Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity (Review Article)
The state-of-art in the field of physics of phenomena occurring at solid/liquid interfaces is presented. The notions of modern physics of wetting are introduced and discussed including: the contact angle hysteresis, disjoining pressure and wetting transitions. The physics of low temperature wetting...
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Bormashenko, Edward 2018-01-18T17:07:51Z 2018-01-18T17:07:51Z 2016 Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity (Review Article) / Edward Bormashenko // Физика низких температур. — 2003. — Т. 42, № 8. — С. 792-808. — Бібліогр.: 84 назв. — англ. 0132-6414 PACS: 68.08.Bc, 68.08.–p https://nasplib.isofts.kiev.ua/handle/123456789/129276 The state-of-art in the field of physics of phenomena occurring at solid/liquid interfaces is presented. The notions of modern physics of wetting are introduced and discussed including: the contact angle hysteresis, disjoining pressure and wetting transitions. The physics of low temperature wetting phenomena is treated. The general variational approach to interfacial problems, based on the application of the transversality conditions to variational problems with free endpoints is presented. It is demonstrated that main equations, predicting contact angles, namely the Young, Wenzel, and Cassie–Baxter equations arise from imposing the transversality conditions on the appropriate variational problem of wetting. Recently discovered effects such as superhydrophobicity, the rose petal effect and the molecular dynamic of capillarity are reviewed. The author is indebted to Dr. Whyman for his longstanding fruitful cooperation in the study of wetting phenomena. His critique and numerous remarks definitely improved the text. I am thankful to Professor R. Pogreb for his contribution in understanding of diversity of wetting phenomena. I want to thank my numerous MSc and PhD students for their research activity and allegiance to a spirit of scientific research. I am grateful to Mrs. Al. Musin for her kind help in editing the review. I am especially indebted to my wife Yelena Bormashenko for her inestimable help in preparing this review. I am greatly thankful to Mrs. Hanna Weiss for her valuable help in English editing of this review. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 75-летию открытия теплового сопротивления Капицы Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity (Review Article) Article published earlier |
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physics of solid–liquid interfaces: from the young equation to the superhydrophobicity (review article) |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The state-of-art in the field of physics of phenomena occurring at solid/liquid interfaces is presented. The notions of modern physics of wetting are introduced and discussed including: the contact angle hysteresis, disjoining pressure and wetting transitions. The physics of low temperature wetting phenomena is treated. The general variational approach to interfacial problems, based on the application of the transversality conditions to variational problems with free endpoints is presented. It is demonstrated that main equations, predicting contact angles, namely the Young, Wenzel, and Cassie–Baxter equations arise from imposing the transversality conditions on the appropriate variational problem of wetting. Recently discovered effects such as superhydrophobicity, the rose petal effect and the molecular dynamic of capillarity are reviewed.
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Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity (Review Article) / Edward Bormashenko // Физика низких температур. — 2003. — Т. 42, № 8. — С. 792-808. — Бібліогр.: 84 назв. — англ. |
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AT bormashenkoedward physicsofsolidliquidinterfacesfromtheyoungequationtothesuperhydrophobicityreviewarticle |
| first_indexed |
2025-11-24T15:49:03Z |
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1850848759928324096 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8, pp. 792–808
Physics of solid–liquid interfaces: from the Young
equation to the superhydrophobicity
(Review Article)
Edward Bormashenko
Ariel University, Natural Science Faculty, Physics Department, P.O.B. 3, 407000, Ariel, Israel
E-mail: edward@ariel.ac.il
Received March 10, 2016, published online June 24, 2016
The state-of-art in the field of physics of phenomena occurring at solid/liquid interfaces is presented. The no-
tions of modern physics of wetting are introduced and discussed including: the contact angle hysteresis, disjoin-
ing pressure and wetting transitions. The physics of low temperature wetting phenomena is treated. The general
variational approach to interfacial problems, based on the application of the transversality conditions to varia-
tional problems with free endpoints is presented. It is demonstrated that main equations, predicting contact an-
gles, namely the Young, Wenzel and Cassie–Baxter equations arise from imposing the transversality conditions
on the appropriate variational problem of wetting. Recently discovered effects such as superhydrophobicity, the
rose petal effect and the molecular dynamic of capillarity are reviewed.
PACS: 68.08.Bc Wetting;
68.08.–p Liquid–solid interfaces.
Keywords: capillarity, interface, superhydrophobicity, rose petal effect, Young equation, wetting.
Contents
1. Introduction .......................................................................................................................................... 793
2. Wetting of ideal surfaces ...................................................................................................................... 793
2.1. Wetting of flat ideal surfaces ........................................................................................................ 793
2.2. Wetting of flat homogeneous curved surfaces .............................................................................. 795
2.3. Considering the line tension ......................................................................................................... 796
2.4. Very thin liquid films and the disjoining pressure ........................................................................ 796
2.5. Considering the influence of absorbed liquid layers and the liquid vapor .................................... 797
2.6. Wetting transitions on ideal surfaces ............................................................................................ 798
3. Wetting of real surfaces ........................................................................................................................ 799
3.1. Contact angle hysteresis ............................................................................................................... 799
3.2. Contact angle hysteresis on smooth homogeneous substrates ...................................................... 800
3.3. Contact angle hysteresis on real surfaces ..................................................................................... 800
3.4. Deformation of the substrate as an additional source of the contact angle hysteresis................... 801
3.5. The dynamic contact angle ........................................................................................................... 801
3.6. Wetting of rough and chemically heterogeneous surfaces: the Wenzel and Cassie models ......... 802
4. Superhydrophobicity and the rose petal effect ..................................................................................... 804
4.1. Superhydrophobicity .................................................................................................................... 804
4.2. Wetting of hierarchical reliefs: approach of Herminghaus ........................................................... 805
4.3. The rose petal effect ..................................................................................................................... 805
4.4. Wetting phenomena and molecular dynamic simulations ............................................................ 806
5. Summary .............................................................................................................................................. 806
References ........................................................................................................................................... 807
© Edward Bormashenko, 2016
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
1. Introduction
The center of mass of research activity of physicists,
chemists and material scientists shifted markedly in the se-
cond half of the XX century from the study of bulk proper-
ties of materials to the processes occurring at interfaces. It
turned out that interfacial processes govern to the much ex-
tent physical phenomena occurring not only in the nearest
vicinity of a surface but also in the bulk of materials. It is
noteworthy, that interfacial phenomena already attracted the
attention of ancient natural philosophers. Plinius the elder,
reported that vegetable oils poured on the surface of a rough
sea have a calming effect on waves [1]. In the realm of clas-
sical physics such giants as B. Franklin, Lord Rayleigh and
W. Thomson studied capillarity and wetting [2–4].
The situation changed, when ideas of modern physics
formed the novel style of the scientific thinking. The field
of wetting remained unattractive for young scientists for a
long time, and this is in spite of the fact that Einstein,
Shrödinger and Bohr devoted their research activity to this
class of effects [5–8]. Einstein treated the origin of surface
waves [6]. Shrödinger and Bohr suggested the witty meth-
ods of measurement of a surface tension of liquids [7,8]. It
has been latently supposed that only physics of particles
and phenomena occurring in a micro-world deserve the
attention of inquisitive minds. Several factors have revived
recently an interest in wetting and wettability. The first of
these was the discovery of the “lotus” effect (or super-
hydrophobicity) by Barthlott and Neinhuis in 1997 [9]. The
second factor was the rapid progress achieved in the field
of wetting by the scientific school leaded by P.G. de
Gennes [10]. It is noteworthy that the main notions of the
modern theory of wetting (such as disjoining pressure,
superhydrophobicity, contact angle hysteresis, wetting
transitions) are younger than the basic ideas of relativity
and quantum mechanics. Hence, the field of wetting phe-
nomena is a rapidly developing field of modern physics,
full of exciting physical insights.
The presented review is devoted to physics of solid-
liquid interfaces, and more particularly to the applications
of variational principles to wetting problems. Exploiting
variational principles allows natural construction of a gen-
eral umbrella enclosing a broad variety of wetting effects
[11,12]. The review demonstrates that the well-known
Young, Neumann–Boruvka, Cassie–Baxter and Wenzel
equations actually represent the boundary transversality
conditions for the appropriate problem of wetting [12].
2. Wetting of ideal surfaces
2.1 Wetting of flat ideal surfaces
Wetting is the ability of a liquid to maintain contact
with a solid surface, resulting from intermolecular interac-
tions when the two are brought together. The idea that wet-
ting of solids depends on the interaction between particles
constituting a solid substrate and liquid has been expressed
explicitly in the famous essay by Thomas Young [13].
