[Paper Review] Making sessile drops easier

Amir H. Fatollahi, Maryam Hajirahimi|arXiv (Cornell University)|Apr 22, 2013

Surface Modification and Superhydrophobicity10 references3 citations

TL;DR

This paper presents a one-parameter shooting method for numerically solving the equilibrium shape of axisymmetric sessile drops by deriving a key identity directly from the Young-Laplace equation, reducing the complexity of traditional multi-parameter approaches. The method achieves high accuracy, validated against experimental data and the ADSA-D method up to Bond numbers of 15, with a provided Mathematica implementation for practical use.

ABSTRACT

Using an identity, directly derived from the Young-Laplace equation, the problem of the equilibrium shape of an axisymmetric sessile drop is reduced to a one-parameter shooting method problem. Based on the method the numerical solutions for drops with Bond number up to 15 are plotted. The agreement between the method and the ADSA-D method as well as the experimental data is tested. A Mathematica code based on the method is presented.

Motivation & Objective

• To simplify the numerical solution of the equilibrium shape of axisymmetric sessile drops by reducing the parameter space.
• To derive and apply a new identity from the Young-Laplace equation that enables a one-parameter shooting approach.
• To validate the method against existing numerical (ADSA-D) and experimental data across a wide range of Bond numbers (up to 15).
• To provide a practical, open-source Mathematica code for reproducible and accessible computation of drop profiles.

Proposed method

• Derives a key identity from the Young-Laplace equation that links curvature and gravitational pressure, enabling reduction of the boundary value problem to a single shooting parameter.
• Transforms the axisymmetric sessile drop problem into a one-parameter shooting method by using the apex curvature as the sole adjustable parameter.
• Solves the resulting ordinary differential equation numerically using NDSolve in Mathematica, with boundary conditions at the apex and contact line.
• Imposes a convergence criterion based on the slope at the apex (|slope| < 0.1) to ensure physical consistency.
• Uses a parametric formulation in terms of angle θ to describe the drop profile r(θ), enabling accurate tracking of the equatorial and apex points.
• Employs a root-finding procedure to locate the equatorial radius and height by solving for the maximum radius in the profile.

Experimental results

Research questions

• RQ1 Can the equilibrium shape of a sessile drop be accurately computed using only one shooting parameter, rather than multiple parameters?
• RQ2 How does the derived identity from the Young-Laplace equation simplify the numerical treatment of the drop shape problem?
• RQ3 What is the accuracy of the one-parameter shooting method when compared to experimental data and the ADSA-D method across varying Bond numbers?
• RQ4 To what extent can this method handle drops with both acute and obtuse contact angles, including high-Bond-number cases?

Key findings

• The one-parameter shooting method achieves excellent agreement with experimental data, with relative errors below 5% for contact radius and height across multiple liquid-surface systems.
• For water on carbon steel (θ = 72°, V = 6.75×10⁻³ cm³), the method computes ρ₀ = 0.1741 cm (0.4% error) and h = 0.1199 cm (4.4% error) compared to experiment.
• For mercury on glass (θ = 136°, V = 1.27×10⁻³ cm³), the method predicts h = 0.1119 cm with a 1.5% error, demonstrating robustness for high-contact-angle drops.
• The method successfully computes drop profiles up to a Bond number of 15, covering both small and large drops where gravity and surface tension are balanced.
• The Mathematica code provided enables direct computation of drop profiles with adjustable parameters, including volume, surface tension, density, and contact angle.
• The method reduces computational complexity and improves convergence by eliminating the need for multi-parameter optimization, making it more efficient than traditional shooting methods.