Capillarity at the Nanoscale: An AFM View

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1

Capillarity at the Nanoscale: An AFM View


F. Mugele*, T. Becker, R. Nikopoulos, M. Kohonen, and S. Herminghaus

Universität Ulm, Abteilung Angewandte Physik, D
-
89069 Ulm, Germany


Abstract

We have used Atomic Force Microscopy (AFM) to image liquid droplets

on solid
substrates. The technique is applied to determine the contact line tension. Compared
to conventional optical contact angle measurements, the AFM extends the range of
accessible drop sizes by three orders of magnitude. We analyze the global shape
of the
droplets and the local profiles in the vicinity of the contact line. These two approaches
show that the optical measurement overestimates the line tension by approximately
four orders of magnitude.




Keywords: contact angle, wetting, line tension,
atomic force microscopy, Young
equation, interface potential









Journal of Adhesion Science
Technology
16
, 951 (2002)



*corresponding author. Electronic address: frieder.mugele@physik.uni
-
ulm.de;

phone: ++49 731 502 2931; fax: ++49 731 502 2958.


2

In
troduction

Since its invention in 1985 [
1
], the atomic force microscopy (AFM) in a variety of
operation modes (contact mode, non
-
contact mode, tapping mode, scanning
polarization force microscopy) has become one of the most widely used experimental
tools i
n surface physics or surface chemistry laboratories. Its unique lateral resolution
has been used for characterization of surfaces in real space down to the atomic scale.
In the context of adhesion and wetting, it is mainly used to measure forces, to measur
e
the surface topography, and to characterize the quality and chemical composition of
substrates with specifically tailored wetting properties. Two specific operation modes
of AFM, namely scanning polarization force microscopy (SPFM) [
2
] and tapping
mode [
3
,
4
], have been applied successfully to image not only the substrates, but also
the topography of partially wetting liquid nanostructures. Both techniques have
revealed details about the molecular forces governing the equilibrium shapes as well
as the ads
orption and spreading of liquids on a scale that was not amenable to
conventional optical techniques.

One of the most heavily debated questions in the field of wetting deals with the
influence of molecular and interfacial forces on the equilibrium contact

angle


of a
liquid droplet with a finite base radius
R

[
5
,
6
]. The influence of the interfacial forces
translates into an excess free energy of the three phase contact line, the line tension

,
which was calculated to be on the order of
10
-
12

to
10
-
10

J/
m
[
7
,
8
,
9
]. In the presence
of a finite line tension, the contact angle of a liquid droplet depends on its size. In a
large number of conventional contact angle goniometry experiments (or its automated
version “axisymmetric drop shape analysis” [
10
]), vari
ations of


with
R

on the order
of 1° have been reported for millimeter
-
sized droplets. These variations were
interpreted in terms of a line tension of




10
-
5

… 10
-
6

J/m
[
5
,
6
,
10
,
11
], four to six
orders larger than predicted on theoretical grounds. Recently, Law and coworkers
obtained a value of




10
-
10

J/m
using optical interferometry [
12
]. In a previous
study, we used an AFM to inv
estigate liquid droplets on patterned substrates [
3
,
4
]. In
that work, we extracted a line tension of the order
10
-
10

J/m

from variations of the
local contact angle with the local
curvature of the contact line. In the present study,

3

we combine the conventional optical method of size
-
dependent contact angle
goniometry on the millimeter scale with AFM measurements using micrometer
-
sized
droplets on homogeneous substrates.

In additio
n to causing variations of the contact angle with droplet size,
interfacial forces are also expected to induce deviations of the droplet profiles from
the ideal spherical cap shape in the vicinity of the three phase contact line. In the
second part of the
present contribution, we analyze individual AFM profiles and
determine the upper limits on the magnitude of the interfacial forces and the line
tension. From both types of analysis we obtain an upper limit of




10
-
10

J/m
. In
particular, we will show that

values of




10
-
5

… 10
-
6

J/m
, as quoted above, are
incompatible with the high
-
resolution AFM measurements.