When a liquid drop is placed on the solid substrate, two
main static scenarios are possible: either liquid spreads
completely, or it sticks to the surface and forms a cap as
shown in Fig. 1(a) (a solid surface may be flat or rough,
homogenous or heterogeneous). Fig. 1(a) also depicts the
apparent contact angle θ, which serves as a natural macro-
scopic parameter of wetting. The precise definition of the
apparent contact angle will be given in the Sec. 3.3; at this
stage we only require that the radius of the droplet should
be much larger than the characteristic scale of the surface
roughness. The observed wetting scenario is dictated by a
spreading parameter
* *ˆ ˆ ˆ( )SA SL LAG G GΨ = − + , (1)
where *ˆ
SAG and *ˆ
SLG are the specific surface energies at the
rough solid/air and solid/liquid interfaces (the asterisk re-
minds us that *ˆ
SAG and *ˆ
SLG do not coincide with the spe-
cific surface energies of smooth surfaces ˆ ˆ, ),SA SLG G and
ˆ
LAG = γ is the specific energy of the liquid/air interface.
When 0,Ψ > total wetting is observed, depicted in
Fig. 1(b). The liquid spreads completely in order to lower
its surface energy (θ = 0). When 0,Ψ < the droplet does
not spread but forms a cap resting on a substrate with a
contact angle θ, as shown in Fig. 1(a). This case is called
partial wetting. When the liquid is water, surfaces demon-
strating /2θ < π are called hydrophilic, while surfaces
characterized by /2θ > π are referred as hydrophobic. One
more extreme, exotic situation is possible, when
cos 1,θ = − such as depicted in Fig. 1(c). This is the situa-
tion of complete dewetting or superhydrophobicity (report-
ed first in Ref. 9) to be discussed in detail in the Sec. 4.1.
When the solid surface is atomically flat, chemically ho-
mogeneous, isotropic, insoluble, non-reactive and non-
stretched (thus, there is no difference between the specific
surface energy and surface tension, the spreading parame-
ter obtains its convenient form (see Ref. 10):
( )SA SLΨ = γ − γ + γ , (2)
Fig. 1. The three wetting scenarios for sessile drops: partial wet-
ting (the apparent contact angle θ is shown) (a); complete wetting
(b); complete dewetting (also called superhydrophobicity) (c).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 793
http://en.wikipedia.org/wiki/Liquid
http://en.wikipedia.org/wiki/Solid
http://en.wikipedia.org/wiki/Surface
http://en.wikipedia.org/wiki/Intermolecular
Edward Bormashenko
where , ,SA SLγ γ γ are the surface tensions at the solid/air
(vapor), solid/liquid and liquid/air interfaces respectively
[10]. When the droplet forms a cap, the line at which solid,
liquid and gaseous phases meet is called the triple (or three
phase) line.
Consider wetting of an ideal substrate in the situation of
the partial wetting when 0.Ψ < When a droplet is deposit-
ed on such an ideal substrate as described in Fig. 2. its free
energy G could be written as:
[ ] 2( , ) (1 ( ) ( )SL SA
S
G h x y h dxdy = γ + ∇ + γ − γ ∫∫ , (3)
where h(x,y) is the local height of the liquid surface above
the point (x,y) of the substrate (it is supposed latently that
there is no difference between surface tensions and surface
energies for ,SLγ ),SAγ and the integral is extended over
the substrate area. The first term of the integrand presents
the capillary energy of the liquid cap and the second term
describes the change in the energy of the solid substrate
covered by liquid.
Now we want to complicate the situation and expose
our droplet to an external field. We restrict ourselves with
an axially symmetrical situation depicted in Fig. 2, thus the
interaction of the droplet with the field is described by the
linear density U(x, h(x)) of the additional energy with the
dimension of (J/m)
( )
0
( , ( )) 2 ( , ) ,
h x
U x h x xw x y dy= π∫ where
w(x,y) is the volume energy density of the droplet in the
external field. The functions w(x,y) and ( ( ), )U h x x are
dictated by the external field and are supposed to be known
(for example, for a uniform gravity field /2,w gy= ρ
2( , ( )) ( /2) ( ),U x h x x gh x= πρ where ρ is the density of the
liquid). Finally, the free energy of the droplet will be given
by:
2( , ) 2 1 2 ( ) ( , )
a
SL SA
o
G h h x h x U x h dx = πγ + + π γ − γ +′ ′ ∫ ,
(4)
where / ,h dh dx=′ and a is the contact radius, shown in
Fig. 2. We also suppose that the droplet does not evapo-
rate, thus the condition of the constant volume V should be
considered as:
2 ( ) const
a
o
V xh x dx= π =∫ . (5)
If we want to calculate the shape of the droplet,
Eqs. (4),(5) will reduce the problem to minimization of the
functional:
0
( , ) ( , , )
a
G h h G h h x dx=′ ′∫ , (6)
2( , , ) 2 1 2 ( )SL SAG h h x x h x= πγ + + π γ − γ +′ ′
( , ) 2U x h xh+ + πλ , (7)
where λ is the Lagrange multiplier to be deduced from
Eq. (5). For a calculation of the droplet's shape we would
have to solve the appropriate Euler–Lagrange equations
[11]. However, we will not focus on the calculation of the
droplet's shape, since our interest is the contact angle θ
corresponding to the equilibrium of the droplet. Now we
make one of the main assumptions of our treatment: we
suppose that the boundary (the triple line) of the droplet is
free to slip along the axis x. It has to be emphasized that
we solve the variational problem with free endpoints [11].
Thus, the conditions of transversality of the variational
problem should be considered [11]. The transversality
condition at the endpoint a yields:
( ) 0h x aG h G =′− =′ ′ , (8)
where hG ′′ denotes the h′ derivative of G [11]. Substitu-
tion of Eq. (7) into the transversality condition, given by
Eq. (8), and taking into account ( ) 0,h a = ( , 0) 0U x a h= = =
will give rise to:
2
2
2
1 0
1
SL SA
x a
hh
h =
γ ′γ + + γ − γ − =′
+ ′
. (9)
Simple transformations yield:
2
1
1
SA SL
x ah =
γ − γ
= γ + ′
. (10)
Considering ( ) tan ,Yh x a= = − θ′ where Yθ is the equilib-
rium (Young) contact angle immediately yields:
cos SA SL
Y
γ − γ
θ =
γ
. (11)
Equation 11 presents the well-known Young equation,
illustrated with Fig. 3 (remarkably it could not be found in
the famous essay by Thomas Young [13]). It asserts that
the contact angle θY is unambiguously defined by the triad
of surface tensions: ,γ ,SLγ ,SAγ as it was stated first
verbally by Sir Thomas Young: “For each combination of
Fig. 2. A cross-section of the spherically-symmetrical droplet
deposited on the ideal solid substrate and exposed to an external
field U(x,h).
794 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
a solid and a fluid, there is an appropriate angle of contact
between the surfaces of the fluid, exposed to the air, and to
the solid” [13]. The Young contact angle θY is supplied by
Eq. (11). The Young contact angle is the equilibrium con-
tact angle that a liquid makes with an ideal solid surface
[14]. It will be shown later that for droplets or surfaces
with very small radii of curvature deposited on the ideal
surfaces the equilibrium contact angle may be different due
to line tension. Equation (11) tells us that the Young angle
depends only on the physicochemical nature of the three
phases and that it is independent on the droplet shape and
external field U under very general assumptions about U,
i.e., U = U(x, h(x)). The external field may deform the
droplet but it has no influence on the Young angle θY.
It should be emphasized that it is not simple to verify
the Young equation experimentally, due to the fact that the
solid/liquid interfacial tension is not well-measured physi-
cal value [10,12]. Moreover, the phenomenon of the con-
tact angle hysteresis (see Sec. 3.1) complicates the experi-
mental establishment of the Young (equilibrium) contact
angle.
There also exist other simple ways of proving the Young
equation by exploiting the principle of virtual works or other
convenient methods of mathematical physics [15]. However,
we preferred the variational approach for two reasons: 1) it
demonstrated the independence of the equilibrium contact
angle on the external fields (this fact is not so intuitively
clear); 2) the variational approach will supply a key to much
more complicated problems.
2.2. Wetting of flat homogeneous curved surfaces
Now consider wetting of flat homogeneous curved sur-
faces. For the sake of simplicity, we start with a 2D wet-
ting problem where a cylindrical drop extended uniformly
in the y direction is under discussion (Fig. 4 depicts the
cross-section of such a drop). We consider the symmetrical
about axis z liquid drop deposited on the curved solid sub-
strate described by the given function f(x) and exposed to
some external field symmetric about axis z. The interaction
of the droplet with the field gives rise to the linear energy
density U(x, h(x)), as it was shown in the previous section.
