APPARENT SURFACE FREE ENERGY OF
SUPERHYDROPHOBIC SURFACES
Emil Chibowski
Department of Physical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska
University, Lublin, Poland
Abstract. Superhydrophobicity results from nano- and/or micro- protrusions present
on a hydrophobic surface. Such surfaces are commonly characterized by advancing
water contact angle, which amounts at least 150o. Contact angle hysteresis also
appears on such surfaces. Generally, two types of wetting are usually considered for
rough surfaces, i.e. the drop suspended on the protrusions (Cassie-Baxter case), or the
collapsed drop (Wenzel case). In the case of arranged posts present on a
superhydrophobic surface some experiments show that the advancing contact angle
does not change while the receding one depends on the post size and spacing.
In this paper is shown that more light can be shed on wetting properties of
superhydrophobic surfaces if their apparent surface free energy is calculated from an
equation relating the probe liquid surface tension and the advancing and receding
contact angles, that is, the contact angle hysteresis. With the help of this equation
mentioned above the two cases of wetting can be distinguished. Some simulated and
experimental results are presented.
INTRODUCTION
Real solid surfaces practically always possess chemical and/or texture
inhomogeneities. They cause contact angle hysteresis, considered as resulting from
pinning of the liquid contact line. However, even on molecularly smooth surface some
amount of hysteresis is found too. During past three decades numerous papers have
been published in which different theoretical models are described to explain nature
of the contact angle hysteresis. Among them, differently patterned model surfaces
were considered. Lately investigations dealing with the hysteresis, including
superhydrophobic surfaces, are of great interest (1-8). It is now well known that
superhydrophobicity appears on hydrophobic surfaces on which micro- and/or nanoprotrusions are present. The superhydrophobic surfaces are commonly characterized
by water contact angle, which has to be at least 150o or larger. Now also the receding
contact angles are more often reported in some papers and two types of wetting
/dewetting processes are considered. Namely, the drop can be suspended on the posts
(i.e. the Cassie-Baxter case) or it fills the interstices between them, which is Wenzel
case (9-11). Some experiments were performed that showed on arranged posts the
advancing contact angle did not change, while the receding one depended on the post
size and spacing (6). Generally, it is considered that in case of suspended drop the
hysteresis can be smaller in comparison to the collapsed drop. However, in 2007 Gao
and McCarthy (12) published a provocatively titled paper: “How Wenzel and Casie
were wrong”, in which, basing on their experimental results, they concluded “that
contact angle behavior (advancing, receding, and hysteresis) is determined by
interactions of the liquid and the solid at the three-phase contact line alone and that
the interfacial area within the contact perimeter is irrelevant”. In consequence several
comments and replies have been published (13-15). For example Nosonovsky (13)
proposed generalized forms of Wenzel and Cassie equations derived by him and
5-18
claimed that they can be applied “when the size of the solid surface
roughness/heterogeneity details are small compared to the size of the liquid-vapor.”
Then Panchagnuala and Vedantman (14) in their comments maintain that Cassie
equation can be used if appropriate surface area fraction is considered, i.e. the fraction
that contact line experience during its advancing should be used in Cassie equation. In
reply, Gao and McCarthy (16) stated that those issues they had published were
actually “restatement of issues that have already been published” and that Wenzel and
Cassie equations “should be used with knowledge of their faults”. They also remained
their papers in which they considered contact line, and not the area fractions, that
helped understanding of hysteresis, the lotus effect and perfectly hydrophobic
surfaces (18,19). However, after that McHale (15) has published a paper entitled
“Cassie and Wenzel: were they really wrong?” in which he discusses the conditions
under which on “more randomly structured surfaces” Cassie-Baxter and Wenzel
equations can be applied. This is possible if the surface fraction from Cassie-Baxter
and roughness parameter from Wenzel equations are taken as global parameters of the
surface and are not defined to the contact area of the droplet. He also argues that local
form of these equations “allow patterning of the surface free energy”. In case of
superhydrophobic surface the local form of Cassie-Baxter equation can be applied for
the suspended on the posts droplet, then the apparent contact angle should result from
minimizing surface free energy by small displacements of the contacting line. If a
penetration of the liquid to the interstices between the posts takes place, then Wenzel
state occurs. Whyman et al. (3) present “rigorous derivation of Young, Casie-Baxter
and Wenzel equations”, in which they presume the free displacement of the triple line.