E
xperimental

For the experiments, we deposited droplets of hexaethylene glycol (HEG) on
silanized silicon substrates. This system was chosen, beca
use HEG has a sufficiently
low vapor pressure at room temperature to prevent any noticeable evaporation on the
time scale of the experiments (several hours). Furthermore, it has a relatively high
surface energy (

lv
=45

mJ/m
2
), which allows for stable imagi
ng conditions with the
AFM. Si substrates (purchased from Wacker) were used for their smoothness and
chemical homogeneity. They were covered with a native oxide layer (thickness
1

nm

<

d
SiO2

<

2

nm). The substrates were ultrasonically cleaned in ethanol an
d
acetone and subsequently left overnight in an oxidizing solution (NoChrom
ix™,
GODAX Laboratories, Inc., Takoma Park, MD, USA) to remove residual organic
contaminants. Each step was followed by extensive rinsing in Millipore water.
Homogeneous coverage of the samples with a molecularly thin layer of
phenyltrichlorosilane (PTCS)
was obtained by exposing the substrates to a saturated
vapor atmosphere for


2 hours and subsequent ultrasonic cleaning in ethanol.

Optical contact angle measurements were performed with a commercial contact
angle measurement system (OCA 15 plus, Dataphy
sics GmbH, Filderstadt, Germany)
using the sessile drop method. Images of the droplets were recorded with a CCD
-
camera. The shape of the liquid
-
vapor interface (droplet contour) was determined by a

4

computer program, which searches for the steepest intensit
y gradient in the images.
Contact angles are obtained by fitting the droplet contour using both Laplace
-
Young
and ellipsoidal fit functions.

To deposit the HEG droplets for the AFM measurements, we produced an
aerosol in a closed container using a standard

vaporizer (Fisher Scientific). The
droplets were allowed to settle down on the substrate by gravity. This procedure leads
to a distribution of droplets with typical diameters between 0.1

µm and 25

µm. AFM
experiments were performed in ambient conditions a
t a room temperature of


22°

C
and a relative humidity between 35% and 45%. Images were recorded with a
commercial stand
-
alone AFM (Bioscope, Digital Instruments Co., Santa Barbara, CA,
USA.) operated in tapping mode. We used Si cantilevers (Nanosensors)

with a typical
resonance frequency of 300 kHz and a tip radius below 10

nm, as specified by the
manufacturer. Before use, the tips were coated with a layer of a perfluorinated
alkylsilane ((Heptadecafluoro
-
1,1,2,2
-
tetrahydrodecyl)
-
dimethylmonochloro
-
silan
e;
ABCR GmbH & Co. KG, Karlsruhe, Germany) from the gas phase, to improve the
stability of the imaging conditions. Non
-
disturbing AFM imaging of liquids is a
delicate process, which requires high stability of the instrument, low damping and
suitable wettin
g properties of the liquid on the tip. We have described a detailed
model of the imaging process in the previous issues of this journal [
13
,
14
]. Briefly, it
involves the formation and rupture of a small liquid neck in each oscillation cycle of
the AFM canti
lever. The presence of the neck leads to an additional intermittent
attractive force close to the lower turning point of the cantilever oscillation. This force
gives rise to the change in oscillation amplitude, which provides the error signal for
the topog
raphy feedback loop. We estimate that the distortion of the liquid surface,
which is associated with the liquid neck, extends only over less than 5

nm [
14
].

Macroscopic approach

Thermodynamic considerations

To

a first approximation, the free energy of a liquid droplet residing on a solid
substrate is given by the sum of the interfacial energies

lv

,

sl

, and

sv

of the liquid
-
vapor, solid
-
liquid and solid
-
vapor interfaces, respectively, each weighted by the a
rea

5

of the respective interface. (We can neglect gravity, because we consider only
droplets smaller than the capillary length.) Minimization of the free energy shows that
the equilibrium shape of the droplet is a spherical cap and that the equilibrium cont
act
angle

Y

is given by the Young equation:



cos (

Y
) = (

sv
-

sl
)/

lv
.

