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Debate - Discussion
Comments on ‘‘An Essay on Contact AngleMeasurements’’ by Strobel and Lyons
Michaela Muller,* Christian Oehr
The potential of contact angle measurements (CAM) as an analytical tool to characterizesurface treatments or modifications is often not fully exploited. Agreeing with Strobel andLyons, comparing contact angles is oftenmuchmore reasonable than comparing deduced datalike surface energies, because the latter are based on models, in turn involving the influenceand knowledge of intermolecular forces at the respective interfaces. For a comprehensivepicture, the measurement of contact angles itself has to be considered together with theappropriate model and the available techniques to carry out CAM. An appropriate measure-ment procedure will be given and a brief discussion of some models to derive free surfaceenergy from CAM.
Preliminary Remarks
In the technical community that deals
with surface modification of materi-
als, contact angle measurements
(CAM) are an important and a power-
ful tool for evaluating effects intended
by any surface treatment. Unfortu-
nately, there are different customs for
carrying out such measurements, on
the one hand,while on the other hand,
different theoretical approaches are
developed for calculating the surface
energy from CAM. Both the use of
different measurement procedures, as
well as the use of different models for
M. Muller, C. OehrFraunhofer Institute for InterfacialEngineering and Biotechnology,Nobelstrasse 12, 70569 Stuttgart,GermanyFax: þ49 711 9704200;E-mail:michaela.mueller@igb.fraunhofer.de
Plasma Process. Polym. 2011, 8, 19–24
� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe
surface energy calculations, often
result in apparent incomparability of
results from different laboratories. In
addition, some of the published mea-
surements are neither useful for dedu-
cing application-relevant properties
(where knowledge of the equilibrium
contact angle is sufficient) nor for
calculating surface energies. These
are the facts about which Strobel
et al. (this issue) complain.
With this present paper, we would
like to support the opinion that the
potential of CAM is often not fully
exploited and, regarding the under-
lying models, deductions are some-
times drawn prematurely. For a com-
prehensive picture, the measurement
of contact angles itself has to be
considered together with the appro-
priate model and the available tech-
niques to carry out CAM. We will first
focus on the contact angle measure-
ment, describing the procedureweuse
and will then briefly discuss some
im wileyonlinelibrary.com
models how free surface energy can be
derived fromthemeasurements. Thus,
the first part of this contribution is
dedicated to CAM, and the second to
the calculation of free surface energy.
Part I Contact AngleMeasurement (CAM)
To underline the problem, a short
description of CAM is now summar-
ized: Placing a droplet of any liquid on
a solid surfacewill in all cases result in
a contact angle between the two,
provided the respective surface ten-
sions of the materials involved are
different. But, it is not only the surface
tensions that determine the resulting
contact angle, because the latter is
additionally influenced by inhomo-
geneities in chemical composition
and/or structure (e.g., roughness) at
the contact line of the three phases
(solid–liquid–gas). Beside this, the
DOI: 10.1002/ppap.201000115 19
20
M. Muller, C. Oehr
purity of test liquid, electrostatic
charge at the surface, as well as
humidity in the gas phase, and sample
preconditioning, have to be controlled.
The inhomogeneities are, by defini-
tion, not representative of the whole
surface area of interest. The actual
contact angle that represents the solid
phase contains information relating
only to 10nm laterally and often less
than 1nm in depth. Thus, it can be
readily understood that uniformity
based on such a restricted volume
cannot be extrapolated for extended
surface areas of technical materials.
For comparison, small-spot XPS draws
its information from volumes that are
10mm laterally and a few nanometers
in depth. Thus, CAM is ‘‘more sensi-
tive’’ to differences in surface chem-
istry and structure, and it should be
taken with comparable care. In our
opinion, one should carry out CAM as
accurately as XPS measurements, for
example, by controlling the (liquidand
gas) phases in contact with the exam-
ined (solid) surface, by avoiding char-
ging, by considering suitable reference
materials, etc. In contrast with other
refined analytical techniques, contact
angles can be measured quite easily,
and people do so; we arewell aware of
this courseofaction fromourownfield
of plasma application: A glow dis-
charge is created with ease, and in all
cases leads to somemeasurable results
on anygiven surface. A complete set of
experimental parameters should be
controlled and quoted if any compar-
ison is to take place, but for the case of
plasma treatments this is often not
done, or at least not accurately docu-
mented.