They have derived an equation relating the potential barrier energy of the line
displacement and the advancing and receding contact angles. The energy is
determined by the liquid adhesion and the solid roughness. Some interesting features
of the system resulted. Namely, the larger hysteresis the larger the energy is (the
Wenzel and Cassie states, respectively). The equation predicts low CAH values for
low contact angles. CAH does not depend on the equilibrium contact angle in a broad
range of the contact angles (50o – 140o), but in case of superhydrophobic surface the
dependence is important. The CAH value is not much dependent on the drop volume,
but in case of small it is, where CAH may overcome the value of equilibrium contact
angle. Finally, in case of a liquid having lower surface tension its CAH should be
larger in comparison to that of higher surface tension.
The cited above papers show that contact angle hysteresis is again a ‘hot issue’, which
now also involves superhydrophobic surfaces. However, the purpose of this paper is
not debate about mechanisms and theoretical description of the contact angle
hysteresis, but it deals with an approach proposed to characterize solid surfaces via
their apparent surface free energy as determined from the apparent contact angle
hysteresis and the liquid surface tension, including superhydrophobic surfaces too. It
is believed that such parameter may deliver more information than the advancing and
receding contact angles alone.
SOLID SURFACE FREE ENERGY
The problems of experimental determination and theoretical description of solid
surface free energy are well known. At present most often used approach is that
proposed by van Oss (20), in which the energy is considered as the sum of apolar
Lifshitz-van der Waals sLW (mostly London dispersion) and Lewis acid-base sAB
interactions, and the sAB is expressed by geometric mean of electron-donor s- and
5-19
electron acceptor s+ contributions. To determine the components one has to measure
contact angle of three probe liquids, two of which have to be polar. Usually for this
purpose diiodomethane (as an apolar liquid), water and formamide are used, for which
arbitrary components of their surface tension are determined assuming for water at
room temperature equality of its acidic and basic interactions, i.e. s+ = s- = 25.5
mJ/m2. However, using different sets of three probe liquids than those mentioned
above the determined for the same solid surface components of the same kind differ
more or less. Among other reasons, it seems important is the ‘hidden assumption’ that
the interactions coming from the solid surface are of the same strength irrespective for
the liquid used. This seems to be debatable, because the acid-base interactions are
mostly due to hydrogen bonding, whose strength varies depending on the origin of H
and O, N or F atoms. They can range from 1-2 kJ/mol to 155 kJ/mol. For example O–
H:O (21 kJ/mol) and N–H:O (8 kJ/mol). Therefore the values obtained from van
Oss’ method are averaged interactions of the solid surface, which are also relative
because of the above assumption for water hydrogen bonding interactions. Actually,
so far we do not have any direct and thermodynamically well supported method for
solid surface free energy determination, and specially its components. This problem
was discussed in details by Lyklema (21). Nevertheless, even these imperfect methods
of the energy determination are useful to track its changes for a surface being in
question.
Recently a new approach to determination of the apparent solid surface free
energy stot has been proposed (22-25). It allows evaluation of the energy form the
advancing a and receding r contact angles of one liquid only, whose surface tension
is l. The equation reads:
 l (1  cos  a ) 2
 stot 
(1)
(2  cos  r  cos  a )
The detailed discussion about its validity has been published elsewhere [23-25]. It
should be mentioned here that the surface energy evaluated with the help of Eq.(1)
depends to some extent on the kind of probe liquid used and the kind of solid surface.
The reasons for this were discussed above. Originally it was considered that when the
three-phase contact line had receded a film of the liquid is left behind the drop, which
reflects in the receding contact angle. The presence of such a film is shown now by
Bormashenko et al. (26). In this paper Eq.(1) is applied for testing apparent surface
free energy changes in aspect of superhydrophobic surfaces. Some simulated results
as well as experimental ones based on the literature contact angles are presented.
