Contact Angle Measurements Using Cellphone Cameras to Implement the Bikerman Method

Abstract

Using the Bikerman equation, the contact angle of a sessile drop can be measured from above. For a known drop volume, the contact angle can be derived from a measurement of the drop diameter. It is shown how this method can be implemented with many currently-available cellphone models. Increased accuracy can be achieved using low-cost close-up lenses and the image can be transmitted to a laptop for subsequent processing. This rapid and straightforward method makes the measurement of contact angles on the shopfloor or in the field, a more attractive proposition.

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1 Methods
Contact angle measurements, by providing informa-
tion on the properties of a surface, such as wettabil-
ity and surface energy, are of growing importance
in countless branches of science and technology.
Unquestionably, the most accurate means of measur-
ing the contact angle of a sessile drop is by compu-
terized drop shape analysis, known as DSA or ADSA,
which has been described in great detail by Neumann
[1] and to which the authors have also made a con-
tribution [2]. It should be noted that while most of
Neumann‘s work relates to drops viewed in profile,
he does also report the use of drop-shape analysis
when viewed from above, i.e. the approach discussed
here. However, the software used by him is not in the
public domain, and it is not clear how to access it.
Over two centuries since the work of Young [3], scores
of methods have been proposed for contact angle
measurement, many of which today are no more
than scientific curiosities. However, a small number
deserve a reappraisal, because the need remains for
easy-to-use, low-cost techniques, not least those which
can be used by operatives on the shop floor. Devel-
opments in open-source computer software and low-
cost digital imaging devices are drivers for such a
reappraisal.
In 1941, Bikerman [4] proposed a novel method of
measuring the contact angle of a sessile drop. This
was based on viewing the droplet from above and
measuring the diameter of the droplet, on known
volume. For small volume spherical drops, he derived
the equation <1>:
d3/v = (24sin3 θ) / (π(2 – 3cos θ + cos3 θ) <1>
Where d is the diameter of the base of the drop,
sometimes referred to as the contact diameter, v is
the volume of the drop, and θ is the contact angle.
No practical application of this method has been
found other than the work of Miller [5] who used the
method to determine whether aircraft fuselages had
been sufficiently cleaned (or excessively so) prior to
painting. Miller, evidently enthused by his success-
ful use of the method, arranged for Lockheed Corp.
to market a kit for a wider use of the idea, under the
name Surfascope. Miller also filed a patent [6] which
covers much the same ground as his publication. This
included a microsyringe, a magnifying glass and a set
of nomograms with finite solutions to the equation
above. Unfortunately, it appears that Miller had not
fully understood Bikerman‘s concept and his publi-
cation embodies this misunderstanding, which was
perpetuated by more recent authors such as Durkee
et al. [7].
In the Bikerman equation, the term d is the diameter
of the droplet base – the circular contact area made
by the drop on the surface on which it rests. For con-
Contact Angle Measurements Using Cellphone Cameras
to Implement the Bikerman Method
By Darren Williamsa, Anselm Kuhnb, Trisha O’Bryona, Megan Konarika and James Huskeya
a Chemistry Department, Sam Houston State University, Huntsville,
Texas/USA
b Publication service Ltd., Stevenage Herts, Great Britain
Using the Bikerman equation, the contact angle of
a sessile drop can be measured from above. For a
known drop volume, the contact angle can be derived
from a measurement of the drop diameter. It is shown
how this method can be implemented with many cur-
rently-available cellphone models. Increased accu-
racy can be achieved using low-cost close-up lenses
and the image can be transmitted to a laptop for sub-
sequent processing. This rapid and straightforward
method makes the measurement of contact angles on
the shopfloor or in the field, a more attractive pro-
position.
Beim Verfahren nach Bikerman wird der Kontakt-
winkel eines Tropfens durch die Betrachtung von
oben. Bei gleichbleibendem Volumen ändert sich mit
dem Kontaktwinkel der Durchmesser des Tropfens.
