''Fig. 7b''' ]]<br />
<br />
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°. <br />
<br />
===Nonsticking Water?===<br />
<br />
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. <br />
<br />
[[Image: Q15.png|thumb|right|400px|'''Fig. 15''' ]]<br />
<br />
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. <br />
<br />
<math>(cos\ \theta_r - cos\ \theta_a) \sim\ \phi_s\ log(1/\phi_s)</math><br />
<br />
'''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. <br />
<br />
'''Second:''' a drop sticking to a vertical rough solid such as microposts will move in a gravity field when:<br />
<br />
<math>\phi_s^{3/2}\ log(1/\phi_s) < R^2 \kappa^2\ ,\kappa = (\rho g/\gamma)^{1/2} = \text{inverse capillary length}</math><br />
<br />
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.<br />
<br />
===Metastable Cassie===<br />
<br />
Optimizing the post density for low drop adhesion leads to a fragile metastable Cassie state, as seen in Figure 16.<br />
<br />
[[Image: Q16.png|thumb|right|400px|'''Fig. 16''' ]]<br />
<br />
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):<br />
<br />
<math>\Delta E = ( \gamma_{SL} - \gamma_{SA}) (r - 1) = \gamma_{LA} (r - 1)\ cos\ \theta</math><br />
<br />
and the energy barrier in terms of the surface geometry is:<br />
<br />
<math>\Delta E \approx\ (2\pi b h)/(p^2 \gamma\ cos\ \theta)</math><br />
<br />
The conclusions we draw from this relation are<br />
<br />
1. Energy barrier for micron scale posts cannot be overcome by thermal energy<br />
2. Higher posts (''h'') increase the Cassie-Wenzel barrier, and ''h'' is a good tuning parameter<br />
<br />
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:<br />
<br />
<math>R^{*} \sim\ p^2/h</math><br />
<br />
'''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.<br />
<br />
<br />
<br />
Generally speaking, the Cassie-Wenzel transition is irreversible. <br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. ''Annu. Rev. Mater. Res.'' 2008. '''38''':71-99 <br />
<br />
2. Gao L, McCarthy TJ. 2006. A perfectly hydrophobic surface. ''JACS'' '''128''':9052–53<br />
<br />
3.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6864Wetting and Roughness: Part 32009-04-21T19:44:55Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
For example, if <math>\theta = 110-120°</math>, and <math>\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). Re entrant designs make more robust Cassie states and allow even hydrophilic surfaces can trap air under liquid!<br />
<br />
<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. <br />
<br />
3.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6863Wetting and Roughness: Part 32009-04-21T19:15:38Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle relation is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s}</math><br />
<br />
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. <br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
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. <br />
<br />
'''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:<br />
<br />
<math>cos\ \theta^{*} = -1 + (1 - \phi_s)\ cos\ \theta</math><br />
<br />
<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. <br />
<br />
3.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_3&diff=6859Wetting and Roughness: Part 32009-04-21T18:15:33Z<p>Aepstein: </p>
<hr />
<div>[[Wetting and Roughness: Part 1]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 3, we examine the sections ''Superhydrophobicity'' and ''Special Properties'' <br />
<br />
==Soft matters==<br />
<br />
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 <br />
<br />
<math>(r - \phi_s)(\gamma_{SL} - \gamma_{SA}) > (1 - \phi_s)\ \gamma_{LA}</math><br />
<br />
and the corresponding critical Young angle is<br />
<br />
<math>cos\ \theta_c = -\frac{1 - \phi_s}{r - \phi_s)</math><br />
<br />
For very rough solids (<math>r \gg 1</math>)<br />
<br />
<br />
[[Image: Q12.png|thumb|right|300px|'''Fig. 12''' ]]<br />
<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. <br />
<br />
3.</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_2&diff=6858Wetting and Roughness: Part 22009-04-21T17:16:48Z<p>Aepstein: /* Pillars or Posts */</p>
<hr />
<div>[[Wetting and Roughness: Part 2]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 2, we look at the sections ''Microtextured Solids'' and ''Hemiwicking'' <br />
<br />
==Soft matters==<br />
<br />
During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.<br />
<br />
[[Image: Q5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
<br />
Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.<br />
<br />
1. Late 1990s research from the Kao Corporation showing large contact angles of liquids <br />
