Bubble Curvature Research Articles

Li–Yau inequalities for the Helfrich functional and applications

We prove a general Li–Yau inequality for the Helfrich functional where the spontaneous curvature enters with a singular volume type integral. In the physically relevant cases, this term can be converted into an explicit energy threshold that guarantees embeddedness. We then apply our result to the spherical case of the variational Canham–Helfrich model. If the infimum energy is not too large, we show existence of smoothly embedded minimizers. Previously, existence of minimizers was only known in the classes of immersed bubble trees or curvature varifolds.

Microtensiometer for Confocal Microscopy Visualization of Dynamic Interfaces.

Adsorption of surface-active molecules to fluid-fluid interfaces is ubiquitous in nature. Characterizing these interfaces requires measuring surfactant adsorption rates, evaluating equilibrium surface tensions as a function of bulk surfactant concentration, and relating how surface tension changes with changes in the interfacial area following equilibration. Simultaneous visualization of the interface using fluorescence imaging with a high-speed confocal microscope allows the direct evaluation of structure-function relationships. In the capillary pressure microtensiometer (CPM), a hemispherical air bubble is pinned at the end of the capillary in a 1 mL volume liquid reservoir. The capillary pressure across the bubble interface is controlled via a commercial microfluidic flow controller that allows for model-based pressure, bubble curvature, or bubble area control based on the Laplace equation. Compared to previous techniques such as the Langmuir trough and pendant drop, the measurement and control precision and response time are greatly enhanced; capillary pressure variations can be applied and controlled in milliseconds. The dynamic response of the bubble interface is visualized via a second optical lens as the bubble expands and contracts. The bubble contour is fit to a circular profile to determine the bubble curvature radius, R, as well as any deviations from circularity that would invalidate the results. The Laplace equation is used to determine the dynamic surface tension of the interface. Following equilibration, small pressure oscillations can be imposed by the computer-controlled microfluidic pump to oscillate the bubble radius (frequencies of 0.001-100 cycles/min) to determine the dilatational modulus The overall dimensions of the system are sufficiently small that the microtensiometer fits under the lens of a high-speed confocal microscope allowing fluorescently tagged chemical species to be quantitatively tracked with submicron lateral resolution.

Concentration profiles around and chemical composition within growing multicomponent bubble in presence of curvature and viscous effects

Abstract The regularities of changing chemical composition and size of a ultra-small multicomponent gas bubble growing in a viscous solution have been analyzed. The full-scale effects of solution viscosity and bubble curvature at non-stationary diffusion of arbitrary number of dissolved gases with any value of gas supersaturations and solubilities in the surrounding liquid solution have been taken into account. The nonuniform concentration profiles of gas species in supersaturated solution around the growing bubble with changing composition have been found as a function of time and distance from the bubble center. Equations describing transition to stationary concentrations of gases in the bubble with increasing radius have been obtained. Analytic asymptotic solutions of these equations for a ternary system have been presented.

A study of bubble growth in the compressible Rayleigh–Taylor and Richtmyer–Meshkov instabilities

Within the framework of modified Layzer-type potential flow theory [V. N. Goncharov, “Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers,” Phys. Rev. Lett. 88, 134502 (2002)], we study bubble growth in compressible Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities. It is known from adiabatic equations that the density ρ and adiabatic index γ are compressibility-related factors for a given static pressure p. Here, we introduce a dynamically varying stagnation point pressure P̃=p±12ρ̃η̇02, which relates time-varying quantities, such as fluid density ρ̃, pressure P̃, and bubble tip velocity η̇0, and then, we analytically derive the governing equations for time evolution of bubbles in the RT and RM instabilities of compressible fluids. For the RT instability, the upper fluid adiabatic index γu and density ρu increase the bubble amplitude and velocity, but they decrease the bubble curvature radius at the early stage, while the lower fluid adiabatic index γl and density ρl have opposite effects on those of γu and ρu, which is consistent with recent results. For the RM instability, γu and ρu decrease the bubble amplitude and velocity, but they increase the bubble curvature radius at the early stage; however, γl and ρl have opposite effects on those of γu and ρu. Moreover, we find a good agreement between our three-dimensional results of the RM bubble amplitude and recent numerical simulations.

