Vanishing Surface Tension Research Articles

Stable bulk nanobubbles can be regarded as gaseous analogues of microemulsions

In our previous work [2022 Phys. Chem. Chem. Phys. 24 9685], we used molecular dynamics simulations to show that bulk nanobubbles can be stabilized by forming a compressed amphiphile monolayer at bubble interfaces. This observation closely matches the origin of stability of microemulsions and inspired us to propose here that, in certain cases, stable bulk nanobubbles can be regarded as gaseous analogues of microemulsions: the nanobubble phase and the bubble-containing solution phase coexist with the external gas phase. This three-phase coexistence is then validated by molecular dynamics simulations. The stability mechanism for bulk nanobubbles is thus given: the formation of a compressed amphiphilic monolayer because of microbubble shrinking leads to a vanishing surface tension, and consequently the curvature energy of the monolayer dominates the thermodynamic stability of bulk nanobubbles. With the monolayer model, we further interpret several strange behaviors of bulk nanobubbles: gas supersaturation is not a prerequisite for nanobubble stability because of the vanishing surface tension, and the typical nanobubble size of 100 nm can be explained through the small bending constant of the monolayer. Finally, through analyzing the compressed amphiphile monolayer model we propose that bulk nanobubbles can exist ubiquitously in aqueous solutions.

A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.

Size-dependent effective shear modulus in a three-phase sphere model incorporating Steigmann-Ogden surface elasticity

We derive a solution for the size-dependent effective shear modulus of a three-phase sphere model incorporating Steigmann-Ogden surface elasticity under a uniform remote deviatoric load. In the three-phase model, the internal spherical inhomogeneity, the intermediate spherical annulus of the matrix and the surrounding infinite region of equivalent homogeneous material are bonded through an inner spherical interface endowed with Steigmann-Ogden surface elasticity (with vanishing surface tension) and an outer perfect spherical interface. The value of the effective shear modulus is equal to that of the matrix for any volume fraction when the three size-dependent interface parameters satisfy a single condition. Furthermore, the neutrality of the composite sphere to a prescribed uniform deviatoric stress field for any volume fraction is accomplished when the three size-dependent interface parameters satisfy two particular conditions. The neutrality is valid only for a Steigmann-Ogden interface and not for a Gurtin-Murdoch interface.

Buckling of lipidic ultrasound contrast agents under quasi-static load.

Collapse of lipidic ultrasound contrast agents under high-frequency compressive load has been historically interpreted by the vanishing of surface tension. By contrast, buckling of elastic shells is known to occur when costly compressible stress is released through bending. Through quasi-static compression experiments on lipidic shells, we analyse the buckling events in the framework of classical elastic buckling theory and deduce the mechanical characteristics of these shells. They are then compared with that obtained through acoustic characterization. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, https://doi.org/10.1002/cpa.3160180124 ). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $$\alpha \ge 0$$ and spatial period $$2\pi /\alpha $$ ; two-dimensional sloshing corresponds to the case $$\alpha = 0$$ . Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $$\alpha > 0$$ . In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $$\alpha \ge 0$$ and shallow radially symmetric containers with m azimuthal nodal lines, $$m = 0, 1$$ . Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $$\alpha = 0$$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.

Laplacian growth with an isolated bubble near a tip.

We construct exact solutions for the Saffman-Taylor problem in a planar Hele-Shaw cell for the case of an isolated bubble near the tip of an expanding bubble in the limit of zero surface tension. This construction utilizes the integrability of the problem in the limit of vanishing surface tension. It exploits the connection, brought by the Schwarz function, between the constants of motion and the shape of the bubbles. The results of this paper provide a step toward the theoretical understanding of the dynamics of Saffman-Taylor problem in a Hele-Shaw cell with randomly distributed tiny bubbles.