on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s.<br />
<br />
2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features <br />
found on hydrophobic plants, such as the lotus. Animal studies followed.<br />
<br />
3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs <br />
to be studied, as inspired by (1) and (2).<br />
<br />
Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.<br />
<br />
===Micro/Nanotexture Inspirations from Nature===<br />
<br />
As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.<br />
<br />
[[Image: Q6.png|thumb|right|400px|'''Fig. 6''' ]]<br />
<br />
Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d). <br />
<br />
====Trying to Synthesize====<br />
<br />
Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.<br />
<br />
===Hemiwicking===<br />
<br />
<br />
[[Image: Q7.png|thumb|right|400px|'''Fig. 7''' ]]<br />
<br />
Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry. <br />
<br />
The simplest example of a hemiwicking microtexture is a groove of width ''w'' and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.<br />
<br />
[[Image: Q8.png|thumb|right|200px|'''Fig. 8''' ]]<br />
<br />
Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math><br />
<br />
The Young equation provides that the liquid progression is favorable (d''E'' < 0) if <math>\theta , \theta_c</math>, where<br />
<math>cos\ \theta_c = \frac{w}{2\delta + w}</math><br />
<br />
The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:<br />
<br />
High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking<br />
Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick<br />
<br />
====Pillars or Posts====<br />
<br />
Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness ''r''. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].<br />
<br />
[[Image: Q9.png|thumb|right|200px|'''Fig. 9''' ]]<br />
<br />
The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With ''dx'' larger than the scale of the post pitch ''p'', the energy of hemiwicking in a forest of posts is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math><br />
<br />
As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous. <br />
<br />
<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math><br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible--the edge of this fried egg is represented in Figure 11. <br />
[[Image: Q11.png|thumb|right|200px|'''Fig. 11''' ]]<br />
<br />
Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d''x'' eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes<br />
<br />
<math>cos\ \theta^{*} = 1 - (1 - \phi_s)\ cos\ \theta</math><br />
<br />
===Hemiwicking dynamics===<br />
<br />
If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on ''r'', and vanishes on a flat surface (''r'' = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting ''x'' as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:<br />
<br />
Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math><br />
<br />
Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant ''D''. For wetting liquids and posts of height ''h'', pitch ''p'', and radius ''b'', the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth ''h'' of the flow and <br />
<br />
<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math> <br />
<br />
which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is<br />
<br />
<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math><br />
<br />
To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. ''Nat. Mater.'' 2007, '''6''', 661-664<br />
<br />
3. C. Ishino, et al. Wicking within forests of micropillars, 2007 ''EPL'' '''79''', 56005</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_2&diff=6857Wetting and Roughness: Part 22009-04-21T17:15:11Z<p>Aepstein: /* Pillars or Posts */</p>
<hr />
<div>[[Wetting and Roughness: Part 2]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 2, we look at the sections ''Microtextured Solids'' and ''Hemiwicking'' <br />
<br />
==Soft matters==<br />
<br />
During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.<br />
<br />
[[Image: Q5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
<br />
Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.<br />
<br />
1. Late 1990s research from the Kao Corporation showing large contact angles of liquids <br />
on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s.<br />
<br />
2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features <br />
found on hydrophobic plants, such as the lotus. Animal studies followed.<br />
<br />
3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs <br />
to be studied, as inspired by (1) and (2).<br />
<br />
Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.<br />