(1)


This approximation includes only the contributions of the two
-
phase interfaces
to droplet energy. However, the molecules close to the perimeter of the droplet, i.e.
close to t
he three phase contact line, are influenced by both the solid
-
liquid and the
liquid
-
vapor interfaces. Therefore, another contribution to the free energy of the
droplet arises, which is given by the product of the length of the contact line and the
line ten
sion

. Taking this term into account, the equilibrium shape of the droplet is
still a spherical cap, but now the equilibrium contact angle

(R)

is given by the
modified Young equation



cos (

(R)) = cos(

Y
)
-

/(

lv
R).

(2)


Hence the equilibrium contact angle i
s expected to increase (decrease) with
decreasing drop size for a positive (negative)

. So far, we introduced


as a purely
phenomenological parameter. Its relation to interfacial forces will be discussed in
detail below.

Optical measurements

The inset of

Fig. 1 shows a picture of a typical droplet along with its mirror
image in the substrate. Viewed from the top, it appears perfectly circular. We
recorded a series of images of HEG droplets with various base radii between 1 and 3.5
mm. The droplet contour
line was determined via the software, as mentioned in the
experimental section. Later, the contour line was fitted using the Laplace
-
Young
fitting formula to determine the contact angle. The average contact angle is 24°.
Within the range of drop sizes inve
stigated, there is an apparent increase in contact
angle of


2° with decreasing R. In the modified Young plot (Fig. 1) we obtain a

6

linear correlation between the cosine of


and the inverse drop radius. At first glance,
it seems natural to interpret th
e slope in this plot as a line tension of


=

9∙10
-
7

J/m. In
this case, the extrapolated contact angle for infinitely large droplets is

Y

=


(R

=


)

=22.9°
. The observed line tension is in qualitative agreement with
other experiments using the same optica
l technique [
10
,
11
]. However, we noticed a
number of problems in the process of data recording and analyzing, which raise
doubts about the accuracy of the result: The contact angle calculated for

each specific
droplet varied by


1°, depending on the setting of the zooms lens on the CCD camera
and on the illumination. We also found that the Laplace
-
Young fitting produced a
systematically increasing fitting error with decreasing drop size. For comp
arison, we
analyzed the same images using ellipsoidal fits of the drop shape. The contact angles
usually agreed with the Laplace
-
Young fits to within

1°, but the apparent line
tension varied by more than one order of magnitude and sometimes even changed
s
ign. For these reasons, we conclude that the accuracy of both the data acquisition and
analysis of this typical commercial contact angle setup are insufficient to analyze
reliably the variations of the contact angle with drop size. Probably, some of the
te
chnical limitations can be improved in a specifically optimized setup. However, the
0.0
0.2
0.4
0.6
0.8
1.0
0.90
0.92

= (9.0
±
0.1)*10
-7
J/m
cos(

)
1/R [mm
-1
]



Fig. 1
Optical contact angle measurements of HEG on coated Si. The cosine of the contact angle versus
the inverse base radius is plotted.
The solid line is a linear fit according to the modified Young
equation. The inset shows a picture of a sessile drop taken with the CCD
-
camera of the contact angle
measurement setup.



7

m
ethod will always be limited to droplet sizes of


100

µm or larger. The most
obvious way to test the reliability of the value of

=

9∙10
-
7

J/m
, however, is to
extrapolat
e the result to smaller drop sizes using eq. (2). For instance, we would
expect a contact angle of 90° for
R

=

22 µm
. In contrast, for a value of




10
-
10

J/m
,
as predicted by theory, we obtain


(R

=

22 µm)
-


(R

=


)

=0.05
°. While it is
impossible to in
vestigate droplets of this size with a conventional contact angle
goniometry setup, it is very simple to distinguish between these two predictions using
AFM.


