Shape of liquid meniscus in open cells of varying geometry: a combined study via simulation and experiment

TL;DR

The paper addresses predicting the shape of the liquid meniscus in open cells of arbitrary geometry to understand evaporation-driven flows relevant to evaporative lithography. It advances a practical workflow that couples experimental boundary measurements (wall wetting height and center thickness) with a hydrostatic-Helmholtz model, producing 3D meniscus profiles by solving $-\sigma(h_{xx}+h_{yy})+\rho g h = C$ under geometry-specific boundary conditions. The key contributions are the experimental protocol for boundary data, the integration of these data into a 3D reconstruction via the Helmholtz equation, and a quantitative comparison across cylindrical, square, and triangular cells showing overall agreement within $\sim$9–14%. This approach enables rapid, geometry-tailored predictions of the liquid surface and evaporation dynamics, with potential to guide deposition-pattern design in evaporative lithography and related microfluidic applications.

Abstract

Evaporative lithography in cells of arbitrary configuration allows for the creation of diverse particle deposition patterns due to the formation of a specific flow structure in the liquid caused by non-uniform evaporation. The latter in turn is determined by the shape of the liquid layer surface and the wetting menisci on the cell walls. Thus, predicting the shape of the wetting menisci can serve as a tool for controlling the process of creating desired particle deposition patterns and evaporation dynamics. Here, we propose a simple and sufficiently accurate methodology for determining the shape of the liquid meniscus in cells of arbitrary geometric shape, based on a combination of mathematical modeling and a series of experimental measurement techniques. The surface profiles of the liquid meniscus in cylindrical, square, and triangular cells were determined by measuring the change in the reflection angle of a laser beam from the free liquid surface while scanning from the cell wall to its center. The height of the wetting meniscus on the inner cell wall and the minimum liquid layer thickness at the center of the cell were measured by analyzing optical images and using a contact method, respectively. 3D meniscus profiles were obtained by numerically solving the Helmholtz equation. The boundary conditions and the unknown constant in the equation were determined based on experimental data obtained for several local points or cross-sections. The simulated meniscus shapes showed satisfactory agreement with the experimental local measurements, with a maximum relative error of less than 14%.

Figures (7)

• Figure 1: The proposed methodology presented as an algorithmic flowchart.
• Figure 2: (a) Picture of the liquid wetting meniscus on the inner border of a square cell; (b) Example of picture processing in Graph2Digit software, red points --- coordinates of the three-phase contact line; (c) Schematic of liquid surface inclination measurement using a laser beam and layer thickness measurement using a micrometric probe (contact method).
• Figure 3: Results of numerical simulation of the meniscus shape in (a) square and (b) triangular cells.
• Figure 4: Results of numerical simulation and experimental measurements of the meniscus shape in a cylindrical cell.
• Figure 5: Shape of the two-phase interface ("liquid--air") (a) along the wall of triangular cell measured by images of the meniscus on the wall, and (b) along its median measured by the laser scanning method.