The Vanishing Surface Tension Limit of the Muskat Problem

The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}(\mathbb{R}^d)$, where $d$ is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension $\frak{s}$ converge to the unique solution of the Muskat problem without surface tension locally in time with the rate $\sqrt{\frak{s}}$ when $\frak{s}\to 0$. This allows for initial interfaces that have unbounded or even not locally square integrable curvature. If in addition the initial curvature is square integrable, we obtain the convergence with optimal rate $\frak{s}$.

Bose–Einstein Condensation as a Deposition Phase Transition of Quantum Hard Spheres and New Relations between Bosonic and Fermionic Pressures

We investigate the phase transition of Bose–Einstein particles with the hard-core repulsion in the grand canonical ensemble within the Van der Waals approximation. It is shown that the pressure of non-relativistic Bose–Einstein particles is mathematically equivalent to the pressure of simplified version of the statistical multifragmentation model of nuclei with the vanishing surface tension coefficient and the Fisher exponent тF = 5/2 , which for such parameters has the 1-st order phase transition. The found similarity of these equations of state allows us to show that within the present approach the high density phase of Bose-Einstein particles is a classical macro-cluster with vanishing entropy at any temperature which, similarly to the system of classical hard spheres, is a kind of solid state. To show this we establish new relations which allow us to identically represent the pressure of Fermi–Dirac particles in terms of pressures of Bose–Einstein particles of two sorts.

Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

Selection of a Hele-Shaw Bubble via Exponential Asymptotics

The well-studied selection problems involving Saffman--Taylor fingers or Taylor--Saffman bubbles in a Hele-Shaw channel are prototypical examples of pattern selection. Exact solutions to the corresponding zero-surface-tension problems exist for an arbitrary finger or bubble speed, but the addition of surface tension leads to a discrete set of solution branches, all of which approach a single solution in the limit in which the surface tension vanishes. In this sense, the surface tension selects a single physically meaningful solution from the continuum of zero-surface-tension solutions. Recently, we provided numerical evidence to suggest that the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results have ramifications for exotic-shaped Saffman--Taylor fingers as well as for the time-dependent evolution of bubbles propagating in Hele-Shaw cells.

Electrorheology of a dilute emulsion of surfactant-covered drops

We investigate the effects of surfactant coating on a deformable viscous drop under the combined action of shear flow and a uniform electric field. Employing a comprehensive three-dimensional approach, we analyse the non-Newtonian shearing response of the bulk emulsion in the dilute suspension regime. Our results reveal that the location of the peak surfactant accumulation on the drop surface may get shifted from the plane of shear to a plane orthogonal to it, depending on the tilt angle of the applied electric field and strength of the electrical stresses relative to their hydrodynamic counterparts. The surfactant non-uniformity creates significant alterations in the flow perturbation around the drop, triggering modulations in the bulk shear viscosity. Overall, the shear-thinning or shear-thickening behaviour of the emulsion appears to be greatly influenced by the interplay of surface charge convection and Marangoni stresses. We show that the balance between electrical and hydrodynamic stresses renders a vanishing surface tension gradient on the drop surface for some specific shear rates, rendering negligible alterations in the bulk viscosity. This critical condition largely depends on the electrical permittivity and conductivity ratios of the two fluids and orientation of the applied electric field. Also, the physical mechanisms of charge convection and surface deformation play their roles in determining this critical shear rate. As a consequence, we obtain new discriminating factors, involving electrical property ratios and the electric field configuration, which govern the same. Consequently, the surfactant-induced enhancement or attenuation of the bulk emulsion viscosity depends on the electrical conductivity and permittivity ratios. The concerned description of the drop-level flow physics and its connection to the bulk rheology of a dilute emulsion may provide a fundamental understanding of a more complex emulsion system encountered in industrial practice.