AFM measurements

We produced micrometer
-
sized HEG droplets, as described above. Fig. 2 shows
an o
ptical micrograph of a sample including the AFM cantilever. For the present
experiments, we imaged a series of isolated droplets with base radii between 0.75

µm
and 8

µm. The base radii of the droplets were obtained from circle fits to the
perimeter of the

droplet in the AFM images. Fig. 3 shows an image of a particularly
small droplet. A few local deformations of the contact line arise from the residual
roughness of the substrate of 1

nm root mean square. The distortions in the contour
lines reflect the no
ise in the measurement. The droplet has a spherical cap shape, as
shown by the profile (open stars) in Fig. 4. Obviously, the contact angle is still finite
for this sub
-
micrometer
-
sized droplet. In Fig. 4, we plot representative profiles for a
series of dr
oplets of varying size. In sharp contrast to the extrapolation using the
optically measured apparent line tension, qualitatively the same contact angle is
obtained for all the droplets. The solid lines are spherical cap fits. For comparison, we
also indica
te the result of the optical measurement in the lower left.

The slope of individual profiles, as shown in Fig. 4, scatters by


2° for profiles
taken at different positions along the contact line of the same droplet. To improve
statistics, we average the
contact angles obtained at each point of the contact line. In
practice, this is achieved by extracting the profiles perpendicular to the local direction
of the contact line at each point. The local contact angles are obtained from the slopes
of parabolic f
its to these profiles at their intersection with the substrate. For the fits,
we excluded the region within 10

nm above the substrate to avoid any possible



8


40µm


Fig. 2

Optical microscope image of HEG drop
lets deposited from an aerosol and the AFM cantilever.



Fig. 3 AFM image of an HEG droplet on PTCS
-

coated Si (R

=

0.38
µm;


=

24.4°)
. The contour
lines are separated by 10

nm. The arrow indicates the position of the profile in Fig. 4.



9

influence of interfacial force or other possible disturbances. The values for


given in
Fig. 4 are obtained by averaging the local contact angles over each entire AFM image.

While the absolute accuracy of the contact angle measurements via the AFM is
similar to the optical technique, the precision of the line tension measurement i
s
improved by more than three orders of magnitude, as shown in the modified Young
plot (Fig. 5). The open symbols are AFM results. The full square at
1/R

=

0 represents
the average of the optical contact angle data (Fig. 1). The dashed line extrapolates th
e
optical results to smaller drop size. Obviously, the slope of the optical measurements
is several orders of magnitude too big to account for the contact angles measured on
the micrometer
-
sized droplets. Instead, a linear fit (solid line in Fig. 5) includ
ing all
the data points yields a value of


=

-
(0.95



1.7)∙10
-
10

J/m. Considering the error we
cannot determine the sign of


from these measurements. The large error arises
mainly from the deviation between the optical and the AFM measurements. We
notice
d that optical measurements on samples, which were exposed to the aerosol
treatment in a closed container, produced


2° smaller values of


than the fresh
samples used for the data presented in Fig. 1. Possibly, a microscopically thin layer of
liquid cov
ering the whole surface forms during the deposition process of the small
droplets and reduces the solid
-
vapor interfacial energy. This effect would explain the
minor deviations between optical and AFM measurements.


0
500
1000
1500
2000
0
100
200
300

=24°
l [nm]
x [nm]
R



7.8 µm 26.4° 2.0°
4.3 µm 27.1° 1.1°
3.5 µm 25.9° 1.6°
3.1 µm 26.7° 1.5°
3.0 µm 25.4° 1.4°
0.38 µm 24.4° 0.4°

Fig. 4
AFM profiles for a series of
HEG droplets with different base
radii. For clarity, only every second
data point was plotted. The solid
lines are spherical cap fits.



10


Microscopic approach

In Fig. 6a we

plot a surface profile for a large droplet (
R

=

7.8

µm
) in the
vicinity of the contact line. On the lateral scale of this plot, the asymptotic spherical
cap profile appears as a straight line. The experimental data closely follow this
asymptotic reference

profile. Systematic deviations between the measured profile and
the reference profile are only observed within 20

nm of the substrate. The maximum
height difference is approximately
5

nm
. To understand the deviations, we have to
consider the influence of
the interfacial forces on the surface profiles. First, we note
that the energy of a thin homogeneous film adsorbed onto a solid substrate depends on
its thickness
l

[
15
]. The excess free energy per unit area of a film with thickness
l

(with respect to
l=

)
is called the effective interface potential
V(l)
. (
V(l)



