Effects of geometrical characteristics of surface roughness on droplet wetting.

Abstract

Surface roughness is known to alter the wettability on a solid substrate. In general, either Wenzel or Cassie-Baxter theory is adopted to describe the apparent contact angle. Following the minimum free energy pathway associated with the imbibition process, we have derived a generalized expression for the apparent contact angle on a textured surface and the liquid-gas contact area within the groove that plays a key role. Depending on the geometrical characteristics of the grooves, the surface wetting falls into three regimes: (i) single stable state which is either Wenzel (completely wetted roughness) or Cassie-Baxter (completely nonwetted roughness) state, (ii) two stable states (Wenzel and Cassie-Baxter) separated by an energy barrier, and (iii) single stable state with partially wetted roughness. The sufficient condition for each regime is derived and several groove geometries are given to show the free energy path. Alteration in the geometric parameters may lead to the wetting crossover. We also show that the Cassie-Baxter can occur at a hydrophilic surface for particular pore shapes.

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Effects of geometrical characteristics of surface roughness
on droplet wetting
Yu-Jane Sheng
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China
Shaoyi Jiang
Department of Chemical Engineering, University of Washington, Seattle, Washington 98195, USA
Heng-Kwong Tsaoa?
Department of Chemical and Materials Engineering, Institute of Materials Science and Engineering,
National Central University, Jhongli, Taiwan 320, Republic of China
?Received 10 September 2007; accepted 10 October 2007; published online 19 December 2007?
Surface roughness is known to alter the wettability on a solid substrate. In general, either Wenzel or
Cassie-Baxter theory is adopted to describe the apparent contact angle. Following the minimum free
energy pathway associated with the imbibition process, we have derived a generalized expression
for the apparent contact angle on a textured surface and the liquid-gas contact area within the groove
that plays a key role. Depending on the geometrical characteristics of the grooves, the surface
wetting falls into three regimes: ?i? single stable state which is either Wenzel ?completely wetted
roughness? or Cassie-Baxter ?completely nonwetted roughness? state, ?ii? two stable states ?Wenzel
and Cassie-Baxter? separated by an energy barrier, and ?iii? single stable state with partially wetted
roughness. The sufficient condition for each regime is derived and several groove geometries are
given to show the free energy path. Alteration in the geometric parameters may lead to the wetting
crossover. We also show that the Cassie-Baxter can occur at a hydrophilic surface for particular pore
shapes. © 2007 American Institute of Physics. ?DOI: 10.1063/1.2804425?
I. INTRODUCTION
The wetting of solid surfaces by a liquid ?water in par-
ticularly? is ubiquitous in everyday lives as well as in indus-
trial processes. Wettability is one of the most important prop-
erties associated with a solid surface and the wetting
behavior is governed by two factors; the chemical composi-
tion and the roughness of the solid surfaces. In terms of the
contact angle ? between the gas-liquid and solid-liquid inter-
faces, the wettability of an ideal flat solid is depicted by
Young’s equation,1
cos ? =?s− ?sl
?l
,
?1?
where ?sl, ?s, and ?lrepresent the interfacial tensions of
solid-liquid, solid-gas, and liquid-gas interfaces, respectively.
In the absence of surface roughness, Young’s equation indi-
cates that the nature of wetting is determined by the relative
affinity of the solid for the liquid or gas phases, as illustrated
by the difference between solid-gas and solid-liquid interfa-
cial tensions in Young’s equation. The interfacial tensions ?sl
and ?sare intrinsic properties associated with a surface and
they can be controlled by chemical modification, such as
fluorination. As ?s−?sl?0, the contact angle is less than 90°
?hydrophilic surface?, whereas ??90° ?hydrophobic surface?
for ?s−?sl?0.
Real solids are actually rough and, thus, their wettability
is significantly influenced by the geometrical structure of the
surface roughness. The earliest work on the wetting of rough
substrates was addressed by Wenzel2and later by Cassie and
Baxter.3Wenzel assumed that the liquid filled up the grooves
on the rough surface and generalized Young’s equation to
obtain the apparent contact angle ?a,
cos ?a= r cos ?,
?2?
where r is termed the “roughness factor” and defined as the
ratio of the actual area of a rough surface to the geometric,
projected area on the horizontal plane. Evidently, the effect
of the roughness results in the improvement of the wetting
for ??90° but enhances the hydrophobicity for ??90°.