The zero surface tension limit of three-dimensional interfacial Darcy flow

We study the zero surface tension limit of three-dimensional interfacial Darcy flow. We start with a proof of well-posedness of three-dimensional interfacial Darcy flow for any positive value of the surface tension coefficient. The primary tool for this well-posedness proof is an energy estimate. The time of existence for these solutions will, in general, go to zero with the surface tension parameter. However, in the case that a stability condition is satisfied by the initial data, we prove an additional energy estimate, establishing that the time of existence can be made uniform in the surface tension parameter. Then, an additional estimate allows the limit to be taken as surface tension vanishes, demonstrating that three-dimensional interfacial Darcy flow without surface tension is the limit of three-dimensional interfacial Darcy flow with surface tension as surface tension vanishes. This provides a new proof of existence of solutions for the problem without surface tension.

How antagonistic salts cause nematic ordering and behave like diblock copolymers.

We present simulation results and an explanatory theory on how antagonistic salts affect the spinodal decomposition of binary fluid mixtures. We find that spinodal decomposition is arrested and complex structures form only when electrostatic ion-ion interactions are small. In this case, the fluid and ion concentrations couple and the charge field can be approximated as a polynomial function of the relative fluid concentrations alone. When the solvation energy associated with transferring an ion from one fluid phase to the other is of the order of a few kBT, the coupled fluid and charge fields evolve according to the Ohta-Kawasaki free energy functional. This allows us to accurately predict structure sizes and reduce the parameter space to two dimensionless numbers. The lamellar structures induced by the presence of the antagonistic salt in our simulations exhibit a high degree of nematic ordering and the growth of ordered domains over time follows a power law. This power law carries a time exponent proportional to the salt concentration. We qualitatively reproduce and interpret neutron scattering data from previous experiments of similar systems. The dissolution of structures at high salt concentrations observed in these experiments agrees with our simulations, and we explain it as the result of a vanishing surface tension due to electrostatic contributions. We conclude by presenting 3D results showing the same morphologies as predicted by the Ohta-Kawasaki model as a function of volume fraction and suggesting that our findings from 2D systems remain valid in 3D.

On Spherical Inhomogeneity With Steigmann–Ogden Interface

The problem of an infinite isotropic elastic space subjected to uniform far-field load and containing an isotropic elastic spherical inhomogeneity with Steigmann–Ogden interface is considered. The interface is treated as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffnesses. The constitutive and equilibrium equations of the Steigmann–Ogden theory for a spherical surface are written in explicit forms. Closed-form analytical solutions are derived for two cases of loading conditions—the hydrostatic loading and deviatoric loading with vanishing surface tension. The single inhomogeneity-based estimates of the effective properties of macroscopically isotropic materials containing spherical inhomogeneities with Steigmann–Ogden interfaces are presented. It is demonstrated that, in the case of vanishing surface tension, the Steigmann–Ogden model describes a special case of thin and stiff uniform interphase layer.

Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension

This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which arises in modeling the motion of such a fluid down an inclined plane, after a coordinate change. We consider the problem both with and without surface tension for horizontally periodic flows. This problem gives rise to shear-flow equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of the equilibria in certain parameter regimes. We prove that there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t=0$ give rise to global-in-time solutions that return to equilibrium exponentially in the case with surface tension and almost exponentially in the case without surface tension. We also establish a vanishing surface tension limit, which connects the solutions with and without surface tension.

Liquid Fuel Evaporation under Supercritical Conditions

Molecular dynamics simulations are performed to study the supercritical mixing process of the n-dodecane/nitrogen binary system. Previous studies have shown the existence of supercritical phenomenon under certain conditions in modern propulsion systems such as diesel engines. However, the physical mechanisms and internal driving forces of this phenomenon are still not well understood. In this paper, we attempt to answer this question through simulating the diffusion and evaporation of gaseous nitrogen and liquid phase n-dodecane. It addresses under what conditions the supercritical transition phenomenon happens and what features the supercritical evaporation process have. A unique configuration is constructed to mimic the evaporation of an n-dodecane thin film in an open nitrogen environment under conditions ranging from subcritical to supercritical. The detailed structure of the liquid-vapor interface during the evaporating process is described and the evaporation rate and the interface thickness are estimated, which show differences between subcritical and supercritical evaporation. Results indicate that under relatively high pressure conditions, the liquid surface transitions into supercritical state, and the liquid-vapor interface expands significantly with vanishing surface tension, leading to a diffusion like mixing process. It is shown that the supercritical evaporation would happen under conditions that correspond to the in-cylinder conditions of a turbo-charged engine.