<br />
===Micro/Nanotexture Inspirations from Nature===<br />
<br />
As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.<br />
<br />
[[Image: Q6.png|thumb|right|400px|'''Fig. 6''' ]]<br />
<br />
Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d). <br />
<br />
====Trying to Synthesize====<br />
<br />
Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.<br />
<br />
===Hemiwicking===<br />
<br />
<br />
[[Image: Q7.png|thumb|right|400px|'''Fig. 7''' ]]<br />
<br />
Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry. <br />
<br />
The simplest example of a hemiwicking microtexture is a groove of width ''w'' and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.<br />
<br />
[[Image: Q8.png|thumb|right|200px|'''Fig. 8''' ]]<br />
<br />
Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math><br />
<br />
The Young equation provides that the liquid progression is favorable (d''E'' < 0) if <math>\theta , \theta_c</math>, where<br />
<math>cos\ \theta_c = \frac{w}{2\delta + w}</math><br />
<br />
The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:<br />
<br />
High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking<br />
Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick<br />
<br />
====Pillars or Posts====<br />
<br />
Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness ''r''. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].<br />
<br />
[[Image: Q9.png|thumb|right|200px|'''Fig. 9''' ]]<br />
<br />
The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With ''dx'' larger than the scale of the post pitch ''p'', the energy of hemiwicking in a forest of posts is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math><br />
<br />
As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous. <br />
<br />
<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math><br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible--the edge of this fried egg is shown represented in Figure 11. <br />
<br />
[[Image: Q11.png|thumb|right|400px|'''Fig. 11''' ]]<br />
<br />
Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d''x'' eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes<br />
<br />
<math>cos\ \theta^{*} = 1 - (1 - \phi_s)\ cos\ \theta</math><br />
<br />
===Hemiwicking dynamics===<br />
<br />
If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on ''r'', and vanishes on a flat surface (''r'' = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting ''x'' as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:<br />
<br />
Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math><br />
<br />
Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant ''D''. For wetting liquids and posts of height ''h'', pitch ''p'', and radius ''b'', the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth ''h'' of the flow and <br />
<br />
<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math> <br />
<br />
which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is<br />
<br />
<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math><br />
<br />
To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. ''Nat. Mater.'' 2007, '''6''', 661-664<br />
<br />
3. C. Ishino, et al. Wicking within forests of micropillars, 2007 ''EPL'' '''79''', 56005</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=File:Q11.png&diff=6856File:Q11.png2009-04-21T17:14:53Z<p>Aepstein: </p>
<hr />
<div></div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_2&diff=6855Wetting and Roughness: Part 22009-04-21T17:14:46Z<p>Aepstein: /* Pillars or Posts */</p>
<hr />
<div>[[Wetting and Roughness: Part 2]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 2, we look at the sections ''Microtextured Solids'' and ''Hemiwicking'' <br />
<br />
==Soft matters==<br />
<br />
During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.<br />
<br />
[[Image: Q5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
<br />
Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.<br />
<br />
1. Late 1990s research from the Kao Corporation showing large contact angles of liquids <br />
on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s.<br />
<br />
2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features <br />
found on hydrophobic plants, such as the lotus. Animal studies followed.<br />
<br />
3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs <br />
to be studied, as inspired by (1) and (2).<br />
<br />
Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.<br />
<br />
===Micro/Nanotexture Inspirations from Nature===<br />
<br />
As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.<br />
<br />
[[Image: Q6.png|thumb|right|400px|'''Fig. 6''' ]]<br />
<br />
Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d). <br />
<br />
====Trying to Synthesize====<br />
<br />
Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.<br />
<br />
===Hemiwicking===<br />
<br />
<br />
[[Image: Q7.png|thumb|right|400px|'''Fig. 7''' ]]<br />
<br />
Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry. <br />
<br />
The simplest example of a hemiwicking microtexture is a groove of width ''w'' and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.<br />
<br />
[[Image: Q8.png|thumb|right|200px|'''Fig. 8''' ]]<br />
<br />
Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math><br />
<br />
The Young equation provides that the liquid progression is favorable (d''E'' < 0) if <math>\theta , \theta_c</math>, where<br />
<math>cos\ \theta_c = \frac{w}{2\delta + w}</math><br />
<br />
The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:<br />
<br />
High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking<br />
Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick<br />
<br />
====Pillars or Posts====<br />
<br />
Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness ''r''. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].<br />
<br />
[[Image: Q9.png|thumb|right|200px|'''Fig. 9''' ]]<br />
<br />
The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With ''dx'' larger than the scale of the post pitch ''p'', the energy of hemiwicking in a forest of posts is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math><br />
<br />
As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous. <br />
<br />
<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math><br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible--the edge of this fried egg is shown represented in Figure 11. <br />
<br />
[[Image: Q11.png|thumb|left|400px|'''Fig. 11''' ]]<br />
<br />
Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d''x'' eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes<br />
<br />
<math>cos\ \theta^{*} = 1 - (1 - \phi_s)\ cos\ \theta</math><br />
<br />
===Hemiwicking dynamics===<br />
<br />
If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on ''r'', and vanishes on a flat surface (''r'' = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting ''x'' as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:<br />
<br />
Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math><br />
<br />
Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant ''D''. For wetting liquids and posts of height ''h'', pitch ''p'', and radius ''b'', the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth ''h'' of the flow and <br />
<br />
<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math> <br />
<br />
which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is<br />
<br />
<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math><br />
<br />
To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. ''Nat. Mater.'' 2007, '''6''', 661-664<br />
<br />
3. C. Ishino, et al. Wicking within forests of micropillars, 2007 ''EPL'' '''79''', 56005</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_2&diff=6854Wetting and Roughness: Part 22009-04-21T17:13:39Z<p>Aepstein: /* Pillars or Posts */</p>
<hr />
<div>[[Wetting and Roughness: Part 2]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 2, we look at the sections ''Microtextured Solids'' and ''Hemiwicking'' <br />
<br />
==Soft matters==<br />
<br />
During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.<br />
<br />
[[Image: Q5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
<br />
Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.<br />
<br />
1. Late 1990s research from the Kao Corporation showing large contact angles of liquids <br />
on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s.<br />
<br />
2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features <br />
found on hydrophobic plants, such as the lotus. Animal studies followed.<br />
<br />
3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs <br />
to be studied, as inspired by (1) and (2).<br />
<br />
Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.<br />
<br />
===Micro/Nanotexture Inspirations from Nature===<br />
<br />
As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.<br />
<br />
[[Image: Q6.png|thumb|right|400px|'''Fig. 6''' ]]<br />
<br />
Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d). <br />
<br />
====Trying to Synthesize====<br />
<br />
Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.<br />
<br />
===Hemiwicking===<br />
<br />
<br />
[[Image: Q7.png|thumb|right|400px|'''Fig. 7''' ]]<br />
<br />
Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry. <br />
<br />