EHD effects on periodic bubble formation and coalescence in ethanol under non-uniform electric field

This paper focus on the effects of electrohydrodynamic (EHD) on bubble dynamic characteristics generated on a submerged capillary in ethanol under non-uniform electric field, using high-speed photography and with a particular interest in the effect of electrical Bond number (BoE) on the bubble behaviors. The obtained results ranges of BoE from zero to 5.52 and Weber number (We) from 0.0006 to 0.1365 show that the bubble formation process associated with three stages at the low electric field strengths, including the nucleation, stable growth and necking stages, while the stable growth stage is absent for the high field strengths. With the increase of both We and BoE, the bubble period and waiting times are shortened rapidly, while the frequency is accelerated remarkably. The bubble dynamic behavior, which is characterized by the bubble volume and curvature radius, depends primarily on the BoE, and the increase in the We slightly raises the bubble volume and curvature radius at the departure time. Furthermore, it is found that with the increasing of BoE, bubbles coalescence evidenced at high We is weakened and eventually disappeared. Nevertheless, this phenomenon has the tendency to subsequently reappear at the higher We and BoE.

A superheat degree driven liquid-vapor phase-change lattice Boltzmann model

This paper proposes a phase-change pseudo-potential lattice Boltzmann model based on the bubble dynamic theory. To maintain the stability of the liquid during the phase change process, the phase change of the model is driven by the superheat degree in the interface area instead of the thermodynamic temperature in the equation of state (EOS). The presented model is verified by the simulations of liquid film evaporation and is applied to the simulation of the bubble growth in superheated fluid and on the wall. The simulation results show that the simulation density ratio over 100 can be reached. The influence of the bubble curvature on the superheat degree in the bubble can also be reflected with the proposed model.

Effect of ζ-Potentials on Bubble-Particle Interactions

Derjaguin and Dukhin (1961) were the first to develop a model for bubble-particle interaction in flotation from first principles by considering the surface forces in wetting (or flotation) films. The model predicted that the energy barriers to bubble-particle attachment arise from the ζ-potentials of fine particles, which corroborates well with an earlier work of Fuerstenau (1957) reported for the flotation of quartz using dodecyl ammonium acetate (DAA) as collector. In the present work, a series of surface force measurements have been conducted by accurately monitoring the changes in bubble curvature during bubble-silica surface interactions. The curvature changes are then used to determine the capillary force, which is the sum of the hydrodynamic and surface forces. By subtracting the former that can also be determined from the curvature changes, one obtains the surface forces (or disjoining pressure). The results obtained as functions of pH and collector concentrations show that control of the ζ-potentials of bubbles relative to those of the silica surfaces is critical for maximizing the negative disjoining pressure, which is conducive to promoting bubble-particle attachment in flotation.

Structural Transition in Liquid Crystal Bubbles Generated from Fluidic Nanocellulose Colloids

AbstractThe structural transition in micrometer‐sized liquid crystal bubbles (LCBs) derived from rod‐like cellulose nanocrystals (CNCs) was studied. The CNC‐based LCBs were suspended in nematic or chiral nematic liquid‐crystalline CNCs, which generated topological defects and distinct birefringent textures around them. The ordering and structure of the LCBs shifted from a nematic to chiral nematic arrangement as water evaporation progressed. These packed LCBs exhibited a specific photonic cross‐communication property that is due to a combination of Bragg reflection and bubble curvature and size.

Structural Transition in Liquid Crystal Bubbles Generated from Fluidic Nanocellulose Colloids.

The structural transition in micrometer-sized liquid crystal bubbles (LCBs) derived from rod-like cellulose nanocrystals (CNCs) was studied. The CNC-based LCBs were suspended in nematic or chiral nematic liquid-crystalline CNCs, which generated topological defects and distinct birefringent textures around them. The ordering and structure of the LCBs shifted from a nematic to chiral nematic arrangement as water evaporation progressed. These packed LCBs exhibited a specific photonic cross-communication property that is due to a combination of Bragg reflection and bubble curvature and size.

Statistical mechanics of two-dimensional foams: Physical foundations of the model.

In a recent series of papers, a statistical model that accounts for correlations between topological and geometrical properties of a two-dimensional shuffled foam has been proposed and compared with experimental and numerical data. Here, the various assumptions on which the model is based are exposed and justified: the equiprobability hypothesis of the foam configurations is argued. The range of correlations between bubbles is discussed, and the mean-field approximation that is used in the model is detailed. The two self-consistency equations associated with this mean-field description can be interpreted as the conservation laws of number of sides and bubble curvature, respectively. Finally, the use of a "Grand-Canonical" description, in which the foam constitutes a reservoir of sides and curvature, is justified.