SIMULATED CALCULATIONS OF APPARENT FREE ENERGY CHANGES
FOR HYDROPHOBIC AND SUPERHYDROPHOBIC SURFACES
Figure 1 shows changes in the apparent surface free energy of a superhydrophobic
solid surface calculated from Eq.1 as a function of the advancing contact angle of
water at different values of the contact angle hysteresis. The range of advancing
contact angles used is from 120o to 170o, and two sets of the contact angle hysteresis
are examined, i.e. a low 5o and 10o, and a large one 50o and 60o. Although the
apparent surface free energy at these contact angles is generally small, especially for
adv > 150o (superhydrophobic surfaces) its changes are distinguishably different for
these two sets of CAH. At the same advancing contact angle and higher hysteresis
(50-60o) visibly lower than at low CAH = 5-10o. Thus, the Cassie-Baxter and Wenzel
cases of the superhydrophobic surfaces are clearly depicted.
5-20
Next, Fig.2 presents changes in the apparent surface free energy as a function of
the hysteresis for several values of the advancing contact angles, again starting with
the angle of 120o. For such a solid if no hysteresis appears, the apparent free energy of
its surface interacting with water is about 18 mJ/m2, which is characteristic for Teflon
surface. If the hysteresis is as large as 70o, the energy drops to about 10 mJ/m2. If the
advancing contact angle is 150o, for zero-hysteresis case the energy amounts only 5
mJ/m2 and decreases to ca. 1.5 mJ/m2 at 70o CAH.
2
3.5
3.0
2.5
10
10
50
o
60
o
o
2.0
1.5
1.0
0.5
0.0
8
150
155
160
165
170
Advancing contact angle, degrees
6
14
12
150
2
0
0
150
160
o
1
160
o
170
0
0
10
20
30 40 50 60 70
Hysteresis, degrees
A = 120
6
2
140
2
80
o
90
100
o
8
4
130
3
10
4
120
4
tot
2
Apparent s , mJ/m
12
tot
16
tot
Apparent s , mJ/m
14
o
Hysteresis
o
5
o
10
o
50
0
60
Apparent surface free energy,s
4.0
5
5
18
4.5
2
16
170
130
o
140
150
0
10
20
30
Advancing contact angle, degrees
40
50
60
70
o
o
80
o
160 o
170
90
100
Hysteresis, degrees
Fig. 1. Apparent surface free energy of a
solid surface versus water advancing
contact angle and different the contact
angle hysteresis.
Fig. 2. Apparent surface free energy of a
solid surface versus water contact angle
hysteresis and different advancing
contact angles.
The general feature of apparent energy vs. hysteresis relationship is the energy
decrease with increasing CAH. However, the relative decrease of the energy depends
strongly on the advancing contact angle value. This is depicted in Fig. 3. As can be
seen, with increasing advancing contact angle the apparent surface free energy
drastically decreases at the same CAH. For example, for A = 120o and CAH = 10o
the energy decreases by 13.6 % only relative its value at zero hysteresis. However, if
A amounts 170o, at the same hysteresis the energy decrease reaches nearly 60 %.
Obviously the decrease in absolute values is large in the former case, i.e. from 18.2 to
15.7 mJ/m2, and from 0.55 to 0.22 15.7 mJ/m2, respectively. This figure also well
illustrates differences between the two cases of wetting, that is suspended or collapsed
drops, for hydrophobic and superhydrophobic surfaces.
0
tot
Relative decrease in s , %
tot
20
Hysteresis, A - R
Apparent s , mJ/m
18
10
20
30
40
A = 120
50
o
130
60
140
70
o
o
150
80
160
90
o
o
170
o
100
0
10
20
30
40
50
60
70
80
Contact angle hysteresis, degrees
Fig. 3. Relative decrease in the apparent surface free energy of a solid versus water
contact angle hysteresis and different values of the advancing contact angles.
5-21
APPARENT SURFACE FREE ENERGY CHANGES OF REAL
SUPERHYDROPHOBIC SOLIDS
For these calculations of apparent surface free energy changes the literature
contact angle data from Öner and McCarthy paper (2) were used. The contact angles
were measured on silane-modified silicon wafer surfaces on which square post of
different sizes and 40 m height were produced by photolithography. The posts were
hexagonally arrayed at the distances equal to their size. Then three different silanes
were used; dimethyldichlorosilane (DMDCS), n-octyl-dimethylchlorosilane
(ODMCS),
and
heptadecafluoro-1,1,2,2-tetrahydrodecyl-dimethylchlorosilane
(FDDSC). The contact angles taken from the paper, together with the contact angle
hystereses are listed in Table 1, where there are also listed contact angles and the
hysteresis measured on smooth surfaces covered with the silanes.
