Transition between superhydrophobic states on rough surfaces.

Abstract

Surface roughness is known to amplify hydrophobicity. It is observed that, in general, two drop shapes are possible on a given rough surface. These two cases correspond to the Wenzel (liquid wets the grooves of the rough surface) and Cassie (the drop sits on top of the peaks of the rough surface) formulas. Depending on the geometric parameters of the substrate, one of these two cases has lower energy. It is not guaranteed, though, that a drop will always exist in the lower energy state; rather, the state in which a drop will settle depends typically on how the drop is formed. In this paper, we investigate the transition of a drop from one state to another. In particular, we are interested in the transition of a "Cassie drop" to a "Wenzel drop", since it has implications on the design of superhydrophobic rough surfaces. We propose a methodology, based on energy balance, to determine whether a transition from the Cassie to Wenzel case is possible.

Full text

Page 1

T ransition between Superhydrophobic States on R ough
Surfaces
Neelesh A. Patankar
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, B224,
Evanston, Illinois 60208-3111
Received March 15, 2004. In Final Form: May 27, 2004
Surface roughness is known toamplify hydrophobicity. It is observed that, in general, twodrop shapes
are possible on a given rough surface. These twocases correspond tothe Wenzel (liquid wets the grooves
oftheroughsurface) andCassie(thedropsitsontopofthepeaksoftheroughsurface) formulas.Depending
on thegeometricparametersofthesubstrate, oneofthesetwocaseshaslower energy.It isnot guaranteed,
though, that a drop will always exist in thelower energy state; rather, thestatein which a drop will settle
depends typically on how the drop is formed. In this paper, we investigate the transition of a drop from
onestatetoanother. In particular, weareinterestedin thetransition ofa “Cassiedrop” toa “Wenzel drop”,
since it has implications on the design of superhydrophobic rough surfaces. We propose a methodology,
based on energy balance, to determine whether a transition from the Cassie to Wenzel case is possible.
1. Introduction
Surface roughness amplifies hydrophobicity.1,2This is
frequently seeninnature1andhasbeendemonstratedfor
microfabricated rough surfaces.3-6
Roughness-inducedsuperhydrophobicity is considered
a viable option for surface-tension-induced drop motion
inmicrofluidicdevices.Another applicationisinspiredby
superhydrophobic plant leaves. Water drops are almost
spherical on some plant leaves and can easily roll off,
cleaningthesurfaceintheprocess.1Thisisusuallyreferred
to as the Lotus effect. It is brilliantly exhibited by the
leavesoftheLotusplant.Therearenumerousapplications
of artificially prepared “self-cleaning” surfaces.
Dropscanexist inmultipleequilibriumstatesonrough
surfaces.7It is now known that there are typically two
prominent states in which a drop can reside on a given
rough surface4-6(Figure 1). The drop either sits on the