Die Methode lässt sich auch mit den heute üblichen
Kameras eines Mobiltelefons anwenden. Eine höhere
Genauigkeit und Archivierbarkeit des Verfahrens
kann durch den Einsatz von kostengünstigen Digi-
talmikroskopen in Verbindungen mit einem Laptop
erzielt werden. Die schnelle und unkomplizierte
Methode erhöht den Anreiz zum Einsatz der Kontakt-
winkelmessung in der chemischen Technik.

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tact angles that are less than 90°, where the drop is a
hemisphere or less, d is readily measured by view-
ing the drop from above. However for contact angles
greater than 90°, where the drop is greater than hemi-
spherical, the maximum girth of the drop will be
greater than its contact diameter, that is to say it will
overhang the contact area and obscure it from view.
Bikerman was well aware of this issue, but neither
Miller [5, 6] nor Durkee et al. [7] mention it. Thus,
the Bikerman method can only be used for contact
angles < 90° unless an alternative means of measur-
ing the contact diameter can be found.
Bikerman proposed several solutions to this problem,
none of them very satisfactory. His first idea was to
allow the drops to evaporate, after which they would
leave a ring-like mark. This appears problematic,
since as the drop evaporates, its volume will contract
and so will the wetted contact area. Whatever causes
a visible mark to be made might depend on changes
in the composition of the liquid. Bikerman suggested
such residue ring-marks might be caused by corrosion
or by a solute being deposited. If there is a change
in solute concentration as the drop evaporates, such
processes would be extremely complex. It is not
believed that these proposals by Bikerman, ingenious
though they are, are of any practical value. A second
idea was to dust the sessile drop with finely-divided
powder to characterize its contact area, but this too,
does not appear to offer a workable solution.
A simple mathematical test identifies situations where
the contact angle is less than 90° and where, in con-
sequence the Bikerman equation can be used with
direct overhead viewing. If the measured diameter is
greater than the 90°-diameter (d90) (Eq. <2>), then
it is valid to use the Bikerman equation.
d90 = (12v/π)1/3 <2>
where d90 is the diameter of a hemisphere of volume v.
Incorporating this validity test (Eq. <2>), the authors
have used computer spreadsheets and cell phone
cameras to implement the Bikerman method with
minimal cost and analysis time. The availability of
computer spreadsheets is perhaps the most impor-
tant factor in making the Bikerman method more
user-friendly. The authors offer a spreadsheet (Tab.
1) that accepts user input of individual volume and
diameter values, calculating the contact angle using
a lookup-table of the Bikerman equation with 0.05°
increments over the range of θ of 0.10° to 90.00°. The
spreadsheet applies the test noted above (Eq. <2>),
warning the user (Tab . 1) if the drop is greater than
hemispherical.
The spreadsheet can also be used to generate the
nomogram sheets laboriously calculated by Miller
(Fig. 1). The validity test of Equation <2> is also
used on this worksheet (Fig. 1). Lastly, if the user pro-
vides uncertainty values, the spreadsheet will com-
pute the uncertainty in contact angle using Equation
Tab. 1: Spreadsheet for calculating the contact angle of a drop of known volume when imaged with a calibration object
The Bold-Italic type face indicates a formula
cell that I not be edited by the user
Volume Properties Image Calibration Diameter
Measurements
Results
Image Analyst
Image Filename
Comments
Drop Description
Drop Volume (v) (cm3)
Drop Volume Uncer-tainity (sv) (cm3)
Hemi-Diameter (d90°) (cm)
Cpx (px)
Scpx (px)
Ccm (cm)
Sccm (cm)
Cali-bration Result (cm/px)
Drop Diameter (dpx) (px)
Drop Diameter Uncer-tainty (Sdpx) (px)
Drop Diameter (dcm) (cm)
Valid Result? Is d > d90°
Contact Angle (0) (deg)
+ Conatct Angle Uncertainty (+s0)
(deg)
- Contact Angle Uncertainty (-s0)
(deg)
D LW Digital Micro-
scope.jpg
al foil
pressed on
3/16 hole
fake 10
uL drop
0.0100 0.0002 0.337 755 4 0.691 0.005 9.15·10-4 525 8 0.480 yes 46.8 1.3 1.4
MMK Digital Micro-
scope.jpg
al foil
pressed on
3/16 hole
fake 50
uL drop
0.0500 0.0002 0.576 555 4 0.691 0.005 1.24·10-3 403 4 0.502 no >90° >90° >90°

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<3>, which was derived using standard propagation
of uncertainty techniques [8].
