## Methods for this measurement.

## Young-Laplace

## Why these Methods?

The Young-Laplace method applies the Young-Laplace equation to calculate a theoretical drop that fit the profile of the experimental one. Once we have a good approximation, we can use the measurement of the theoretical data to provide a precise result of surface tension.

## Surface Tension measurement using a smartphone

To proceed with surface tension measurement, an image of a pendant drop (generated using a needle connected with a syringe) is obtained with the smartphone camera. The outline of the drop (experimental profile) in an image is detected by using a variety of image analysis methods.

The profile of a pendant drop can also be calculated (theoretical profile) by solving the differential equation, Laplace equation, [1] with knowing the physical properties of the liquid drop, i.e. density difference between two fluids Δρ, the gravity acceleration (g), surface tension (γ), length of drop profile (s), and the Laplace pressure difference (ΔP). For each measurement, the values of Δρ and g are needed as the initial input. Initial guesses for γ, s, and ΔP are also needed. By comparing the theoretical and experimental profiles the summation (F) of the Euclidean distance between experimental profile points and the theoretical profile can be calculated. The liquid surface tension of this liquid is the guess value which gives the minimum value of F.

In this instrument, the physical size of each image pixel is obtained by an automatic calibration system which measures the number of pixels in the capillary/needle seen in the image (the needle diameter is needed as an input). Since, in the drop shape methods, gravity is always applied in the vertical direction and parallel to the axis of symmetry of the drop [7,18]. However, for one to use the smartphone to take a drop picture, it is very likely the phone will not be perfectly leveled. Therefore, an offset angle (α) can exist between the gravity (and axis of symmetry) and the vertical direction in the drop image which can significantly affect the measurement results. To minimize the off-set angle error, a self-correction system is programmed into the smartphone. This is done using the accelerometer of the smartphone.