peaks of the rough surface or it wets the grooves (to be
referred to as a wetted contact), depending on how it is
formed. The drop that sits on the peaks has “air pockets”
alongitscontactwiththesubstrate;hence,itwillbetermed
a “composite” contact.
The apparent contact angle of the drop that wets the
grooves, θr
w, is given by Wenzel’s formula8
whereristheratiooftheactualareaofliquid-solidcontact
totheprojected area on thehorizontal planeand θeis the
equilibrium contact angle (which is typically a value
between the advancing and receding contact angles) of
theliquiddropon theflat surface. Wedonot consider the
separate cases of advancing and receding angles on a
surface. Theapparent contact angleof a drop that sits on
the roughness peaks, θr
c, is given by Cassie’s formula9
where φsis the area fraction on the horizontal projected
planeof theliquid-solid contact and rwis theratioof the
actual area tothe projected area of liquid-solid contact.
This will be referred to as the Cassie or composite drop.
If φs) 1, wehaverw) r, in which casetheCassieformula
becomes the Wenzel formula.
A Cassie drop shows significantly less hysteresis10
comparedtoa Wenzel dropduetolow resistancefromthe
airpockets.Hence,itisimperativethatacompositecontact
isachievedfor applicationssuch asself-cleaning surfaces
or surface-tension-induced droplet motion. The rough
surfaceshouldbesuch that it prefers toform a composite
contact rather than have the liquid wet its grooves.
Our recent theoretical analysis11hasshownthat, ofthe
twopossible states of a drop on a rough surface, one has
lower energy (or a lower apparent contact angle). Geo-
metric parameters of the surface roughness determine
whether a Cassie or a Wenzel drop has lower energy. We
verified our predictions by performing experiments.5On
the basis of this analysis, we can propose that, toensure
theformationofacompositecontact,thesurfaceroughness
parameters should be chosen such that the Cassie drop
(1) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1-8.
(2) Hazlett, R. D. J . Colloid Interface Sci. 1990, 137, 527.
(3) Onda, T.; Shibuichi, N.; Satoh, N.; Tsuji, K. Langmuir 1996, 12,
2125-2127.
(4) Bico, J .; Marzolin, C.; Que ´re ´, D. Europhys. Lett. 1999, 47, 220-
226.
(5) He, B.; Patankar, N. A.; Lee, J . Langmuir 2003, 19, 4999.
(6) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K.
Langmuir 2002, 18, 5818.
(7) J ohnson, R. E.; Dettre, R. H. Adv. Chem. Ser. 1963, 43, 112.
(8) Wenzel, T. N. J . Phys. Colloid Chem. 1949, 53, 1466.
(9) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11.
(10) Lafuma, A.; Que ´re ´, D. Nat. Mater. 2003, 2, 457.
(11) Patankar, N. A. Langmuir 2003, 19, 1249.
cos θr
w) r cos θe
(1)
F igure1. Twostatesofwater dropsonthesameroughsurface.
Theleft sideshowsacompositedrop,whiletheright sideshows
a wetted drop (courtesy of J . Lee and B. He).
cos θr
c) rwφscos θe+ φs- 1 (2)
7097 Langmuir 2004, 20, 7097-7102
10.1021/la049329e CCC: $27.50© 2004 American Chemical Society
Published on Web 07/22/2004