<3>
Where sd3/v is the uncertainty in the d3/v term, sd is
the uncertainty in the diameter measurement (d), and
sv is the uncertainty in the drop volume (v). The cali-
bration object (C) is measured in pixels and in cm.
The px subscripts in Equation <3> indicate an image
analysis measurement in pixels. The image analysis
will be further explained by a description of the vari-
ous measurement methods.
There are slight differences in the positive and nega-
tive uncertainties of contact angle because of the non-
linear nature of the Bikerman equation. To account
for this, the uncertainty in contact angle is calculated
by looking up the positive and negative deviations
separately using the Bikerman lookup-table (Tab. 1 ).
2 Experimental
To implement this method one merely needs an accu-
rate drop delivery system such as a Hamilton microsy-
ringe (v and sv in Eq. <3>) and a digital camera
with a macro focus capability. In the present study,
a piece of aluminum foil (Reynolds, Heavy Duty,
25 µm thick) was pressed with a finger into a 0.475 cm
(3/16 in.) hole of a gage card to produce a non-evap-
orating standard drop shape. After three attempts, a
wrinkle-free simulated drop was produced (Fig. 2).
Fig. 1: Image of the authors‘ nomogram creation worksheet. The “#N/A” values in the contact angle column indicate
that the contact angle is > 90° for the given drop diameters

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Tab. 2: Contact angles of an Imperial drill gage card
for various drop volumes
Hole Diameter
(in.)
Drop Volume
(µL)
1 2 5 10 15
1/16 0.062 89.2 > 90° > 90° > 90° > 90°
5/64 0.078 60.9 89.4 > 90° > 90° > 90°
3/32 0.093 40.6 67.8 > 90° > 90° > 90°
7/64 0.109 26.6 48.5 86.0 > 90° > 90°
1/8 0.125 18.0 34.3 69.1 > 90° > 90°
9/64 0.140 12.9 25.2 55.0 83.7 > 90°
5/32 0.156 9.4 18.5 42.6 70.3 86.9
11/64 0.171 7.2 14.2 33.6 58.7 75.7
3/16 0.187 5.5 10.9 26.3 48.1 64.4
13/64 0.203 4.3 8.5 20.9 39.3 54.3
7/32 0.218 3.5 6.9 17.0 32.6 46.1
15/64 0.234 2.8 5.6 13.8 26.8 38.6
1/4 0.250 2.3 4.6 11.4 22.3 32.4
17/64 0.265 2.0 3.9 9.6 18.8 27.6
9/32 0.281 1.7 3.3 8.0 15.9 23.5
19/64 0.296 1.4 2.8 6.9 13.7 20.2
5/16 0.312 1.2 2.4 5.9 11.7 17.4
Tab. 3: Contact angles of a metric drill gage card
for various drop volumes
Hole Diam.