0

for
l




.
) Its derivative leads to the interfacial forces or, more precisely, defines the
disjoining pressure

=
-
dV/dl
. Close to the contact line, the local thickness
l(x,y)
is
small and |
V(l(x,y
))|


becomes noticeably larger than zero (x and y are the lateral
coordinates on the substrate). Hence, there is a contribution to the free energy of the
whole droplet that depends on
V

and on the equilibrium shape of the surface profile

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.70
0.75
0.80
0.85
0.90
0.95
1.00


= -9.5*10
-11
J/m


= +9 *10
-7
J/m
cos(

)
1/R [µm
-1
]

Fig. 5 Cosine of contact angle
versus the inverse base radius. Open
circles are AFM data, the filled
square represents the average of all
the data point in Fig. 1. The dashed
line is an extrapolation of the
optical results to smaller drop size.
The so
lid line is a fit through all the
data points. The indicated values of


correspond to the slopes of the
dashed and solid lines, respectively.



11




within t
his range. The equation for the equilibrium profile is obtained by minimizing
the free energy of the droplet. In the case of an axisymmetric drop, it reads [
16
,
17
]



.

(3)


Here, r is the radial coordinate and
p
c

=2

lv
/R
3D

is the Laplace
-
pressure. (
R
3D

is
the radius of curvature of the liquid
-
vapor interface.) Primes denote the derivatives
with respect to r. As we saw already in Fig. 6a, the average curvature of the liquid
-
vapor interface could be neglected in the vi
cinity of the contact line for sufficiently
large droplets. In this case both
p
c

and the second term in the parentheses on the left
-
hand side of eq. (3) can be omitted. We obtain the simplified one
-
dimensional analog
of eq. (3)

[
8
]




.

(4)






dl
dV
x
l
x
l
lv






2
/
3
2
)
1
(










c
lv
p
dl
dV
r
l
r
r
l
r
l
r
l



















2
/
1
2
2
/
3
2
)
1
(
)
1
(


0
10
20
30
40
50
60
-4
-2
0
2
V [mJ/m
2
]
thickness,
l
[nm]

800
900
1000
1100
0
20
40
60
80
x=x
d
thickness,
l
[nm]
x [nm]
Fig. 6 a)Experimental AFM profile for a large HEG droplet (R=7.8

µm) close to the contact line. x
d

indicates the intersect
ion of the asymptotic profile (solid line) with the substrate . b) Experimental
effective interface potential
V

calculated from the experimental profile in a).


a)

b)


12

Here we replaced
r

by
x

to be consistent with our previous notation [
4
]. Now,
we calculate the effective interface potential
V

by integrat
ing eq. (4) using the
numerical derivatives
l´(x)

and
l´´(x)

of the experimental profile
l(x)




(5)


We choose the constant to obtain
V



0

for
l




. The open symbols in Fig. 6b
show the result. The zero position of the thickness scale was taken as the

average
substrate height far away from the droplet. Like any value of thickness quoted
throughout this work, the thickness in this graph is relative to the unknown thickness
l
0

in that region. However,
l
0

is typically smaller than the roughness of the sub
strate,
which gives rise to the scatter in the data points around
l=0

in Fig. 6b. The curve in
Fig. 6b meets two important requirements. First,
V

remains fairly constant at a large
film thickness, which confirms that the approximations leading to eq. (5) a
re fulfilled.
Second, the depth of the minimum of
V

at
l

=

l
0

for a partially wetting liquid in
thermodynamic equilibrium is given by

V

=

V(

)
-
V(l
0
)

=

lv

+

sl

-

sv

=


lv
(1
-
cos(

Y
))
. Using the asymptotic slope in Fig. 6a, we obtain

V

=4.4 mJ/m
2
, in
agreement with Fig. 6b. This second criterion can be used as an independent test to
identify possible tip
-
induced distortions of the profi
le.

