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The rigorous derivation of Young, Cassie–Baxter and Wenzel
equations and the analysis of the contact angle hysteresis phenomenon
Gene Whyman, Edward Bormashenko
*
, Tamir Stein
The Ariel University Centre of Samaria, P.O. Box 3, Ariel 440700, Israel
Received 28 August 2007; in final form 13 November 2007
Available online 19 November 2007
Abstract
The rigorous derivation of Young, Cassie–Baxter and Wenzel equations carried out in the framework of the unified thermodynamic
approach is presented. Wetting of rough surfaces controlled with external stimuli is treated. Areas of validity of Cassie–Baxter and Wen-
zel approaches are discussed. General properties of the contact angle hysteresis are investigated on the same thermodynamic basis.
2007 Elsevier B.V. All rights reserved.
1. Introduction
Wetting phenomena are essential and ubiquitous in a
variety of natural and technological processes. Wetting of
solids has been exerted to intensive experimental and theo-
retical activities in two past decades [1–4]. An interest to
the wetting has been also increased due to the superhydro-
phobicity phenomenon, revealed recently in a variety of
natural and artificial objects [5–10].
Wetting of smooth and rough solids is governed by
Young, Wenzel and Cassie–Baxter equations [1,2,10–13].
The Young equation results from the equilibrium of forces
acting on the triple line [1]. Wenzel and Cassie–Baxter
equations supplying the value of apparent contact angles
inherent to rough surfaces for a long time were based on
the semi-empirical arguments. The rigorous derivation of
these equations has been carried out recently basing on
the principle of virtual work and concepts supplied by non-
extensive thermodynamics [14,15]. The factor complicating
the understanding of wetting phe nomena is the effect of the
hysteresis of the contact angle; actually at the same (even
smooth) surface co-exist various contact angles, the mini-
mal and maximal of which are called receding and advan c-
ing contact angles respectively [16–24].
The profound microscopic understanding of the phe-
nomenon of the contact angle hysteresis could be picked
up from the recent work by Yaminsky [18]. According to
Yaminsky the phenomenon of the contact angle hysteresis
resembles the well-known effect of the static friction, when
the triple line is pinned to the solid substrate due to the
long-range forces or roughness of the surface [18]. Thus
the spectrum of contact angles becomes possibl e [1,16–24].
However the drop when deposited on the surface dem-
onstrates usually contact angles dictated by Young, Wenzel
or Cassie–Bax ter equations; thus the open question is: why
the drop ‘selects’ certain contact angle from the spectrum
of possible ones. We demonstrate in our communication
that contact angles derived from Young, Wenzel or
Cassie–Baxter equations supply minimum to the free
energy of the drop. Thus one more rigorous derivation of
these equations based on pure thermodynamic arguments
becomes possible.
2. Derivation of the Young formula
Let us start with the drop of the radius R deposited on
the ideally flat surface. If h denotes contact angle we have
the volume V and surface S of the liquid–air interface
expressed as (Fig. 1)
V ¼
pR
3
3
ð1 cos hÞ
2
ð2 þ cos hÞ; ð1Þ
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2007.11.033
*
Corresponding author. Fax: +972 3 9066621.
E-mail address: Edward@ariel.ac.il (E. Bormashenko).
www.elsevier.com/locate/cplett
Available online at www.sciencedirect.com
Chemical Physics Letters 450 (2008) 355–359
Author's personal copy
S ¼ 2pR
2
ð1 cos hÞ: ð2Þ
The Gibbs free energy of the drop is expressed by Eq. (3)
(within an additive constant)
G ¼ cS þ pðR sin hÞ
2
ðc
SL
c
SV
Þ¼cS p ðR sin hÞ
2
a; ð3Þ
where c, c
SL
, c
SV
are the surface tensions at the liquid/
vapor, solid/liquid and solid/vapor interfaces, respectively,
S is the spherical liquid/air interface area. Having in mind
further applications we have introduced here the constant
a, which in the special case of flat homogeneous substrates
is defined as
a ¼ c
SV
c
SL
: ð4Þ
The general form of the dependence in the right-hand part
of Eq. (3) remains true in many other cases of physical
interest.
