Science - USA (2021-07-16)

Cassie-Baxter relation can be rewritten in terms
of the spacing ratio as

cosq
¼ 1 þ

1

D

sinqEþ

ðÞpqEcosqE ð 4 Þ

Increasing values ofD correspond to higher
surface porosity, and correspondingly higherq

and generally lowerDqvalues.
Pbrcan be expressed in terms of another
dimensionless parameterA,calledtherobust-
ness factor ( 27 ). For a cylindrical morphol-
ogy ( 24 , 30 )

A
¼

Pbr

Pref

¼

‘cap

RDðÞ
 1

ðÞ 1 cosqE

ðÞD
 1 þ2sinqE

ð 5 Þ

wherePref=2gLV/‘capis the reference pressure,
which is close to the minimum possible pres-
sure differential across the liquid-vapor inter-
face for a millimeter-sized liquid droplet or

puddle;‘cap¼

ffiffiffiffiffi

gLV

rg

q

is the liquid capillary

length;ris the liquid density; andgis the
acceleration due to gravity. The robustness
factor thus incorporates properties of the solid
texture (R,D), the contacting liquidð‘capÞ, and
qE.SurfacesforwhichA*≤1 for a given con-

tacting liquid cannot support a composite in-

terface and transition to the Wenzel state,

whereas values ofA*>> 1 imply a robust com-

posite interface ( 24 , 27 ).

Ideal nonwetting surfaces would enable

D*>> 1 andA*>>1 with a contacting liquid

to simultaneously display both high appar-

ent contact angles and high breakthrough

pressures. However, it is evident from Eq. 5

that increasingD*would result in decreasing

the magnitude ofA*. There are several design

strategies to break this trade-off: (i) For a given

surface porosity,A*values can be increased

without loweringD*by lowering the substrate

surface energy or by changingymin( 24 ) (Fig.

2A). Figure 2A showcases how surface tex-

ture can be used to create air-infused liquid-

repellent surfaces using materials with widely

differing surface energies, and thereby surface

chemistries. The design of air-infused omni-

phobic or superomniphobic surfaces based on

the micro-hoodoo (ymin≈0°) ( 23 , 27 )andthe

doubly reentrant ( 31 )(ymin≈−90°) geometries

are particularly worth mentioning in this con-

text (Fig. 2A, inset). (ii) For a given surface

chemistry, the magnitude of the robustness

factorA*can be increased while maintain-

ing the same spacing ratioD*(and thereby

the apparent contact angles) by decreasing the

length scale of the features comprising the

solid texture. This strategy essentially allows

us to move along theyaxis in the design chart

showninFig.2B( 32 ). (iii) For a given surface

composition, the spacing ratioD*can be in-

creased while maintaining the values forA*by

developing surfaces with hierarchical scales of

texture. This strategy allows us to move along

thexaxis of the design chart shown in Fig. 2B

( 32 ). One related challenge is the long-term sta-

bility of air pockets in the Cassie-Baxter state

for completely submerged nonwetting surfaces.

Here again, the solid texture length scale is

important, as thermodynamic analysis has

shown that submicrometer texture spacing

is required to sustain air pockets on surfaces

submerged underwater at atmospheric pres-

sure ( 33 ).

Another approach to liquid repellency is the

one adopted by the carnivorousNepenthes

pitcher plant, which locks in a lubricating layer

of water within the porous surface texture of

its rim. Any insect or even small rodents that

try to walk along the rim cannot gain traction

and slide off into the stomach of the plant.

Basedinpartontheunderstandingofsuch

structures, a variety of lubricant-infused omni-

phobic surfaces have been developed ( 34 Ð 37 ).

On such surfaces, a liquid lubricant, typically a

silicone or fluorinated oil, is stabilized within

a textured or porous solid through capillary

Dhyaniet al.,Science 373 , eaba5010 (2021) 16 July 2021 3 of 13

Fig. 2. Design charts for
air-infused liquid-repellent
surfaces.(A) The variation of
the robustness factor (A)
as a function of the solid
surface energy. The different sur-
face energies for common surface
chemistries (fluorinated organic,
nonfluorinated organic, metal
oxide, etc.) are also shown.
Three different regimes for liquid
repellency are drawn based
on the minimum value of the
texture angleymin= 90°
( 147 , 152 , 154 , 155 ),ymin<90°
( 3 , 5 , 23 , 27 , 29 , 125 , 148 –
151 , 156 – 158 ), andymin<0°.
The data illustrate that a lower
value ofyminenables a higher
value of the robustness factor and
allows for a wider range of surface
chemistries to be utilized for
developing liquid-repellent
surfaces. For a given texture angle, a reduction in the surface energy leads to a higher value of the robustness factor. Example SEM images for different air-infused,
liquid-repellent surfaces are included as insets. These surfaces include vertical pillars ( 147 ), electrospun fabrics ( 23 ), and micro-hoodoos ( 23 , 31 ). The orange
data points represent oil-repellent surfaces, whereas the blue data points represent water-repellent surfaces. Image credits: vertical pillars, reprinted (adapted) with
permission from the American Chemical Society, copyright (2000) ( 147 ); electrospun fabrics, adapted from ( 23 ); micro-reentrant structure, adapted from ( 23 );
micro-double-reentrant structure, adapted from ( 31 ). (B) The variation ofA as a function of the spacing ratio (D) for octane on cylindrical textures [e.g.,
micromeshes ( 5 , 149 )]. Values ofA< 1 indicate the formation of the Wenzel state and complete wetting. For the same value ofD, it is possible to increaseA values,
and thereby the breakthrough pressure, by making the features on a finer length scale. Additionally, for the same value ofA, it is possible to increaseD values,
and thereby the apparent contact angles, by fabricating surfaces with a hierarchical texture. Image credits: micro-meshes from ( 5 ) reproduced with permission from
Springer Nature; hierarchical micromeshes, adapted from ( 149 ).

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