Analyzing Sessile Drop Evaporation: Contact Angle Measurement Review

This literature review provides an overview of sessile drop evaporation, focusing on contact angle measurement techniques and their evolution. It discusses the significance of sessile drops in everyday life and various industrial applications, highlighting the importance of understanding heat transfer and evaporation processes near the contact line. The review covers different methods for measuring contact angles, including Axisymmetric Drop Shape Analysis (ADSA) and other numerical strategies, along with their limitations and advantages. It also touches upon the influence of surface characteristics like roughness and wettability on the behavior of sessile drops, emphasizing the ongoing research and the need for refined studies on mass transfer and heat transfer phenomena. The project aims to analyze the spreading behavior of liquids on aluminum surfaces with varying roughness and evaluate the evaporation patterns of sessile drops formed on solid surfaces.

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Introduction
Sessile drops are a common encounter in everyday life among them drops of rainwater on
raincoats, espresso spills and even water on a cooking plate. For numerous years now, a lot of
research has been carried out over the sessile drop to explore the contact phenomenon and the
surface including dynamic spreading, contact angle, characteristics of the surface as well as
evaporation. Among the surfaces, characteristics that have been studied include roughness of the
surface and wettability1. Various strategies were deployed in the quantification and visualization
of the sessile drop as the study progressed including the Captive bubble methods/ Axisymmetric
Drop Shape Analysis, Goniometry method, and even the Wilhelmy plate method.
There is a relatively large variation in the characteristics of sessile drop when it comes into
contact with different surfaces of solids since some of the liquids have the capability of wetting
the surface tony to some extent and thus creating an intermediate drop shape at a specific contact
angle. On the other hand, other liquids are able to spread over the surface of the solid to form a
film. The characteristic of a sessile drop in which it tries to keep contact with the solid surface
refers to wetting2. A contact angle is the angle formed when between a tangent that is in line
with the liquid and the solid surface at the point of intersection of the three interfaces as shown in
figure 1. The three interphases include the solid-vapor, solid-liquid as well as the liquid-vapor
interfaces and their intersection is called the contact line. Contact angle defines the microscopic
representations of the phenomenon that is microscopic in natures among them surfaces

roughness, surface coating as well as surface energies all of which have an important role to play
in the wettability of materials for any provided liquid.
For a long time, the challenge with evaporation has been established and this recently once again
attracted the interest of scholars. This problem is quite defined in micro fluids in which
evaporation take part of the size of a small droplet. A deposited drop on the surface of a solid
substrate creates together with gas and solid phases the triple contact line2. A comprehension of
the heat transfers and evaporation process in a drop of a liquid and close to the contact line is
quite a very fundamental concept in numerous industrial applications among the ink-jet printing,
micro-electronics, nano and micro-fabrication among other applications.
There are abundant studies both in the theoretical and experimental dimensions on evaporating
droplet with the small size of the capillary in literature. The contact angle is a determinant of
diffusive evaporation. Investigations on the effect of the substrate nature are still vital. The
model f Larson and Hu have successfully and sufficiently defined the evaporation of sessile
drops from pure liquids with the aid of diffusive evaporation mass flux add in the case of natural
evaporation at an ambient temperature1. A review of Erbil has been used to illustrate the recent
attainments on the topic. Moreover, there is still need to conduct more refined research on the
problems of mass transfer and could heat transfer as such findings are fundamental in the
comprehension of enhancement of heat transfer and the dynamics of the contact line.
As common as the evaporation of a sessile droplet on a surface, it is a common situation that is
experienced in different situations and thus has received sufficient attention and interest in
literature. A shape in the form of a wedge is formed close to the contact line when a sessile drop
forms on a flat source. This wedge-like shape forms a capillary flow that is controlled by

evaporation. This capillary flow works by sucking the drop to the contact line and as a result of
the wedge shape of the evaporating droplet, there is an increase in the rate of evaporation
towards the wedge. The evaporations result in mass-loss from the phase of the liquid and this
change in the profile of the drop; either through a lowering the base radius or decreasing the
contact angle or even a blend of the two.
A hypothetical and trail examination was spearheaded by Picknett and Bexon about the
evaporation of small drops in which there was neglect in the effects of gravity and instead a
spherical cap approximation was for the shape of the drop was adopted. The results illustrated
the closeness of the three different modes of drop evaporation: constant contact angle, constant
contact radius or a mixed mode2. Numerous years after the turning point of this work, the
various modes of evaporation were explained and attributed to the wetting behavior by different
scholars. There is a decrease in the contact angle with time as the evaporation starts on
hydrophilic surfaces while the contact radius remains unchanged. This project aimed at
analyzing the spreading behavior of numerous liquids on an aluminum surface with various
roughnesses and to evaluate the pattern of evaporation of the sessile drop that is formed on the
surface of a solid.
1.1 Brief description of the previous research on the measurement of the contact angle
Measurements of contact angle have been achieved through the development of various drop
shape techniques as well as the artificial liquid tension derived from the sessile drop shape. The
technique and approach used in measuring the shape if a sessile drop depends on the similarity of
the theoretical profile that is calculated from the numerical integration of Laplace equation. As
soon as the principal radii of the curvature of the sessile drop and the surface tension are