Now we will make the main assumption of our treat-
ment, namely we assume the constant volume of the drop ,
V = constant. Substitution of formulae (1) and (2) into Eq.
(3) yields
G ¼
9pV
2
ð1 cos hÞð2 þ cos hÞ
2
"#
1=3
ð2c að1 þ cos hÞÞ: ð5Þ
Now G is a function of only one independent variable h
that is the contact angle.
The straightforward differentiation gives
dG
dh
¼
9V
2
p
ð1 cos hÞ
4
ð2 þ cos hÞ
5
"#
1
3
2ða c cos hÞ sin h: ð6Þ
It is clear that
dG
dh
ðh ¼ h
E
Þ¼0 is fulfilled when
a ¼ c cos h
E
; or ð7Þ
cos h
E
¼
c
SV
c
SL
c
: ð8Þ
Thus the well-known Young equation for the equilibrium
contact angle on flat homogeneous surfaces is obtained.
3. The Cassie–Baxter and Wenzel equations
Now consider, following Wenzel [1] , the rough surface
(Fig. 1B) characterized by the roughness f > 1, i.e. by the
ratio of the real surface in contact with liquid to its projec-
tion onto the horizontal plane. This means that the area of
liquid–solid interface is equal to p(Rsin h)
2
f and we have for
the free energy equation (3) with
a ¼ f ðc
SV
c
SL
Þ: ð9Þ
From Eq. (7) then follows the Wenzel equation for the
equilibrium contact angle h
*
called the apparent contact
angle
cos h
¼
a
c
¼ f
ðc
SV
c
SL
Þ
c
¼ f cos h
E
: ð10Þ
Let suppose that the surface under the drop is flat but con-
sists of n sorts of materials randomly distributed over the
substrate. Each material is characterized by its own surface
tension coefficients c
i, SL
and c
i, SV
, and by the fraction U
i
in
the substrate surface, U
1
+ U
2
+ + U
n
= 1. Analogously
to the above treatment we sim ply put in Eq. (3)
a ¼
X
n
1
U
i
ðc
i;SV
c
i;SL
Þð11Þ
and obtain the Cassie–Baxter apparent contact angle
c cos h
¼
X
n
1
U
i
ðc
i;SV
c
i;SL
Þ: ð12Þ
The most frequently n =2
c cos h
¼ U
1
ðc
1;SV
c
1;SL
Þþð1 U
1
Þðc
2;SV
c
2;SL
Þ: ð13Þ
A more special case is important when air remains trapped
under the drop (Fig. 1C). If the air surface fraction under
the drop is 1 U, then c
2,SV
= 0 and c
2,SL
= c (S ” V)
a ¼ c cos h
¼ Uðc
SV
c
SL
Þð1 UÞc: ð14Þ
Let find the equilibrium surface energy of the drop on the
substrate. Substituting Eq. (7) in the form a = c cosh
*
into
Eq. (5) one obtains
G
0
¼ c½ð3V Þ
2
pð1 cos h
Þ
2
ð2 þ cos h
Þ
1=3
ð15Þ
The graph of G
0
/c as a function of cos h
*
= a/c is presented
in Fig. 2 . The value of G
0
at h
*
= p is equal to ð36V
2
pÞ
1=3
c ¼ 4pR
2
0
c that is the surface energy of the spherical drop
of the radius R
0
which is in contact with vapor only (before
depositing on a substrate). The parameter a is defined by
the geometry of the substrate and by the surface tension
coefficients. Since G
0
is a decreasing function of a, a system
with flexible parameters will tend to increase a. Practically
the wetting flexibility of the system could be achieved with
external stimuli, such as temperature, external fields or
vibration of the drop; this was already realized in the elec-
trowetting or the recently reported vibration experiments
[25,26]. The additional restriction is given by Eq. (12):
Fig. 1. Schemes of different wetting regimes. A – flat substrate; B – rough substrate, the Wenzel regime; C – rough substrate with air trapped under the
drop, the Cassie–Baxter regime.
356 G. Whyman et al. / Chemical Physics Letters 450 (2008) 355–359
Author's personal copy
a 6 c. For example, if the fraction U of wetted surface of
the system described by Eq. (13) may change, the system
will proceed to the state with a = min(c, c
SV
c
SL
) with
the maximal possible U supplying maximum to a in Eq.