Once
V

is determined, we can calculate


within the framework of the interface
displacement model [
8
]



Here,
x
d

denotes the intersection between the asymptotic reference profi
le
a(x)

and
l=l
0
.
a´=0

for
x

<

x
d

and
a´=tan

Y

for
x

>

x
d

is the slope

of the reference profile.
There are two contributions to

. The term in first set of brackets arises directly from
the interface potential
V
. The one in the second set of brackets aris
es from the
additional liquid
-
vapor interfacial area that is created due to the (interface potential
-
induced) deformation of the profile. In both cases the respective contribution of the



























































2
2
0
1
1
x
a
x
l
dx
dxV
l
dxV
x
l
dxV
lv
x
x
d
d




(6)











.
~
)
~
1
(
~
~
2
/
3
2
const
x
d
x
l
x
l
x
l
x
l
V
x
lv












13

undisturbed profile in the absence of
V

is subtracted [
18
]. For the pr
esent data set, we
obtain


=

1.7∙10
-
11

J/m
. The absolute value is very sensitive to the contact angle of
the reference profile. An error of


=

0.5°

leads to an uncertainty of


=



3∙10
-
11

J/m
. Comparing a series of randomly chosen profiles on the same droplet, we found
variati
ons of


=



7∙10
-
11

J/m
. Hence, we can give an upper limit of
|

|

=

10
-
10

J/m

based on the profile analysis. Within the error,

could be either positive or negative.



To demonstrate that a value of


>>

10
-
10

J/m

is incompatible with the AFM
profiles,

we calculated surface profiles using a model function for the effective
interface potential. As an example that mimics the partial
-
wetting situation of HEG on
silanized Si, we used a repulsive (i.e. wetting) van der Waals contribution with a
positive Hama
ker constant
A

and an exponential short
-
range contribution with a
prefactor
C

<

0.





(7)


We introduced a cutoff length
d
0

=

0.1

nm

to prevent the divergence of the van
der Waals term for
l



0
. We chose a decay length


=

1.5

nm

and adjusted the value
of
C

to match the experimental contact angle of 24°. The model function was inserted
in eq. 4 and the corresponding droplet profiles were computed numerically using a
Runge
-
Kutta algorithm. To obtain profiles for various v
alues of

, we varied

A

between
10
-
19

J

and
10
-
16

J.
(For HEG on silanized Si, we expect
A=1.8∙10
-
19

J,

based
on the refractive indices.) For each value of
A
, we adjusted
C

to maintain the contact
angle constant. The results are shown in Fig. 7 along with
the corresponding values of

. As expected, the deviations from the asymptotic reference profile increase with
increasing

. For


= 6.0

∙10
-
10

J/m

the distance between
x
d

and the intersection of the
surface profile is

x



40

nm
. Such large deviations are

clearly incompatible with the
experimental profile. The exact values of both the line tension and

x

depend on the
specific choice of the model function. Qualitatively, however, other model potentials
that we used for comparison also produced values of

x

>>

10

nm

whenever the
parameters were adjusted to obtain
|



>>

10
-
10

J/m
.






2
0
mod
12
exp
d
l
A
l
C
l
V
el















14

While this result is consistent with the macroscopic approach, the calculated
profile for
A=10
-
19

J

does not fit the experimental data in Fig. 7. If van der Waals
interactions
were dominant in this system, we would expect deviations of the profile
only for

l

<

5

nm

(see solid curve in Fig. 7). However, the HEG absorbs significant
amounts of water under ambient conditions [
19
]. Thus the liquid is a two
-
component
system that can di
splay concentrations gradients in the vicinity of the liquid
-
solid
interface. In this case, osmotic effects produce long
-
range profile distortions and a
long
-
range effective interface potential. Given the poor control of these effects under
ambient conditi
ons, we do not attempt to fit or explain this aspect quantitatively.

