Equilibrium Shapes Of Droplets Research Articles

Droplet dynamics: A phase-field model of mobile charges, polarization, and its leaky dielectric approximation

This paper presents a Poisson–Nernst–Planck–Navier–Stokes–Cahn–Hillard (PNP–NS–CH) model for an electrically charged droplet suspended in a viscous fluid under an external electric field. Our model incorporates spatial variations in electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model: a dynamic model for the net charge (assuming unchanged conductance) and a leaky-dielectric model (assuming unchanged conductance and net charge). For the leaky-dielectric model, we perform a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are conducted to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electro-coalescence. Our numerical results from these two approximation models reveal that the polarization force, induced by the spatial variation in electric permittivity perpendicular to the external electric field, consistently dominates the Lorentz force arising from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Moreover, in the presence of interfacial capacitance, a local variation in effective permittivity results in the accumulation of counter-ions near the interface, leading to a reduction in droplet deformation. Our numerical solutions also confirm that the leaky-dielectric model is a reasonable approximation of the original PNP–NS–CH model when the electric relaxation time is sufficiently short. Both the Lorentz force and droplet deformation decrease significantly when the diffusion of net charge increases.

Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned Surfaces

The equilibrium shape of droplets on surfaces, functionalized with stripes of alternating wettability, have been investigated using simulations employing a finite element method. Experiments show that a droplet deposited on a surface with relatively narrow hydrophobic stripes compared to the hydrophilic stripes adopts a strongly elongated shape. The aspect ratio, the length of the droplet divided by the width, decreases toward unity when a droplet is deposited on a surface with relatively narrow hydrophilic stripes. The aspect ratio and the contact angle parallel to the stripes show unique scaling behavior as a function of the ratio between the widths of the hydrophobic and hydrophilic stripes. For a small ratio, the contact angle parallel to the stripes is low and the aspect ratio high, while for a large ratio, the contact angle parallel is high and the aspect ratio low. The simulations exhibit similar scaling behavior, both for the aspect ratio of the droplets and for the contact angles in the direction parallel to the stripes. Two liquids with different surface tensions have been investigated both experimentally and in simulations; similarities and differences between the findings are discussed. Generally, three parameters are needed to describe the droplet geometry: (i) the equilibrium contact angles on the hydrophilic and (ii) hydrophobic areas and (iii) the ratio of the widths of these chemically defined stripes. Furthermore, we derive a simple analytical expression that proves to be a good approximation in the quantitative description of the droplet aspect ratio.

JNURBS: An object-oriented, symbolic framework for integrated, meshless analysis and optimal design

In this paper, we propose a design-analysis integrated CAD framework termed jNURBS. jNURBS, developed using the Java language, is an extensible, Object-Oriented framework that enables meshless analysis of physical behavior and optimal design. The geometry as well as the analysis fields (displacement, temperature, etc.) are described mathematically using a common representation, namely the Non-Uniform Rational B-Spline (NURBS). Thus, NURBS serves to design the geometry as well as carryout meshless analysis thereby integrating the design and analysis in an efficient manner. The program kernel provides tools to symbolically describe complex multi-physics problems, methods to manipulate the NURBS entities, a set of primitive NURBS entities, and an iterative optimization solver. The problems are symbolically defined using a newly developed high-level, natural language description through an interface termed JNS (jNURBS Script). A number of example problems, selected from meshless structural analysis, material microstructure simulation, shape optimal design, and equilibrium shape of droplets, are presented to demonstrate the effectiveness and versatility of the framework.

Equilibrium shapes of droplets across grain boundaries

The equilibrium shape of a crystal in an infinite or a semi-infinite system has been known for decades. Here, we consider shapes of droplets which straddle two semifinite systems, joint at a planar, common boundary, such as a grain boundary of twinned crystals. For arbitrary anisotropic surface tensions, this problem is solved, in two dimensions. Some examples and special limits are shown.