Journal of Colloid and Interface Science 327 (2008) 477–479
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Journal of Colloid and Interface Science
www.elsevier.com/locate/jcis
Short Communication
Mechanical derivation of the Wenzel and Cassie equations using a statisticalinterpretation of drop dispensation
Miguel A. RodríguezValverde
∗
Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, Campus of Fuentenueva, E18071 Granada, Spain
a r t i c l e i n f o a b s t r a c t
Article history:
Received 16 May 2008Accepted 15 August 2008Available online 3 September 2008
Keywords:
Contact anglesRoughnessHeterogeneityMechanical equilibriumWenzel–Cassie equation
The global mechanical equilibrium condition of a liquid on a rough and chemically heterogeneous surfacewas derived for threedimensional situations from a statistical outlook of dispensation of many dropsand the assumption of local mechanical equilibrium. Unlike the conventional thermodynamic derivationsfrom variational methods, the current proof is based on vector algebra rather than differential geometry.The mechanicsbased derivation becomes less intricate although the minimum energy condition is notestablished. An effective contact angle is computed from the directional sampling of threephase linesafter local drop dispensations. The ﬁnal expression is a combined mechanical version of the Wenzel andCassie equations.
©
2008 Elsevier Inc. All rights reserved.
1. Introduction
One of the main challenges in surface thermodynamics is toquantify meaningfully surface energetics of solids from multiplemeasured contact angles [1]. This variability in contact angle arises
from the nonideal features of surfaces. There is just one measurable contact angle closely related to the concerning solid interfacialenergies (material properties), at least on average. However, unlessvery particular conditions, this angle is hardly experimentally accessible, not even recognizable if the contact line is corrugated.Instead, different effective contact angles have been proposed inthe literature [2] in order to estimate thermodynamically meaningful contact angles. In this paper, inspired in the work of Swainand Lipowsky [3], a new effective contact angle is formulated and
further related to the Wenzel and Cassie angles, i.e. the equilibriumcontact angles on roughhomogeneous and smoothheterogeneoussurfaces [4,5], respectively.
Swain and Lipowsky [3] derived the thermodynamic equilibrium condition for rough and chemically heterogeneous surfacesusing an elegant formalism based on energy minimization, differential geometry and spatial averaging. However, the Wenzel andCassie equations can be intuitively derived [6], although less rigorously. Variational calculus has been frequently used to obtain thethermodynamic equilibrium conditions involved in a solid–liquid–vapour system. In this framework, the Young–Laplace equation describes the shape of the liquid–vapour interface and the Youngequation [7] rises as its “natural” boundary condition due to the
*
Fax: +34 958 24 32 14.
Email address:
marodri@ugr.es.
solid phase contact. The Young equation is locally fulﬁlled provided that the triple contact line is unconstrained, i.e. when theYoung equation becomes a Neumanntype boundary condition.The Young–Laplace equation is indistinctly interpreted either asthe mechanical or thermodynamic equilibrium condition for mostliquid–ﬂuid interfaces (isotropic phases) due to the fast moleculesdiffusion between phases (i.e. chemical equilibrium). However, although Thomas Young srcinally formulated his famous equationfrom mechanistic arguments, the Young equation is not indeedthe mechanical equilibrium condition of a solid–liquid–vapour system because of the absence of diffusion equilibrium at commonsolid–liquid interfaces [8,9]. Both thermodynamic and mechanical
equilibrium conditions will coincide just on an
ideal
solid surface:smooth, chemically homogeneous, chemically inert and rigid surface; where the solid interfacial tensions numerically coincide withthe respective speciﬁc interfacial energies. Hence, on moderatelyrigid solids and at experimental time scales, a metastable mechanical conﬁguration is usually visualized at the threephase line ratherthan a global thermodynamic equilibrium state [10]. Thermodynamic equilibrium state will be truly attained when the wholesolid–liquid interface is accordingly deformed or molecularly arranged, at very long times. This picture stimulates the currentcontroversy about the applicability of the Wenzel and Cassie equations: does the attainable global equilibrium state depend on thesurface conditions underneath the drop or just at its threephasecontact line?From the local mechanical equilibrium condition of a solid–liquid–vapour system, i.e. the local balance of interfacial forces,the current note provides a combined mechanical version of theWenzel and Cassie equations, if there is no trapped gas inside theconcave grooves of solid surface. In order to probe all surface fea
00219797/$ – see front matter
©
2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2008.08.048
478
M.A. RodríguezValverde / Journal of Colloid and Interface Science 327 (2008) 477–479
tures over the entire solid area, a gedanken dispensation of manydrops allowed applying the concerning spatial averaging.