Page 2

has lower energy than the Wenzel drop. This may be
regarded as a conservative design condition.
Previous experiments have shown that even if the
Wenzel drop has lower energy a Cassie drop can be
obtained if, for example, thedrop is formed by depositing
it gently on the rough surface.5Available experimental
data4-6imply that work has to be done in order to
transitiontheCassiedroptoaWenzeldropoflowerenergy.
Various means of doing this have been reported, for
example, depositing thedropfrom someheight,5pushing
thedrop,4orusingaheavydrop.6Thisobservationsuggests
that, in the experiments above, the Cassie and Wenzel
states represent local energy minima separated by an
energy barrier. Due to this barrier, the Cassie drop can
beobserved even if it has higher energy than theWenzel
drop.Inourpreviouswork,webrieflydiscussedthenature
of this energy barrier for the particular roughness
geometry of square pillars in a periodic array.11In this
paper,wewill considerthisissueindetail.Wewill consider
the effect of various surface geometries on the nature of
theenergy barrier betweentheCassieandWenzel states.
This will leadtoa methodology, basedon energy balance,
to determine whether a transition from the Cassie to
Wenzel case is possible.
In this paper, we will formally present the theoretical
arguments to predict transition between the superhy-
drophobic states, namely, the Cassie and Wenzel states.
The theoretical arguments will be applied to the experi-
mental results reported in the literature. In particular,
we will consider the experiments of Yoshimitsu et al.6
2. E nergy Balance to Determine the Condition
for T ransition
For the applications of our interest, the objective is to
havea Cassiedropor a compositecontact. Hence, wewill
focus on the transition of a Cassie drop toa Wenzel drop
and on how that could be avoided. One underlying
assumption in this work is that the drop size is much
bigger than the size of the surface roughness. This
assumption allows us to use the Cassie and Wenzel
formulas for the apparent contact angles. We will first
consider,insection2.1,aroughnessmadeofsquarepillars
in a periodic array. In section 2.2, we will consider the
implications of other roughness geometries.
2.1. Surface R oughness of Periodically Placed
Pillars. Let the rough surface be made of square pillars
ofsizea ×a,height H,andspacingbarrangedinaregular
array (Figure2). Thecontact angle, θe, on theflat surface
is assumedgiven. Thegeometricparameters in eqs 1 and
2 are
ConsideraCassiedroponaroughsubstrate.Theenergy,
Gc, of the drop is represented by
whereScistheareaoftheliquid-vapor contact ofaCassie
drop, Acisthearea ofcontact withthesubstrateprojected
on the horizontal plane, σlvis the liquid-vapor surface
energy per unit area (or surfacetension), and weassume
gravity tobe negligible. In our discussion, the liquid can
be considered tobe water and the vapor is air. If gravity
isnegligible,thenthedropcanbeassumedtobeaspherical
cap. Substituting appropriate expressions for Scand Ac,
eq 4 becomes11
where V is the drop volume. The left-hand side denotes
nondimensional energy.
IftheCassiedroptransitionstoa Wenzel drop, thenew
energy, Gw, of the drop is given by eq 5 but with θr
replaced by θr
hand side of eq 5 is a monotonically increasing function
of θ for 0° e θ e 180°. Hence, if θr
drop will have higher energy than the Wenzel drop and
vice versa. The liquid-vapor surface area, Sw, and the
projected area of contact, Aw, of the Wenzel drop are
different from those of the Cassie drop.
We consider the case where θr
drop is at higher energy. This implies Ac< Awfor a drop
of given volume, V.11We ask the following question: Is
it necessary that a Cassie drop that has higher energy
will always transition toa Wenzel drop at lower energy?
In an attempt to answer this question, we will first
hypothesize how the energy of the drop might change if
it transitions from a Cassie to a Wenzel drop.
Consider a microscopicview of thecontact between the
dropandtheroughsurface.For aCassiedrop,theliquid-
vapor interface hangs from corner tocorner at the top of
the grooves (Figure 3a). This interface does have a
curvatureof theorder of theradius of thedrop, but it can
beignoredat thescaleofthesurfaceroughness(notethat
wehaveassumedthedropsizetobemuch larger than the
roughnesssize).Theaboveconclusionsareconsistent with
the assumption of a rounded corner with a very small
radius of curvature.12The local contact angle of the
interface at the corner is equal to θe to satisfy local
equilibrium.12It can be easily verified that any hydro-
phobic (>90°) contact angle can find an equilibrium
position at some point on a 90° corner.12
The actual details of the transition from composite to
wetted contact are not well understood. Different pos-
c
w.11It can be easily verified that the right-
c> θr
w, then the Cassie
c> θr
w, that is, the Cassie
(12) Oliver, J . F.; Huh, C.; Mason, S. G. J . Colloid Interface Sci.
1977, 59, 568.
rw) 1
φs)
1
((b/a) + 1)2
r )(1 + 4φsH
a)}
(3)
Gc) Scσlv- cos θr
cAc
(4)
F igure 2. Side and top views of a roughness geometry of
periodically placedpillars (filledblack) of squarecross section.
The pillar cross-sectional size is a × a. In the figure, we show
one period.
Gc
?
39πV2/3σlv
) (1 - cos θr
c)2/3(2 + cos θr
c)1/3
(5)
7098 Langmuir, Vol. 20, No. 17, 2004 Patankar