The free energy of the droplet is supplied by:
2( , ) 1 ( )
a
SL SA
a
G h h h
−
= γ + + γ − γ ×′ ′∫
21 ( , ( ))f U x h x dx× + +′ , (12)
where h(x) is the local height of the liquid surface above
the point x of the substrate (the profile of the droplet h(x) is
assumed to be a single-valued and even function). Condi-
tion of the constant area S has also to be taken into ac-
count:
[ ]( ) ( ) const
a
a
S h x f x dx
−
= − =∫ . (13)
Note that this is equivalent to the constant volume re-
quirement in the case of cylindrical “drops” (extended in
the y direction; h is independent of y).
Equations (12), (13) reduce the problem to minimiza-
tion of the functional:
( , ) ( , , )
a
a
G h h G h h x dx
−
=′ ′∫ , (14)
2 2( , , ) 1 ( ) 1SL SAG h h x h f= γ + + γ − γ + +′ ′ ′
( , ) ( )U x h h f+ + λ − , (15)
where λ is the Lagrange multiplier to be deduced from
Eq. (13). The constant terms in Eq. (15) could be omitted
when the functional G is minimized; however, they turn
out to be important for the analysis of the situation at the
boundary. As mentioned above, we focus on the calcula-
tion of θ and ignore the calculation of the droplet's shape.
As for flat surfaces the variational problem with free end-
points is solved; i.e., it is suggested now that the endpoints
of the drop x = ±a are not fixed and are free to move along
the line f(x). Without the loss of generality, we suggest that
the curve f(x) and the entire problem are symmetrical
around the vertical axis. Thus, the transversality condition
in this case obtains the form [11]:
( ) 0h x a
G G f h′ =
+ − =′ ′ ′
, (16)
Fig. 3. Scheme illustrating the Young equation. The triad of sur-
face tensions: γ, γSL, γSA may be interpreted as forces acting on
the unit length of the triple (three-phase line).
Fig. 4. Scheme of the section of a cylindrical drop deposited on a
flat homogenous curved substrate.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 795
Edward Bormashenko
where hG ′′ denotes the h′ derivative of G . Substitution
of Eq. (15) into the transversality condition, supplied by
Eq. (16), and considering ( ) ( )h a f a= , ( , ( )) 0U a h a =
gives rise to:
2 2
2
( )1 ( ) 1 0
1
SL SA
x a
h f hh f
h =
γ −′ ′ ′γ + + γ − γ + + =′ ′
+ ′
.
(17)
Simple transformations yield:
2
2
1 ( ) 1 0
1
SL SA
x a
h f f
h =
+ ′ ′γ + γ − γ + =′
+ ′
. (18)
Considering ( ) tanh x a= = − θ′ , where θ is the slope of
the liquid-air interface at x = a, and ( ) tan ,f x a= = − θ′
where tan− θ is the slope of the solid substrate in x = a,
( /2)θ < π immediately gives:
cos( ) SA SLγ − γ
θ − θ =
γ
. (19)
The Young equation (compare with Eq. (11)) is recog-
nized. It is reasonable to define the equilibrium (Young)
contact angle as .θ − θ The re-defined Young angle is in-
sensitive to an external field, meeting the conditions:
( , ), ( ),U U x h U U h= ≠ ′ ( , ( )) 0.U a h a =
Three dimensional flat homogeneous axially symmet-
rical surfaces are treated in a similar way. The free energy
functional G supplying the free energy of the droplet as-
sumes the form
0
( , ) ( , , )
a
G h h G h h x dx=′ ′∫ , where:
2 2( , , ) 2 1 2 1 ( )SL SAG h h x x h x f= πγ + + π + γ − γ +′ ′ ′
( , ) 2 ( )U x h x h f+ + πλ − , (20)
where λ is the Lagrange multiplier. The simple transfor-
mations akin to already presented result again in the Young
equation for the 3D wetting of ideal surfaces [12].
2.3. Considering the line tension
Surface tension is due to the special energy state of the
molecules at a solid or liquid surface [16]. Molecules lo-
cated at the triple (three-phase) line where solid, liquid and
gaseous phases meet are also in an unusual energy state.
The notion of line tension has been introduced by Gibbs.
Gibbs stated: “These (triple) lines might be treated in a
manner entirely analogous to that in which we have treated
surfaces of discontinuity. We might recognize linear densi-
ties of energy, of entropy, and of several substances which
occur about the line, also a certain linear tension” [17]. In
spite of the fact that the concept of line tension is intuitive-
ly clear, it remains one of the most obscure and disputable
notions of the surface science [17]. The researchers disa-
gree not only about the value of the line tension but even
about its sign. Experimental values of a line tension Γ in
the range of 10–5–10–11 N were reported [10,12,16,17].
Very few methods allowing experimental measurement of
line tension were developed [18–20]. Marmur estimated a
line tension as 4 cot ,m SA YdΓ ≅ γ γ θ where dm is the
molecular dimension, ,SAγ γ are surface energies of solid
and liquid correspondingly, and θY is the Young angle.
Marmur concluded that the magnitude of the line tension is
less than 95 10 N,−⋅ and that it is positive for acute and
negative for obtuse Young angles [21]. However, research-
ers reported negative values of the line tension for hydro-
philic surfaces [19]. As to the magnitude of the line tension
the values in the range 10–9–10–12 N look realistic. Large
values of Γ reported in the literature are most likely due to
contaminations of the solid surfaces [10].
Let us estimate the characteristic length scale l at which
the effect of line tension becomes important by equating
surface and “line” energies: / 1 100 nm.l ≅ Γ γ = − It is
clear that the effects related to line tension can be im-
portant for nano-scaled droplets or for nano-scaled rough
surfaces [10,12].
Let us estimate the influence of the line tension on the
contact angle of an axisymmetric droplet. The free energy
functional supplying its free energy while also considering
the line tension is given by:
0
( , ) ( , , ) ,
a
G h h G h h x dx=′ ′∫
where:
2( , , ) 2 1 2 ( )SL SAG h h x x h x= πγ + + π γ − γ +′ ′
( , ) 2 2U h x xh+ + πλ + πΓ . (21)
For the sake of simplicity, Γ is anticipated as constant.
Substitution of Eq. (21) into the transversality condition,
defined by Eq. (8) yields:
cos SA SL
a
γ − γ Γ
θ = −
γ γ
. (22)
where a is the contact radius of the droplet [22]. Equation
(22) represents the well-known Boruvka–Neumann for-
mula considering the effect of line tension [17,22].
2.4. Very thin liquid films and the disjoining pressure
Consider very thin liquid films (typically with a thick-
ness less than 10 nm) deposited on ideal solid surfaces. If
we place a film of thickness e (see Fig. 5) on an ideal solid
substrate its specific surface energy will be .SLγ + γ How-
ever, if the thickness e tends to zero we return to a bare
solid with a specific surface energy of SAγ (see Refs. 10,
23,24). It is reasonable to present the specific surface ener-
gy of the film ˆ /G G S= (S is area) as:
ˆ ( ) ( )SLG e e= γ + γ + Ω , (23)
796 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
where ( )eΩ is a function of the film defined in such a way
that lim ( ) 0
e
e
→∞
Ω = and
0
lim ( ) SA SL
e
e
→
Ω = Ψ = γ − γ − γ
[10]. It could be shown that when the molecules of solid and
liquid interact via the Van der Waals interaction (see
Refs. 23,24), ( )eΩ obtains the form:
2( )
12
Ae
e
Ω =
π
, (24)
where A is the so called Hamaker constant, which is in the
range of 19 2010 10 JA − −≅ − [10,23,24]. The Hamaker
constant could be expressed as:
2 ( )L S LA = π ωα α − α , (25)
where ,L Sα α are specific volume polarizabilities of liquid
and solid substrate respectively, ω is a constant that de-
pends very little on the nature of solid and liquid [10].
It could be seen from Eq. (25) that the Hamaker con-
stant could be positive or negative. It will be positive when
the solid has higher polarizability than the liquid
( ).S Lα > α This situation can happen on high-energy sur-
faces; the opposite case occurs on low-energy surfaces
( )S Lα < α [10]. It could be seen from Eq. (23) that, when
( ) 0,eΩ < it diminishes the specific surface energy of the
solid/thin liquid film system thus the Van der Waals inter-
action will thin the film trying to cover as large a surface
of the substrate as possible [10].
The negative derivative of ( )eΩ is called the disjoining
pressure:
3( )
6
d Ae
de e
Ω
Π = − =
π
, (26)
introduced into surface science by B.V. Derjaguin [25,26].