2. Deﬁnitions of contact angle
In the context of Gibbs’ dividing surface, “interfaces” are treatedas mathematical surfaces, continuous and differentiable; thereforetheir normal unit vectors can be deﬁned analytically. In order toavoid misleading interpretations, every angle involved in the threephase contact line of a solid–liquid–vapour system must be conveniently deﬁned (Fig. 1):
(1)
Actual
contact angle,
θ
, is the local angle formed by the normalunit vectors to the solid surface
N
and to the liquid–vapourinterface
N
lv
at a point
s
of the threephase line:cos
θ(
s
)
≡
N
lv
(
s
)
·
N
(
s
).
This angle is also referred to as microscopic contact angle.(2)
Apparent
contact angle,
θ
app
, is the local angle formed by theapparent unit normal of the solid surface (i.e. the vertical direction
e
z
) and the unit normal of the liquid–vapour interface
N
lv
at a point
s
of the threephase line:cos
θ
app
(
s
)
≡
N
lv
(
s
)
·
e
z
.
The apparent or macroscopic contact angle is experimentallyaccessible by optical techniques rather than the actual contactangle.(3) Roughness angle or surface pitch,
α
, is the local angle formedby the vertical direction
e
z
and the unit normal to the solidsurface
N
(
s
)
at a point
s
of the threephase line:cos
α
(
s
)
≡
N
(
s
)
·
e
z
.
(4) Angle
β
is the planar angle that points out about the absenceof coplanarity of the triplet
e
z
,
N
and
N
lv
:cos
β(
s
)
≡
(
ˆ
N
lv
(
s
)
× ˆ
e
z
)
·
(
ˆ
N
(
s
)
× ˆ
e
z
)
 ˆ
N
lv
(
s
)
× ˆ
e
z
 ˆ
N
(
s
)
× ˆ
e
z

.
The mentioned triplet will be coplanar when
β
=
0
,
π
.Applying vector algebra [11], the following identities can be readily
proved:
ˆ
N
lv
· ˆ
N
−
(
ˆ
e
z
· ˆ
N
lv
)(
ˆ
e
z
· ˆ
N
)
=
(
ˆ
N
lv
× ˆ
e
z
)
·
(
ˆ
N
× ˆ
e
z
),
1
−
(
ˆ
e
z
· ˆ
N
lv
)
2
1
−
(
ˆ
e
z
· ˆ
N
)
2
=  ˆ
N
lv
× ˆ
e
z
 ˆ
N
× ˆ
e
z

,
and substituting the abovementioned deﬁnitions into them, thelocal angles involved in the threephase contact line are relatedamong themselves as follows:cos
θ(
s
)
=
cos
θ
app
(
s
)
cos
α
(
s
)
+
sin
θ
app
(
s
)
sin
α
(
s
)
cos
β(
s
).
(1)This expression can be also derived from spherical trigonometrybecause the vectorial triplet
e
z
,
N
and
N
lv
(Fig. 1) draws a sphericaltriangle. Indeed, Eq. (1) is the identity between the surface tensionforce directly projected on the solid surface and the addition of the projections of the horizontal and vertical components of thesurface tension force on the solid surface. It is worth to emphasizethat this equality is not a condition of mechanical equilibrium. Thisexpression can be rewritten as:cos
θ(
s
)
=
cos
θ
app
(
s
)
−
α
(
s
)
cos
2
β(
s
)
2
+
cos
θ
app
(
s
)
+
α
(
s
)
sin
2
β(
s
)
2
Fig.1.
Layout of unit vectors (spherical triangle) and local angles involved in a pointof threephase contact line over a rough heterogeneous surface.