(mm)
Drop Volume
(µL)
1 2 5 10 15
2.00 59.7 > 90° > 90° > 90° > 90°
2.50 35.05 60.7 > 90° > 90° > 90°
3.00 21.15 39.8 76.15 > 90° > 90°
3.50 13.5 26.3 56.85 85.55 > 90°
4.00 9.1 17.95 41.6 69.05 85.7
4.50 6.4 12.75 30.55 54.4 71.25
5.00 4.7 9.3 22.75 42.45 58.05
5.50 3.55 7 17.3 33.15 46.8
6.00 2.75 5.4 13.4 26.1 37.65
6.50 2.15 4.25 10.6 20.8 30.4
7.00 1.75 3.45 8.5 16.8 24.75
7.50 1.4 2.8 6.95 13.75 20.35
8.00 1.15 2.3 5.7 11.35 16.9
2.2 Pass-Fail Images
In the case of Miller’s use of the Bikerman method [5,
7], a wettable surface with a water contact angle less
than 72.8° was required for 90 % paint adhesion to
occur. The authors’ spreadsheet may be used to deter-
mine that a 3.884-mm diameter 10-µL drop would
exhibit a contact angle of 72.8°. A washer with a
4-mm inner diameter may be used directly as a sec-
ondary standard. Drops of 10-µL with diameters larger
than 4 mm indicate a surface that passes the wettabil-
ity test, and vice versa. A visual comparison to the
4-mm washer is all that is needed, but a cell phone
camera could be used for documentation purposes.
2.3 Bracket Images
Further efficiency can be achieved by employing a
drill gage card, which can be purchased from most
tool suppliers. Instead of using a calibration washer,
this method quickly determines the approximate con-
tact angle by placing a known volume (1, 2, 5, 10,
or 15 µL) of liquid onto a surface and then selecting
the best matching gage hole on the card (Fig. 2). The
size selected on the card and the drop volume is then
referenced in Table 2 (Imperial) or Table 3 (metric) to
determine an approximate contact angle. While this
method is only an estimate, it is useful for bracketing
Using the spreadsheet and Equation <1>, a 10-µL
drop with this diameter would express a 48.1° contact
angle. This standard drop shape was viewed from
above and analyzed using various measurement meth-
ods, a digital microscope, four cell phone cameras and
two types of cell phone macro lenses.
2.1 Calibrated Reticle
Contact angle measurement using a magnifying eye-
piece with a calibrated reticle is a tempting option
because of its portability. However, most magnifying
eyepieces are constructed to view flat objects, and are
unable to image a sessile drop without unacceptable
distortion. A telescope-style eyepiece was constructed
that accepts collimated light from the sample that
passes through the calibrated reticle before magnifi-
cation. Even then, parallax effects made it impossible
to obtain an acceptable reading. Additionally, the cost
of these eyepiece components approaches that of a
small digital microscope, which is much more useful
even though it is tethered to a computer.

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the contact angle. Figure 2 shows that the diameter of
the aluminum contact angle standard is a best match
to the 0.187-inch (4.75 mm) hole, which for a 10-µL
drop would be a contact angle near 48°. Without image
analysis it is difficult to specify the exact contact
angle, but this image shows that the drop diameter
is certainly between the next larger (0.203 inch) and
smaller (0.171 inch) holes – a contact angle range of
39° to 59° (Tab . 2).
2.4 Digital Microscope
A digital microscope (DinoLite, AM411T) was used
to capture an image of the foil contact angle standard
(Figs. 2 and 3), and the spreadsheet was used to cal-
culate the actual contact angle as if it were a 10-µL
drop. Adjacent to the drop, and included in the same
image, was an object of a similar size, in this case
a metal washer (Fig. 3) with dimensions of 0.691
± 0.005 cm (Ccm and sccm in Eq. <3>) measured by
five separate persons using a vernier micrometer.
The uncertainty term sccm is the standard deviation
of the five measurements. The resulting image was
then analyzed using a freely available image analysis
package (Meazure) [9]. To obtain data from Meazure,
a circle was fitted to the inner diameter of the washer
to calibrate the scale (Cpx and scpx in Eq. <3>). A
circle was also fitted to the outline of the drop from
which the diameter (W in Fig. 3, dpx and sdpx in Eq.
<3>) was read. The advantage of this approach is
that it allows the user to test the circularity of the
drop and, should it not be truly circular, to derive
mean, minimum, and maximum values for d and by
extension for θ. The uncertainties in pixels (scpx and
sdpx) were conservatively estimated using the range of
repeated measurements in pixels.