Page 3

sibilities may be hypothesized. Consider a possibility
where the liquid-vapor interface, initially at the top of
the groove, starts moving down; that is, it starts wetting
the sides of the pillars (Figure 3b). The transition is in
general a nonequilibrium process wherethelocal contact
angle is not necessarily equal to the equilibrium value.
The surface energy will change during transition. As-
suming that the filling of the grooves happens below the
projected area, Ac, of the Cassie drop, the energy at an
intermediate stage is given by
where y is the location of the interface from the bottom
of thegroove(Figure3b) andGyis thedropenergy at that
state. As the liquid fills the grooves, Scand Acshould
changebecausesomevolumeismovingoutofthespherical
liquidcapabovethesubstrate.Thisinturnshouldchange
the energy, Gc, but we will assume these changes to be
negligible compared to the surface energy changes (the
second expression on the right-hand side of eq 6) due to
the wetting of the grooves. This assumption is justified
because the surface area per unit volume is much larger
in the grooves compared to the spherical liquid cap. Gc
and Acare therefore assumed constants in eq 6.
Equation 6 implies that the energy of the drop at
intermediate states is larger than Gcfor θe> 90°. The
maximum value is reached at y ) 0 when the liquid has
filledthegroovesbuttheliquid-solidcontactatthebottom
of the valley is yet to be formed (Figure 3c). When the
liquid wets the bottom of the valley, the corresponding
change in energy is given by -(1 + cos θe)σlv(1 - φs)Ac;
that is, the energy of the system decreases. The liquid
would then proceed towet a greater area of thesubstrate
(Aw> Ac) to eventually reach the equilibrium shape of a
Wenzel drop at the energy Gw. Since we are assuming θr
> θr
Themaximum energy state(Gyat y ) 0) among all the
intermediate states can be used toobtain an estimate of
the barrier energy for the transition of a Cassie drop to
a Wenzel drop, GB1.
c
w, we have Gc> Gw, as argued above.
Even if the Cassie drop, on a surface roughness made of
pillars, istransitioning toa Wenzel stateat lower energy,
it has to go through a higher intermediate energy state.
Hence, energy must be provided to the drop to enable
transition.Thetransition can beenabled, for example, by
depositing the drop from some height,5by pushing the
drop,4or simply due to its own weight.6
A different estimate for the energy barrier can be
obtained by considering the Wenzel drop but with the
liquid-solid contact yet tobeformed at thebottom of the
valleys. The barrier energy of this state, GB2, is given by
The actual wetting of the grooves may not occur as
hypothesizedhere.It isquitepossiblethat theliquiddoes
not wet the grooves all at once, as supposed above, but
doessoinparts.Ifthat isthecase,thentheenergy barrier
will be less than that estimated in eqs 7 and 8. Another
way of looking at this is the following. At the end of the
transition, there is a net decrease in the energy of the
drop equal to Gc- Gw(since we are considering a lower
energy Wenzel state). We could then assume that part of
this energy is available to overcome the energy barrier
estimated above.
Wehaveassumedthatgravitydoesnotplayasignificant
role in determining the shape of the drop either in the
Cassie state or in the Wenzel state. This assumption is
reasonable if the drop radius, R, is ,acap, where acap)
(σlv/Fg)1/2isthecapillary length ofa liquidofdensity Fand
g is the gravitational acceleration. For water, acap) 2.7
mm; a spherical water drop of radius acapweighs 82 mg.
Thus, the water drop should be smaller than 82 mg for
thegravityeffectstobeoflittlesignificanceindetermining
theshapesoftheCassieandWenzel states.Wecanrestate
the above condition in terms of a nondimensional pa-
rameter. We define the Bond number as Bo ) (V/Vcap)2/3
) (m/mcap)2/3, where m is the mass of the drop and mcapis
the mass of a drop of radius acap. V’s denote the corre-
sponding volumes. Bo denotes the square of the ratio of
the length scale of the drop tothe capillary length scale.
The drop shape is almost spherical if Bo , 1. Even if
gravity playsaninsignificant roleindeterminingthedrop
shape, it canplay animportant roleinthetransitionfrom
a Cassie to a Wenzel drop. This will be discussed next.
At small Bovalues, the effect of gravity is tomake the
drop shape only slightly nonspherical. This lowers the
center of mass of the drop by some height, δ, compared
toa perfectly spherical drop. Thecorresponding decrease
in the potential energy of a drop, that has a superhydro-
phobic contact with the surface, can be estimated as13
where g is the gravitational acceleration.
Next, we estimate the potential energy change when a
droptransitionsfromaCassietoaWenzel state.Sincewe
are considering a lower energy Wenzel state, which has
a lower apparent contact angle, the center of mass of the
Wenzel drop is lower. The change in the gravitational
potential energy during transition is given by
(13) Mahadevan, L.; Pomeau, Y. Phys. Fluids 1999, 11, 2449.
F igure 3. (a) Side view of the liquid on top of the pillars. The
liquid interface hangs from edge to edge of the pillars. (b) An
intermediatestatein which theliquid is entering thevalley as
a drop transition from a Cassie to a Wenzel state. (c) An
intermediate state during transition where the liquid has not
yet wetted the bottom of the valleys.
Gy) Gc-(1 -y
H)(r - 1)cos θeσlvAc
(6)
GB1) Gy)0) Gc- (r - 1) cos θeσlvAc
(7)
GB2) Gw+ (1 - φs)(1 + cos θe)σlvAw
(8)
mgδ ∼(mg)2
σlv
(9)
Transition between Superhydrophobic StatesLangmuir, Vol. 20, No. 17, 2004 7099

