this state, the apparent contact angleq(the
contact angle on a textured surface) can be
determined as
cosq∗¼rcosqE ð 2 ÞHereris the surface roughness, defined as
the ratio between the actual and the projected
surface area. Note thatr≥1. Thus, roughness
always enhances the inherent wetting (q<<
90°, ifq< 90°) or nonwetting (q>> 90°, ifq>
90°) characteristics of surfaces in the Wenzel
state. Additionally, textured surfaces in the
Wenzel state typically display high–contact
angle hysteresis, owing to the large solid-
liquid interfacial area ( 22 ).
By contrast, in the Cassie-Baxter state, the
contacting liquid droplet does not fully pen-
etrate within the surface texture and sits
partially on pockets of entrapped air. The
contacting liquid penetrates into the surface
texture only until the local texture angle (y)
for the solid-liquid-vapor three phase con-
tact line becomes equal to the equilibrium
contact angle (qE)( 23 , 24 ). The apparent
contact angle (q) in the Cassie-Baxter state
is given as
cosq¼rffscosqEþfs 1 ð 3 Þwhererfis the roughness of the wetted area
andfsis the areal fraction of the liquid-air in-
terfaceoccludedbythesurfacetexture(Fig.1A).
Here, we will also refer to the Cassie-Baxter
state as the air-infused composite interface,
to distinguish it from the lubricant-infused
surfaces discussed later. Small values offsin
the Cassie-Baxter state result in higher contact
angles and smaller contact angle hysteresis,
and this is the preferred state for the design
of superhydrophobic and superoleophobic
surfaces. Additionally, unlike the Wenzel state,
in the Cassie-Baxter state, it is possible to ob-
tainq*>> 90°, even ifqE< 90° ( 23 , 24 ).Design principles for liquid-repellent surfaces
There are numerous natural superhydropho-
bic surfaces, including a variety of plant leaves,
insect legs, and insect wings ( 25 ) (Fig. 1A). By
contrast, only one natural oleophobic surface
has been identified thus far–the skin of var-
ious springtails (Fig. 1A, inset). Springtails are
arthropods that live in soil, decaying organic
matter, or plant leaves and can thus come into
contact with various low–surface tension liquids
such as different plant oils. The springtails
breathe through their skin and would suf-
focate if their body were soaked by any con-
tacting liquid. Reentrant surface texture (i.e.,
surfaces that possess an overhang or bend
back on themselves) ( 26 , 27 ) (Fig. 1A, inset)
allows the springtail skin to form an air-infused
composite interface with a wide range of polar
and nonpolar organic liquids, in addition to
water. Both experimental and theoretical work
has described the necessity of reentrant tex-
ture to support a composite interface with low–surface tension contacting liquids, as there is
no known chemistry to enableqE> 90° with
low–surface tension liquids such as methanol
and octane ( 23 , 27 ). Surfaces with reentrant
texture possess local texture anglesy< 90°
(Fig. 2A) and allow for the possibility of form-
ing a composite interface even with these ex-
tremely low–surface tension liquids as long as
qΕ≥ymin( 27 , 28 ). Here,yminis the minimum
possible local texture angle on a given surface
geometry (Fig. 2). Of note, the oleophobic prop-
erties of springtail skin were identified after the
first synthetic oleophobic surfaces had already
been fabricated ( 23 , 24 , 27 , 29 ).
Inspired in part by natural nonwetting sur-
faces, a range of different air-infused superhydro-
phobic, superoleophobic, and superomniphobic
surfaces have been fabricated over the last dec-
ade(Fig.1B).Thesystematicdesignofsuch
nonwetting surfaces requires the maximiza-
tion of two important physical properties for
a composite interface: (i) the magnitude of
the apparent contact angle (q)and(ii)the
magnitude of the breakthrough pressure (Pbr),
i.e., the pressure required to force a transi-
tion from the Cassie-Baxter state to the Wenzel
state ( 24 ).
As discussed above,qvalues are a function
of the surface porosity. For surfaces with a pre-
dominantly cylindrical morphology, we can
define a dimensionless measure for porosity
called the spacing ratio, given asD* =(R+D)/R
( 23 , 24 ). Here,Ris the radius of the cylin-
der, and 2Dis the intercylinder spacing. TheDhyaniet al.,Science 373 , eaba5010 (2021) 16 July 2021 2 of 13
Fig. 1. State-of-the-art liquid-
repellent surfaces.(A) The
relation between the cosine of the
equilibrium contact angle and
the cosine of the apparent contact
angle. The different wettability
domains on rough surfaces
[Cassie-Baxter State ( 138 ),
Wenzel state ( 139 ), Hemi-wicking
state ( 140 )] are also shown.
The oleophobic springtail with
reentrant features ( 141 ), and
synthetic doubly reentrant
surface textures ( 31 ), are
included as insets. Image credits:
springtail and scanning electron
microscopy (SEM) image on
its skin, adapted from ( 141 ) under
https://creativecommons.org/
licenses/by-nc/4.0/; micro-
reentrant structure, adapted from
( 23 ); micro-double-reentrant structure, adapted with permission from The American Association for the Advancement of Science (AAAS) ( 31 ). (B) The contact angle
hysteresis versus the apparent contact angle for three different state-of-the-art liquid-repellent surfaces. The different approaches for liquid repellency include
lubricant-infused surfaces ( 9 , 35 , 49 , 50 , 63 ), where a textured surface is infused with a liquid lubricant, a liquid-like brush regime ( 39 , 122 , 142 – 146 ) where a
liquid-like monolayer is covalently attached to the underlying substrate, and an air-infused regime ( 4 , 23 , 125 , 147 – 153 ) that repels liquids by trapping pockets of air
underneath the contacting liquid. The data points and the inset images are adapted from previous work. The orange data points represent oil-repellent surfaces,
whereas the blue data points represent water-repellent surfaces. Image credits: lubricant-impregnated nanotextured surfaces, adapted with permission from
the American Chemical Society; copyright (2012) ( 49 ).
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