The simplest example of a hemiwicking microtexture is a groove of width ''w'' and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.<br />
<br />
[[Image: Q8.png|thumb|right|200px|'''Fig. 8''' ]]<br />
<br />
Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math><br />
<br />
The Young equation provides that the liquid progression is favorable (d''E'' < 0) if <math>\theta , \theta_c</math>, where<br />
<math>cos\ \theta_c = \frac{w}{2\delta + w}</math><br />
<br />
The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:<br />
<br />
High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking<br />
Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick<br />
<br />
====Pillars or Posts====<br />
<br />
Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness ''r''. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].<br />
<br />
[[Image: Q9.png|thumb|right|200px|'''Fig. 9''' ]]<br />
<br />
The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With ''dx'' larger than the scale of the post pitch ''p'', the energy of hemiwicking in a forest of posts is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math><br />
<br />
As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous. <br />
<br />
<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math><br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible. <br />
<br />
Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d''x'' eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes<br />
<br />
<math>cos\ \theta^{*} = 1 - (1 - \phi_s)\ cos\ \theta</math><br />
<br />
===Hemiwicking dynamics===<br />
<br />
If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on ''r'', and vanishes on a flat surface (''r'' = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting ''x'' as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:<br />
<br />
Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math><br />
<br />
Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant ''D''. For wetting liquids and posts of height ''h'', pitch ''p'', and radius ''b'', the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth ''h'' of the flow and <br />
<br />
<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math> <br />
<br />
which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is<br />
<br />
<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math><br />
<br />
To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. ''Nat. Mater.'' 2007, '''6''', 661-664<br />
<br />
3. C. Ishino, et al. Wicking within forests of micropillars, 2007 ''EPL'' '''79''', 56005</div>Aepsteinhttp://soft-matter.seas.harvard.edu/index.php?title=Wetting_and_Roughness:_Part_2&diff=6853Wetting and Roughness: Part 22009-04-21T17:11:48Z<p>Aepstein: /* Hemiwicking dynamics */</p>
<hr />
<div>[[Wetting and Roughness: Part 2]]<br />
<br />
Authors: David Quere<br />
<br />
Annu. Rev. Mater. Res. 2008. 38:71–99<br />
<br />
====Soft matter keywords====<br />
microtextures, superhydrophobicity, wicking, slip<br />
<br />
By Alex Epstein<br />
----<br />
<br />
===Abstract from the original paper===<br />
We discuss in this review how the roughness of a solid impacts its wettability.<br />
We see in particular that both the apparent contact angle and the contact angle<br />
hysteresis can be dramatically affected by the presence of roughness. Owing<br />
to the development of refined methods for setting very well-controlled<br />
micro- or nanotextures on a solid, these effects are being exploited to induce<br />
novel wetting properties, such as spontaneous filmification, superhydrophobicity,<br />
superoleophobicity, and interfacial slip, that could not be achieved<br />
without roughness.<br />
<br />
In Part 2, we look at the sections ''Microtextured Solids'' and ''Hemiwicking'' <br />
<br />
==Soft matters==<br />
<br />
During the past decade, the use of microtextured solids and more recently nanotexturing has been popular to induce surface wetting properties that cannot otherwise be obtained. Roughness of the surface changes the unique Young angle to a range of possible angles and generates an apparent angle in the surface plane that is different from the local angle at the contact line.<br />
<br />
[[Image: Q5.png|thumb|right|300px|'''Fig. 5''' ]]<br />
<br />
Quere asserts that three factors are responsible for the sudden resurgence of interest in this area.<br />
<br />
1. Late 1990s research from the Kao Corporation showing large contact angles of liquids <br />
on fluorinated rough surfaces. Note that this was similar to results reported in the 1940s.<br />
<br />
2. Papers by Neinhuis and Barthlott in Germany reporting the variety of surface features <br />
found on hydrophobic plants, such as the lotus. Animal studies followed.<br />
<br />