Page 4

where Hcand Hware the heights of the center of mass of
the Cassie and Wenzel drops (assumed to be spherical),
respectively. Rcand Rware the radii of curvature of the
Cassie and Wenzel drops, respectively, and V is the drop
volume. The R cos θ term in eq 10 is dominant. Thescale
of mg(Hc- Hw) is at least mgV1/3.
Using eqs 9 and10, weconcludethat mgδ/mg(Hc- Hw)
∼Bo,1;that is,thechangeinthegravitational potential
energy due to the transition is much more compared to
mgδ. Thus, gravity is important for transition even if it
doesnot significantly changethespherical shapeofadrop.
All or part of the gravitational potential energy during
transitionmaybeavailabletoovercometheenergybarrier.
Weformulatevarious conditions for thetransition of a
Cassie drop to a Wenzel drop. These are listed below.
(1) The barrier energy is GB1, and the source of energy
to overcome the energy barrier is gravitational:
(2) The barrier energy is GB1, and the source of energy
is gravitational and Gc- Gw:
(3) The barrier energy is GB2, and the source of energy
is gravitational:
(4) The barrier energy is GB2, and the source of energy
is gravitational and Gc- Gw:
GB1-GcandGB2-Gcaregivenbyeqs7and8,respectively.
Thebasearea ofa CassiedropisAc)πRc
of a Wenzel dropis Aw) πRw
given in eq10. Gc- Gwcan befound using eq5. In all the
criteria, above, we have included the gravitational term.
We could have set up a criterion where the only gain in
energy comes from Gc- Gw. In that case, the transition
would be independent of the drop mass, a prediction
contrary to the observations of Yoshimitsu et al.6(to be
discussedbelow).Therefore,wedidnotsetupsuchcriteria.
In setting up the above criteria, we are not quantifying
the viscous dissipation that will consume some energy
during the transition process.
Totest the above criteria, we will consider the experi-
ments of Yoshimitsu et al.6They studied the effect of
surfacegeometry onthehydrophobicstatesofdrops.They
fabricated surfaces that had a roughness geometry of
periodically placed square pillars. In their experiments,
θe) 114°. Thereweretwocases that areof interest tous:
(A) They considered a 1 mg drop on substrates with
pillar dimensionsgivenby a )50µmandb)100µm.The
height of thepillars was changed by fabricating different
surfaces. They reported that a pillar height of 35 µm was
2sin2θr
c, andthat
2sin2θr
w, whereRcandRware
required to obtain a Cassie drop. For lesser heights, a
Wenzel drop was formed.
(B) In another set of experiments, they considered a
substrate with a ) 50 µm and b ) 50 µm. Two pillar
heights,14µmand53µm,wereconsidered.They observed
that all drops (they haddrops uptoa maximumof 35 mg)
werein theCassiestateon thesurfacewith a 53µmpillar
height. On the surface with a 14 µm pillar height, they
observed that a 7 mg drop was in the Cassie state while
12 mg and heavier drops were in the Wenzel state.
We will now check the transition criteria (eqs 11-14)
by comparing them with the above observations.
Equations11-14canbeusedtoobtainthecritical mass
for thedrop totransition totheWenzel state. Weplot the
critical mass as a function of H/a with the rest of the
parameters given. If the mass of the drop is greater than
the critical mass at a given value of H/a, then the theory
predicts transition totheWenzel state; below thecritical
mass, a Cassie drop should be formed. Figure 4 shows a
plot of the critical mass versus H/a for case A where b/a
) 2 (from eq 3, φs) 0.11) and θe) 114°.
All thecriteria show a general trend wherethecritical
mass increases with H/a. As H/a increases, more energy
is required towet the deeper grooves; that is, the energy
barrier is larger. A heavier drop is required toovercome
thelarger energy barrier. This trend continues such that
the critical mass tends to infinity as H/a f 2.92. At H/a
)2.92,wehaveθr
and Cassie drops are the same (Gc) Gw). There is no
change in the gravitational potential energy (Hc) Hw).
Thus, thereis nosourceof energy toovercometheenergy
barrier whichisnonzero.A transitionisthusnot possible.
For H/a > 2.92, the Cassie drop is at lower energy. The
center of mass of the Wenzel state is higher due to its
c)θr
w;that is,theenergiesoftheWenzel
mg(Hc- Hw) ) mg{
π
4V(Rc
(Rccos θr
3V
c)2(2 + cos θr
3V
π(1 - cos θr
4sin4θr
c- Rw
4sin4θr
w) -
c- Rwcos θr
c)}
w)}
w)}
Rc){
Rw){
π(1 - cos θr
1/3
w)2(2 + cos θr
1/3 }
(10)
mg(Hc- Hw) g GB1- Gc
(11)
mg(Hc- Hw) + Gc- Gwg GB1- Gc
(12)
mg(Hc- Hw) g GB2- Gc
(13)
mg(Hc- Hw) + Gc- Gwg GB2- Gc
(14)
F igure 4. Plots of thetheoretical estimateof thecritical mass
(in milligrams) of a drop, for the transition from a Cassie toa
Wenzel state, as a function of H/a for caseA oftheexperiments
of Yoshimitsu et al.6(a) Plots according to the four criteria
hypothesized for transition. (b) A close-in view of the plot in
part a along with the experimental data points.
7100Langmuir, Vol. 20, No. 17, 2004Patankar