in order to gather the geometrical ﬁndings of Shuttleworth andBailey [12]:
θ
=
θ
app
∓
α
for cylindrical interfaces
(β
=
0
,
π
)
.In addition to the mentioned angles, there is another “angle”known as
intrinsic
contact angle. In fact, this angle is an abstract,unlocated angle. The intrinsic contact angle at a point
(
x
,
y
)
of thesolid surface is deﬁned by the following expression:cos
θ
i
(
x
,
y
)
≡
γ
sv
(
x
,
y
)
−
γ
sl
(
x
,
y
)
γ
lv
,
(2)where
γ
lv
,
γ
sv
and
γ
sl
symbolize the interfacial tensions of eachinterface, respectively. Strictly,
θ
i
is not a geometrical angle because it is deﬁned on the entire solid surface, regardless of theexistence of a threephase contact line. In this paper, the variations in
θ
i
(
x
,
y
)
are considered smooth with a macroscopic lengthscale and so, the longrange interactions acting in the core region are negligible. The assumptions of weak chemical contrast anddifferentiable surface proﬁles (i.e. no sharp edges) allow avoidingcontact line pinning. Otherwise, the contribution from line tensionwas neglected.According to the local mechanical equilibrium condition for unpinned contact lines, once a liquid drop placed on a real surfaceattains a stable conﬁguration (not necessarily static), the actualcontact angle
θ
at a point
s
of the contact line
(
x
c
(
s
),
y
c
(
s
))
mustbe equal to the concerning intrinsic contact angle:cos
θ(
s
)
=
cos
θ
i
x
c
(
s
),
y
c
(
s
)
.
(3)This equality would be the precursor one to the Young equation before the thermodynamic equilibrium conditions were established. Although the interfacial tensions involving the solid phase
γ
si
(
i
=
l
,
v
)
do not generally coincide with the respective speciﬁcinterfacial energies
σ
si
, the local difference
γ
sv
(
x
,
y
)
−
γ
sl
(
x
,
y
)
canbe roughly approximated to
σ
sv
(
x
,
y
)
−
σ
sl
(
x
,
y
)
[9], provided thatthe chemical equilibrium is guaranteed at the liquid–vapour interface [8]. Consequently, Eq. (3) becomes
numerically
equivalent tothe Young equation [7].Substituting Eq. (3) into Eq. (1), the following local equality is
hold over all contact line:cos
θ
i
x
c
(
s
),
y
c
(
s
)
=
cos
θ
app
(
s
)
cos
α
(
s
)
+
sin
θ
app
(
s
)
sin
α
(
s
)
cos
β(
s
).
(4)Marmur derived the twodimensional version of Eq. (4) for roughhomogeneous surfaces by variational methods [13] and from theYoung equation (i.e. using energies instead of tensions). Alike, using their generalized Young equation, Swain and Lipowsky reportedEq. (4) although in differential form (see Eq. (9.1) in [3]).
3. Statistical interpretation of drop dispensation
Experimentally it is not feasible to scan all possible stableconﬁgurations (i.e. metastates) of a system in a given pseudo
M.A. RodríguezValverde / Journal of Colloid and Interface Science 327 (2008) 477–479
479
Fig. 2.
Geometrical interpretation of the statistical dispensation of drops around aﬁxed surface point.
thermodynamic state (e.g. temperature, volume and chemical potential). Instead, in principle, we might dispense a great number of drops, with ﬁxed volume, whose stable contact lines pass over agiven surface point in any direction. The dispensation probabilitywould be the same one because the surface is considered uniformly rough and heterogeneous, everywhere. In practice, a scalelength of inhomogeneities much lower than the drop size is equivalent to evenly distributed inhomogeneities. It is worthy to mention that Swain and Lipowsky used the same statistical approachin their working deﬁnition of effective angle (Eq. (6.8) in [3]). Geometrically, the multiple dispensation of drops (or scanning of dropconﬁgurations) involves the rotation of the unit normal
N
lv
aroundthe vector
N
ﬁxed at a surface point
(
x
c
(
s
),
y
c
(
s
))
, for a given
θ
i
value (Fig. 2). Since the local vector
N
lv
will depend on the localsurface point,
s
, and the rotating angle,
ψ
, the apparent contactangle,
θ
app
, and the
β
angle will also depend on them. However,these two angles are mutually independent.Averaging over all possible conﬁgurations in the considered“ensemble,” i.e. all drops with constant volume having locally anactual angle equal to
θ
i
and which contact lines pass over the local surface point
s
, the mean cosine value of the local apparentcontact angles will be given by:cos
θ
app
(
s
)
≡
12
π
2
π
0
cos
θ
app
(
s
,ψ)
d
ψ.