3. Developments in micro/nanofabrication techniques that allowed more sophisticated designs <br />
to be studied, as inspired by (1) and (2).<br />
<br />
Briefly, the Kao experiment plotted relation of the apparent contact angle <math>\theta^{*}</math> of various liquids on a rough fluorinated surface against the expected Young angle of each liquid on a flat fluorinated surface. The S-shaped curve, seen in Figure 5, describes the amplifying effect of roughness on hydrophilicity and hydrophobicity. The first, steeper slope on the right side follows Wenzel's roughness closely. The second, smaller slope is the superhydrophilic regime, in which Wenzel breaks down because hemiwicking of surrounding surface cavities (as considered below) leads to the droplet sitting on both solid and liquid.<br />
<br />
===Micro/Nanotexture Inspirations from Nature===<br />
<br />
As early as AD 77, Pliny the Elder reported the beading of water drops on woolly plant leaves, the first reported observation of superhydrophobicity. Truly systematic studies of natural microtextures, however, have only happened in the last decade.<br />
<br />
[[Image: Q6.png|thumb|right|400px|'''Fig. 6''' ]]<br />
<br />
Plant leaves often features bumps on the scale of 10-50 <math>\mu</math>m, and some, including the now famous lotus, also have a finer 100 nm scale of features (Fig. 6b). The fractal geometry appears to contribute to the superhydrophobicity and self-cleaning ability of the leaf. The mechanism of the fractal surface is debated still, but phenomenologically it provides both a high contact angle and a low hysteresis. Thus drops on the leaves have very high mobility and roll off with ease. Remarkably, the rice leaf has an anisotropic arrangement of papillae that direct the flow of water along preferred directions. Other noteworthy example of superhydrophobic surfaces in nature are the feathers of pigeons and ducks; cicada, butterflies; and the leg setae of water striders in Fig. 6c that rest on trapped air (c.f., Leidenfrost effect) as they travel on water. Mosquitoes' eyes are completely drying due to a pattern of 100 nm bumps (Fig. 6d). <br />
<br />
====Trying to Synthesize====<br />
<br />
Quere notes that we can make a superhydrophobic surface by a very crude technique in the garage: take a piece of glass to a sooty flame, and the dark soot coating will provide microroughness and plenty of carbon to repel water. Obviously most of today's research uses sophisticated techniques to create surface patterns.<br />
<br />
===Hemiwicking===<br />
<br />
<br />
[[Image: Q7.png|thumb|right|400px|'''Fig. 7''' ]]<br />
<br />
Quere introduces the term "hemiwicking" to describe an imbibition phenomena in rough surfaces that is similar but different from classical wicking. As a liquid film progresses through surface micro/nanostructures such as those shown in Figure 7, at least one additional side is exposed to air. Thus, we have another liquid-air interface. This is different from the Wenzel model, in which the cavities of the surface are filled as though they are capillary tubes while areas not under the drop remain dry. <br />
<br />
The simplest example of a hemiwicking microtexture is a groove of width ''w'' and depth <math>\delta</math>, shown in Figure 8. Such grooves can be exploited, as in the rice leaf, to achieve directional wetting.<br />
<br />
[[Image: Q8.png|thumb|right|200px|'''Fig. 8''' ]]<br />
<br />
Hemiwicking will occur if the solid is wetting (<math>\gamma_{SL} < \gamma_{SA}</math>) and if this energy change overcomes the additional liquid-vapor interface formed on top. This liquid-vapor interface will be flat to minimize area, so the surface energy change is:<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(2\delta + w )\ dx + \gamma_{LA}\ w\ dx</math><br />
<br />
The Young equation provides that the liquid progression is favorable (d''E'' < 0) if <math>\theta , \theta_c</math>, where<br />
<math>cos\ \theta_c = \frac{w}{2\delta + w}</math><br />
<br />
The right-hand side of this equation varies from 0 to 1 depending on the groove cross-section aspect ratio <math>\delta / w</math>. Important conclusions:<br />
<br />
High aspect ratio (narrow and deep) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> high <math>\longrightarrow</math> spontaneous hemiwicking<br />
Low aspect ratio (shallow and wide) groove <math>\longrightarrow</math> critical angle <math>\theta_c</math> low <math>\longrightarrow</math> difficult to hemiwick<br />
<br />
====Pillars or Posts====<br />
<br />