calculated through the use of drop shape technique, integration of the Laplace equation can be
done in order to measure the contact angle1.
One of the earliest works in Axisymmetric Drop Shape Analysis was done by Adams and
Bashforth who came up with the drop profiles of various surface tensions and established the
numerical solution of the equilibrium shape of the interfaces of the axisymmetric fluids. The
findings from these calculations were represented in the forms of tables. It was thus possible to
determine the contact angle and the surface tension using the actual profile by the use of linear
interpolation of the values that are provided and obtained from their tables. The same approach
was conducted by Hartley and Harland who were able to present the numerous solutions in a
form a modified table that was used in finding the tension of the interface of the various shapes
of axisymmetric drop2. A computer program which was a contribution by FORTRAN was used
in the integration of the appropriate form of Laplace equation.
Through the FORTRAN computer, it is possible to automatically evaluate the drop profile in
which the results are thereafter presented in the forms of tables. Data acquisition acted as the
main source of error in this method. The FORTRAN computer program was accompanied by
numerous limitations and disadvantages. One of such is that the drop interface in determined
through taking measurements of the few preselected critical points from the entire drop surface
and because there location of this point is associated with such high criticality and they are in
correspondence with special characteristics including inflection surface point, a determination
must be done with utmost precision. Still, the method is confined to a specific size and thus for a
sessile drop that has a low contact angle say 20⁰, it turns out to be very challenging to measure
the contact angle using this method2.

Following the aforementioned limitations and disadvantages of the method that was developed
by Hartley and Hartland, a new method of analysis and graphical representation was developed
by Malcolm and Paynter that was used in taking measurements of contact angle as well as
interfacial tension using the axisymmetric sessile drops profile. The method is based on an
assumption that the sessile drop formed through a common fluid that is on a common solid
substrate is axisymmetric and hence will have identical contact angle, length of capillary and
interfacial tension. The method calls for the measurement of the equatorial diameter of the drop
as well as its height from the apex to the solid substrate to numerous drops so as to increase the
levels of accuracy in processing of the data and interpolation. This method is only valid and thus
usable for sessile drops that have their contact angles greater than 90⁰ and this forms the only
limitation.
A numerical strategy was developed by Maze and Burnet. This strategy adopts the use of
numerical integration of the Laplace equation as well as the use of least-square optimization
which are non-linear in nature. The numerical strategy allows taking a measurement of contact
angle and interfacial tension for whichever shape of a sessile drop as long as its contact angle is
more than zero1. The method is ideally composed of a calculation involving an anon-linear
regression procedure where a Laplace curve or a calculated drop shape is made is such a way
that it fits the shape of the measured drop through optimizing two values until the best of the fits
is achieved.
Numerous arbitrary coordinates are chosen in measuring the shape of the drop from a drop
profile of an experiment and thus the methods turn out to be measuring the profile of the drop
using a set of coordinate points of which none of the points has any unique importance. This is
one of the important advantages of this method. The fitting procedure can only initiate through

the appropriate measures of the size and shape of the drop. The first measurements were derived
using the values from the tables that were provided by Bashforth and Adam’s. This method also
comes with limitations despite the numerous benefits attached to it. The error function which is
the difference between the experimental profile and theoretical profile is one such limitation.
The error function is the sum of all the squares of the horizontal distance between the
experimental profile and the theoretical curve. Sessile drops are greatly under the influence of
gravity and thus the definition of error function does not seem sound reasonable for sessile drops.
On the other hand, there is a tendency of compression at the peak of the drop in large drops that
have low surface tension1. In this case, the possibilities of an error resulting from any data that is
close to the peak are high in as much as the apex point is located near the theoretical curve.
Another limitation of the method is that the summit of the drop is predetermined. It is a
requirement that the summit point must be known before the method is applied so as to be used
as a point of initiation for all the curves that need to be calculated. Maze and Burnet escaped the
task and challenge of situating the origin through changing their original program. This is despite
the fact that only the vertical displacement worked while the horizontal displacement was not
fitting for the program.
A procedure like the one developed by Maze and Burnet was developed by Huh and Reed. In
this development, the apex was still required to be established and known beforehand but the
objective normal was used in approximating the normal distance. This methods developed by
Huh and Reed was only usable in sessile drops that had a contact angle greater than 90⁰.
A technique for processing image in shaping the drop was developed by Girault et al. in which
their task was mainly composed of a digitizer of a video image profile that was used in obtaining

