Most techniques for measuring contact angles rely on direct optical methods. One common approach involves measuring the tangent angle at the edge of a droplet resting on a surface (a “sessile drop”). Another widely used method fits the droplet’s shape to a geometric profile—such as a circle, ellipse, or polynomial curve—to calculate the angle. In this study, we introduce a novel software tool developed in Visual Basic that uses image analysis to directly evaluate contact angles from sessile drop images. This tool provides a user-friendly and efficient way to perform accurate contact angle measurements.
Software subsystem
Contact angle and surface free energies
The wettability of a substrate is directly related to the surface energies and surface energies are linked through Young equation
\begin{equation}
\gamma_{s} = \gamma_{sl} + \gamma_{l} Cos \theta
\end{equation}
This equation contains four independent parameters. The contact angle \(\theta\) and the surface free energy of the liquid are the only measurable parameters, \(\gamma_{s}\) and \(\gamma_{sl}\) can only be determined by the so-called adhesive tension or wetting tension (“\(\gamma_{s}\)” and “\(\gamma_{sl}\)” ).
Equation also shows that, the high solid surface free energy, low interfacial free energy and low liquid surface free energy make surfaces more wettable.
Method of measuring contact angles
The measurements of contact angle of solder droplets were done by the developed software. This software allows user to fit a polynomial curve or an ellipse a to the droplet phase boundary by constrained nonlinear regression methods.
For the contact angle measurement, the droplet image has to be loaded to the software in any common digital image formats (JPEG, PNG). Then the base line for the droplet has to define by the user. User has to select the fitting method; a) Polynomial fitting or b) ellipse fitting and the program converts these data into equations. Equations generated by the program are solved automatically to find the roots, thereby it obtains the accurate contact positions and then calculate the slope at these contact points.
For the contact angle measurement, the droplet image has to be loaded to the software in any common digital image formats (JPEG, PNG). Then the base line for the droplet has to define by the user. User has to select the fitting method; a) Polynomial fitting or b) ellipse fitting and the program converts these data into equations. Equations generated by the program are solved automatically to find the roots, thereby it obtains the accurate contact positions and then calculate the slope at these contact points.
Polynomial root finding algorithm is based on the Jenkings/Traub method. Lang et al.(1994) examined this method and found that the Jenkings/Traub method is one of the best root finding methods.
The ellipse fitting is the other fitting method included in the program. Here user has to define the base line and three distinct points on the curved droplet surface for the program to generate the ellipse.
The typical formula of the equation of an ellipse with center (h, k) and major axis parallel to the x-axis is,
\begin{equation}
\frac{(x-h) ^{2}}{a^{2}} + \frac{(y-k) ^{2}}{b^{2}} = 1
\end{equation}
Where the length of the major axis is 2a and length of minor axis is 2b.Drop shape fitting procedure by the developed programme is illustrated in below figure.
Mathematical theory behind the software
Elliptic solution to the Young–Laplace differential equation
The Young–Laplace equation differential form can be solved under the elliptic representation for a fluid–fluid interface in the range (0 to 90). This method can significantly simplify the young laplace equation to analytical relation between the curvature radius and the elliptic parameters that yields the surface tension in the range 0.125 <\(\beta\) < 100. Therefore, it can be used to fit a drop shape by combining with the ellipse fitting methods. Finding the surface tension of a three-phase system will be much easier. The Young–Laplace equation that relates the surface tension to the curvature confirms this geometric description,
\begin{equation}
[\frac{1}{R_{1}} + \frac{1}{R_{2}}]\sigma = \delta P
\end{equation}
where \(\delta P\) is the pressure difference between the concave and the convex side of the surface, \(\sigma\) is the surface tension, and \(R_{1}\) and \(R_{2}\) are the maximum radii of curvature; the reciprocal of the last ones, \(\frac{1}{R_{1}}\) and \(\frac{1}{R_{2}}\) , are the curvatures in shift. Under an external field, as gravity g,
\begin{equation}
[\frac{1}{R_{1}} + \frac{1}{R_{2}}]\sigma = \frac{2 \sigma}{b} + \rho gz
\end{equation}
The parameter \(\beta\) is a shape factor that contains the properties of the fluid, the dimensionality and the value of the surface or interfacial tension, and the relative effects of the gravitational and capillary forces. Thus, knowledge of \(\beta\) allows evaluation of the surface tension.
If the correct elliptic profile is overlapping the drop profile it is possible to fit an elliptic equation to solve the Young–Laplace equation.
The ellipse equation centered at the origin is,
\begin{equation}
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1
\end{equation}
where a and b are the lengths of the semimajor and seminar axes, respectively. From the first and second derivative, we calculate the first and second curvature,
\begin{equation}
\frac{1}{R_{1}} = \pm \frac{b^{2}}{(a^{4}y^{2} +b^{4}x^{2})^{1/2}}
\end{equation}
\begin{equation}
\frac{1}{R_{2}} = \pm \frac{b^{4}a^{4}}{(a^{4}y^{2} +b^{4}x^{2})^{1/2}}
\end{equation}
Applying the boundary conditions for the above equation,
\begin{equation}
\frac{a}{b} + \frac{a^{3}}{b^{3}} = 2 + \beta^{*} a^{2}
\end{equation}
This final equation allows calculation of \(\beta^{*}\) from the Young–Laplace equation using only two points from the drop profile. The points include the equator radius that is equal to the first curvature radius and the second radius from the equator distance to the apex.
Top view contact angle measurements
From two measurable parameters (the droplet volume V and the related diameter D=2r), one can calculate the height (h) of the spherical cup, through the simple relation:
\begin{equation}
\pi h^{3} + (3 \pi a^{2})h – 6V = 0
\end{equation}
Third order equations can be solved through different analytical methods
\begin{equation}
R = \frac{a^{2} + h^{2}}{2h}
\end{equation}
Finally, the contact angle \(\theta\) comes out, with the precision derived from the measuring chain:
\begin{equation}
\theta = Cos^{-1} \frac{(R-h)}{R}
\end{equation}
Validation of the program
A comparison test was carried out to examine the accuracies of the results obtained from the developed program. Two proof bodies with known contact angles were used in this test. Contact angles of these bodies were calculated by using the developed software. The results obtained from the program were compared with the angles found from geometrical methods. This validation process was constructed on the basis of the procedure established by kiyomura et al. Three individuals were asked to analyze each image of proof bodies. Based on the results MAE (Mean absolute error) was calculated.
Contact angles measured by developed method and the respective established values of sessile droplets were compared in the below table.
Calculated mean absolute error (MAE) value for the conducted comparison test appeared to be around 1%.
Hardware subsystem
To properly handle the contact angle measurement process, additional hardware is needed. Below figure illustrates the arrangement of the hardware. The equipment comprises three essential parts, a base plate for holding the Camera, a stage that can level (Vertically) the surface, and a light source. Every single part is associated with two bars, which permit the user to modify the horizontal position of the Camera to satisfy distinctive experimental prerequisites
A digital microscope [Magnification:50x, eight light-emitting diodes, USB 2.0 compatible] Was selected as the solder drop images capturing device.
It must be pointed out that this system is not limited to the digital microscope used in this particular case. Some other cameras with appropriate magnification capabilities can also be used for this purpose.
As the figure shows, the essential components of the hardware can be easily disassembled by unscrewing the rods. This specific design allows user to easily assemble new components to the hardware.