Formeaningful CAM, the conditions
of the subsequent application as well
as temperature, electric fields, and
surface charges have to be taken into
account; furthermore, polymeric sur-
faces (reorientation of surface func-
tionalities) will adapt their surface
properties according to the environ-
ment, if the material is not highly
cross-linked. Due to this complex
situation it is of importance to mea-
Plasma Process. Polym. 2011, 8, 19–24
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sure the static advancing and receding
contact angles, which represent the
upper and lower limits of the ‘‘true’’
(meaning ‘‘equilibrium,’’ or Young’s)
contact angle, a value which lies
between the two (near the advancing
angle). In addition, it is generally
agreed that the advancing angle is
more strongly influenced by low-
energy domains in the surface, while
the receding angle is more influenced
by domains of higher surface energy.
This, again, represents valuable infor-
mationformanyapplicationareasand
it should be used. If the observed
contact angle hysteresis is small, this
signals that the surface is quite homo-
geneous with respect to chemical
composition and/or structure. The
interpretation of contact angle hyster-
esis must be done carefully since also
regular nanostructured surfaces can
show very low contact angle hyster-
esis, e.g., those which follow Cassie–
Baxter regime of wettability.
Strobel and co-workers[1] prefer the
Wilhelmy plate technique, which of
course is less ‘‘operatordependent,’’ but
with this method both sides of the
given sample must first undergo an
identical treatment, something that is
oftennot guaranteed. Second, theback-
side of the second sample is supposed
to have exactly the same composition
and structure as the untreated mate-
rial, but that is also not easy to
guarantee. Third, this method is
restricted to flat sheets or single fibers,
but it is not applicable to three-dimen-
sional samples. Fourth, fortunately an
often-fulfilled requirement is that the
contribution of the edges be insignif-
icant. Nevertheless, all of the methods
used do have some difficulties or other.
Therefore, it is of great importance that
the measurement procedures be used
in a standardized way if meaningful
comparisons are intended.
In our laboratory, we proceed with
the sessile drop technique as follows:
Prior to measurements, the samples
are storedandequilibratedunderwell-
defined humidity conditions. In the
case of insulating materials, possible
im
electrostatic charges are neutralized,
e.g., by treatment with an antistatic
gun.Asmall dropletof the selected test
liquid is dosed at the tip of a hollow
needle connected to a syringe and
brought into contact with the test-
surface. The droplet volume is then
carefully raised to about 10mL, so that
the contact line moves across the
substrate surface. This addition of
liquid is halted and when the contact
angle has stablilized, it is taken as the
value of advancing contact angle,
considering both sides of the projected
drop shape. Care must be taken to
avoid possible disturbance of the drop
shape by the inserted needle. In some
cases, the needle must even be with-
drawn prior to CAM. Then the liquid is
withdrawn until no further change in
the value of contact angle is observed,
and this is the measured receding
contact angle. This procedure is carried
out for each sample and each test
liquid at aminimumof three-different
positions on the sample’s surface.
Sometimes, one can observe different
angles on either side of the projected
droplet due to local differences in
surface composition and/or structure.
Therefore, we take both angles (left
and right side of the droplet) and use
the arithmetic mean value of 2. In
some cases, the duration between the
instant when the droplet volume no
longer changes to that when the
contact angle is read is also important,
especially when kinetic hysteresis has
to be considered. In this case, some
possible dissolution processes at the
surface or some swelling must be
taken into account. Such situations
should be avoided, because the final
measuredangleswill notbe stable and
true equilibrium will not have been
reached. Therefore, the test liquids
have to be chosen in a proper way
such that no swelling, dissolution, or
reorganization at the solid surface
occurs, at least none that are relevant
to the time-frame of the measure-
ment’s duration.
For surfaces which are employed in
contact with humid or fluid environ-
DOI: 10.1002/ppap.201000115
Comments on ‘‘An Essay on Contact Angle Measurements’’ . . .
ments, and/or which are subject to
thermodynamicandkinetichysteresis
(surfaceswhichareoftenof interest for
biomedical applications), the captive
bubble method[2] should be used,
instead. These samples are completely
immersed in the test liquid, with the
side to bemeasured facing downward.