Page 5

higher apparent contact angle. Both mg(Hc- Hw) and Gc
- Gware negative (i.e., the respective energies increase)
and cannot act as the source of energy to overcome the
barrier. Once again, transition tothe Wenzel state is not
possible unless there is a different energy source. There-
fore, in Figure 4, we show values of H/a only up to 2.92
(not including 2.92).
Thetheory hasassumedthat gravity doesnot influence
the Wenzel and Cassie drop shapes significantly. As
argued above, only those results in Figure 4 where Bo,
1, that is, where the critical mass, m, is ,mcap()82 mg),
are consistent with this assumption.
Figure 4a shows the trend of theoretical predictions
based on all criteria. Thecritical mass based on criterion
3 is much larger than any other criteria. The values are
also much higher than the experimental observations.
Criterion 3, thus, does not appear realistic. This is not
surprising,sincetheenergy barrierisbasedontheWenzel
drop. Criterion 3 is based on the contact area Awwhich
islarger thanAcfor theparametersbeingconsideredhere.
This results in a larger value for the energy barrier.
Figure4bshows a close-in view oftheplot in Figure4a.
The critical mass predicted by criteria 2 and 4 is zeroup
to a certain value of H/a. This is because Gc - Gw is
sufficient to overcome the energy barrier. In criterion 1,
we assume that Gc- Gwis not available for transition;
hence, thecritical mass is zeroonly at H/a ) 0. Figure4b
alsoshowstwoexperimental datapointsfromYoshimitsu
et al.,6both of which aredrops with a mass equal to1 mg.
A Wenzel drop was formed on a substratewith H/a ) 0.2
(H ) 10 µm), and a Cassie drop was formed when H/a )
0.72 (H ) 36 µm). Thus, a mass of 1 mg was claimed to
be critical for H/a ) 0.7 (H ) 35 µm).
Criterion 2 predicts a Wenzel drop for both the data
points.It underestimatesthecritical mass.Criteria1and
4 appear to provide the most reasonable predictions,
criterion 1 being in best agreement with theobservations
of Yoshimitsu et al.6
In Figure 5, we compare the observations in case B
with the theoretical predictions. Here, b/a ) 1 (from eq
3, φs)0.25) andθe)114°. For this case, thecritical mass
tends to infinity as H/a f 1.09. At H/a ) 1.09, we have
θr
veryhigh;thatis,aCassiedropshouldbeformedaccording
to all criteria. This is in agreement with the data of
Yoshimitsuet al.6whoobservedaCassiedropfor all cases.
Their maximum dropmass was 35 mg. For H/a ) 0.28 (H
) 14 µm), a 7 mg drop was in the Cassie state while a 12
mg drop was in the Wenzel state. Both these data points
are shown in Figure 5. While none of the criteria predict
this exactly, once again criterion 1 is closest to the
observation.
c) θr
w. For H/a ) 1.06 (H ) 53 µm), the critical mass is
Very littleisunderstoodoftheactual transitionprocess,
but the above comparisons suggest that criterion 1
probably provides the best predictions. This conclusion