Surface tension driven processes densify and retain permeability in magma and lava

We offer new insights into how an explosive eruption can transition into an effusive eruption. Magma containing >0.2 wt% dissolved water has the potential to vesiculate to a porosity in excess of 80 vol.% at atmospheric pressure. Thus all magmas contain volatiles at depth sufficient to form foams and explosively fragment. Yet gas is often lost passively and effusive eruptions ensue. Magmatic foams are permeable and understanding permeability in magma is crucial for models that predict eruptive style. Permeability also governs magma compaction models. Those models generally imply that a reduction in magma porosity and permeability generates an increased propensity for explosivity.Here, our experimental results show that surface tension stresses drive densification without creating an impermeable ‘plug’, offering an additional explanation of why dense magmas can avoid explosive eruption. In both an open furnace and a closed autoclave, we subject pumice samples with initial porosity of ∼70 vol.% to a range of isostatic pressures (0.1–11 MPa) and temperatures (350–950 °C) relevant to shallow volcanic environments. Our experimental data and models constrain the viscosity, permeability, timescales, and length scales over which densification by pore-scale surface tension stresses competes with density-driven compaction. Where surface tension dominates the dynamics, densification halts at a plateau connected porosity of ∼25 vol.% for our samples. SEM, pycnometry and micro-tomography show that in this process (1) microporous networks are destroyed, (2) the relative pore network surface area decreases, and (3) a remaining crystal framework enhances the longevity of macro-pore connectivity and permeability critical for sustained outgassing. We propose that these observations are a consequence of a surface tension-driven retraction of viscous pore walls at areas of high bubble curvature (micro-vesicular network terminations), and that this process drives bulk densification and permits continued outgassing. We propose a regime diagram of the relative dominance of surface tension and gravitational compaction that illustrates the interplay between viscosity, permeability, lengthscale and timescale. We contend that surface tension-driven magma densification is an as-yet overlooked phenomenon that extends our volcanological, geothermal and hydrothermal knowledge of how gas can escape densifying volcanic plugs and why dense lavas remain permeable.

Positron study of surface tension reduction in liquid alkane at high pressure of argon

Surface tension σ of sub-nanometric bubbles in n-nonadecane + argon system as a function of pressure was estimated using the positron lifetime spectroscopy. The surface tension is higher than for flat surface (due to negative bubble curvature). The coefficient (∂σ/∂p)p=0 is negative but much smaller than for flat gas-liquid dividing surface. It can be explained by the lack of adsorbed argon layer on the bubble surface. In the narrow range of temperatures, where nonadecane melts by application of argon pressure, the concentration of argon in the liquid is found smaller than at higher temperatures and probably a kind of intermediate structure is formed.

Effects of head loss on the growth of the Rayleigh–Taylor and the Richtmyer–Meshkov instabilities

Within the Zufiria’s potential flow theory generalized by Sohn (2007), we investigate the effects of head loss on the growth of the Rayleigh–Taylor and the Richtmyer–Meshkov instabilities. The nonlinear asymptotic solutions including the effects of head loss are obtained analytically for the bubble amplitude, velocity and curvature. We find that head loss depresses the bubble amplitude and velocity but enhances the bubble curvature. With an increase in the loss coefficient ζ, the bubble amplitude and velocity decease rapidly, whereas the bubble curvature increases rapidly. A maximum of ζ is found for the Rayleigh–Taylor instability, i.e., ζ→ζmax=0.5 which is independent of the Atwood number A. But for the Richtmyer–Meshkov instability, ζmax depends on the Atwood number and ζmax=0.625-0.875 for A=0-1. Our results of the bubble amplitude are found to be in good agreement with experimental data when we choose the suitable loss coefficients.

Effects of surface tension on Rayleigh-Taylor instability

In this paper, Layzers model which has a simple velocity potential, and Zufirias model are extended to the case of non-ideal fluids, and the effects of surface tension on Rayleigh-Taylor instability are investigated. Firstly, the analytical expressions for the asymptotic bubble velocity and curvature are obtained in the two models. Secondly, the effects of surface tension on Rayleigh-Taylor instability are studied systematically. Finally, the two models are compared with each other and the comparisons with numerical simulation are made as well. The results indicate that the surface tension depresses the bubble velocity, but does not affect the bubble curvature. The Layzers model with the simple velocity potential gives a smaller bubble velocity than that predicted by the Layzers model with a complex velocity potential. But the bubble velocity predicted by the Layzers model with the simple velocity potential is larger than that obtained by Zufirias model. Both Layzers models lead to the same bubble velocity when the Atwood number is A = 1.

HIGH-ORDER POTENTIAL FLOW MODELS FOR HYDRODYNAMIC UNSTABLE INTERFACE

We present two high-order potential flow models for the evolution of the interface in the Rayleigh-Taylor instability in two dimensions. One is the source-flow model and the other is the Layzer-type model which is based on an analytic potential. The late-time asymptotic solution of the source-flow model for arbitrary density jump is obtained. The asymptotic bubble curvature is found to be independent to the density jump of the fluids. We also give the time- evolution solutions of the two models by integrating equations numerically. We show that the two high-order models give more accurate solutions for the bubble evolution than their low- order models, but the solution of the source-flow model agrees much better with the numerical solution than the Layzer model.