An air bubble is then brought in
contact with the solid surface from
below. Depending on the wettability,
the bubble must be fixed in place by
the position of the needle. After a few
seconds, the static contact angle near
the triple phase line ismeasured.With
this method only the displacement of
liquid from the solid–liquid interface
by air can be analyzed, which corre-
sponds to the receding contact angle
by the sessile drop method.
Unfortunately, lessand lessworkers
now resort to measurements of both
advancing and receding contact
angles. Instead, they prefer to take
only readings (at least, hopefully) at
different surface locations; neverthe-
less, they obtain more-or-less stochas-
tic values between those of the advan-
cingand recedingangles, but these can
oftencover ranges that farexceed10 8Con technical surfaces. Unfortunately,
in Germany this tendency is encour-
aged by precisely those authorities
that define industrial standards (DIN).
In their recently published normative
papers,[3] they advise users to employ
static contact angle devices to observe
sets of ten droplets and to calculate a
mean angular value from these data,
with no mention whatever of advan-
cing or receding angles. It is merely
stated that within an area of
10 cm� 10 cm, three droplets should
Table 1. Important molecular interaction for
Interaction force
Coulomb
Keesom–van der Waals
Debye–van der Waals
London–van der Waals
Hydrogen bonding
Plasma Process. Polym. 2011, 8, 19–24
� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe
be placed and thus three (randomly)
taken angles should be measured.
With this type of developments in
mind, it is important to restate the
meaning and utility of CAM from time
to time.
An often not considered topic is that
which deals with the segment-size of
sample surface responsible for the
observed contact angle.[4] This should
be clarified before certain ideas regard-
ing the influence of dimensions of
inhomogeneities be discussed. Again,
the contact angle represents only the
equilibriumof forces at the triple-phase
contact line,andnotofanysurfacearea.
In order to characterize an area, the
contact linemust bemoved, preferably
at very low capillary numbers, and the
angle has to be measured during this
constantmovement.Models have been
developed for this case, and they have
been used in industry experienced in
thedepositionof thinorganicfilms, e.g.,
for photographic purposes.[5] Here, the
dynamic advancing anglewould be the
appropriate one for the calculation of
surface energy.
On the microscopic level, the con-
tact line is determined by the inter-
molecular forces (Table 1) acting at the
point were the three phasesmeet, and
it is influencedby therangeofactionof
these forces. Beside the Coulomb
interaction, which scales with r�1
(r being the distance between inter-
acting molecules/atoms), all other
forces of interaction scale with r�6
and are therefore very short-ranged.
The interrelation between line energy,
advancing, receding, and Young’s con-
tact angles has recently been well
described in a paper by Tadmor.[6]
ces (r represents intermolecular distance).
Type of interaction
Electrostatic interaction
Dipole–dipole interaction
Dipole-induced dipole interaction
Dispersive interaction
im
In summary, measured contact
angles are dependent on how the
measurement is carried out. To com-
pare results of different working
groups it is therefore essential to
describe the measurement conditions
andprocedure indetail. Eachmeasure-
mentmethodmentioned, sessile drop,
tilting angle, and Wilhelmy plate
(static or dynamic), has some merits
and some disadvantages. But, if mea-
surements are carried out in a stan-
dardizedmanner, theywill be compar-
able. In order to obtain valuable
information about a surface by CAM,
the advancing and receding angles
from several locations on the surface
should be recorded, so as to ensure
some statistical significance. The
operator should be aware that his
measurement is even more surface-
sensitive than comparable techniques
like XPS, etc. and that sample prepara-
tion therefore has to be done at least
with the samedegree of accuracy. If all
this is respected, papers published in
Plasma Processes and Polymers will
maintain the high ranking of the
journal. Accurate measurements with
some statistical validation are in all
cases sufficient if the main topics of a
paper are depositionof filmsor surface
treatment and their application, on
the one hand, and demonstration of
the potential of plasma processes to
createnewsurfaces, on theotherhand.