The disjoining pressure plays a primary role in the theory
of thin liquid films deposited on solid surfaces, however
one of the most amazing examples is discovered when liq-
uid helium is deposited on a solid surface. The polari-
zability of liquid helium is lower than that of any solid
substrate; thus the Hamaker constant given by Eq. (25) will
be positive (this corresponds to the repulsive Van der
Waals film force across an adsorbed helium film), and the
disjoining pressure will thicken the film so as to lower its
energy. Let us discuss the liquid helium film climbing a
smooth vertical wall, depicted in Fig. 6, and derive the
profile of the film e(z). The components of the free energy
of the unit area of the film depending on its thickness are
supplied by:
2
ˆ ( )
12
AG e ghe
e
= + ρ
π
. (27)
The equilibrium corresponds to ˆ / 0,G e∂ ∂ = which yields
the thickness profile:
1/3
( )
6
Ae h
gh
= πρ
. (28)
Considering the disjoining pressure becomes important
for very thin angstrom-scaled films; however, when the
liquid is water, the range of the effects promoted by the
disjoining pressure could be as large as 100 Å, due to the
Helmholtz charged double layer [10,23,24]. The electrical
double layers give rise to the disjoining pressure described
by an expression different from Eq. (26), i.e.
( ) exp ( )EDL e D eΠ = −χ , (29)
where1/ 100nm,χ ≈ and D is the characteristic parameter
of the system, which can be either positive or negative
[27]. Yet another component of the disjoining pressure
SΠ is the so-called structural component caused by orien-
tation of water molecules in the vicinity of the solid sur-
face or at the aqueous solution/vapor interface [23,26].
Only a semi-empirical equation for the structural compo-
nent of the disjoining pressure exists for today:
exp( )S eΠ = Λ −ν , (30)
where Λ and ν are constants, 1/ 10 15 Åν ≈ − [26,27].
2.5. Considering the influence of absorbed liquid layers
and the liquid vapor
Up to this point we neglected two important factors:
layers of absorbed liquid molecules which may be present
on the solid substrate (still supposed to be ideal), and the
impact of the gaseous phase. Consideration of these factors
Fig. 5. Scheme illustrating the origination of the disjoining pres-
sure. e is the thickness of a liquid film.
Fig. 6. Film of liquid helium climbing upwards due to the disjoin-
ing pressure.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 797
Edward Bormashenko
was carried out recently by Starov and Velarde [27] They
imposed three obvious conditions of the thermodynamic
equilibrium of a droplet/substrate/vapor system. When the
drop is in equilibrium, the chemical potentials of the liquid
molecules in the ambient vapor phase and the liquid inside
the droplet should be equal. The latter results in Kelvin's
equation inside the drop:
lnL
ML S
RT pp
V p
= , (31)
where liq vap ;Lp p p= − vap liq,p p are the pressures in the
vapor and the liquid phases respectively, Lp is the Laplace
pressure, MLV is the molar volume of the liquid, Sp is the
pressure of the saturated vapor at the temperature T above
the flat liquid surface, R is the gas constant, and p is the
vapor pressure, which is in equilibrium with the drop (for a
detailed derivation and explanation of Kelvin's equation see
Ref. 24). The Kelvin equation (Eq. (31)) was the first re-
quirement of thermodynamic equilibrium, imposed on the
problem of wetting of solid surface by a drop in Ref. 27.
Starov and Velarde also suggested that the solid sub-
strate is covered by a thin layer of a thickness e of ab-
sorbed liquid molecules, as depicted in Fig. 7 [27]. The
thermodynamic equilibrium requires equality of chemical
potentials of molecules in the vapor phase and in the ad-
sorbed layer. This was the second condition. And the third
condition was a minimum of the excess free energy of a
droplet. These conditions combined with use of the appa-
ratus of transversality conditions of the variational problem
of wetting leading to the following equation defining the
contact angle θ:
1cos 1 ( )
e
e de
∞
θ ≈ + Π
γ ∫ , (32)
where ( )eΠ is the disjoining pressure introduced in the pre-
vious paragraph. Emergence of ( )eΠ in Eq. (32) predicting
the contact angle is natural, the thickness of the adsorbed
liquid layer is supposed to be nano-scaled [27]. It should be
stressed that the contact angle θ needs redefinition, because
the droplet cap does not touch the solid substrate, as shown
in Fig. 7. Starov and Velarde defined the contact angle in
this case as an angle between the horizontal axis and the
tangent to the droplet cap profile at the point where it touch-
es the absorbed layer of molecules (which is also called the
precursor film) [27]. It is seen that Eq. (32) represents the
obvious alternative to the Young Eq. (11), relating the Young
angle to the disjoining pressure and not to the interplay of
interfacial tensions.
Let us estimate the disjoining pressure in the absorbed
layer according to 3( ) /6 .e A eΠ = π If we will assume
19 2010 –10 J, 1 nmA e− −≈ = we obtain giant values for
the disjoining pressure: 4 5( ) 5 10 –5 10 Pa.eΠ ≅ ⋅ ⋅ For
10nme = we obtain much more reasonable values of the
disjoining pressure: 2( ) 50–5 10 Pa,eΠ ≅ ⋅ however, they
are still larger or comparable to the Laplace pressure in the
drop. For 1mmr ≈ we have 2 / 140Pa.p r= γ ≅ How is the
mechanical equilibrium possible in this case? Perhaps it is
due to the negative curvature of the droplet at the area
where the cap touches the absorbed layer, shown in Fig. 7.
Moreover, if we take for the disjoining pressure Eq. (26),
we obtain from Eq. (32):
2
1cos 1 ( ) 1 1
12e
Ae de
e
∞
θ ≈ + Π = + >
γ πγ∫ ,
which corresponds to complete wetting [27]. The latter con-
dition implies that at oversaturation no solution exists for an
equilibrium liquid film thickness e outside the drop. If we
take A < 0, there is a solution for an equilibrium liquid film
thickness e but such an equilibrium state is unstable [27].
In order to understand how the partial wetting is possi-
ble in this case, Starov and Velarde discussed more com-
plicated forms of disjoining pressure, comprising the Lon-
don–van der Waals, double layer and structural contri-
butions given by Eqs. (29), (30). They considered more
complicated disjoining pressure isotherms, such as those
depicted in Fig. 8 (curve 2) [27]. The development of
Eq. (32) yielded:
1cos 1 ( ) 1
e
S S
e de
∞
− +−
θ ≈ + Π ≈ −
γ γ∫ , (33)
where S− and S+ are the areas depicted in Fig. 8. Obvi-
ously the partial wetting is possible when S S− +> [27].
Thus, when a droplet is surrounded by a thin layer of liquid
the possibility of partial wetting depends according to
Starov and Velarde on the particular form of the Derjaguin
isotherm [23].
2.6. Wetting transitions on ideal surfaces
The surface tension of liquids is temperature-sensitive;
SAγ and SLγ are also temperature sensitive. What will be
observed when both the droplet and substrate are heated?
At a certain point, it may be that the sum of the solid-liquid
and the liquid-air (vapor) surface tensions becomes equal
to the solid–air (vapor) interfacial tension; then the spread-
ing parameter ( )SA SLΨ = γ − γ + γ will equal zero, and
the transition from partial wetting to complete wetting will
occur (see Fig. 1). The wetting transition is the transition
Fig. 7. Droplet of the radius r surrounded by the thin layer of
liquid of the thickness e governed by the disjoining pressure.
798 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
between a partial and a complete wetting state [28]. The
temperature of transition is called the wetting temperature
TW. The order of the wetting transition is determined — in
the same manner as for a bulk phase transition — by the
discontinuities of the surface free energy. If a discontinuity
occurs in the first derivative of the free energy, the transi-
tion is said to be of the first order, and will take place in a
discontinuous way. If the first derivative of the free energy
is continuous at a phase transition point, then that indicates
that it is a higher-order phase transition. For the wetting of
a liquid drop on a substrate the relevant free energy is the
surface tension of the substrate-air (vapor) interface .SAγ
Let us rewrite the Young equation in this a way:
( ) (1 cos )SA SL Yγ = γ + γ − γ − θ . (34)
Since the term proportional to (1 cos )Yγ − θ is the
part that is going to zero at the wetting transition to
complete wetting, it is the critical part of the specific
free energy to be examined to determine the critical ex-
ponents. According to the definition of the critical expo-
nent, this part of the specific free energy approaches zero
following ˆ2(1 cos ) ( ) ,Y WT T −α− θ ∞ − where α̂ is the
specific heat exponent, determining the order of the wet-
ting transition. For ˆ 1α = , the first derivative of cos θY,
and therefore the first derivative of the specific surface
free energy is discontinuous with respect to temperature
(cosθY = 1, for )WT T≥ and so the wetting transition is
of the first order [28].
