The shape of two-dimensional and three-dimensional drops on flat and curved hydrophilic substrates: variational, numerical and molecular dynamics simulation investigations

M Foroutan and MT Rad and A Boudaghi and H Ataeizadeh, JOURNAL OF THE IRANIAN CHEMICAL SOCIETY, 19, 423-433 (2022).

DOI: 10.1007/s13738-021-02309-6

In this paper, we analyze the shape of two-dimensional and three- dimensional drops on flat and curved substrates by minimizing free energy with the variational approach. The process initiated from the flat substrate and has proved the drop shape equation. Then, the drop shape equation for these models has been determined by extending the method to curved substrates. The constraints in 2D and 3D models are constant surface area and volume, respectively. The Euler-Lagrange differential equation can be solved analytically in the two-dimensional case and the drop height as a function of x has been driven which is the equation of a circle. In the 3D model, a fundamental partial differential equation (PDE) has been obtained but it cannot be solved analytically and has been solved through a numerical method. This PDE becomes a first-order ordinary differential equation (ODE) after two reduction steps and is solved by the Euler explicit method. Six different systems are simulated and a logical algorithm is introduced to compare the results of the simulation, numerical, and analytical. The simulation results determined there is an iso-density curve or surface that is closest to the analytical result. It has revealed the solution of the three-dimensional case is not a sphere. The spherical cap approximation (SCA) has been widely used in many wettability studies. Comparison of numerical and SCA results have determined this assumption is valid only for contact angles far from 90 degrees. The present results are determined all the drops on each surface can be described by only one equation if the contact angle is measured for the line (in the case of a 2D drop) or the passing plate (in the case of a 3D drop) from the drop edges. The substrate characteristics at the three-phase contact point (TPCP) and the three-phase contact line (TPCL) are the only factors determining the drop shape. The other points and areas away from TPCP and TPCL have not influenced the drop shape.

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