Part II Surface Free Energy
If thermodynamic properties and
molecular kinetics of surfaces are the
focus of a paper, then certain addi-
Interaction range
�r�1
�r�6
�r�6
�r�6
�r�6
www.plasma-polymers.org 21
22
M. Muller, C. Oehr
tional aspects have to be taken into
account. In such a case, contact angles
are used to derive some material
properties that may be expected from
theoretical considerations. Thus, the
surface tension of a solid material can
be derived by starting with a macro-
scopic thermodynamic approach (top–
down approach), as well as by con-
sidering intermolecular forces on a
microscopic level (where both bot-
tom–up and top–down approaches
are used). Accordingly, to the authors’
knowledge, these different concepts
are not unified and are the objects of
controversial discussion. This situa-
tion explainswhy, for example, for the
samemeasured contact angles and for
the same triplet of (gas–liquid–solid)
materials that coincide at a contact
line, different surface energy values
might be calculated. This is due to the
use of different algorithms with dif-
ferentbasic analytical concepts,which
may result in different values of sur-
face free energies. The reason lies in
the dilemma that the surface tension
of a solid is not directly accessible. To
decide which calculation model is
more appropriate than the others,
one has to make assumptions regard-
ing intermolecular forces acting at
interfaces. The main molecular inter-
action forces are London-, Debye-,
Keesom-, Coulomb-, and hydrogen
bonding forces already mentioned in
Table 1. These forces originate from
chemical entities (OH-groups, CF3-
groups, etc.) at the interface, which
contribute to the total surface free
Table 2. Concepts to separate contributions
Concept Split-up of fr
Fowkes[11] gtotal ¼ gdispersiv
van Oss et al. [13] gtotal ¼ gLifshitz�
Plasma Process. Polym. 2011, 8, 19–24
� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe
energy. To derive a complete picture
and to calculate amacroscopic contact
angle, the chemical functions respon-
sible for these forces should be known,
more specifically their densities, their
distributions, and orientations at the
surface. Furthermore, one has to take
into account the additional fact that
technical surfaces are rough, at least
on the nanometer scale, and that the
mentioned functionalities will be
mobile, depending on cross-linking/
mobility of the supporting network
structure. One can readily envisage
that complete information regarding
surface composition and structure is
usually not available.
Since the beginning of the past
century, going back to Einstein’s first
paper,[7] when people were convinced
it was possible to deduce macroscopic
behavior via a bottom–up approach
starting with a material’s stoichiome-
try, several attempts were made to
calculate surface tension.One concept,
developed during the 1920s by Sud-
gen,[8] defines the so-called parachor,
which correlates the density differ-
ence of two phases with the surface
tension. The parachor of a molecule,
which has the dimensions of volume,
was divided into volume equivalents
of segments of the molecule. This
approach was successful in a practical
way, but its theoretical foundation
was very weak. Due to strong criti-
cisms[9] in the 1960s, this approach
now appears to have been forgotten,
but a comprehensive overview on this
topic has appeared[10]; however, other
of molecular interaction forces to the surface f
ee surface energy g Commentary
e þ gpolar gdispersive is th
Waals forces
subsumes all
vanderWaals þ gacid�base gacid�base cont
(electrostatic
gLifshitz�vanderW
Debye- and Lo
im
molecularly oriented approaches,
dividing a calculated surface energy
in portions due to intermolecular
forces, seem to be more effective.
Thereby, surface energy is divided into
dispersive and polar components
(Fowkes,[11] Girifalco/Good, Owens/
Wendt[12]); later, Fowkes added Lif-
shitz/van der Waals and acid/base
components, the acid /base compo-
nent then being further analyzed by
van Oss et al.[13]
These top–down approaches result
in more and more complex mathema-
tical formulae (Table 2) and at least
three-different test liquids have to be
used to obtain the respective contribu-
tions to the overall value of surface
energy. Here, the main challenge is to
choose the appropriate organic liquids
that will not cause swelling of the
polymer surface, but inwhichall of the
various contributions (dispersive, acid
base contribution, etc.) nevertheless
differ among one another. Thus, the
liquids for the measurements must be
carefully chosen so as not to interact
with the solid, and must be used only
in a fresh state because any deteriora-
tion (e.g., due to oxidation, relevant for
the often-used methylene diiodide)
will change the test liquid’s polar
component. In Table 3 are shown the
methods we mainly use in our own
laboratory.