Cassie and Baxter considered the wettability of a composite
surface, composed of two types of homogeneous patches that
have different solid-fluid interfacial tensions. The apparent
contact angle is then given by
cos ?a= f1cos ?1+ f2cos ?2,
?3?
where fiand ?irepresent the surface area fraction and the
contact angle of patch i, respectively.
For porous or corrugated surfaces, the roughness is
mainly filled with air. The openings of the pores can be re-
garded as nonwetting patches with ?2=180°. Since f2=1
−f1, Eq. ?3? becomes
cos ?a= f1?1 + cos ?1? − 1.
?4?
In accord with Eq. ?4?, if surface hydrophobicity ??1? and
surface roughness ?f1? are appropriately combined, a water
droplet deposited on such a superhydrophobic surface can
remain nearly spherical. A beautiful example is the leaves of
a?Electronic mail: hktsao@cc.ncu.edu.tw.
THE JOURNAL OF CHEMICAL PHYSICS 127, 234704 ?2007?
0021-9606/2007/127?23?/234704/7/$23.00© 2007 American Institute of Physics
127, 234704-1
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the lotus plant. Owing to the geometrical structure of solid
surfaces alone, the contact angles for water on those surfaces
can range from 140° to as much as 174°.4,5Recent advent of
microfabrication techniques allow us to design microscale
structures on a solid surface and thus to control the wettabil-
ity by roughening it, even without altering any surface chem-
istry. Inspired by the so-called lotus effect, superhydrophobic
surfaces have been created by decorating a homogeneous
substrate with an array of pillars.6–8
Besides the roughness factor ?r? and the wetted area
fraction ?f?, the Wenzel and Cassie-Baxter theories are es-
sentially independent of the geometrical characteristics of the
roughness ?the shape of the pore?. For a given roughness
geometry, the Wenzel and Cassie-Baxter theories may pre-
dict different apparent contact angles. The wetting state is
usually believed to be in either Wenzel or Cassie-Baxter
state. However, it is not decisively clear on which theory
should be employed and when. A simple criterion is that the
wetting state corresponds to the one with a lower free energy.
In general, the Wenzel state ?wetted groove? prevails for ?
??c, while the Cassie-Baxter state ?air pocket? dominates for
???c, where ?cdenotes the critical point and ?c??/2.6,8
Recently, it is experimentally showed that even if the Wenzel
drop possesses the lower energy, a Cassie-Baxter drop can be
observed. In fact, there could be two apparent contact angles
on the same rough surface, depending on how a drop is
formed.6,8For example, when pressing a drop on a hydro-
phobic surface decorated with spikes, a sharp change in the
apparent contact angle from 170° to 130° ?±5°? is observed.6
Another example is that the Cassie-Baxter state is formed if
the droplet is deposited gently on a hydrophobic surface with
square pillars but the Wenzel state is formed if dropped from
a height.8The possibility of multiple free energy minima for
droplet states has been proposed to explain the phenomenon
of multiple apparent contact angles.8–11Therefore, work
must be done to cause the transition from the Cassie-Baxter
to Wenzel state because of the energy barrier between them.
The actual details of the crossover from the Cassie-
Baxter to Wenzel state are not well understood and there may
exist many pathways. A possible pathway has been consid-
ered: the liquid enters the valleys and wets the sides of the
pillars by maintaining the location of the liquid-gas interface
at the same height from the bottom of the groove ?imbibition
in parallel?.10,11In fact, the extent of penetration into the
roughness grooves is initially unknown and has to be deter-
mined by the minimization of the free energy.12As a result,
the role of the liquid-gas interfaces within the roughness
grooves is essential. If the imbibition of liquid into rough-
ness pores in parallel corresponds to the minimum free en-
ergy pathway in the free energy landscape, one is able to
write down a generalized free energy formulation and finds
out the possible stable states for wetting on a textured sur-
face. In this paper, we focus on surfaces with roughness
pores and predict the existence of a stable state with partially
wetted roughness in addition to the Wenzel ?completely wet-
ted roughness? and Cassie-Baxter ?completely nonwetted
roughness? states. In addition, the influences of the geometri-
cal characteristics associated with roughness pores on the
wetting state are examined.