the drop profile. The method needed the calculation of the inflection plane as well as the volume
of the drop. Still, this method is only applicable to pendant drops.
The Axisymmetric Drop Shape Analysis, (ADSA) was a method that was first developed by
Rotenberg et al. in which the measured profile of the drop was fitted to a Laplacian curve
through the use of the procedure of non-linear regression. In this method, the objective function
is provided in terms of the sum of the squares of the normal distances that occur between the
measured point and the calculated curve which is then deployed in the analysis of the difference
between the actual profile and the theoretical Laplacian curve. The assumption made in this
method is that the location of the apex of the drop is not known and that the apex coordinates are
regarded as the independent variable of the objective function2. This is one of the features that
make the use of ASDA method different from the other methods of measuring the shape of the
drop making measuring the shape of the drop convenient from any view or frame.
The initial ASDA method deploys that method of incremental loading that concurs with the
Newton-Raphson method in the numerical minimization of the objective functions as well as
offering a scheme that is used in approaching the solution from a remote situation. The method
does not depend on any tables that are generated by Adams and Bashforth and there are no initial
values that are needed to optimize any of the parameters such as the radius of curvature of the
apex, the apex coordinates and the surface tension which are fully working. The only limitation
of ASDA method is its failure to a large flat sessile drop due to the limitations of the Newton
methods especially when the initial parameters are found to be distant from the solution.
There were attempts by Jennings and Pallas to come up with a similar method. In their method,
Jennings and Pallas ensured that the method was more efficient in terms of computations, an

achievement they made through introducing numerical implications to the method. This was in
as much as the numerical simplification had effects on the accuracy of the methods. This method
makes use of a technique of optimization and is this found to be superior relative to the Newton-
Raphson technique. This methods need an assumption to be made for the initial parameters and
contribute to the list of its limitations.
Axisymmetric Drop Shape Analysis-Diameter (ADSA-D) was developed by a team of scholars
in which the computations of the contact angle was on the basis of the volume and the diameter
of the sessile drop that had a known surface tension through viewing the drop from above. This
method was mainly designed for measuring low contact angles drop on surfaces that were
hydrophobic or non-ideal1. The difference in the density across the interface of the two liquids,
the drop volume, gravitational acceleration, and surface tension of the fluid and the equatorial
diameter of the drop are the only parameters that are needed for the ADSA-D method.
Cheng et al. used synthetic drop to establish the performance of the ASDA methods that were
developed by Rotenberg et al. sessile drop and pendant. In their study, they examined the choice
of the data as the input in which the test was done by the collection of data at five various
location of the drop profile individually in order to establish the influence of each of the location
on the findings. From the results, it was noticed that there was a greater impact on the results
from the data that was collected close to the liquid-solid interface of the sessile drop relative to
the points from other locations. The first generation ASDA method was found to be more
reliable in terms of accuracy expect in the case of the very large sessile drops in which the
method was found to be unsuccessful.

Another challenge with this method is the difficulty of obtaining a perfect camera alignment with
a plumb line which is manually obtainable on the computer screen using a mouse. These errors
that are realized in the first generation ASDA were corrected and refined improvements made in
the second generation ASDA that was developed by Neuman and Rio who applied more efficient
algorithms. In the resolutions, the radius of curvature was adopted in the optimization of the
parameters and instead the curvature of the apex was adopted.
ADSA-Constrained Sessile Drop was introduced by Yu et al. which was usable in the
measurement of very low surface tension. Constrained Sessile Drop is a drop configuration that
is made up of a sessile drop on a pedestal and has an edge that is in the form of a sharp knife.
The edge aids in the prevention of spreading or leakage of a film as in the case of insoluble
monolayers. The liquid is pushed through a hole that is inside the pedestal to form the drop as is
the case with measuring the ADSA contact angle.
A modified version of ADSA known as ADSA-Captive Bubble was developed by Zuo et al. to
be used for measuring the surface tension of configuration of a captive bubble. This development
fosters pulmonary surfactant studies1. A complex technique for analysis of images alongside the
Cunny edge detection was adopted in this method to eliminate noise that comes from the image
resulting from the opacity of the surface suspension during captive bubble experiments.
The assumption made by the standard version of the ADSA is that the only external force that is
acting on the drop is gravity. A new version, ADSA-Electric Fields has been developed to
enhance an extension of the application of ADSA method in the measurement of the interfacial
properties of liquids, especially in an electric field. ADSA-Electric Fields makes use of sessile
drop and constrained sessile drop configurations in achieving this. In as much as ADSA-Electric