(14) (usually c
SV
c
SL
> 0). If c > c
SV
c
SL
then
a = c
SV
c
SL
, U = 1 that means that air is completely re-
moved from under the drop and the contact angle receives
its (finite) Young value. In the opposite case c < c
SV
c
SL
,
a = c, h
*
= 0 and the substrate is completely wetted by the
liquid but air pockets of total area 1 U remain trapped
under the liquid film:
1 U ¼
c
SV
c
SL
c
c
SV
c
SL
þ c
: ð16Þ
The similar techni que based on Eqs. (9)–(14) may be ap-
plied for the analysis of wetting the surfaces of a compli-
cated structure, superhydrophobicity, etc.
It has to be emphasized that the derivation of the Cassie–
Baxter and Wenzel eq uations presented in our communica-
tion suffers from the same shortc oming like those carried
out by Bico et al. based on the virtual work principle
[14]. Both our approach and application of the virtual
work principle presumes the free displacement of the triple
line, however in the situation depicted in Fig. 3 when water
fills the grooves of the surface relief, state A is statically
impossible. The same is true for the situation of heteroge-
neous wetting, when air pock ets are formed (see Fig. 3D
and E). In this case the equilib rium of the triple line
becomes possible only when the drop sits on solid islands
only (see Fig. 3F). The equilibrium in states D and E is
impossible. The drop can sit on the air pocket but the triple
line cannot. Thus free displacement of the triple line
becomes impossible and both application the virtual work
principle and variation of contact angle are at least prob-
lematic. In this situation the detailed study of the fine struc-
ture of triple line, taking into account formation of
precursor film surrounding the drop is necessary [27,28].
On the other hand the approach presented in our com-
munication explains why the drop ‘selects’ certain contact
angle from the variety of angles, permitted by the contact
angle hysteresis phenomenon. The equilibrium contact
angle supplies minimum value to the free energy of the
drop.
4. Contact angle hysteresis
Actually, the experimentally established apparent contact
angle h
*
is confined within a certain range h
rec
< h
*
< h
adv
,
where h
adv
and h
rec
are the so-called advancing and reced-
ing contact angles [1,16–21]. The difference between h
adv
and h
rec
is called the contact angle hysteresis (CAH, see
Section 1). Some general properties of this phenomenon
may be understood on the basis of the above used simple
thermodynamic model.
The straightforward differentiation of Eq. (6) immedi-
ately yields
d
2
G
dh
2
h¼h
E
¼
9V
2
p
ð1 cos h Þ
4
ð2 þ cos h Þ
5
"#
1
3
2c sin
2
h: ð17Þ
The increase in the energy in Eq. (5) is for small variations
dh of the contact angle
d
2
E ¼
1
2
d
2
G
dh
2
h¼h
E
dh
2
: ð18Þ
Now we apply the approach proposed by Shanahan for
consideration of water drop evaporation [22]. It has to be
emphasized that the CAH has been observed experimen-
tally in two different experimental situations, namely, when
the contact line is pinned [18,23,24] and when contact line
moves [22,26]. We will restrict our treatment with the CAH
due to displacement of triple line. We also suppose that this
displacement is slow, thus effects due to ‘dynamic contact
angle phenomena’ are neglected [1]. Let U be the height
0
0.5
1
1.5
2
2.5
-1.0 -0.5 0.0 0.5 1.0
cosθ*
G
0
/γ (3V)
2/3
0
0.5
1
1.5
2
2.5
h (θ*)
Fig. 2. The equilibrium surface energy of the sessile drop (solid line) and
geometrical factor h(h) of the contact angle hysteresis (dashed line) versus
the apparent contact angle.
Fig. 3. Various situations of wetting for the Wenzel (A,B,C) and Cassie–Baxter (D, E, F) states.