Discussion

In previous AFM measurements, we investigated droplets of HEG on substrates
covered with a periodic pattern of hydrophilic and hydrophobic stripes [
4
]. This
wettability pattern induced a modulation of the three phase contact line along with a
variation of the local contact angle. In those experiments, the line tension was
calculated by analyzing the dependence of the local contact a
ngle on the local
curvature of the contact line. Equation (3) was interpreted in terms of a local balance
of forces at the three phase contact line. The line tension was obtained from slope of
the linear regions in the modified Young plot. This was an eleg
ant way to cover a
large range of curvature in a single measurement. With the present approach using
homogeneous substrates, we need to image several droplets to obtain the same
information. While this procedure is considerably more time consuming, the dat
a
0
20
40
60
80
100
0
20
40
increasing
A


[10
-10
J/m]
-0.26
-0.15
+0.72
+3.6
+6.0
+0.17
±
0.3
thickness,
l
[nm]
x [nm]

Fig. 7 Equilibrium profiles (lines)
calculated using eq
. 4 for the model
interface potential (eq. 7) with a
series of parameters A

=

10, 100,
1000, 5000, and 10000


10
-
20

J
corresponding to the indicated
values of


.


increases in the
same order as
A
. Open squares:
experimental data (cf. Fig. 6a).



15

analysis is more straightforward. First, it is technically simpler to determine the
contact angle, because there are no variations of the local curvature of the contact line,
which induce distortions of the profiles. Second, the interpretation of the res
ults is
simply a continuation of the macroscopic optical approach. Hence it allows for a
direct comparison to the optical measurements, as presented in Fig. 5. The
discrepancy between the dashed line in this figure, which is based on the optical
measuremen
ts alone, and the AFM results shows that it is not correct to interpret the
trend in the optical measurements in terms of a line tension, as it appears in the
modified Young equation. This conclusion is now obtained on the basis of the
modified Young equat
ion in its simplest form, without invoking its local
interpretation.

In spite of this conclusion, the optical contact angle measurements clearly
produced a trend in the contact angle with changing drop size. We already mentioned
some of the technical problems in the data analysis that in our opinion cause the trend
in the d
ata. An alternative explanation, which was suggested in the literature, is the
so
-
called pseudo
-
line tension. Small
-
scale heterogeneities on the substrate cause local
deformations of the contact line and thereby increase the area of the liquid
-
vapor
interf
ace. The associated modulation of the interface decays exponentially with
increasing distance from the contact line [
20
]. This increases the total surface energy
depending on the amplitude and wavelength of the contact line roughness. From the
roughness of
the contact line in the AFM measurements we find an upper limit of


20

nm for the amplitude. The wavelength of the deformations is also


20

nm. For
these values, we obtain an increase in capillary energy per wavelength, i.e. a pseudo
-
line tension, of the

order 10
-
9

J/m. This value is approximately three orders of
magnitude smaller than the slope in Fig. 2. Hence, we have no explanation for this
trend, except for the experimental problems discussed earlier.


C
onclusions

The possibility to image liquid stru
ctures on the nanometer scale by an AFM
extends the range of optical contact angle goniometry by three orders of magnitude. In

16

this study, we have demonstrated the use of this new technique to determine the line
tension of a liquid
-
solid
-
vapor three phase
contact line and have compared it to the
conventional technique of optical contact angle measurements. The AFM
measurements revealed that the strong drop size dependence of the contact angle,
which is frequently observed in optical contact angle goniometry
, is not caused by the
line tension. This result is obtained directly in a macroscopic approach by analyzing
the global shape of the liquid
-
vapor interface of micrometer
-
sized droplets.
Furthermore, in the system investigated here, the specific long
-
range
interfacial forces
allow for a microscopic analysis of the force
-
induced distortions of the liquid
-
vapor
interface in the immediate vicinity of the contact line. Their magnitude confirms the
upper limit for the line tension of




10
-
10

J/m

that was obtain
ed from the global
shape of the surface. We expect that similar results can be obtained for other organic
liquids. Future experiments should also address questions related to substrate
heterogeneities and contact angle hysteresis.

A
cknowledgements

We would

like to thank the Deutsche Forschungsgemeinschaft (DFG) for
financial support under Grant No. He 2016/5 within the Priority Program “Wetting
and Structure Formation at Interfaces”.



17

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In principle, the relation between


and
V

and
l(x)

includes non
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local contributions. However, a
detailed comparison between non
-
local density functional calculations and the local approximation (Eq.
6) showed, that the non
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experiments [
9
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18






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