is, however, basedon limiteddata. Moreexperiments are
desirable to understand the transition better. The main
conclusion from the above analysis is that, even if the
Wenzel state is at a lower energy, it does not necessarily
mean that a drop will transition to that state. It will do
soonly if it can overcome the energy barrier. Criterion 1
may beusedtopredict whether thetransition is possible.
Thisinformationisuseful indesigningsuperhydrophobic
self-cleaning surfaces where a Cassie drop is more
desirable since it has less hysteresis.
Analysessuchastheoneabovecanbeenvisagedfor the
cases of pushing the drop4and releasing the drop from
some height5to achieve transition. Similarly, different
geometries can also be analyzed.
2.2.Other Surface R oughness Geometries.Most of
the experimental data on the Wenzel and Cassie states
on fabricatedsurfaces haveconsidereda pillar geometry.4-6
As discussed in the previous section, in this case, there
isanenergy barrier betweentheCassieandWenzel states
for hydrophobic contact (θe > 90°). This is argued
theoretically andevidencedby experimental observation.
We now ask the question if an energy barrier will exist
between the Cassie and Wenzel states for any surface
geometry.
Specifically, wewill onceagain consider a Cassiestate
with higher energy compared to the Wenzel state. We
wishtoexploreifthesetwostatesarenecessarilyseparated
byanenergybarrierforanysurfacegeometry.Ifanenergy
barrier is not present, then a Cassie drop will always
transitiontothelower energy Wenzel state.It will become
evident from the discussion below that there can be
examples (in addition to the particular case considered
below)where,infact,anenergybarriermaynotbepresent.
This is consistent with the conclusions of Marmur.14The
methodology of Marmur14can be used to analyze the
energy change for different roughness geometries.
Figure6showsaparticular geometry wherethesurface
roughness is similar tothat in section 2.1 except that the
pillars now have an inclined surface instead of vertical
faces. The geometric parameters are as defined in the
figure. Note that a Cassie drop will be formed only if a
liquid-vaporinterfacecanhangfrompillar-to-pillar.This
interface can find equilibrium at the top corners of the
pillars only if θse θee 180°.12For other values of θe, a
Cassie drop is not possible and the valley will always be
filled. In thefollowing discussion, wewill consider a case
(14) Marmur, A. Langmuir 2003, 19, 8343.
F igure 5. Plots of thetheoretical estimateof thecritical mass
(in milligrams) of a drop, for the transition from a Cassie toa
Wenzel state, as a function of H/a for caseB oftheexperiments
of Yoshimitsu et al.6
F igure 6. Side and top views of a roughness geometry of
periodically placed pillars with inclined side walls.
Transition between Superhydrophobic StatesLangmuir, Vol. 20, No. 17, 20047101