II. FREE ENERGY FORMULATION
We start by considering the free energy of a droplet sit-
ting on a rough, chemically heterogeneous substrate. For
simplicity, we assume that the radius of the drop is much
greater than the separation of asperities. For a sufficient
small drop, where the change in hydrostatic pressure with
height can be negligible, it can be shown that the solutions of
the axisymmetric Laplace equation yield a gas-liquid inter-
face that has the shape of a spherical cap with the radius R of
curvature.13Therefore, the free energy can simply be ex-
pressed by
*− ?s*? + 2?R2?1 − cos ?a??l,
*and ?s*denote the effective interfacial tensions for
solid-liquid and solid-gas contacts, respectively, due to the
heterogeneity of surfaces. Since the volume V of the drop is
constant, the drop radius R of curvature is related to apparent
contact angle cos ?aby
V =?R3
3
F = ??R sin ?a?2??sl
?5?
where ?sl
?1 − ??2?2 + ??,
?6?
where ?=cos ?a. Equation ?6? is justified when the liquid
volume within the roughness ?pores? is small compared to
the total volume. The minimization of the free energy,
?F/??=0, yields a relation similar to Young’s equation,
?lcos ?a= ?s*− ?sl
*.
?7?
In terms of the apparent contact angle, the free energy can be
rewritten as
=?3V
F???
??l
??
2/3
??1 − ??2?2 + ???1/3.
?8?
Since F decreases monotonically with increasing ?, this re-
sult reveals that a droplet with a lower apparent contact angle
possesses a lower free energy.
Now the goal is to determine the effective interfacial
tensions associated with a composite surface. For a homoge-
neous surface, one has ?i*=?iand Eq. ?7? simply reduces to
Young’s equation. Without loss of generality, we consider a
textured surface with two types of roughness. As shown in
Fig. 1, the first type of surface ?convex surface?, possessing
actual area A1and interfacial tensions ?sand ?sl, is always
wetted by the liquid. The second type of surface ?concave
surface? is associated with pores or spikes, which may form
air pockets. Its actual area and interfacial tensions are de-
FIG. 1. Schematic representation of a heterogeneous surface containing two
types of roughness. The first type is always wetted and possesses actual area
A1, interfacial tensions ?sand ?sl, and projected area fraction f1. The second
type can form air pockets ?grooves? and has the corresponding properties
?A2,?s?,?sl? ,f2?. A2? depicts the wetted area within the grooves.
234704-2Sheng, Jiang, and Tsao J. Chem. Phys. 127, 234704 ?2007?
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noted by A2, ?s?, and ?sl?, respectively. Note that these two
different types of surfaces can have different interfacial ten-
sions. The total projected area is Apand the area fraction of
the type i corresponding to Apis fiwith f1+f2=1. The
roughness factor of the type i surface ?ri? is defined as
ri=
Ai
Apfi
? 1.
?9?
The effective solid-gas interfacial tension ?per unit projected
area? ?s*is simply the sum of the contributions originating
from all types of surfaces in the absence of liquid.
?s*= f1?r1?s? + f2?r2?s??.
?10?
The effective solid-liquid interfacial tension ?sl
complicated. It may involve liquid-gas and solid-gas contacts
within the air pockets. If the liquid wets the pores partially,
then the wetted area of the second type ?A2?? can be related to
the projected area by r2?=A2?/Apf2with 0?r2??r2. In terms
of r2?, ?sl
*= f1?r1?sl? + f2?r2??sl? + ?r2− r2???s? + h?r2???l?.
*is more
*can be expressed by
?sl
?11?
Besides the contribution from the first type of surface, the
liquid may wet the second type of surface partially with area
A2? in the air pockets and the rest of the second type of
surface with area ?A2−A2?? is still in contact with air. When
the pores are partially wetted, the liquid is also in contact
with air. The liquid-gas contact area within the groove ?Ah? is
related to the projected area by h=Ah/Apf2. Note that this
scaled liquid-gas contact area ?h? may vary with the wetted
area in the pores and relates to the minimum free energy
pathway along the “wetting coordinate” r2?, 0?h?r2???1.
Owing to the two extreme conditions, nonwetting and com-
plete wetting of the pores, one has h?r2?=0?=1 and h?r2?
=r2?=0.