The accumulated experimental data and much theoreti-
cal work carried out in the field confirm the fact that wet-
ting transitions are generally of the first order, as shown in
Fig. 9. In this case, if one measures the thickness of the
absorbed film beside the droplet, at the wetting transition a
discontinuous jump in film thickness occurs from a micro-
scopically thin to a thick film [28]. This is true for a broad
range of liquid/solid pairs ranging from liquid helium to
room temperature binary liquids and high temperature me-
tallic systems. There were also several exceptions reported,
for which a discontinuity in a higher derivative of the spe-
cific surface free energy was observed. Such a behavior
was reported for liquid/air pairs governed by the long-
range Van der Waals interactions [28].
3. Wetting of real surfaces
3.1. Contact angle hysteresis
The Young equation given by Eq. (11) predicts a sole
value of the contact angle for a given ideal solid/liquid
pair. As it always occurs in reality, however, the situation
is much more complicated. Let us deposit a droplet onto an
inclined plane, as described in Fig. 10 in the situation of
partial wetting; it is latently supposed that the spreading
parameter Ψ 0.< The inclined plane is supposed to be
ideal, i.e. atomically flat, chemically homogeneous, iso-
tropic, insoluble, non-reactive and non-deformed. We will
nevertheless recognize different contact angles θ1, θ2, as
shown in Fig. 10. This experimental observation definitely
contradicts the predictions of the Young equation. Moreo-
ver, a droplet on an inclined plane could be in equilibrium
only when contact angles θ1, θ2 are different [10,12]. If we
increase the inclination angle α, contact angles θ1, θ2 will
change, and at some critical angle α the droplet will start to
slip. This critical contact angle is called the sliding angle.
We conclude that a variety of contact angles can be ob-
served for the same ideal solid substrate/liquid pair.
Let us perform one more simple experiment. When a
droplet is inflated with a syringe as shown in Fig. 11 we
observe the following picture: the triple line is pinned to
Fig. 8. Disjoining pressure (Derjaguin’s isotherms): isotherm
corresponding to the complete wetting, only the London–Van der
Waals component is considered (1); isotherm comprising Lon-
don, double layer and structural contributions and corresponding
to the partial wetting (2).
Fig. 9. Typical dependence of the cosine of the contact angle on
the temperature, illustrating wetting transitions on flat substrates
as established for liquid helium on cesium substrate; cosθ goes
linearly to unity at the temperature of transition, indicating that
the wetting transition is of the first order.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 799
Edward Bormashenko
the substrate up to a certain volume of the droplet. When
the triple line is pinned the contact angle increases till a
certain threshold value θA beyond which the triple line
does move. The contact angle θA is called the advancing
contact angle [10,12]. When a droplet is deflated as de-
picted in Fig. 11, its volume can be decreased to a certain
limiting value; in parallel the contact angle decreases till a
threshold value θR, known as the receding contact angle
[10,12]. When θ = θR, the triple line suddenly moves. Both
θA and θR are equilibrium, although metastable contact
angles [10,12]. The difference between θA and θR is called
the contact angle hysteresis.
Both measurement and understanding of the phenomenon
of the contact angle hysteresis remain challenging experi-
mental and theoretical tasks. It is customary to attribute the
phenomenon of the contact angle hysteresis to physical or
chemical heterogeneities of the substrate [10,12]; however,
even ideal substrates discussed in the previous Sections
demonstrate significant contact angle hysteresis. We'll start
our discussion from the physical reasons of the contact angle
hysteresis on ideal substrates.
3.2. Contact angle hysteresis on smooth homogeneous
substrates
Contact angle hysteresis has been registered even for sili-
con wafers which are regarded as atomically flat rigid sub-
strates, and are considered very close to be ideal ones.
Extrand et al. studied the contact angle hysteresis for various
liquids, including water, ethylene glycol, methylene iodide,
acetophenone and formamide, deposited on silicon wafers
with a tilted plane method [29]. Contact angle hysteresis
(defined as )A Rθ − θ as high as 14° was established for the
water/silicon wafer and methylene iodide/silicon wafer
pairs. It should be mentioned that the contact angle hystere-
sis on the order of magnitude of 5–10° has been reported for
other silicon wafer/liquid pairs [29]. High contact angle hys-
teresis has been observed also for atomically smooth poly-
mer substrates. Lam et al. used polymer-coated silicon wa-
fers for study of the contact angle hysteresis, and reported
the values of contact angle hysteresis on the order of tens of
degrees [30] The question is: how is such dispersion of con-
tact angles possible, in contradiction to the predictions of the
Young equation?
The explanation of the contact angle hysteresis ob-
served on smooth surfaces becomes possible if we consider
the effect of the pinning of the triple line. The intermolecu-
lar forces acting between molecules of solid and those of
liquid, which pin the triple line to the substrate, are respon-
sible for the contact angle hysteresis. Yaminsky developed
an extremely useful analogy between the phenomena oc-
curring at the triple line with the static friction [31].
I quote: “… for a droplet on a solid surface there is a static
resistance to shear. It occurs not over the entire solid-liquid
interface, but only at the three-phase line…”. This paradox
is easily resolved once one realizes that the liquid–solid
interaction is in fact not involved in the process of over-
flow of liquids above solid surfaces. A boundary condition
of zero shear velocity typically occurs even for liquid-
liquid contacts… But even given that the strong binding
condition does apply to solid–liquid interfaces, this does
not prevent the upper layer of the liquid from flowing
above the “stagnant layer” of a gradient velocity. The
movement of the liquid over the wetted areas occurs in the
absence of static resistance. Interaction in a manner of dry
friction occurs only at the three-phase line [31].
Thus, the contact angle hysteresis on ideal surfaces is
caused by the intermolecular interaction between mole-
cules constituting a solid substrate and a liquid; this inter-
action pins the triple line and gives rise to a diversity of
experimentally observed contact angles [30,31].
3.3. Contact angle hysteresis on real surfaces
Real surfaces are rough and chemically heterogeneous.
The macroscopic parameter describing wetting of surfaces
is the apparent contact angle [14]. The apparent contact
angle is an equilibrium contact angle measured macroscop-
Fig. 10. (Color online) Drop on the inclined plane. Difference
between contact angles θ1, θ2 prevents the droplet sliding; α is the
inclination angle.
Fig. 11. Inflating and deflating of a droplet. Advancing θA and
receding contact angles θR are shown.
800 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
ically on a solid surface that may be rough or chemically
heterogeneous [10,12,14]. The detailed microscopic topog-
raphy of a rough or chemically heterogeneous surface can-
not be viewed with regular optical means; therefore this
contact angle is defined as the angle between the tangent to
the liquid-vapor interface and the apparent solid surface as
macroscopically observed [10,12,14]. Actually the spec-
trum of apparent contact angles is observed on real surfac-
es. A diversity of physical factors contributes to the contact
angle hysteresis, including the pinning of the triple line,
liquid penetration and surface swelling, deformation of the
substrate, etc [12,30]. It should be emphasized that the
contact angle hysteresis turned out to be a complicated,
time-dependent effect. The contact angle hysteresis, as it
seen from the phenomenological point of view, is due to
the multiple minima of the free energy of a droplet depos-
ited on the substrate. These minima are separated by poten-
tial barriers [32]. Contact angle hysteresis is strengthened
by the roughness and chemical heterogeneity of a substrate
[33]. The comprehensive review of the contact angle hyste-
resis is supplied in Refs. 10,12.
3.4. Deformation of the substrate as an additional source
of the contact angle hysteresis
Let us take a closer look at the Young Eq. (11) and
Fig. 3. The Young equation could be interpreted as the
balance of horizontal projections of forces acting on the
triple line. However, the vertical balance is still neglected.
The component of the liquid surface tension sinγ θ per-
pendicular to the plane of the solid (see Fig. 12) must be
equilibrated, and this leads necessarily to some distortion
of the substrate near the triple line, called the “wetting
ridge” [34,35]. This distortion is negligible for rigid sub-
strates such as glass or steel, but it should be considered
for soft substrates such as rubbers (elastomers) [36]. This
wetting ridge (depicted in Fig. 12) leads to additional pin-
ning of the triple line and strengthens the contact angle
hysteresis.
The problem of elastic deformation of a substrate by a
droplet was treated in Refs. 34–36. The scaling dimension-
less parameter δ, relating contributions of surface tension
and elastic terms, could be introduced according to:
SA
d
γ
δ =
µ
, (35)
where µ is the elastic (shear) modulus of the solid, and d
is the depth (thickness of the substrate) [34–36]. For dis-
tances much larger than the thickness d the vertical dis-
placement ζ (see Fig. 12) decays exponentially:
sin sin exp
x x
k
γ θ
ζ ≅ − µ κ ′
, (36)
where x is a distance measured from the triple line parallel
to the undisturbed surface (see Fig. 12), and , kκ ′ are char-
acteristic lengths of the order d [34]. At intermediate dis-
tances d x dδ < < the deformation ζ is given by:
sin ln
2
d
x
γ θ
ζ ≅
πµ
. (37)
Equation 37 is true for x > Є, where Є is a cutoff
length, below which the solid no longer behaves in a line-
arly elastic manner (typically on the order of a few na-
nometers for an elastomer) [34–36]. At short distances
( )x d< δ the vertical displacement ζ is estimated as:
1 sin 1ln
2
γ θ
ζ ≅
π µ δ
. (38)
For the details of the solution of a problem of distortion of
a soft substrate by a droplet see Ref. 34–36. Anyway, this
distortion is not negligible for soft materials such as elas-
tomers and it contributes essentially to the contact angle
hysteresis.
