Bubble Formation in a Finite Cone: More Pieces to the Puzzle

Abstract

We investigate the stability of bubble formation, starting with a convex or a concave meniscus, from a liquid solution (of water and a dissolved gas) inside a finite cone at constant temperature and constant liquid pressure (above the saturation pressure of the pure solvent). It is assumed that the dissolved gas (nitrogen) forms a dilute solution at equilibrium, which can be described by Henry's law. The number and nature of equilibrium states are determined with Gibbsian composite-system thermodynamics, both from the intersection of the equilibrium Kelvin radius with the geometry radius and from the extrema in the plot of free energy of the system versus size of the new phase. Bubble stability is studied along the whole growth path, as the bubble grows inside, gets pinned, and grows further outside the finite cone. The changes in the concentration of the liquid bulk phase and the vapor phase during the growth of the bubble are carefully incorporated in the equations. The effects of various parameters, including cone apex angle, cone half mouth radius, contact angle, total number of moles, and initial degree of saturation, on the stability of the bubble are also investigated. Stability of bubble formation from a liquid solution inside a confined geometry such as a finite cone is of interest in areas such as restoring underwater superhydrophobicity and adhesion of particles to the roughness of synthetic biomaterials.

Gibbsian Surface Thermodynamics

“…Ward's and Levart's treatment of curved interface phase equilibrium in a dilute liquid−gas solution was used by Zargarzadeh and Elliott for nucleation of drops and bubbles at solid interfaces with a variety of geometries. 61,62 With the goal of elucidating the effect of interface curvature on multicomponent vapor−liquid equilibrium phase diagrams across the complete composition range of the liquid and the vapor, Shardt and Elliott incorporated activity coefficients to capture the thermodynamic nonideality of the liquid phase (while still treating the vapor phase as an ideal gas) and derived forms of the Kelvin equation describing the vapor-phase pressure at the bubble point 13,63…”

“…9 In Figure 8, the free energy is examined for a bubble growing out of a water−nitrogen solution where the nucleation happens inside a finite conical pit in a solid. 62 In the case examined, there are two equilibrium states. There is an unstable equilibrium at a small bubble size indicating the energy barrier that must be overcome for the bubble to nucleate inside the cone, and there is a stable equilibrium at a larger bubble size that occurs when the bubble is pinned at the cone mouth.…”

Underwater Fast Bubble Generating on Pitaya Thorn and Enhanced Biomimetic Gas Collection

“…For the given experimental condition (Temperature: 21 ± 0.7 °C, atmospheric pressure), R U is the universal gas constant, γ LG is the water–gas ( L – G ) interfacial tension, ≈72 mN m −1 , p ∞ is the saturation pressure of pure water, ≈3 kpa, k H is Henry's law constant, for dissolved oxygen in water it is 4.15× 10 4 atm. [ 26,27 ] n 1 is the number moles of water, n 2 is that of gas, in this model they are, n 1 ≈ 8.7 × 10 −5 mol, n 2 = 1.5 × 10 −9 mol (see the Supporting Information). V G is the volume of gas column and bubble.…”

Thermodynamic Investigation of the Effect of Electric Field on Solid–Liquid Equilibrium

“…Shardt and Elliott applied Gibbsian composite-system thermodynamics to derive the equilibrium conditions for the wetting of rough surfaces resulting in a line-fraction form of the Cassie–Baxter equation and a new line-roughness-controlled Wenzel equation . Gibbsian composite-system thermodynamics has been used to study the nucleation and thermodynamic stability of new fluid phase (liquid or vapor) formation at solid surfaces in several geometries, − including a thermodynamic description of surface nanobubbles . Eslami and Elliott used Gibbsian composite-system thermodynamics to show why nucleation at a fluid surface occurs more readily than at a rigid surface .…”