Data-driven time-dependent state estimation for interfacial fluid mechanics in evaporating droplets

Abstract

Droplet evaporation plays crucial roles in biodiagnostics, microfabrication, and inkjet printing. Experimentally studying the evolution of a sessile droplet consisting of two or more components needs sophisticated equipment to control the vast parameter space affecting the physical process. On the other hand, the non-axisymmetric nature of the problem, attributed to compositional perturbations, introduces challenges to numerical methods. In this work, droplet evaporation problem is studied from a new perspective. We analyze a sessile methanol droplet evolution through data-driven classification and regression techniques. The models are trained using experimental data of methanol droplet evolution under various environmental humidity levels and substrate temperatures. At higher humidity levels, the interfacial tension and subsequently contact angle increase due to higher water uptake into droplet. Therefore, different regimes of evolution are observed due to adsorption-absorption and possible condensation of water which turns the droplet from a single component into a binary system. In this work, machine learning and data-driven techniques are utilized to estimate the regime of droplet evaporation, the time evolution of droplet base diameter and contact angle, and level of surrounding humidity. Droplet regime is estimated by classification algorithms through point-by-point analysis of droplet profile. Decision tree demonstrates a better performance compared to Naïve Bayes (NB) classifier. Additionally, the level of surrounding humidity, as well as the time evolution of droplet base diameter and contact angle, are estimated by regression algorithms. The estimation results show promising performance for four cases of methanol droplet evolution under conditions unseen by the model, demonstrating the model's capability to capture the complex physics underlying binary droplet evolution.

Figure 1

(a) Schematic of the experimental setup with macroscopic parameters of droplet shown in the inset; (b) regime map of droplet evaporation (top-left) under various relative humidity (RH) of surrounding and substrate temperature (T), evolutions of nondimensional contact angle, volume, and diameter versus time for evaporation-dominated (bottom-left), transition (top-right), and condensation-dominated (bottom-right) regimes. Each scale bar in droplet images represents a length of 1 mm. The schematic is made using free and open-source software Inkscape (Harrington).

Figure 2

Results of regime classification: (a) correlation matrix for parameters in droplet evaporation; (b) results of test set for regime detection illustrated with confusion matrices for Naïve Bayes (NB) and decision tree (DT) algorithms; (c) point-by-point validation results with NB and DT classifiers for regime detection with experimental data of droplet evaporation for RH = 80% and T = 35 ${}^{\circ }$C; (d) point-by-point results of estimation set with NB and DT classifiers for regime detection with experimental data of droplet evaporation for experiment #4 (RH = 75% and T = 25 ${}^{\circ }$C) in estimation set. The colors in (c,d) correspond to the regime colors used in Fig. 1b.

Figure 3

Relative humidity regression results: (a) test set; (b) validation; (c) estimation. Markers for all colors in (b) represent different temperatures as shown in legend. Markers and colors in (c) are the same as in (a).

Figure 4

Diameter (D*) and contact angle $({\theta }^{\ast })$ regression results: (a) test set; (b) validation set; (c) estimation set; (d) diameter estimation with quadratic regression for E3 (top) and contact angle estimation with third-order polynomial regression for E1 (bottom).