3.5. The dynamic contact angle
Until this Section we have discussed only the statics of
wetting. Now we’ll consider a much more complicated
situation: when the triple line moves. When the triple line
moves the dynamic contact angle Dθ does not equal the
Young angle as shown in Fig. 13. It may be larger or
smaller than the Young angle (see Fig. 13). The excess
force pulling the triple line is given by (see Ref. 10):
( ) cosD SA SL DF θ = γ − γ − γ θ . (39)
As we already mentioned in the previous section the effect
of contact angle hysteresis complicates the study of wet-
ting even in a static situation. The movement of the triple
line introduces additional difficulties, so the reproducibility
of the results of the measurements of dynamic contact an-
Fig. 12. Scheme of the wetting ridge. ζ is the vertical displace-
ment caused by the vertical component of surface tension γ sin θ.
Є is the cutoff distance for linear elastic behavior.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 801
Edward Bormashenko
gles becomes a challenging task [10]. We’ll start from the
theoretical analysis of dynamic wetting on ideally smooth,
rigid, non-reactive surfaces.
Now we find ourselves in the realm of hydrodynamics.
Systematic study of the problem of the dynamics of wet-
ting has been undertaken by Voinov [37]. When the iner-
tia-related contributions are neglected (and this is the case
in the model proposed by Voinov) the only dimensionless
number, governing the flow is the capillary number Ca,
defined as:
vCa η
=
γ
, (40)
where v is the characteristic velocity and η is the viscosity
of the liquid. The capillary number describes the interplay
between the viscosity and surface tension induced effects.
Voinov also phenomenologically introduced the angle of the
free surface slope mθ at the height of the limiting scale hm:
,D m mh hθ = θ = . (41)
Voinov noted that mθ is unknown beforehand and should
be determined during the solution of the problem [37]. The
accurate mathematical solution of the hydrodynamic prob-
lem of wetting yielded for the dynamic contact angle:
1/3 1/3
3 3( ) 9 ln 9 lnD m m
m m
v h hh Ca
h h
η
θ = θ + = θ + γ
. (42)
Equation (42) is referred as the Cox–Voinov law, and it is
valid for 3 /4Dθ < π (see Ref. 37). Hoffmann has shown
that the experimental dependence ( )D Caθ is represented
by a universal curve (corrected with a shifting factor) for a
diversity of liquids [38]. A detailed discussion of the vali-
dity and applicability of the Cox–Voinov law is supplied in
Ref. 39. It is seen from Eq. (42) that the slope varies loga-
rithmically with the distance from the triple line. Thus, it is
impossible to assign a unique dynamic contact angle to a
triple line moving with a given speed [39]. Hence, Fig. 13
depicts an obvious oversimplification of the actual dynam-
ic wetting situation. It is also noteworthy that Dθ depends
slightly on the cut-off length hm, however, it depends
strongly on the microscopic angle mθ . For a detailed dis-
cussion of actual values of mθ and hm, see Ref. 39.
3.6. Wetting of rough and chemically heterogeneous
surfaces: the Wenzel and Cassie models
In this Section present models describing the wetting of
rough and chemically heterogeneous surfaces, i.e., the
Wenzel and Cassie models. Recall that wetting of rough or
chemically heterogeneous surfaces is characterized by the
apparent contact angle, introduced in the Sec. 3.3. The
Cassie and Wenzel models predict the apparent contact
angle, which is an essentially macroscopic parameter. This
fact limits the field of validity of these models: they work
when the characteristic size of a droplet is much larger
than that of the surface heterogeneity or roughness. The
use of the Wenzel and Cassie equations needs a certain
measure of care; numerous misinterpretations of these
models are found in the literature.
Let us start from the Wenzel model, introduced in 1936,
which deals with the wetting of rough, chemically homo-
geneous surfaces and implies total penetration of a liquid
into the surface grooves, as shown in Fig. 14 [40,41].
When the spreading parameter 0Ψ < (see Sec. 2.1 and
Fig. 1(a)), a droplet forms a cap resting on the substrate
with an apparent contact angle θ*. If the axisymmetric
droplet is exposed to an external field U(x,h), the free en-
ergy of G could be written as:
2( , ) 2 1 2 ( ) ( , ) ,
a
SL SA
o
G h h x h x r U x h dx = πγ + + π γ − γ +′ ′ ∫
(43)
where h(x,y) is the local height of the liquid surface above
the point (x,y) of the substrate, U(x,h(x)) is the linear density
Fig. 13. Origin of the dynamic contact angle θD. The dynamic
contact angle θD is larger than the Young angle θY (a). The oppo-
site situation: the dynamic contact angle θD is smaller than the
Young angle θY (b).
Fig. 14. Wenzel wetting of a chemically homogeneous rough
surface: liquid completely wets the grooves. θ* is the apparent
contact angle.
802 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
of interaction of the droplet with the external field with the
dimension of J/m, a is the contact radius, and the integral is
extended over the substrate area (see Sec. 2.1). Equation
(43) is very similar to Eq. (4), the only difference being
parameter r which is the roughness ratio of the wet area;
in other words, the ratio of the real surface in contact with
liquid to its projection onto the horizontal plane. Parameter
r describes the increase of the wetted surface due to
roughness, and obviously 1r > takes place. We also sup-
pose that the volume of a droplet is constant (see Eq. (5)).
Equations (5), (43) reduce the problem to the minimization
of the functional:
0
( , ) ( , , )
a
G h h G h h x dx=′ ′∫ , (44)
where:
2( , , ) 2 1 2 ( )SL SAG h h x x h x r= πγ + + π γ − γ +′ ′
( , ) 2U x h xh+ + πλ , (45)
where λ is the Lagrange multiplier to be deduced from
Eq. (5).
The use of the conditions of transversality of the varia-
tional problem, akin to that, presented in detail in Sec. 2.1
yields:
*cos cos Yrθ = θ . (46)
Equation (46) presents the famous Wenzel equation
[40,41]. Three important conclusions follow from Eq. (46):
—Inherently smooth hydrophilic surfaces ( /2)Yθ < π
will be more hydrophilic when riffled: *
Yθ < θ due to the
fact, that 1.r >
—Due to the same reason, inherently hydrophobic flat
surfaces ( /2)Yθ < π will be more hydrophobic when
grooved: * .Yθ > θ
—The Wenzel apparent contact angle, given by Eq. (46),
is independent on the droplet shape and external field U
under very general assumptions about U, i.e., U = U(x, h(x)).
Wenzel wetting of chemically homogeneous curved
rough surfaces is discussed in Ref. 42.
The Cassie–Baxter wetting model, introduced in
Refs. 43,44, deals with the wetting of flat chemically heter-
ogeneous surfaces. Suppose that the surface under the drop
is flat, but consists of n sorts of materials randomly distrib-
uted over the substrate as shown in Fig. 15. This corre-
sponds to the assumptions of the Cassie–Baxter wetting
model [43,44]. Each material is characterized by its own
surface tension coefficients ,i SLγ and , ,i SAγ and by the
fraction fi in the substrate surface, f1 + f2 + … + fn = 1. The
free energy of an axisymmetric drop of a radius a exposed to
an external field U(x, h) will be given by the following ex-
pression (analogous to Exp. (5.1)):
2( , ) 2 1
a
o
G h h x h= πγ + +′ ′∫
, ,
1
2 ( ) ( , )
n
i i SL i SA
i
x f U x h dx
=
+ π γ − γ +∑ . (47)
Equation (5), demanding the constancy of the droplet
volume, again introduces the Lagrange multiplier λ. Anal-
ogously to the above treatment we obtain for G :
2
, ,
1
( , , ) 2 1 2 ( )
n
i i SL i SA
i
G h h x x h x f
=
= πγ + + π γ − γ +′ ′ ∑
( , ) 2U x h xh+ + πλ . (48)
Substitution of Exp. (48) into the transversality condi-
tion, given by Eq. (8) and transformations akin to afore-
mentioned ones yield the famous Cassie–Baxter equation:
, ,
* 1
( )
cos
n
i i SA i SL
i
f
=
γ − γ
θ =
γ
∑
, (49)
predicting the so-called Cassie apparent contact angle *θ
on flat chemically heterogeneous surfaces [12,43,44]. We
demonstrated convincingly that the Cassie apparent contact
angles are also insensitive to external fields [12,22]. When
the substrate consists of two kinds of species, the Cassie–
Baxter equation obtains the form:
*
1 1 2 2cos cos cosf fθ = θ + θ , (50)
which is widespread in the scientific literature dealing with
the wetting of heterogeneous surfaces [10,12,24,43,44].