G. Whyman et al. / Chemical Physics Letters 450 (2008) 355–359 357
Author's personal copy
of the potential barrier connected with the motion of the
triple line along a substrate. This quantity should obviously
be related to the unit length of this line. Thus the move-
ment will start after the energy increase reaches the poten-
tial barrier U
d
2
E
2pR sin h
¼ U : ð19Þ
The natural estimation of the CAH is h
adv
h
rec
=2dh,
thus from this and Eqs. (17)–(19), (1), the final result
follows:
h
adv
h
rec
¼
8U
cR
0
1=2
hðh
Þ;
hðh
Þ¼
ð1 cos h
Þ
1=12
ð2 þ cos h
Þ
2=3
2
1=3
ð1 þ cos h
Þ
1=4
; ð20Þ
where R
0
=(3V/4p)
1/3
is the initial radius of the spherical
drop before deposition on the substrate. The function
h(h) of the equilibrium contact angle is shown in Fig. 2.
Some important qualitative conclusions follow from this
result. First of all, the CAH does not depend explicitly on
the constant a, i.e. wetting constants and roughness. The
larger CAH in the Wenzel state compared to the Cassie
one is connected with the larger value of U in the former
case [29]. It should be emphasized that the result in Eq.
(20) follows from the geometric consideration and physical
information is mostly concentrated in the magnitude of the
potential barrier U. In turn, the value of U is determined by
both adhesion of liquid by solid and roughness of soli d
[20].
In the broad region of values, approximately from 50 to
140, the CAH does not depend on the equilibrium contact
angle h
*
(see Fig. 2, h(h) 1.3). But it becomes more
pronounced for angles close to 180 (so-called superhydro-
phobicity situation). Ultra-low hysteresis reported recently
for superhydrophobic surfaces is due to specific effects con-
nected with hierarchical roughness of the relief that may
also be explained by small U values [7,10].
For low contact angles Eq. (20) predicts very low CAH
(Fig. 2). Probably this is the reason for the lack of CAH
experimental data for water on metals. Fig. 2 also shows
that both the surface energy and the potential barrier tend
to zero for h
*
! 0.
The CAH dependence on the drop volume is very weak,
however for small drops the CAH h
adv
h
rec
becomes lar-
ger than h
*
itself. This supplies one of the possible explana-
tions why in the end of drop evaporation the contact angle
approaches to zero [22]. Another consequence of Eq. (20) is
that the CAH of liquids with a low surface tension should
be larger when compared to those with a high one, all other
conditions being equal, i.e. U 6¼ U(c), h(h) @ constant.
The dimension of U is noteworthy. It is J/m and coin-
cides with the dimension of linear surface tension debated
intensively in the past decade [1,30,31]. Pompe and Her-
minghaus established U as 10
10
–10
11
J/m for water drop-
lets deposited on the quartz glass substrate [30]. These
values yield according to formula (20) very small CAH of
few tenths of a degree. This pr ediction coincides with
experimental findings for small CAH values observed on
the atomically flat silica substrates [32,33]. However, the
experimental data concerning CAH values inherent for
water/flat silica system are contradictory, and CAH as
much as 10 [32] and even 20 have been also reported
[34]. Large values of U corresponding to mentioned above
CAH values U 10
7
–10
6
J/m are perhaps due to con-
taminants, surface roughness or dynamic contact angle
effects. It as also noteworthy that straightforward applica-
tion of formula (20) for calculation of U needs some pre-
cautions, since it neglects various factors stipulating
CAH, including precursor film surrounding the film, heter-
ogeneity of the surface, etc.
Recently, the alternative calculation of the actuat ing
force of the drop motion on super hydrophobic textured
surfaces was done [35], which gave for U the values in
the range of 10
7
–10
5
J/m for CAH of 10–20 in agree-
ment with the result of Eq. (20). Moreover, the depen dence
of this force on the equilibrium contact angle [35] accords
with that of Eq. (20). In particular, the potential barrier
turns to zero for h
*
! 180 as U 1/h
2
(h
*
), h(h
*
) !1
(Fig. 2).
5. Conclusions
Several wetting phenomena have been considered on the
single mathematical basis. The expression for the free
energy of a sessile drop on the solid substrate was used
for derivation of the Young, Wenzel, and Cassie–Baxter
equations describing different wetting processes. Advanta-
ges and shortcomings of the existing wetting models have
been discussed. General consequences are deduced for the
contact angle hysteresis, which grows with the decrease of
the surface tension and drop volume.
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