Our description up to this point is
based on the assumption that no
motions of molecules or functional
groups are involved on a scale of some
nanometers or belowat the interfaces.
ree energy.
e contribution of the London–van der
to surface free energy, and gpolar
other interaction force contributions
ains the strong interaction forces
and hydrogen bond interactions),
aals the much weaker Keesom-,
ndon-van der Waals forces
DOI: 10.1002/ppap.201000115
Table 3. Some frequently applied models for calculation of surface free energy.
Model/authors Equationsa) Commentary
Separation between polar and
dispersive contributions;
harmonic mean; Wu[14]
1þ cos#ð Þg l ¼ 42gd
sgdl
gds þ gd
l
þ 2gps g
pl
gps þ g
pl
!From our own experience,
suitable for non-ionic
surfaces with surface free
energies below ca. 35mNm�1
Separation between polar and
dispersive contributions;
geometric mean; Owens,
Wendt, Rabel and Kaelble[12]
1þ cos#ð Þg l
2ffiffiffiffiffigdl
q ¼ffiffiffiffiffigds
qþ
ffiffiffiffiffigps
q ffiffiffiffiffigpl
gdl
sFrom our own experience,
suitable for non-ionic
surfaces with surface free
energies above ca. 35mNm�1
Separation between
Lifshitz–van der Waals and
acid–base contributions;
van Oss et al. [13]
1þ cos#ð Þg l ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigLWs � gLW
l
qþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigþs � g�
l
pþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig�s � gþ
l
q� �Suitable for surfaces
with acid–base contributions
a)W, contact angle; g , surface (free) energy; subscripts l, s, surface energy of liquid and solid; superscripts d, p, LW, þ, �, dispersive, polar,
Lifshitz–van der Waals, acid, and base parts of surface energy.
Comments on ‘‘An Essay on Contact Angle Measurements’’ . . .
But this is not valid in many cases,
especially for polymer surfaces. A
polymer surface will rearrange its
structure and composition depending
on the properties of the contacting
phase, its chain mobility, internal free
volume, and on the temperature.
Beside these more and more com-
plex microscopic models, the purely
thermodynamic approach is also still
in use.[15]
Due to rapid advances in computer
science, elaborate simulations can
now be done, and as a consequence
researchers in that field attempt to
simulate macroscopic phenomena
starting with ab initio calculations at
the molecular level. Thus, contact
angles of liquids on plasma-deposited
fluorcarbon surfaces[16] were found to
be in good agreement with simula-
tions of ‘‘droplets’’ comprising some
1000watermolecules.[17] This simula-
tion is still far from being able to
describe a real droplet, but we hope to
obtain new insights about interac-
tions at solid–liquid interfaces via
molecular dynamics.
Finally, it should be stated that
phenomena become ever more com-
plex with regard to friction, hydro-
Plasma Process. Polym. 2011, 8, 19–24
� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe
dynamics, and molecular kinetics, if
one considers the physics of moving
wetting lines. For such an approach,
the reader is referred to the paper by
Blake.[5]
Summarizing, numerous attempts
have been made since more than a
century to calculate the surface free
energy from measured data. At the
present time, all of these approaches
still coexist, and each one is used in
more or less restricted areas. Consid-
ering that the portion of surface
science that deals with surface free
energy is important in several very
different fields, a variety of scientific
disciplines are involved and come to
prominence depending on the parti-
cular focal issue: ‘‘Scientists tend to
think in terms of their most familiar
models. It is not accidental that the
earliest descriptions of the moving
wetting line and its associated con-
tact angle were in terms of displaced
equilibrium (chemists), friction (phy-
sicists), and viscous bending of the
liquid–vapor interface (engineers and
mathematicians).[5]’’ This sentenceby
Blake characterizes the contemporary
situation in the field of surface free
energy.
im
Summary
We agree with Strobel and Lyons that
comparing contact angles is often
much more reasonable than compar-
ing deduced data like surface energies,
because the latterarebasedonmodels,
in turn involving the influence and
knowledge of intermolecular forces at
the respective interfaces. Especially
regarding the interface between
liquids and solids, it is difficult to
consider and assess all of the forces
present, and to avoid complicating
effects like swelling, penetration, or
dissolution at the nanometer scale.