The presented derivation demonstrates explicitly that the
Cassie–Baxter apparent contact angle is insensitive to ex-
ternal fields of a very general form, i.e., U = U(x, h(x)).
The peculiar form of the Cassie–Baxter equation, given
by Eq. (50), was successfully used for the explaining the
phenomenon of superhydrophobicity, which will be dis-
cussed in detail in the next Section. Jumping ahead, we
admit that in the superhydrophobic situation, a droplet is
partially supported by solid substrate and partially by air
cushions, as shown in Fig. 16. Consider a situation where
Fig. 15. (Color online) Cassie–Baxter wetting of flat chemically
heterogeneous surfaces (various colors correspond to different
chemical species). θ* is the apparent contact angle.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 803
Edward Bormashenko
the mixed surface is comprised of solid surface and air
pockets, with the contact angles Yθ (which is the Young
angle of the solid substrate) and π respectively. We denote
by fs and 1 Sf− relative fractions of solid and air respec-
tively. Thus, we deduce from Eq. (50):
*cos 1 (cos 1)S Yfθ = − + θ + . (51)
As it always takes place in nature, the pure Wenzel and
Cassie wetting regimes introduced in previous Sections are
rare in occurrence. More abundant is a so-called mixed
wetting state, depicted schematically in Fig. 17, introduced
in Ref. 45 and discussed in much detail in Ref. 46 In this
situation the use of transversality conditions of the varia-
tional problem of wetting yields for the apparent contact
angle:
*cos cos 1S Y Srf fθ = θ + − . (52)
Obviously for 1,r = we return to the usual Cassie air-
trapping described by Eq. (51). Equation (52) is extremely
useful for understanding the phenomenon of superhydro-
phobicity to be discussed in detail in the next Section. For
considering the role of line tension in wetting of chemical-
ly heterogeneous surfaces see Refs. 47,48.
4. Superhydrophobicity and the rose petal effect
4.1. Superhydrophobicity
The phenomenon of superhydrophobicity was revealed
in 1997, when W. Barthlott and C. Neinhuis studied the
wetting properties of a number of plants and stated that the
“interdependence between surface roughness, reduced par-
ticle adhesion and water repellency is the keystone in the
self-cleaning mechanism of many biological surfaces” [9].
They discovered the extreme water repellency and unusual
self-cleaning properties of the “sacred lotus” (Nelumbo
nucifera) and coined the notion of the “lotus effect”, which
is now one of the most studied phenomena in surface sci-
ence. Afterwards, the group led by W. Barthlott studied a
diversity of plants and revealed a deep correlation between
the surface roughness of plants, their surface composition
and their wetting properties (varying from superhydro-
phobicity to superhydrophilicity) [49–51].
The amazing diversity of the surface reliefs of plants ob-
served in nature was reviewed in Ref. 49. Barthlott et al.
also clearly understood that the micro- and nano-structures
of the plants surfaces define their eventual wetting proper-
ties, in accordance with the Cassie–Baxter and Wenzel
models (discussed in detail in the previous Section) [49–51].
Since Barthlott et al. reported the extreme water repellency
of the lotus, similar phenomena were reported for a diversity
of biological objects: water strider legs, as well as bird and
butterfly wings [52–55].
All the phenomenon was called superhydrophobicity and
natural and artificial surfaces characterized by an apparent
contact angle larger than 150° are referred to as super-
hydrophobic [9,10,52–55]. It should be emphasized that
high apparent contact angles observed on a surface are not
sufficient for referring it as superhydrophobic [56–58]. True
superhydrophobicity should be distinguished from the pseu-
do-superhydrophobicity inherent to surfaces exhibiting the
“rose petal effect” to be discussed later. The pseudo-super-
hydrophobic surfaces are characterized by large apparent
contact angles accompanied by the high contact angle hyste-
resis. In contrast, truly superhydrophobic surfaces are char-
acterized by large apparent contact angles and low contact
angle hysteresis resulting in a low value of a sliding angle: a
water drop rolls along such a surface even when it is tilted at
a small angle. Truly superhydrophobic surfaces are also self-
cleaning, since rolling water drops wash off contaminations
and particles such as dust or dirt. Actually, the surface
should satisfy one more demand to be referred as super-
hydrophobic: the Cassie–Baxter wetting regime on this sur-
face should be stable [59].
The Cassie–Baxter equation (Eq. (51)), developed for
the air trapping situation where the droplet is partially sup-
ported by air cushions (see Fig. 16), supplies the natural
explanation for the phenomenon of superhydrophobicity.
Indeed, the apparent contact angle *θ in this situation giv-
en by: *cos 1 (cos 1)S Yfθ = − + θ + ultimately approaches
π when the relative fraction of the solid fs approaches zero.
This corresponds to complete dewetting, discussed in the
Sec. 2.1 and illustrated by Fig. 1(c). Note that the apparent
contact angle also approaches π when the Young angle
tends to π. However, this situation is practically unachiev-
able, because the most hydrophobic polymer, polytetra-
Fig. 16. The particular case of the Cassie wetting: a droplet is
partially supported by solid and partially by air cushions. θ* is the
apparent contact angle.
Fig. 17. The mixed wetting state. The droplet is partially support-
ed by the solid and partially by air cushions.
804 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
fluoroethylene (Teflon) demonstrates an advancing angle
smaller than 120°, and this angle is always larger than the
Young one. Hence, it is seen from the Cassie–Baxter equa-
tion that the apparent contact angles could be increased by
decreasing the relative fraction of the solid surface under-
neath a droplet. However, there exists a more elegant way
to manufacture surfaces characterized by ultimately high
apparent contact angles: producing hierarchical reliefs
[12,59,60], and this is the situation observed in natural
objects such as lotus leaves and birds’ wings [9,55].
Note, that the Wenzel equation (Eq. (46)) also predicts
high apparent contact angles approaching π for inherently
hydrophobic surfaces ( /2),Yθ > π when 1.r >> However,
the Wenzel-like wetting, depicted in Fig. 14, is character-
ized by the high contact angle hysteresis, whereas superhy-
drophobicity accompanied by self-cleaning calls for the
contact angle hysteresis to be as low as possible.
4.2. Wetting of hierarchical reliefs: approach of
Herminghaus
Herminghaus developed a very general approach to the
wetting of hierarchical reliefs, based on the concept of the
effective surface tension of a rough solid/liquid interface
eff
SLγ [61]. It is reasonable to suggest phenomenologically
that this surface tension is increased when compared to that
of the flat solid surface ,SLγ due to the roughness.
Herminghaus treated indented surfaces; however, his ap-
proach keeps its validity for bumpy ones as well. The effec-
tive surface tension of a rough surface with a single-scale
roughness is given by:
eff
01 (1 ) ( )L SL L SASL f f gγ ≅ − γ + γ + γ , (53)
where Lf is the fraction of free liquid surfaces suspended
over the indentations of the relief, 0 1g ≥ is the geomet-
rical factor describing the total surface area of the indenta-
tion, SAγ is the surface tension of the flat solid surface/air
interface, and the subscript 1 in eff
1SLγ denotes the single-
scale type of the roughness. It is seen from Eq. (53) that an
indented interface has a larger effective surface tension
than a flat one. This warrants the apparent contact angle
1θ larger than Yθ inherent to the flat surface, but does not
explain the exceptionally large apparent contact angles
observed on many biological objects [52–55]. In order to
explain the extreme apparent contact angles Herminghaus
analyzed hierarchical reliefs, such as those depicted in
Fig. 18. For such a double-scaled relief, the effective sur-
face tension will be given by:
eff
1 1 1 1 02 (1 ) ( (1 ( 1) ))L SL L SA LSL f f g g fγ ≅ − γ + γ + γ + − . (54)
For hierarchically indented substrates, Herminghaus
deduced the following recursion relation:
1cos (1 )cosn Ln n Lnf f+θ = − θ − , (55)
where n denotes the number of the generation of the indenta-
tion hierarchy. A larger n corresponds to a larger length scale.
According to Eq. (55), 1cos cos (1 cos ) 0,n n Ln nf+θ − θ = − + θ <
so that the sequence represented by Eq. (55) is monotonic.