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell.

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

Temperature-Tuned Faceting and Shape Changes in Liquid Alkane Droplets.

Recent extensive studies reveal that surfactant-stabilized spherical alkane emulsion droplets spontaneously adopt polyhedral shapes upon cooling below a temperature Td while remaining liquid. Further cooling induces the growth of tails and spontaneous droplet splitting. Two mechanisms were offered to account for these intriguing effects. One assigns the effects to the formation of an intradroplet frame of tubules consisting of crystalline rotator phases with cylindrically curved lattice planes. The second assigns the sphere-to-polyhedron transition to the buckling of defects in a crystalline interfacial monolayer, known to form in these systems at some Ts > Td. The buckling reduces the extensional energy of the crystalline monolayer's defects, unavoidably formed when wrapping a spherical droplet by a hexagonally packed interfacial monolayer. The tail growth, shape changes, and droplet splitting were assigned to the decrease and vanishing of surface tension, γ. Here we present temperature-dependent γ(T), optical microscopy measurements, and interfacial entropy determinations for several alkane/surfactant combinations. We demonstrate the advantages and accuracy of the in situ γ(T) measurements made simultaneously with the microscopy measurements on the same droplet. The in situ and coinciding ex situ Wilhelmy plate γ(T) measurements confirm the low interfacial tension, ≲0.1 mN/m, observed at Td. Our results provide strong quantitative support validating the crystalline monolayer buckling mechanism.

The vanishing surface tension limit for the Hele-Shaw problem

In this paper, we show the existence of solutions of the Hele-Shaw problem in two dimensions in the presence of surface tension and for a general class of initial data. The limit problem from nonzero to zero surface tension will also be investigated. In the case of injection and when volume conservation holds, for a sufficiently small surface tension, we prove the existence and uniqueness of perturbed solutions with nonzero surface tension near solutions with zero surface tension. We also show that solutions with nonzero surface tension exist up to a finite time before a possible singularity occurs in which solutions with zero surface tension are well defined. In addition, in the finite time interval, we prove that the solutions with nonzero surface tension approach the solutions with zero surface tension as the surface tension coefficient goes to zero. In the case of suction, for sufficiently small surface tension, we prove the existence of perturbed solutions near solutions with zero surface tension in any initially smooth domains. In this case, the local existence time depends on the surface tension coefficient.

Atomization and Dense-Fluid Breakup Regimes in Liquid Rocket Engines

Until recently, modern theory has lacked a fundamentally based model to predict the operating pressures where classical sprays transition to dense-fluid mixing with diminished surface tension. In this paper, such a model is presented to quantify this transition for liquid-oxygen–hydrogen and n-decane–gaseous-oxygen injection processes. The analysis reveals that respective molecular interfaces break down not necessarily because of vanishing surface tension forces but instead because of the combination of broadened interfaces and a reduction in mean free molecular path. When this occurs, the interfacial structure itself enters the continuum regime, where transport processes rather than intermolecular forces dominate. Using this model, regime diagrams for the respective systems are constructed that show the range of operating pressures and temperatures where this transition occurs. The analysis also reveals the conditions where classical spray dynamics persists even at high supercritical pressures. It demonstrates that, depending on the composition and temperature of the injected fluids, the injection process can exhibit either classical spray atomization, dense-fluid diffusion-dominated mixing, or supercritical mixing phenomena at chamber pressures encountered in state-of-the-art liquid rocket engines.