Fields has the same structure as ADSA, it is more complex due to the electric fields components.
Calculation of the distribution of electric field along the drop surface has been made possible
through the incorporation of a new module in the ADSA-Electric Fields. There is need to
perform significant changes to this new model in order to explain the effect of electric current in
the generation of theoretical profiles.
It is not possible to accurately measure the surface tension of drops that are nearly specific using
the drop shape methods described above. The source of this limitation was investigated by
Hoorfar et al. through the scrutiny of the whole ASDA technique inclusive of the hardware and
the software. Hoorfar et al. modularized the original ASDA as it was firm faced in order to
enable the implementation of alternative techniques for numerical analysis for the various
experimental situations such as poor images. Hoorfar et al. also introduced the shape parameter
criterion which was important in the provision of prior knowledge of the accuracy of measuring
surface tension using ADSA without including ADSA calculations. This shape parameter was
defined using a dimensionless parameter that specified the difference in the shape between an
inscribed circle whose diameter is equivalent to the apex radius curvature and an experimental
drop profile2. A critical shape parameter which is the value of the below shape parameter
provides the readings of the surface tension that is less accurate relative to the expected
tolerance.
To summarize, ADSA has two roots relative to the strategy and structure. In terms of the
structure, ADSA has three major modules: image analysis which provides the profile coordinates
from the drop image as the first module, the optimization procedure that is used in determining
the best fit of the theoretical Laplace curves and the third module that produces the theoretical
profiles through numerical integration of Laplace equation in known values of surface tension.

The best fit is used in the identification of the interfacial tension of the liquid-liquid interface,
surface area, contact angle, drop volume, a radius of curvature of the apex as well as the radius
of the contact circle that exists between the liquid and the solid for the ax of sessile drops. In
terms of strategy, ADSA calls for these inputs which are physically necessary such as gravity,
drop image, the difference in density with the selection of the experimental drop profile points in
an arbitrary manner as opposed to depending on specific profile points.
Numerous method for drop shape has been developed since the development of the ADSA
method most of which if not all have maintained the structure of ADSA in their development.
The Theoretical Image Fitting Analysis technique was recently developed which has a dissimilar
structure and approach from ADSA. In this technique, calculations of the interfacial properties
are achieved through having the entire theoretical image fitted to the drop experimental image.
Numerical solution of the Laplace equation is used in arriving at the theoretical image which is
black and white. The experimental image gradient is used as opposed to the raw image in the
minimization of the impacts of lighting and contrast conditions.
1.1 Research Progress to a Situation in which evaporation occurs from the surfaces
Going by the predictions of Good’s model and Neumann, energy barriers are available on
heterogeneous or coarse surfaces of solids which have the capability to inhibit spontaneous
relaxation of the liquid drop to a state of its global minimum energy. There are two portions that
make the difference in the contact angle as shown in figure 2. The first part is their existence of a
jump in the contact angle when the velocity is zero. This jump which is called contact angle
hysteresis defines the difference in the contact angle that is ignited by the effects of the surface
including coarseness and heterogeneity2. It can be perceived as the difference between the

highest and the lowest contact angles in which there is a minimum in the Gibbs free energy. In as
much as these angles never attain full stability, the energy inhibitors to attain the minimum in the
free energy are often very great that make it possible to see the maximum and the minimum
angles only.
A drop can be knocked in any metastable state which corresponds to the local minimum state of
free energy should it be placed on any solid surface randomly. The minimum in the free energy
is often very great that makes it possible to see the maximum and the minimum angles only. An
assumption made is through theoretical studies establish that the receding and advancing angles
are the angle that also experienced the lowest barrier to energy and thus generally speaking the
east vibration has the ability to shift the angle to a minimum that is proximate to the global
minimum in energy, but hardly to the minimum of itself. In this regard, the hysteresis that was
established experimentally may be found to be less than the theoretical values.
There is very high stability in the static hysteresis such that it is possible to deposit a droplet in a
state such that it jeep the contact angle between the two limiting angles without undergoing any
form of motion or relaxation throughout which it will be assumed that there was no evaporation
that took place. as a result of the deformations that occur to the substrates and are induced by the
surface normal component of the surface tension of the surface that is not balanced, there exists a
retention time which depends on the contact angle hysteresis. It is worth noting that such static
hysteresis wholly depends on the surface that is exactly under the contact line. The adoption of
an over the liquid-solid interface average is allowed for the cases where there is need to establish
contact lines that are long over random surfaces1.