The pixel count measured by the Meazure program [9]
is dependent upon the software magnification, so care
was taken to measure the washer and the drop at the
Fig. 3: The use of the image analysis software (Meazure) [9] to calibrate the digital microscope image using a metal washer
(left) and to measure the diameter (W = 379 px) of the aluminum foil contact angle standard (right)
Fig. 2: Comparison of the aluminum foil contact angle
standard with the gage card that was used to produce it

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same magnification. For this reason, it was preferable
to measure the inner diameter of the washer, since
bringing the outer diameter of the washer into view
would reduce the pixel count across the drop (Fig. 3).
2.5 Cell Phone Macro Photography
Using Auxiliary Lenses
The digital microscope is preferred if the samples
can be analyzed in the laboratory, but for shop-floor
or field data collection the use of cell phones holds
promise. Cell phones are not made to take close-up
photos, but one may purchase snap-on macro lenses
for most camera models [10, 11]. Camera alignment
is not critical since the calibration object placed next
to the drop serves as an internal optical standard. The
only requirements are a close-up image with a substan-
tial number of pixels across the drop and the calibra-
tion washer, and a crisply-focused image which aides
the measurement of the drop and washer diameters.
A variety of cell phone models [12–15] were employed
with and without the snap-on lenses [10, 11] to take
pictures of the aluminum foil standard and the cali-
bration washer. The photos were then analyzed by
multiple persons using the authors’ spreadsheet.
The two smart phones (HTC [14] and iPhone [15])
yielded images which were sufficiently crisp and clean
without requiring any auxiliary lenses. These smart
phone models are equipped with auto zoom and auto
focus features that allow the capture of images with
adequate pixel resolution for analyzing 10-µL drops.
More basic cell phones (Samsung [12] and Black-
berry [13]) were not equipped with these features,
and thus, the close-up photos appeared out of focus.
The magnetically mounted macro lens [10] is shipped
with an adhesive-backed mounting washer and a
magnetic ring on the back of the lens so that it can
be added and removed at will. However, when using
this lens, the user must remove any protective cover-
ing or case around the phone. The lens is designed to
be attached to the phone, and any form of covering
will push the lens too far from the photo sensor. The
magnetic adhesion to the phone allows the user to use
both hands to steady the camera phone thus reducing
image blurring.
The Jelly Lens [11] is so named because it uses a
tacky gel polymer ring to adhere the lens to the phone
body. There are some drawbacks to using the Jelly
Lens. The tacky adhesive did not stick well to the
Fig. 4: A montage sample of images showing the crisp detail (or not) of the Digital Microscope, the HTC-Macro Lens, the
Blackberry-Macro Lens, the iPhone-Macro Lens, the Samsung-No Lens, and the Samsung-Jelly Lens configurations listed
left-to-right and top-to-bottom

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phones in this study, and the user was required to hold
the lens in place to keep it from falling while a second
person positioned the phone to capture the image.
Furthermore, the lens itself is contained in a bulky
plastic casing that often obscured parts of the image.
This lens functions best when the camera face has
a flat and texture-free surface. The Jelly Lens has a
very short focal length requiring the user to hold the
camera very close to the drop, thus, making difficult
to obtain an image that contained both the drop and
the calibration washer.
3 Results
The digital microscope was very easy to use because
the on-board LED illumination was always sufficient,
the microscope mounting was stable, and the fine
adjustment provided crisp photos with good resolution.
This image is seen in the top-left panel in Figure 4.
Modern smart phones can acquire very crisp photos
when used with a makeshift hand rest. Some difficulty
was encountered when trying to get the camera auto
focus to lock onto the objects. Using the magnetic
Macro Lens, the pictures were very close in quality
to those obtained with the digital microscope (Fig. 4).
The Macro Lens improved the quality of the images
taken with the basic model phone also, but the qual-
ity did not match those taken with the smart phone
models.