Micro/nanoposts are the bread and butter surfaces that we study and apply in the Aizenberg Group. So it is helpful to realize that similar arguments hold for a solid decorated with posts (Fig. 7a) as with grooves. The two characteristics of this surface are pillar density <math>\phi_s</math> and roughness ''r''. Hemiwicking through a forest of posts is seen in Figure 9: a film of ethanol permeates the posts in a circle beyond the footprint of a drop. In some cases, the film conforms to the symmetry of the post pattern and will be square or hexagonal, as explored by the Stone Group [2].<br />
<br />
[[Image: Q9.png|thumb|right|200px|'''Fig. 9''' ]]<br />
<br />
The impregnating front propagates as shown in Figure 10. Liquid coats the solid on an area proportional to <math>r - \phi_s</math>, while the liquid-vapor interfacial area is proportional to <math>1 - \phi_s</math>. With ''dx'' larger than the scale of the post pitch ''p'', the energy of hemiwicking in a forest of posts is:<br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
<math>dE\ = (\gamma_{SL} - \gamma_{SA})(r - \phi_s)\ dx + \gamma_{LV}(1 - \phi_s)\ dx</math><br />
<br />
As with grooves, there is a maximum Young angle <math>\theta_c</math> below which progression of the liquid is spontaneous. <br />
<br />
<math>cos\ \theta_c = (1 - \phi_s)(r - \phi_s)</math><br />
<br />
[[Image: Q10.png|thumb|left|400px|'''Fig. 10''' ]]<br />
<br />
The take-away message here is that we can tune the invasion of liquid by controlling the geometry of the forest. For rare defects (small <math>\phi_s</math>), <math>cos\ \theta_c \approx 1/r</math>: the rougher the substrate, the more favorable is hemiwicking. In comparison to grooves, the liquid front in a forest of posts must somehow be activated to achieve the jumps in Fig. 10. For wetting liquids, this is facilitated by the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a metastable Wenzel state, and additional external energy such as vibration is need to nucleate contact with the next row of posts. This means that an equilibrium "fried egg" configuration of a drop coexisting with a finite ring of film is possible. <br />
<br />
Looking back at Fig. 5, the second low-slope regime (<math>\theta < \theta_c</math> results from the hemiwicking effect. The low slope of the apparent angle is due to the fact that the drop sits on a composite surface consisting mainly of liquid; therefore the solid roughness has little effect on <math>\theta^{*}</math>. In this situation, moving the contact line by d''x'' eliminates solid-air interfaces on a surface fraction <math>\phi_s</math>, and liquid-air interfaces are eliminated on a fraction <math>1 - \phi_s</math>. This tweaks the earlier energy statement, and the apparent angle becomes<br />
<br />
<math>cos\ \theta^{*} = 1 - (1 - \phi_s)\ cos\ \theta</math><br />
<br />
===Hemiwicking dynamics===<br />
<br />
If wetting of the liquid is complete (<math>S > 0, \theta = 0</math>), a molecular precursor film propagates first, and then the hemiwicking film follows to lower the total surface energy by reducing liquid-vapor interface. Then the driving force for hemiwicking is <math>\gamma(r -1)</math>, depends only on ''r'', and vanishes on a flat surface (''r'' = 1). Since the surface defects are small, the main competing force is viscous, and it scales as <math>\eta V x</math>, denoting ''x'' as the impregnated distance and <math>\eta</math> as the viscosity. The force balance recovers the same time scaling as the Washburn law inside a porous medium:<br />
<br />
Hemiwicking distance <math>x\ \propto\ \sqrt{t}</math><br />
<br />
Equivalently, we can apply the classic diffusion law <math>x^2 = Dt</math> to quantify the diffusion constant ''D''. For wetting liquids and posts of height ''h'', pitch ''p'', and radius ''b'', the wicking force scales as <math>\gamma b h/p^2</math>. The situation is slightly different for low and high aspect ratio posts. For short posts, the viscous resistance is fixed by the depth ''h'' of the flow and <br />
<br />
<math>\text{Low aspect ratio posts:}\ D \sim\ (\gamma /\eta)(b h^2 /p^2)</math> <br />
<br />
which is easily tuned by the post height. [3] For tall posts, most of the viscous resistance is around the posts and the coefficient is<br />
<br />
<math>\text{High aspect ratio posts:}\ D \sim \gamma b/ \eta</math><br />
<br />
To conclude: Fixing the height of the pillars does not only influence the film dynamics, it also selects its thickness, since surface energy favors a film thickness which matches the pillar height.<br />
<br />
===References=== <br />
1. Quere, David. Wetting and Roughness. Annu. Rev. Mater. Res. 2008. 38:71-99 <br />
<br />
2. Courbin L, et al. Imbibition by polygonal spreading on microdecorated surfaces. ''Nat. Mater.'' 2007, '''6''', 661-664<br />
<br />
3. C. Ishino, et al. Wicking within forests of micropillars, 2007 ''EPL'' '''79''', 56005</div>Aepstein