Table 1. Advancing and receding contact angles of water and their hystereses
measured on silanized silicon surface possessing different size square posts.
silicon
DMDCS
surface
A/R,
with posts deg
ODMCS
hysteresis
deg
A/R,
deg
hysteresis
deg
A/R,
deg
hysteresis
deg
smooth
2 m
8 m
16 m
32 m
64 m
128 m
5
35
39
27
26
58
36
102/94
174/141
173/139
174/134
170/132
114/64
95/58
8
33
34
40
38
50
37
119/110
170/146
170/140
168/145
170/146
149/100
131/93
9
24
30
23
24
49
38
107/102
176/141
173/134
171/144
168/142
139/81
116/80
FDDCS
Apparent surface free energy, mJ/m2
As can be seen, the contact angles and hystereses sharply increase for the surfaces
possessing the posts in comparison to the smooth surfaces. For the posts between 2
m and 32 m both the contact angles and hystereses values are similar. The greatest
values of the hysteresis appear if 64 m posts are present. However, from the contact
angles and the hystereses it is difficult to distinguish any differences between these
three surfaces. But, much clearer picture is obtained if the apparent surface free
energies are calculated from Eq. (1). Thus calculated energies are presented in Fig.4.
30
1
25
20
15
2
10
3
5
0.4
0.3
1- ODMCS
2- DMDCS
3- FDDCS
0.2
0.1
0.0
0
20
40
60
80
Size of square post, m
100
120
Fig. 4. Changes of apparent surface free energy of silane-modified silicon surface
possessing different size square posts 40 m high. Three different silanes were used
for the surface modification (see text).
5-22
In the figure there are also shown the energy values calculated for smooth surfaces.
Depending on the kind of the silane used the free energy of smooth surface ranges
from 16.5 to 26.5 mJ/m2. The greatest apparent free energy possesses the smooth
surface covered by n-octyldimethylchlorosilane –Si(CH3)2{(CH2)7CH3}, while the
lowest one the smooth fluorinated surface covered with Si(CH2)2{(CF2)7CF3}. In
fact the energy of the surface covered by =Si(CH2)2 (DMDCS) is very close (24.3
mJ/m2) to that covered by ODMCS. However, if the posts have been produced on the
surface, whose cross-sections are between 2 and 32 m, the apparent surface free
energy drops to extremely low values, almost to 0 - 0.15 mJ/m2. If the post sizes are
64 -128 m the energy increases, but still it is lower than that of smooth surface,
respectively. Fig.4 also shows that in the case of 40 m high posts their cross sections
up to 32 m are not important for the superhydrophobic effect. Moreover, from
practical point of view also the kind of silane is not important for the resulting
apparent surface free energy. But, this is not the case if a larger size posts are present
on the surface, and the energy increase is the largest in the case of noctyldimethylsilane, where for the posts 128 m it approaches the value of the smooth
surface. This must be due to the presence of relatively long n-octyl chains, because
the energy of the surface with only dimethyl groups is much lower, if 128 m posts
are present (DMDCS, Fig.4).
0.18
ODMCS surface
1- 16x16 m
2- 32x32 m
Surface free energy, mJ/m
2
0.16
0.14
0.12
0.10
0.08
2
0.06
0.04
0.02
1
0.00
20
40
60
80
100
120
140
Post height, m
Fig. 5. Apparent surface free energy changes of silicon/ODCMS covered surface
possessing square 16  16 m or 32  32 m posts of different heights.
Effect of the post height on the apparent surface free energy can be also
interesting. Using the contact angles published by the same authors (2) , which were
measured on square posts 16  16 m or 32  32 m but of different height, on the
surface covered by ODCMS silane, the calculated apparent surface free energy is
presented in Fig. 5. As could be expected from the contact angle values the energy
values are extremely low, but they are a bit large for the large-surface posts. With
increasing height of the posts the energy changes are not monotonic. The fluctuations
could be ascribed to the experimental error of contact angle measurements. On the
other hand, the changes of the energy on these two surfaces run in a very similar way,
but shifted each another. Therefore one can consider them as real ones being due to
tiny changes in the wetting mechanism.
