Liquid drops on compliant and non-compliant substrates: an ellipsoid-based fitting for approximating drop shape and volume

We present a novel method of 3-dimensional surface fitting of a droplet using ellipsoids such that the droplet is a combination of segments of two to four distinct ellipsoids. Further, this fitting method has been used to develop an analytical model estimating the volume of a droplet resting over compliant as well as non-compliant substrate. Here, we have used Glass and Poly (methyl methacrylate) (PMMA) substrates as rigid, and Polydimethylsiloxane (PDMS) free-hanging thin membranes (with thickness ranging from 20–40 µm) as compliant substrates. The analytical model considers the base length, width, height, and contact angles of the droplet captured from the experiment and estimates the droplet volume. The proposed analytical model could predict the volume correctly for droplets resting over compliant as well as non-compliant substrates with a maximum deviation of 16.6% for the volume range of 5–70 µL. Further, the predictions from the proposed analytical model are compared with the spherical cap-based model for droplets placed over compliant as well as non-compliant substrates. While the spherical cap-based model failed to accurately estimate droplet volume over a compliant substrate with an error of over 50%, the ellipsoid-based model proposed in this study could predict droplet volume accurately with a maximum error of 16.6%. Also, the proposed analytical model estimates the volume of droplets even at high contact angle hysteresis (> 50°) where the droplet has high azimuthal asymmetry. Further, the study also illustrates how Artificial Neural Networks (ANNs) can be used to forecast droplet width and contact angle hysteresis (CAH). The droplet width predicted from ANN could be used to eliminate the requirement of measuring droplet width from the top view experimental image. The volume of the droplet can thus be predicted from its side profile alone when utilized in conjunction with the theoretical model. Further, we developed an ANN model which predicts the CAH of the droplet by considering the length scales of the droplet. The developed ANN models performed a very good prediction with an R-value of > 0.98 .