Herminghaus stressed that 0θ corresponding to Yθ must
only be finite, but need not exceed /2π for obtaining high
resulting apparent contact angles on hierarchical surfaces.
Herminghaus also considered fractal surfaces and estimated
the Hausdorf dimension of such surfaces. Generally, the
model proposed by Herminghaus successfully explained high
apparent contact angles observed on a diversity of biological
objects [52–55].
It is noteworthy, that hierarchical surfaces enable de-
sign of superoleophobic surfaces (depicted in Fig. 19),
which repel no only water but also organic oils, which is
an important technological task [62–65].
4.3. The rose petal effect
It was already mentioned in the Sec. 4.1 that high ap-
parent contact angles are necessary but not sufficient for
true superhydrophobicity accompanied by self-cleaning
properties of a surface. Low contact angle hysteresis and
high stability of the Cassie wetting states are also neces-
sary (the problem of the stability of the Cassie wetting
state was studied recently in detail in Refs. 66–71).
Jiang et al. reported that rose petal surfaces demonstrate
high contact angles attended with extremely high contact
angles hysteresis [72]. The surface of the rose petal is built
from hierarchically riffled “micro-bumps” resembling
those of lotus leaves [9,72]. At the same time, the wetting
of rose petals is very different from that of lotus leaves.
The apparent angles of droplets placed on a rose petal are
high, but the droplets are simultaneously in a “sticky” wet-
ting state; they do not roll or slip, when put on an inclined
plane (it seems that the first systematic treatment of the
physics of droplets, placed on inclined planes was carried
out by the distinguished physicist Yakov Ilyitch Frenkel in
Ref. 73; restoring historical justice calls for mentioning
that Frenkel first clearly demonstrated that the Young
equation is actually the boundary condition of the problem
of wetting (see also Secs. 2.1, 2.2, and 3.6). Water droplets
on rose petals kept the spherical shape and did not slide
Fig. 18. Scheme of wetting of a hierarchical relief (for details see
Ref. 61).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8 805
Edward Bormashenko
even when the surface was turned upside down, Jiang
called this phenomenon the “rose petal effect” [72].
Later, very similar wetting behavior was revealed on
surfaces built from lycopodium particles [74]. Lycopo-
dium particles, which are spores of the plant Lycopodium
clavatum, are microscopically scaled porous balls with
the external diameter of 20–30 µm, and they are charac-
terized by a pronounced hierarchical structure [74].
Lycopodium particles comprise pores with a characteris-
tic size of 3–5 µm, thus according to the approach devel-
oped in the Sec. 4.2, these particles are well expected to
exemplify pronounced superhydrophobicity. Indeed, sur-
faces built from these particles demonstrate apparent con-
tact angles as high as 150° [74]. However, droplets de-
posited on these surfaces did not slide when the surface
was tilted; moreover, they were steadily attached even
when the surface was turned upside down, as shown in
Ref. 74. Artificial surfaces demonstrating the “rose petal
effect” have been also reported [75].
The natural explanation for the “rose petal effect” is
supplied by the Wenzel model (see Sec. 3.6). Inherently
hydrophobic flat surfaces may demonstrate apparent con-
tact angles approaching π when rough [76]. Wenzel wet-
ting is characterized by the high contact angle hysteresis.
However, the Wenzel model does not explain the existence
of the “rose petal effect” for inherently hydrophilic surfac-
es. Bhushan and Nosonovsky demonstrated that wetting of
hierarchical reliefs may be of a complicated nature, result-
ing in the “rose petal effect”, as shown in Fig. 20 [77].
Various wetting modes are possible for hierarchical reliefs:
it is possible that a liquid fills the larger grooves, whereas
small-scaled grooves are not wetted and trap air as shown
in Fig. 20(a). The inverse situation is also possible, in
which small-scaled grooves are wetted and large scale ones
form air cushions (see Fig. 20(b)). According to Ref. 77
the larger structure controls the contact angle hysteresis,
whereas the smaller (usually nanometric) scale is responsi-
ble for high contact angles [77,78]. Thus, the relief depict-
ed in Fig. 20(a) will demonstrate high contact angles at-
tended by high contact angle hysteresis. This hypothesis
reasonably explains the “rose petal effect” [77]. However,
it is clearly seen that a broad variety of wetting modes is
possible on hierarchical surfaces [77,78]. When the smaller
scale is nanometrically scaled the effects due to the disjoin-
ing pressure (see Sec. 2.4) are already non-negligible.
It should be stressed that various stimuli such as pres-
sure, temperature or external fields may promote the transi-
tion from the low friction Cassie wetting state to the sticky
Wenzel state, which are usually separated by the potential
barrier [66–70]. The physics of these so called Cassie-
Wenzel transitions is reviewed in Ref. 71.
4.4. Wetting phenomena and molecular dynamic
simulations
Till now, we kept pure a macroscopic approach. New in-
sights came to the physics of wetting from computer simula-
tions, recently reported by several groups for study of capil-
larity; in particular computer simulations are useful for study
of the stability of the Cassie-like wetting [66,71,79–81].
Savoy et al. calculated the energy barriers separating the
Cassie and Wenzel states [82,83]. It is well accepted that for
macroscopic droplets, the energetic barrier separating the
Cassie state and Wenzel state is extremely large compared to
thermal fluctuations. However, it was shown with molecular
simulations for nano-droplets that this barrier becomes com-
parable to the energy of thermal fluctuations and ranges
from 8 Bk T to a few tenths of Bk T [84].
5. Summary
Wetting phenomena starting from Plinius the elder via
B. Franklin, Th. Young, W. Thomson and Lord Rayleigh
in the epoch of the classical physics and afterwards via
Einstein, Shrödinger, Bohr and de Gennes in the realm of
modern physics are continuing to inspire fascinating exper-
imental and theoretical studies.These effects are important
from both fundamental and applicative points of view
while governing spreading of liquids on natural and artifi-
Fig. 19. (Color online) The phenomenon of superoleophobicity.
Castor-oil droplet with a volume of 8µL placed on the
superoleophobic surface. High apparent contact angle is clearly
seen. The image is taken in the Laboratory of Polymers of the
Ariel University.
Fig. 20. Schemes of various wetting scenarios possible on a hier-
archical relief (for details see Ref. 77).
806 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 8
Physics of solid–liquid interfaces: from the Young equation to the superhydrophobicity
cial surfaces, numerous biological events and technological
processes such as painting, printing and gluing. The review
presents the general physical approach to capillarity ef-
fects, based on the variational treatment of the appropriate
problem of wetting. It is demonstrated that basic equations
describing wetting of ideal and real (rough and chemically
heterogeneous) surfaces, namely the Young, Wenzel and
Cassie–Baxter equations, arise at the transversality condi-
tions of variational problems of wetting, which are defined
as “problems with free endpoints”. Thus, general thermo-
dynamic approach to the effects occurring at solid/liquid
interfaces becomes possible. The low temperature wetting
events, such as wetting transitions and spreading of liquid
helium are treated. The phenomena of contact angle hyste-
resis and disjoining pressure are discussed. The particular
emphasis is put on recently discovered effects of super-
hydrophobicity and superoleophobicity.
Acknowledgements
The author is indebted to Dr. Whyman for his
longstanding fruitful cooperation in the study of wetting
phenomena. His critique and numerous remarks definitely
improved the text. I am thankful to Professor R. Pogreb for
his contribution in understanding of diversity of wetting
phenomena. I want to thank my numerous MSc and PhD
students for their research activity and allegiance to a spirit
of scientific research. I am grateful to Mrs. Al. Musin for
her kind help in editing the review. I am especially indeb-
ted to my wife Yelena Bormashenko for her inestimable
help in preparing this review. I am greatly thankful to Mrs.
Hanna Weiss for her valuable help in English editing of
this review.
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1. Introduction
2. Wetting of ideal surfaces
2.1 Wetting of flat ideal surfaces
2.2. Wetting of flat homogeneous curved surfaces
2.3. Considering the line tension
2.4. Very thin liquid films and the disjoining pressure
2.5. Considering the influence of absorbed liquid layers and the liquid vapor
2.6. Wetting transitions on ideal surfaces
3. Wetting of real surfaces
3.1. Contact angle hysteresis
3.2. Contact angle hysteresis on smooth homogeneous substrates
3.3. Contact angle hysteresis on real surfaces
3.4. Deformation of the substrate as an additional source of the contact angle hysteresis
3.5. The dynamic contact angle
3.6. Wetting of rough and chemically heterogeneous surfaces: the Wenzel and Cassie models
4. Superhydrophobicity and the rose petal effect
4.1. Superhydrophobicity
4.2. Wetting of hierarchical reliefs: approach of Herminghaus
4.3. The rose petal effect
4.4. Wetting phenomena and molecular dynamic simulations
5. Summary
Acknowledgements
References
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