To show usefulness of Eq.(1) for investigation of the apparent surface free energy
changes caused by a solid surface treatment, below some other results are presented.
Fig. 6 shows changes of the energy of a silicon surface grafted with different silanes
(27). As can be seen, in the case of dimethylchlorosilane in the range of 1 to 18
carbon atoms the energy is practically independent on the R chain length. However, it
strongly depends in the case of dichloro- and trichloro- silane. But, if the R chain
5-23
consists of 6 and more carbon atoms the apparent surface free energy is in fact very
similar for all three silanes used (Fig.6). Next, in Fig. 7 are plotted the energy changes
of gold and glass covered from the gaseous phase with poly(-methylstyrene)
possessing average molecular weight 685 or 1300 (28). Again, the curves show that if
the film thickness is 100 nm and more the kind of substrate on which the film is
deposited has no meaning for the film surface energy. The energy is about that
characteristic for a solid paraffin wax. It is worth to pay attention that the apparent
surface free energies for the silicon grafted with the silanes (Fig. 6) and sufficiently
thick poly(-methylstyrene) films (Fig. 7) are comparable.
40
Apparent surface free energy, mJ/m
2
Apparent surface free energy, mJ/m
2
Silicon grafted with the silanes
38
- R(CH3)2SiCl
36
- R CH3 SiCl2
34
- RSiCl3
32
30
28
26
24
22
20
38
Mn=1300, on gold
36
Mn =1300, on glass
34
Mn=685, on glass
32
30
28
26
24
22
20
0
2
4
6
8
10
12
14
16
18
20
Number of carbon atoms in R chain
0
50
100
150
200
250
300
PMS film thickness, nm
Fig. 6. Apparent surface free energy of
silane grafted silicon surfaces as a
function of the number of methylene
groups in the R chain.
Fig. 7. Apparent surface free energy of
gold and glass surfaces covered with
poly(-methylstyrene) film depending on
its thickness.
It may be also interesting to learn about apparent surface free energy of the same
surface, but determined from the contact angles of different liquids, for example
highly polar water, nonpolar n-hexadecane, and almost nonpolar diiodomethane. Such
results are presented in Fig.8 for silicon surface grafted with 12 different
monochlorosilanes (1). They possessed various aliphatic hydrocarbon chains (for
details see ref. 24). If hexadecane was used as the probe liquid, the apparent free
energy is the same for the all surfaces, about 25 mJ/m2, which results from London
dispersion interactions. If diiodomethane was the probe liquid, the apparent surface
free energy is bigger and slightly depends on the alkane chains structure, and it ranges
from 33 to 40 mJ/m2. While the surface tension of n-hexadecane is 26.4 mN/m, so
that of diidomethane amounts 50.8 mN/m. This liquid may also slightly interact as the
electron-acceptor, if in contact with an electron-donor site on bare silicon surface
patches. However, the apparent energy calculated from water contact angles strongly
depends on the silane structure bonded to the silicon surface. Except for two silanes
(Fig.8), the energy values are grouped into two groups, 21-26 mJ/m2 and 38-42
mJ/m2. The structure of the silanes within each of these two groups have some
similarities (see the table at Fig.8). The higher values of the energy indicate that the
film structure is not very compact and some water can penetrate to bare silicon
surface, and the lower values of the energy suggest a tight structure of the films. The
lowest value of the energy, 15 mJ/m2, is obtained for the surface possessing exposed
three methyl groups, ‘stretching’ from the surface at a distance of three –C–. The
groups interact with water by the dispersion forces only, and probably they well
5-24
Surface free energy, mJ/m
2
screen the silicon surface from water molecules. It is worth to note that the dispersion
interactions of water (21.8 mJ/m2) are weakest ones among these three probe liquids.
It is also interesting that the silanes possessing long aliphatic chains, –C8H17, and –
C18H37 show the highest apparent surface free energy.