Knowing ?s*and ?sl
contact angle and the extent of imbibition, which follows the
minimum free energy pathway. Substituting Eqs. ?10? and
?11? into Eq. ?7? yields
*gives the relation between apparent
cos ?a= r1f1cos ? + ?r2? cos ?? − h?r2????1 − f1?,
?12?
where cos ??=??s?−?sl??/?l. This generalized expression can
be reduced to the Wenzel and the Cassie-Baxter theories. As
cos ??=cos ? and r2?=r2?complete wetting of the pores?, Eq.
?12? becomes
cos ?a=?A1+ A2
Ap?cos ? = r cos ?,
?13?
which is simply the result of the Wenzel theory. On the other
hand, as r2?=0 ?nonwetting of the pores?, one has
cos ?a= r1f1cos ? − ?1 − f1?.
?14?
If r1=1, the above equation reduces to the result of the
Cassie-Baxter theory. Equation ?14? indicates that the appar-
ent contact angle of a hydrophobic surface is amplified by
both the surface roughness of the wetted, convex area and
the liquid-air contact at the openings of the air pockets. It
contains main characteristics associated with both Wenzel
and Cassie-Baxter theories and can be used in both hydro-
philic and hydrophobic regions. Equation ?12? also reveals
that if r2cos ???−1 the free energy of the Wenzel state is
lower than that of the Cassie-Baxter state. For r2cos ???
−1, one has the opposite result. However, this information
alone is not enough to judge what the stable state is.
III. ENERGY BARRIER AND PARTIALLY WETTED
ROUGHNESS
The importance of the scaled gas-liquid contact area as-
sociated with partial wetting of the pores has been disclosed
in Eq. ?12?, which manifests the free energy path associated
with the imbibition process. Along the wetting coordinate r2?,
h?r2?? is assumed to follow the minimum free energy pathway
from the Cassie-Baxter scenario ?nonwetting of the pores
and r2?=0? to Wenzel scenario ?complete wetting of the pores
and r2?=r2? or vice versa. That is, for a simple geometry of
roughness such as circular cones, h?r2?? corresponds to the
minimum liquid-gas contact area at a given r2? ?wetting area
in the pores?. If one knows the wetting/dewetting path and
the geometry of the pores, then the relation between h and r2?
can be determined. Consequently, the free energy path
F???r2??? can be obtained for a given set of physical proper-
ties ??,??,r1,r2,f1?. After locating the free energy minima or
the energy barrier ?maximum?, the stable droplet shape ?or
apparent contact angle? can be inferred.
The actual detail of the wetting path is not well under-
stood and it depends on the wetting process. Although dif-
ferent possibilities may be hypothesized, the simplest sce-
narios for the wetting/dewetting process are wetting/
dewetting all pores in parallel10,11or in series. We assume
that imbibition into all pores proceeds in parallel and the
effect of meniscus in the grooves is neglected. For a speci-
fied geometry of the pores, the variation of the free energy
with increasing the wetting area can be evaluated by Eq. ?8?,
?????
?F
?r2?=?F
where the free energy is scaled by ??l?3V/??2/3. The pos-
sible stable states of wetting can be decided by Eq. ?15? for
−1???1. If ?F/?r2??0 for 0?r2??r2, then one has a bor-
der minimum corresponding to a single stable state. When
?F/?r2??0 ???/?r2??0?, the Wenzel state is the only stable
state because the free energy declines with increasing the
wetting area in the roughness. On the contrary, when
?F/?r2??0 ???/?r2??0?, the Cassie-Baxter state is the only
stable state since wetting the grooves results in the increment
in free energy.
If ?F/?r2?=0 at r2?=r2*with the condition 0?r2*?r2, then
there exists the extremum corresponding to a possible wet-
ting state in the free energy landscape. The stability of the
state F?r2*? has to be determined by the sign of the second
derivative. When ?2F/?r2?2?r2*??0, F?r2*? denotes the local
maximum and thus both the Wenzel and Cassie-Baxter states
are the border minima. That is, there exists two stable states
and the energy barrier from the Cassie-Baxter to Wenzel
state is ?Fb=F?r2*?−F?r2?=0?. The energy barrier is able to
resist a certain extent of external disturbances and prevents
the crossover from one state to another. On the contrary, for
?r2??= −
?1 + ??
??1 − ???2 + ??2?1/3???
?r2??,
?15?
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?2F/?r2?2?r2*??0, F?r2*? denotes the local minimum and,
therefore, the stable wetting state is associated with partially
wetted roughness with the penetration extent 0?r2*/r2?1.