The Jelly Lens did not perform well with the smart
phones. Its magnification appeared to be too strong
for the auto zoom and auto focus features which then
worked against their proper function. Most of the
images were poorly defined, lacking the crispness of
those obtained using the more advanced cell phones
with or without auxiliary lenses. However, the Jelly
Lens proved to be an ideal tool for use with the basic
model phone (Samsung) to obtain almost the same
quality of image as with the smart phones (Fig. 4).
The accuracy, or bias, of this method was tested by
calculating the mean of the contact angle results
obtained by four analysts measuring the same images.
Also calculated, was the absolute error in contact
angle (θ – 43.1°) where 48.1° is the contact angle of
a nominal 10-µL drop with the diameter of the alu-
minum contact angle standard. As seen in Table 4,
the most accurate phone-lens configurations were the
Samsung-Jelly Lens, HTC-Macro Lens, Blackberry-
No Lens – all with absolute errors that fall within the
experimental uncertainty values. The iPhone, Black-
berry, and HTC cameras performed slightly better
than the microscope without any additional lenses.
The precision of this method (sθ) was tested by cal-
culating the standard deviation of the contact angle
results obtained by the four analysts (Tab. 4 ). This
captures the variability of the user-dependent image
analyses. The uncertainties are quite good consider-
ing that each analyst determined on their own the best
fit of the circles to the objects in each of the images.
3.1 Wider Status of the Bikerman Method
Bikerman‘s approach appears to be almost unknown
and unused. Neumann, arguably the leading authority
in the field, while clearly aware of the method, notes
it but without comment. Interestingly, however, one
recent patent [16], though without naming or acknowl-
edging Bikerman, has, one might say, re-invented the
method, setting out an equation essentially identical
to Eqation <1>. While using the Bikerman approach,
the patent addresses a rather special case, where the
sessile drop rests on a transparent surface (in the con-
text of fingerprint recording sensor). This 2D sensor
array allows a direct imaging of the underside of the
drop, thereby removing the restriction noted above, as
regards droplets of greater than hemispherical size.
Tab. 4: Contact angle results, standard deviations,
and error from the nominal value (θ – 48.1°) for
various camera and lens configurations
Lens Phone θ
(°)
sθ
(°)
Error
(°)
Error
(%)
Jelly Samsung 47.8 1.1 -0.3 -0.7
Macro HTC 46.9 1.8 -1.2 -2.5
None Blackberry 46.4 2.5 -1.7 -3.5
Macro iPhone 46.0 2.0 -2.1 -4.3
Macro Blackberry 44.6 1.6 -3.5 -7.4
None HTC 44.2 2.2 -3.9 -8.2
None iPhone 43.9 0.4 -4.2 -8.7
None Microscope 43.5 2.3 -4.6 -9.6
Macro Samsung 41.1 4.2 -7.0 -15
Jelly iPhone 33.1 10.8 -15 -31
None Samsung 32.7 3.3 -15 -32
Jelly Blackberry 29.2 4.1 -19 -39
Jelly HTC 24.5 1.2 -24 -49

4 Conclusion
In conclusion, apart from the restriction noted above,
the Bikerman method is admirably simple and low-
cost. Direct application of Bikerman for all contact
angles would require a view from below via transpar-
ent samples or the very special case noted above. In
all other cases overhead viewing of less-than-hem-
ispherical drops is facile. Overhead viewing allows
measurements to be made on large surface areas,
where it is more difficult to view sessile droplets in
profile. The digital microscope is the tool of choice
for many, however, the microscope is tethered to a
computer or laptop – a relatively bulky device when
compared to a cell phone. The cell phone has the
additional advantage that, if required, the measure-
ment result can be instantly transmitted to remote
locations. The cell phone camera enables the lab to
go to the sample when used in conjunction with high-
quality microsyringes. This is therefore a method that
can be used both for simple pass-fail analyses to pro-
vision of accurate and precise contact angle values.
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