Sessile Droplet Analysis: Mechanical Engineering Project at Curtin

This mechanical engineering report focuses on the analysis of sessile droplets, particularly the quantitative measurement of wetting a solid surface by a liquid through contact angle analysis. It delves into the fundamental equations governing contact angles, including the Laplace equation of capillarity and Young's equation, explaining the relationship between interfacial tensions and equilibrium contact points. The report also explores the concept of contact angle hysteresis, discussing its causes such as surface irregularity and chemical heterogeneity, and examines early theoretical studies by Wenzel and Cassie-Baxter. Furthermore, it investigates different wetting modes, including complete wetting, highlighting the dependence on liquid and surface properties, and concludes by emphasizing the ubiquitous nature of wetting behavior and its significance in various applications. This document is useful for students looking for solved assignments and project reports and is available on Desklib.

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The quantitative measure of wetting a solid surface by a liquid is called the contact angle.
Geometrically the contact angle can be defined as the angle formed by a liquid at the three-phase
contact line where there is the intersection of a liquid, solid and vapor. When the contact point is
applied along the interface of the liquid gas in the droplet file, then it can be said that the tangent
line has been acquired geometrically.
The shape of a liquid droplet is determined by the liquid surface [5] tension over a solid surface.
Inside the liquid droplet, every molecule inside the bulk is pushed uniformly to all sides by the
adjacent liquid molecule which results in a net force that is equal to zero. Seemingly, one has to

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understand that the liquid molecule that is near the outside surface does not have the adjacent
particles which are responsible for providing a net force that is well balanced. On that note, one
has to understand that the adjacent molecules will be dragged inward which will result in the
creation of an internal force inside the liquid droplet. On the other hand, there is the maintenance
of the lowest surface that is energy free because it is responsible for contacting the surface area
by the liquid molecules. Surface tension refers to the intermolecular forces that are inside the tiny
droplet that contract the surface area.
Figure 1: The surface tension [5] and the Schematic of a sessile drop contact.
1.1.1 The Contact angle basic equation
The two most essential equations relating to contact angles are the Laplace equation of capillarity
and Young’s equation.
The most basic equation of the capillarity is the Laplace equation which is given by the Laplace
that is responsible for mathematically governing the distance between the external forces and the
surface tension like gravity. The shape of the sessile drop is determined by the Laplace equation
which achieves by relating it to the pressure of the capillary. It relates the difference in pressure
across a vapor-liquid interface to the curvature of the surface tension and the interface.
γlv ( 1
R1
+ 1
R2 )=∆ P

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The liquid-vapor interfacial tension is the γlv while R1 and R2 represent the two principal radii of
the curvature of the surface and the difference in pressure across the interface is represented by
the∆ P. The linear function of the elevation without the external forces apart from the gravity is
called the pressure difference.
∆ P=∆ P0 +(∆ ρ) gz
The pressure difference ∆ P0 at the reference plane is represented by g which is the gravitational
acceleration and ∆ ρ is the difference between the most bulk phases while z represents the
vertical height of the steady drop that is attained from the leveled reference. Hence, the drop
shape can be measured for the known value of the physical properties like the difference in
density and gravity and also the known geometrical quantities like the two principal radii of the
surface curvature of a known liquid vapor tension interface which is represented byγlv. It is also
possible to inverse the principle like the liquid-vapor interface γlv can be determined from the
shape of the fluid but is a tedious procedure.
Another important equation is the Youngs equation that relates to the interfacial tensions stated
by Thomas Young in the year 1805. He described it as the liquid drops contact line on an ideal
surface that is solid which can be simply derived from the mechanical equilibrium of the three
interfacial tensions.
γlv cos θe=γsv −γ sl
Where Yiv, Ysv, and Ysl are the solid-liquid, solid vapor and the liquid-vapor interfacial tensions
respectively while the theatre e is the equilibrium contact point. The Young's equation is derived

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based on the assumption that the solid surface is an ideal surface that is inert, smooth, chemically
homogeneous and isotropic. You must find a good approximation of the equilibrium contact
angle to use Young's equation of the solid-vapor interface and the solid, liquid interface tensions.
1.2 Hysteresis of the Contact angle
It is one of the most classic elements and important regarding wetting a liquid on solid surfaces.
There exist several metastable states when a drop of the liquid rests on a horizontal surface, and
hence it is noticed that contact angles are not equal to θe. The wetting of the liquid on the solid
surface makes it appear more than a static state because when the liquid moves, it exposes a fresh
surface which is finally wetted. Therefore, it is not adequate to measure a single static contact
angle that can categorize the wetting behavior. The three-phase contact line produces a contact
angle in motion is called the dynamic contact angle. The contact angle that results from
expanding the liquid is referred to us the Advancing contact angle θa while receding contact
angle θr is the one formed due to the contracting of the liquid.
Figure 2: The Illustration of the Advancing Contact Angle and the Receding.
Contact angle hysteresis is defined as the difference between the receding contact angle and the
advancing contact angle for a contact line stirring in reverse direction at the same velocity [7]:

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H=θa−θr (4)
The contact angle hysteresis arises because of the surface irregularity and chemical
heterogeneity. For some non-homogenous surfaces, there exists a domain that blocks the motion
of the contact line. For example, hydrophobic surfaces will smidgen the movement of the liquid
front as it advances; causing an upsurge in the contact angle and reverse it will hold back the
contracting motion of the fluid front when the liquid recedes, this is the reason for the reduction
in a contact angle. Thus, the surface coarseness plays a crucial role in producing contact angle
hysteresis. Interpreting such contact angle data regarding Young's equation can be misleading
because it Young's equation fails to consider the surface topography.
1.1.1 Early theoretical studies on contact angle hysteresis
The measurement of the contact angle and its interpretation on a perfect surface, i.e., chemically
homogenous, flat, insoluble, rigid and non-reactive is straightforward. In situations of rough
surfaces, the measurement of contact angle hysteresis and contact was first recognized by
Wenzel, who resolved that the shallow contact angle on uneven surfaces is related to the Young
perfect contact angle [8]. Due to the convolution to combine the chemical heterogeneity and
surface roughness, it is advisable to distinct heterogeneity and roughness by concentrating into
(1) rough surface that is chemically homogeneous and (2) smooth surface with chemical
heterogeneity.

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Figure 2: Schematic of Wenzel and Cassie-Baxter sessile drop method for contact angles of the
coarse surface[9]
By considering the hard surface coarseness and supercilious that surface is wetted by the fluid in
contact as shown in the figure, Wenzel propound that the geometric interfacial area is less than
the actual solid-liquid interfacial area and can be defined as the average surface coarseness ratio
as [6],
r = actual surface area
geometric surface area (5)
Presenting Wenzel contact angle, θw into Young’s equation, in respect of the coarseness value,
r(>1) the equation obtained was:
γlv cos θw=r ( γ ¿¿ sv−γ sl) ¿ (6)
The Wenzel contact angle is the shallow contact angle restrained on the geometric level of the
rough surface. Due to the existence of metastable states, the Wenzel contact angle acquiescent to
experimental measurements. In particular, the regular price of the Wenzel contact angle depends
upon the method applied when the liquid drop is formed on the solid. Also, whether the decline
is proceeding or retreating, the contact angle formed by a proceeding drop is larger compared to
the contact angle formed by receding reduction.

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For a plane but chemically different solid surface, the wetting might be estimated by Cassie-
Baxter theory. According to the theory, the solid surface comprises two different chemical
composition, with occupying fraction f 1and f 2 and having contact angle θ1 and θ2 respectively.
For a two-component system, the setting is described by Cassie-Baxter equation and is described
as:
cos θCB=f 1 cos θ1 + f 2 cos θ2 (7)
In relation to Young’s equation,
γlv cos θCB=f 1 ¿ ¿ (8)
The relation between f 1and f 2 is f 1+ ¿ f 2=1. For a hydrophobic surface, it was assumed that the
air is entrapped in the voids of a rough surface. For the porous surface, f 2 represents the fraction
of air entrapped in the voids, which corresponds to θ2=1800 for a non-wetting condition.
Therefore, the Cassie-Baxter equation of a leaky surface was reduced to:
cos θCB=f 1 cos θ1−f 2 (9)
The Cassie and Wenzel equations are pragmatic or symmetrical equations from basic models,
which have been commonly used to forecast wetting behavior on actual surfaces. However, for
non-patterned, coarse and leaky material it is hard to apply Wenzel’s equation since it is difficult
to measure the correct surface area of such materials. Cassie equation can be applied to leaky
elements, but it becomes hard to use this approach when the material has non-patterned