Interfacial rheology and stability of air bubbles stabilized by mixtures of hydrophobin and β-casein

Class II hydrophobin (HFBII) is a highly surface active molecule and in the context of aeration can be considered to be an air structuring protein conferring exceptional stability to foams for periods far in excess of that obtained with any other commonly used protein. This is of interest to the food industry, since producing shelf stable foams in food formulations is very difficult. Although HFBII has proven to be very promising in terms of foam stability when used alone, it is still unknown whether HFBII will be able to maintain its functionality when other surface active agents are present, such as in real food systems. The surface rheology: surface shear viscosity (ηs) and surface dilatational elasticity (ε), for HFBII and β-casein mixes at various ratios is described in this paper and how this relates to bubble stability. The addition of β-casein up to a certain ratio seems to increase ηs significantly, whilst ε is less affected. This is accompanied by improved stability of air bubbles to coalescence and allows the formation of very small air bubbles that remain extremely stable to disproportionation. Overall, there is the suggestion of some kind of synergy between the two proteins. The exact nature of this interaction is unknown. Measurements of the ζ-potential of the proteins suggest that electrostatic interactions are probably not important at the pH investigated (pH 7). Confocal microscopy of individual bubbles over prolonged periods of time, stabilized by HFBII and a fluorescently labeled β-casein, suggests that the enhanced stability is due to highly unusual and complicated interfacial packing phenomena plus local bubble curvature effects that require further investigation.

High-resolution vortex simulations of the Richtmyer-Meshkov instability

We perform numerical simulations for the Richtmyer-Meshkov instability by employing an adaptive vortex method, and we present the late-time nonlinear evolution of the unstable interface. The adaptive vortex method takes much longer times than previous vortex simulations and gives results for the highly distorted interfaces in fine resolutions. For a small density jump, only the inner core of the interface is distorted on a small scale, yielding a secondary instability, but the interface maintains a global structure of the uniform roll-up. For a moderate density jump, the interface has stronger distortions on the roll-up overall and develops to a more complex structure. We also compare the numerical results with solutions of theoretical models and examine disagreements between the theoretical models for the Richtmyer-Meshkov instability. The numerical results are in good agreement with a potential source-flow model and show convergence of the bubble curvature to a constant limit.

Asymptotic bubble evolutions of the Rayleigh–Taylor instability

We examine the multiple harmonic model for the single-mode Rayleigh–Taylor instability, and present a new class of the asymptotic solution for the bubble evolution. Previously reported solutions for the bubble curvature and velocity from the model were quantitatively different from other theoretical models and numerical results, for small density jumps. The discrepancy between the theoretical models is resolved by our new approach to the model. Our solution agrees with the Layzer–Goncharov model, and gives the independence of the bubble curvature on the density ratio.

A molecular dynamics study of nano-bubble surface tension

The main purpose of this paper is to numerically investigate the effect of bubble curvature on the surface tension. The effect of surface wettability on the bubble shape is also investigated due to its direct effects on the bubble curvature. Also, the validity of Young–Laplace (Y–L) equation to describe the bubble characteristics in nano order is examined using the molecular dynamics (MD) simulations method. A computer code based on the MD method is developed. The code carries out a series of simulations to generate bubbles of various radii between two planar solid surfaces. In our simulation, the total volume of the simulation box as well as the surface wettability affects the bubble curvature. The pair potential for the liquid–liquid and liquid–solid interaction is considered using the Lennard-Jones model. Pressure tensor and density profile are locally calculated. Furthermore, liquid pressure is evaluated far from the interface using the virial theorem and gas pressure is obtained using an equation of state. It is observed that the gas pressure is almost independent of the bubble radius. However, the liquid pressure becomes more negative as the radius decreases. Surface tension is computed using the Y–L equation and compared with the surface tension directly computed from the MD simulation. The amount of surface tension increases with a decrease in the radius.

Nonlinear evolution of the Richtmyer–Meshkov instability

We report analytical and numerical results describing the dynamics of the two-dimensional coherent structure of bubbles and spikes in the Richtmyer–Meshkov instability for fluids with a finite density ratio. The theory accounts for the non-local properties of the interface evolution, and the simulations treat the interface as a discontinuity. Good agreement between the analytical and numerical results is achieved. To quantify accurately the interface dynamics in the simulations, new diagnostics and scalings are suggested. The velocity at which the interface would move if it were ideally planar is used to set the flow time scale as well as the reference point for the bubble (spike) position. The data sampling has high temporal resolution and captures the velocity oscillations caused by sound waves. The bubble velocity and curvature are both monitored, and the bubble curvature is shown to be the relevant diagnostic parameter. According to the results obtained, in the nonlinear regime of the Richtmyer–Meshkov instability the bubbles flatten and decelerate, and the flattening of the bubble front indicates the multiscale character of the coherent dynamics.