This equilibrium state is neither the Wenzel nor Cassie-
Baxter state and only a part of the pore surface is wetted.
Two examples for the minimum free energy pathway are
illustrated in Fig. 2. The wetting surface with two stable
states can be achieved from hemispherical pores while that
with partially wetted grooves is obtained from pores created
by the revolution of y=x1/3about the y axis.
The above analysis indicates that depending on the wet-
ting characteristics of the grooves h?r2??, the wetting state
falls into one of the three regimes: ?i? single stable state
which is either Wenzel or Cassie-Baxter state, ?ii? two stable
states ?Wenzel and Cassie-Baxter? separated by an energy
barrier, and ?iii? single stable state with partially wetted
roughness. Clearly, the “phase diagram” of the wetting states
on a textured rough surface has to be determined by the
scaled liquid-gas contact area function h?r2??, which is, in
turn, dependent on the geometric characteristics associated
with the surface roughness. According to Eqs. ?12? and ?15?,
the free energy is closely related to h?r2?? through
??
?r2?= ?1 − f1??cos ?? −?h
?r2??.
?16?
The sufficient condition to have the Wenzel state is ?F/?r2?
?0 at r2?=r2. According to Eq. ?16?, one has
cos ?? ??h
?r2??r2? = r2??17?
for the Wenzel regime.
On the other hand, the sufficient condition to have the
Cassie-Baxter state is ?F/?r2??0 at r2?=0. Using Eq. ?16?
leads to
cos ?? ??h
?r2??r2? = 0??18?
for the Cassie-Baxter regime.
These two equations indicate that the possible wetting
state is the result of the competition between the energy re-
duction by wetting the surface of the roughness ?cos ??? and
the energy increment due to the increase of the liquid-gas
contact area within the groove ??h/?r2??. When the former
dominates over the latter, one has the Wenzel state. On the
contrary, the Cassie-Baxter model is the stable state as the
latter is dominant. For a given pore shape, Eqs. ?17? and ?18?
provide the boundaries in the phase diagram. The overlapped
domain between the Wenzel and Cassie-Baxter regimes rep-
resents the regime with two stable states. Conversely, the
domain, which is neither the Wenzel or Cassie-Baxter re-
gimes, denotes the stable state with partially wetted rough-
ness.
IV. EXAMPLES OF PHASE DIAGRAM
The phase diagram is determined by the liquid-gas con-
tact area within the grooves h?r2??, which is a function of the
geometrical characteristic of the roughness. In general, the
shape of the pore can simply be classified into linear ?e.g.,
circular cone?, convex ?e.g., hemisphere?, or concave ?e.g.,
revolution of asteroid? functions. In order to demonstrate the
three regimes, we consider an example for each case.
A. Single stable state: Wenzel or Cassie-Baxter
We consider a smooth surface with linear pores, which
are modeled as circular cones. The radius of the opening is a
and the depth is h. We assume that r1=1 and the area fraction
occupied by pores is 1−f1. The total second type surface
area is represented by the roughness factor r2=?al/?a2?1,
where l=?a2+h2?1/2. When the liquid-gas contact area in the
pore is depicted by a circle with radius b, the wetted area
within the grooves is r2?=?al?1+b/a??1−b/a?/?a2=?1
−?b/a?2?r2. As a result, the liquid-gas contact area is
h?r2?? =?b
a?
2
= 1 −?r2?
r2?.
?19?
According to Eq. ?12?, the apparent contact angle is
cos ?a= f1cos ? +?r2??cos ?? +1
r2?− 1??1 − f1?,
?20?
which varies linearly with the wetting area r2? with the slope
?1−f1? ?cos ??+1/r2?. Since ??/?r2??0, there exists no ex-
treme value for 0?r2??r2. As a result, only one stable state
?either at r2?=0 or r2?=r2? is possible and the stable state
?minimum free energy? is determined by the value of
FIG. 2. The minimum free energy pathway along the wetting coordinate r2?.
The free energy is scaled by ??l?3V/??2/3. The two-state regime with an
energy barrier is drawn for hemispherical ?convex? pores while the stable
state with partially wetted roughness is obtained from concave pores of
depth 0.5, created by the revolution of y=x1/3about the y axis. The used
parameters are cos ?=cos ??=−0.5, r1=1, and f1=0.4.