Otherwise, the mean value of the product sin
θ
app
(
s
,ψ)
cos
β(
s
,ψ)
will be proportional to the product of each mean value:sin
θ
app
(
s
)
cos
β(
s
)
because both variables are uncorrelated. Sincethe
β(
s
,ψ)
angle scans as the rotating angle,
ψ
, the entire range
[
0
,
π
]
then cos
β(
s
)
will be practically zero. Applying these resultsto Eq. (4), the following expression is derived:
cos
θ
app
(
s
)
=
sec
α
(
s
)
cos
θ
i
x
c
(
s
),
y
c
(
s
)
,
(5)which might be interpreted as the local Wenzel equation, i.e.the macroscopic condition of local mechanical equilibrium at thethreephase line regardless of the drop shape.Since a rough heterogeneous surface locally manifests differentchemical and topographic features, the drop dispensation shouldbe extended over the entire surface. Thus, after the concerningspatial averaging,
1
Eq. (5) is rewritten in terms of surfaceaveragedvalues as follows:
cos
θ
app
xy
=
r
cos
θ
i
A
,
(6)where
r
≡
sec
α
xy
is the Wenzel factor and the following property was applied:
f
sec
α
xy
=
sec
α
xy
f
A
.
1
Surface average over the nominal area:
·
xy
≡
1
A
z
·
dxdy
and surface averageover the actual area:
·
A
≡
1
A
·
dA
.
As Swain and Lipowsky suggested [3], an effective measurable an
gle
θ
eff
might be deﬁned as:cos
θ
eff
≡
cos
θ
app
xy
that substituting it into Eq. (6) provides:cos
θ
eff
=
r
cos
θ
i
A
.
(7)The effective measurable angle
θ
eff
represents, of all possible conﬁgurations of a drop on a rough heterogeneous surface, the conﬁguration of global mechanical equilibrium. In fact, Eq. (7) is a uniﬁedmechanical version of the Wenzel and Cassie equations [4,5] forthreedimensional situations. Hence, accordingly, for a smooth heterogeneous surface
(
α
=
0
)
, Eq. (7) is reduced to:cos
θ
eff
=
cos
θ
i
xy
and for a rough homogeneous surface
(θ
i
=
θ
i
(
x
,
y
))
, Eq. (7) drawsto:cos
θ
eff
=
r
cos
θ
i
.
There is no extra term in Eq. (7) related to gravity and root meansquare roughness, as found in Eq. (9.2) of the paper of Swain andLipowsky [3]. Furthermore, this term has been no more reported
in the rest of literature. This term indeed depends on the
β
angleand it does not vanish if a single spatial average
·
xy
is applied.
4. Summary
In this note, the global mechanical equilibrium condition of aliquid on a rough heterogeneous surface was derived for threedimensional situations from a statistical view of dispensation of many drops and the assumption of local mechanical equilibrium.Unlike the derivation of Swain and Lipowsky [3], based on the en
ergy minimization but the same statistical arguments and effectiveangle than the current proof, this is based on vector algebra instead of differential geometry. Moreover, the concerning surfacesaverages were correctly applied from the directional sampling of threephase lines after local drop dispensations. The mechanicalderivation becomes less intricate although the minimum energycondition is not established. The expression (7) is a combined mechanical version of the Wenzel and Cassie equations.
Acknowledgments
This work was supported by the “Ministerio Español de Educación y Ciencia” (contract “Ramón y Cajal” 1808463B750), theEuropean Social Fund (ESF) and the “Junta de Andalucia” (projectFQM02517).
References
[1] A. Marmur, Soft Matter 2 (1) (2006) 12–17.[2] S.D. Iliev, N.C. Pesheva, Langmuir 19 (23) (2003) 9923–9931.[3] P.S. Swain, R. Lipowsky, Langmuir 14 (1998) 6772–6780.[4] R. Wenzel, J. Phys. Chem. 53 (1949) 1466–1467.[5] A. Cassie, Discuss. Faraday Soc. 3 (1948) 11–16.[6] G. McHale, M.I. Newton, Colloids Surf. A Physicochem. Eng. Aspects 206 (2)(2002) 193–201.[7] T. Young, Philos. Trans. R. Soc. 95 (1805) 65–87.[8] A.I. Rusanov, Pure Appl. Chem. 64 (1) (1992) 111–124.[9] P. Roura, J. Fort, J. Colloid Interface Sci. 272 (2) (2004) 420–429.[10] N. Eustathopolous, M. Nicholas, B. Drevet, Fundamental Equations of Wetting,Elsevier Science, Oxford, 1999, Chap. 1, pp. 16–17.[11] J.W. Gibbs, E.B. Wilson, Vector Analysis: A TextBook for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs, YaleUniversity Press, New Haven, 1929.[12] R. Shuttleworth, C. Bailey, Discuss. Faraday Soc. 3 (1948) 16–22.[13] A. Marmur, Adv. Colloid Interface Sci. 50 (1994) 121–141.