45
Water
///-Si
///-Si
CH2I2
40
///-Si
///-Si
///-Si
35
n-C16H34
///-Si
///-Si
30
///-Si
///-Si C8H17
///-Si C18H37
25
///-Si
///-Si
20
15
20
25
30
35
40
45
50
55
2
Liquid dispersion surface free energy, mJ/m
Fig. 8. Apparent surface free energy of silicon surface grafted with different
chlorosilanes, whose structures are shown. The values are calculated from advancing
and receding contact angles of water, n-hexadecane and diidodomethane.
Finally, it may be interesting to depict the relationship between total apparent
surface free energy (calculated from the contact angle hysteresis) and the dispersion
part of the energy sd, calculated from Fowkes equation and the advancing contact
angle a.
WA = l (1+cosa) = 2(sd ld)1/2
(2)
Where WA is the work of adhesion, and ld is the dispersion component of probe
liquid surface tension. In case of an apolar liquid its total surface free energy l equals
ld. The discussed above stot and sd values for the same silanes are plotted in Fig.9,
where the contact angles of n-hexadecane and diiodomethane were applied. It is
obvious that sd component cannot be larger than the apparent total surface free
energy stot. In the case of apolar probe liquid the relationship between total apparent
surface free energy of the solid and its dispersion component reads (24, 25):
γ tot
γ sd  s 2  cos θ a  cos θ r 
(3)
4
From Eq.3 results that if a = r = 90o, sd should amount ½ stot, while if a = r = 0o,
sd would be equal to stot.
As is seen in Fig.9 the dispersion component determined from both hexadecane
and diiodomethane advancing contact angles is smaller than the apparent surface free
energy determined from the appropriate contact angle hysteresis. If the energies were
determined from hexadecane contact angles the differences between stot and sd are
smaller (0–1.5 mJ/m2) than corresponding differences if the energies were calculated
from diiodomethane contact angles (5-8 mJ/m2). This must result from the different
strength of the interactions of these two apolar liquids, whose surface tensions are
26.4 mN/m (hexadecane) and 50.8 mN/m (diiodomethane).
5-25
27.0
26.0
25.5
///-Si
2
///-Si
3
25.0
d
24.5
s
24.0
s
2
Surface free energy, mJ/m
1
From n-C16H34 contact angles
26.5
4
///-Si
///-Si
tot
5
///-Si
6
///-Si
7
///-Si
23.5
23.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
40
From CH2I2 contact angles
38
8
///-Si
36
34
9
///-Si
10
///-Si
32
30
28
26
24
0
1
2
3
4
5
6
7
8
9
10
11
s
d
s
tot
12
11
///-Si C18H37
12
///-Si C8H17
13
Type of silane
Fig. 9. Apparent total surface free energies and their dispersion components of the
silanes bonded to silicon surface determined from n-hexadecane and diiodomethane
contact angles.
CONCLUSIONS
Although hydrophobic and superhydrophobic surfaces can be characterized by the
value of water advancing contact angle, their wetting properties are better depicted if
the apparent surface free energy is calculated, where also the receding contact angle is
applied. This also allows to distinguish between Cassie and Wenzel cases of the
surface wetting, as well the time-dependent changes in the energy can be evaluated.
REFERENCESS
1. Fadeev Y, A, McCarthy T, J: ‘Trialkylsilane Monolayers Covalently Attached to
Silicon Surfaces: Wettability Studies Indicating that Molecular Topography
Contributes to Contact Angle Hysteresis, Langmuir, 1999, 15, 3759-66.
2. Öner D, McCarthy T, J: ‘Ultrahydrophobic Surfaces. Effects of Topography Length
Scales on Wettability’, Langmuir, 2000, 16, 7777-82.
3. Whyman G, Bormashenko E, Stein T: ‘The rigorous derivation of Young, CassieBaxter and Wenzel equations and the analysis of the contact angle hysteresis’,
Chem. Phys. Lett. 2008, 450, 355-359.
4. Gao L, McCarthy J: ‘”Artificial Lotus Leaf” Prepared Using a 1945 Patent and a
Commercial Textile’, Langmuir, 2006, 22, 5998-00.
5. Li W, Amirfazli: ‘A thermodynamic approach for determining the contact angle
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