234704-4Sheng, Jiang, and TsaoJ. Chem. Phys. 127, 234704 ?2007?
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?r2cos ??+1?. As r2cos ???−1, r2?=0 yields the lowest free
energy ?Cassie-Baxter state?. On the contrary, for r2cos ??
?−1, r2?=r2corresponds to the largest apparent contact angle
?Wenzel state?. That is, even for hydrophobic pores ?cos ??
?0?, the pores can be completely wetted as long as the pore
is shallow enough ?small l?. If cos ??=−1/r2, the apparent
contact angle maintains the same regardless of the wetting
area. This consequence indicates that for a given cos ???0,
the Cassie-Baxter state can be achieved by increasing r2. For
example, increasing the depth of the pore ?h? large enough
leads to the formation of air pockets. Nevertheless, the high-
est apparent contact angle that can be achieved depends on
the area fraction of solid surface f1, i.e., cos ?a=f1cos ?
−?1−f1?. Note that for such a geometry with a given r2or
?a,h?, it is impossible to observe the crossover between the
Cassie-Baxter and Wenzel states by external disturbances.
B. Single stable state: partially wetted roughness
When the liquid-gas contact area in the pore declines
fast enough with increasing the wetting area, the minimum
free energy may take place at neither complete nonwetting
nor complete wetting of the pores. In the second case, we
consider a smooth surface with concave pores. They are cre-
ated by the revolution of y=c2/3x1/3about the y axis, as de-
picted in Fig. 3. The depth of the pore is ac. The analysis is
simplified if all lengths are scaled by c, i.e. y=x1/3with depth
a. In terms of roughness factor, the area of the second type of
surface is r2=??1+9a4?3/2−1?/27a6. Since imbibition into all
pores in parallel is a reasonable assumption, the liquid-gas
contact area is
h?r2?? =??1 + ??1 + 9a4?3/2− 1?
??1 −r2?
According to Eq. ?16?, the solution ?0?r2*?r2? satisfy-
ing cos ??=?h/?r2???F/?r2?=0? gives an extremum of the free
energy at
r2??
2/3
− 1?
3/2?27a6.
?21?
r2*= r2− ??sin ???−3− 1?/27a6.
The stability of this state can be examined by ?2?/?r2?2?0
??2F/?r2?2?0? at r2*. Since the second derivative of ? at r2*is
given by
?2?
?r2?2= ?1 − f1??−?2h
?r2?2?= − ?1 − f1?9a6sin3??
?cos ???? 0,
?22?
this extremum corresponds to a minimum of the free energy
and a maximum of the apparent contact angle. This result
indicates the stable, wetting state with partially wetted
roughness. According to Eqs. ?17? and ?18?, the Wenzel and
Cassie-Baxter regimes are cos ???0 and cos ???−3a2/?1
+9a4?1/2, respectively. The wetting state depends mainly on
the contact angle of the pore ?cos ??? and the geometrical
parameter ?a?. The phase diagram is shown in Fig. 3. For
hydrophilic surfaces ?cos ???0?, the Wenzel state is pre-
ferred. For hydrophobic surfaces ?cos ???0?, however, the
Cassie-Baxter state exists only for shallow pores. The rough-
ness becomes partially wetted for deep apertures.
C. Two stable states: Wenzel and Cassie-Baxter
In the third case, we consider square pillars of size 2a,
height H, and spacing 2b arranged in a regular array. The
pillar is based on a square pyramid with slant height l??b? as
illustrated in Fig. 4. This geometry can be regarded as con-
vex pores. Assume that r1=1 and the area fraction associated
with the top surface of the pillar is f1. The area of the second
type of surface is r2=2aH/??a+b?2−a2?+l/b. Assume imbi-
bition into grooves in parallel and then the liquid-gas contact
area is
h?r2?? =?
1,
?1 +2a + b
for x ? H
l?r2? − r2+l
b???1 −b
l?r2? − r2+l
b??, for x ? H,?
?23?
FIG. 3. The phase diagram for concave pores created by the revolution of
y=x1/3about the y axis. Depending on the intrinsic contact angle of the
roughness ?cos ??? and the depth of the pore ?a?, the wetting state may be in
the regime of Wenzel ?W?, Cassie-Baxter ?CB?, or partially wetted rough-
ness ?P?. The two-state regime does not exist for such geometry. The phase
boundaries are determined by Eqs. ?17? and ?18?.
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