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coarseness. The Wenzel and Cassie method is lack of rigorous analysis of thermodynamic
principles, and therefore the results obtained are not consistent with the theoretical predictions.
For example, based on the projection of the Wenzel equation, if the seeming angle is more
significant than 900 on a hard surface, increasing the surface coarseness will decline the contact
angle [6]. However, Neuman[10] experimental results showed that the progressing contact angle
always rises and receding contact angle still drops by increasing the surface coarseness.
Bartell Extrand [11] questioned Wenzel’s theory, and the outcomes exhibited that the contact
angles of the droplet on the surfaces highlighting coarseness within the contact line were similar
to those of even surfaces [9]. Bartell prepared the covers with a chemically different coating that
showed the contact angle identical to surfaces with no coat. Extrand[11] showed that the contact
angle is not controlled by the interface that is a solid-liquid which is beneath the droplet at the
three-phase contact line. From the Gao and McCarthy[9] work it can be concluded that the
hysteresis and the contact angle are a function of contact line structure in a manner that the
kinetics of the droplet movement, rather than thermodynamics, dictate the wettability [9].
Therefore, Cassie and Wenzel's model has limited applicability in practice, and their equation
should be used with the information of their innate fault.
The contact angle necessity on the size of the sessile droplet still faces many challenging
questions. The work of Viswanathan and Chase[12] explained the drop-size dependent on solid-
liquid contact angle and concluded that on increasing the size of the drop the contact angle
decreases. They, however, found that the receding and advancing contact angles are difficult to
relate to the initial contact angle of sessile drops

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1.2 Wetting modes
Wetting behavior is ubiquitous. The act of liquid that is in contact with a solid surface is known
as wetting or spreading. The action of the fluid is different on different surfaces. The wetting
behavior is seen whenever a surface is exposed to liquid, and the liquid is expected to spread on
the surface until it reaches equilibrium state [13]. The wetting allows the fluid to spread covering
the surface or remain as a drop adopting the spherical shape. The wetting process is depended on
the physical properties of the liquid (surface tension and viscosity) and properties of the surface
(mainly surface roughness) and also depended upon some external features such as pressure,
temperature, relative humidity [13]. The contact angle defines typically the wettability of a liquid
on a solid surface.
1.2.1 Complete wetting
The complete wetting occurs when the surface energy per unit area of the dry substrate is higher
than that of the wetted surface. Therefore, the wetting behavior of liquids is the combination of
solid-liquid and liquid-vapor surface tensions and, in such case, it will not be enough to
overcome the solid-vapor surface tension [14]. Liquids with low surface tension tend to spread
entirely on surfaces, resulting in a thin film of liquid.
1.2.2 Non-wetting
The non-wetting occurs when the interactions of liquid molecules are stronger than of the solid-
fluid interactions [14]. In such case, the solvent molecules are tightly packed, and liquid will
slide like a ball without sticking on the surface due to its active surface tension and adhesion and
forms a globule shape. Such fluids are used in thermometry for example mercury which can slide

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down easily. Similar behavior is observed when the water drop is resting on the hydrophobic
surface.
1.2.3 Partial Wetting
Partial wetting occurs when the surface energy per unit area of the dry substrate is less than that
of the wetted substrate the liquid will not wholly spread [14]. In such case, the liquid will form
an intermediate drop shape, which will have a non-zero contact angle at the contact line.
Figure 3: Schematic diagram of wetting regimes depending on the surface and the liquid[14]
1.4 The techniques applied in the measuring of angles
Several methods serve in the grading of contact angle while making use of a thin substrate
1.4.1 Wilhelmy plate method
The checking of the surface tensions of liquids or surfaces was primarily done by making use of
the densitometry. Tensiometry majorly used was characterized by its ability to apply the force
balance method and offered great help in the calculation of the contact angle of liquids that their
surface tensions have already being identified. The technique makes use of a plate of the plate
that has been partially immersed vertically in a pool of fluid. This is done until the fluid is raised

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towards the plate. After that, there will be the measuring of the vertical force acting on the plater;
this is done by placing the plates on balance, as shown in figure 4. The measuring process will
have to neglect the weight on the plate.
Figure 4: Wilhelmy plate tensiometry[1]
Making use of the method will allow the general force balance equation that Wilhelmy does give
to be used in the calculation of the contact angleθ.
γ cosθ= F
p
In the formula F serves as are representation of the vertical force acting on the plate that was
recorded after the measuring. On the other hand, p represents the wetted perimeter of the plate
and the surface tension is represented by y. In working to advance or recede the contact, one will
have to push or pull out the plate in the pool of fluid. The method is preferred over the
conventional optical method citing several advantages. Some of them include the reduction of
the measurement angle to being the length and weight and more so, can be done with high levels
of accuracy. Despite being good, the method has several disadvantages. There is need to carry
out the test observing a uniform cross-section. An ideal sample is the test surface with the known