Papers which do not consider
advancingandrecedingcontactangles
are less informative, and they fall far
short of the potential offered by the
CAM methodology.
We also agree that no additional
information is provided if the surface
energy is calculated from both advan-
cingandrecedingangle. Specifying the
algorithm one uses and a single
calculation suffices. Moreover, if the
sample material and the treatment
procedure are given, the reader may
himself choose the algorithm he pre-
fers.
www.plasma-polymers.org 23
24
M. Muller, C. Oehr
According to published findings, we
would now like to place procedures for
measuring contact angles into an
hierarchical order: The lowest-priority
procedure comprises measuring an
arbitrary-taken contact angle from a
sessile droplet. Such data should be
avoided altogether. The second level of
priority will be the use of more than
one droplet for statistical purposes
(that is, the DIN procedure). On the
same level is the use of ‘‘test inks’’ for
surface tensionmeasurements (a ‘‘hor-
ror’’ for people who think in terms of
theoretical concepts). This approach is
tolerable forprocess control, but just to
indicate relative changes. The third
priority level is the use of advancing
and receding CAM (at different loca-
tions on the surface). On the fourth
level of priority, at least two or three
(also a subject for discussion) different
liquids are used to formulate an
impression about dispersive and polar
contributions to the surface free
energy. The fifth level comprises the
use of multiple liquids in order to
separate the surface tension into its
constituent parts, whereby more
information related to interaction of
Plasma Process. Polym. 2011, 8, 19–24
� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe
the liquid with the solid becomes
available.
Finally, this branch of surface
science is still under development,
and it might be enhanced by enligh-
tened argumentation from different
sub-disciplines. Beside thermody-
namics, hydrodynamics (in case of
moving water contact lines), and
molecular kinetics can also shed some
additional light upon the observed
physical effects. Meanwhile, it may be
possible to carry out some new bot-
tom–up approaches.
Received: August 25, 2010; Revised:September 30, 2010; Accepted: October 1,2010; DOI: 10.1002/ppap.201000115
Keywords: advancing; calculation models;interaction forces; receding contact angle;surface free energy; wetting
[2] J. D. Andrade, R. N. King, D. E. Grego-
[1] J. Park, C. S. Lyons, M. Strobel, M. Ulsh,M. J. Kinsinger,M. J. Prokosch, J. Adhes.Sci. Technol. 2003, 17, 643.
nis, D. L. Coleman, J. Polym. Sci. Symp.1979, 66, 313.
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[3] DIN 55660-2: Paints and varnishes –Wettability – Part 2: Determination ofthe free surface energy of solid sur-faces by measuring the contact angle;Draft version 2009-07-06.
[4] L. Gao, T. J. McCarthy, Langmuir 2009,25, 7249.
[5] T. D. Blake, J. Colloid Interface Sci.2006, 299, 1.
[6] R. Tadmor, Langmuir 2004, 20,7659.
[7] A. Einstein, Ann. Phys. 1901, 4, 513.[8] S. Sudgen, J. Chem. Soc. 1924, 125,
1177.[9] O. Exner, Collect. Czech. Chem. Com-
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2008, 94(12), 1650.[11] F. M. Fowkes, J. Phys. Chem. 1962, 66,
382.[12] [12a] D. K. Owens, R. C. Wendt, J. Appl.
Polym. Sci. 1969, 13, 1741; [12b]W. Rabel, Farbe und Lack 1971, 77,997; [12c] D. H. Kaelble, J. Adhes.1970, 2, 66.
[13] C. J. van Oss, R. J. Good, M. K. Chaudh-ury, Langmuir 1988, 4, 884.
[14] S. Wu, J. Polym. Sci. 1971, C34, 19.[15] J. K. Spelt, D. R. Absolom, A. W. Neu-
mann, Langmuir 1986, 2, 620.[16] J. Barz, M. Haupt, U. Vohrer, H. Hilgers,
C. Oehr, Surf. Coat. Technol. 2005, 200,453.
[17] J. M. Knaup, C. Kohler, M. Hoffmann,P. H. Konig, T. Frauenheim, Eur. Phys. J.Spec. Top. 2007, 149, 127.
DOI: 10.1002/ppap.201000115
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