Analytical solutions in fluid dynamics can be used to elucidate the physics of complex flows and to serve as test cases for numerical models. In this work, we present the analytical solution for the acoustic boundary layer that develops around a rigid sphere executing small amplitude harmonic rectilinear motion in a compressible fluid. The mathematical framework that describes the primary flow is identical to that of wave propagation in linearly elastic solids, the difference being the appearance of complex instead of real valued wave numbers. The solution reverts to well-known classical solutions in special limits: the potential flow solution in the thin boundary layer limit, the oscillatory flat plate solution in the limit of large sphere radius and the Stokes flow solutions in the incompressible limit of infinite sound speed. As a companion analytical result, the steady second order acoustic streaming flow is obtained. This streaming flow is driven by the Reynolds stress tensor that arises from the axisymmetric first order primary flow around such a rigid sphere. These results are obtained with a linearization of the non-linear Navier-Stokes equations valid for small amplitude oscillations of the sphere. The streaming flow obeys a time-averaged Stokes equation with a body force given by the Nyborg model in which the above mentioned primary flow in a compressible Newtonian fluid is used to estimate the time-averaged body force. Numerical results are presented to explore different regimes of the complex transverse and longitudinal wave numbers that characterize the primary flow.
E. Klaseboer, D. Y. C. Chan
On the derivation of the Smoluchowski result of electrophoretic mobility
Journal of Colloid and Interface Science 568, 176-184 (2020). pdf, DOI
One of the most enduring, broadly applicable and widely used theoretical results of electrokinetic theory is the Smoluchowski expression for the electrophoretic mobility. It is a limiting form that holds for any solid particle of arbitrary shape in an electrolyte of any composition provided the thickness of the electrical double layer is "inﬁnitely" thin compared to the particle size and the particle has uniform surface potential. The familiar derivation of this result that is a simpliﬁed version of the original Smoluchowski analysis in 1903, considers the motion of the electrolyte adjacent to a planar surface. The theory is deceptively simple but as a result much of the interesting physics and characteristic hydrodynamic behavior around the particle have been obscured. This paper provides a derivation of this key theoretical result by starting from Smoluchowski’s original 1903 analysis but brings out overlooked details of the hydrodynamic features near and far from the particle that have not been canvassed in detail. The objective is to draw together all the key physical features of the electrophoretic problem in the thin double layer regime to provide an accessible and complete exposition of this important result in colloid science.
Q. Sun, E. Klaseboer, A. J. Yuffa, D. Y. C. Chan
Robust Field-Only Surface Integral Equations: Scattering from a Perfect Electric Conductor
Journal of the Optical Society of America 37, 276-283 (2020). pdf
A robust ﬁeld-only boundary integral formulation of electromagnetics is derived without the use of surface currents that appear in the Stratton–Chu formulation. For scattering by a perfect electrical conductor (PEC), the components of the electric ﬁeld are obtained directly from surface integral equation solutions of three scalar Helmholtz equations for the ﬁeld components. The divergence-free condition is enforced via a boundary condition on the normal component of the ﬁeld and its normal derivative. Field values and their normal derivatives at the surface of the PEC are obtained directly from surface integral equations that do not contain divergent kernels. Consequently, high-order elements with fewer degrees of freedom can be used to represent surface features to a higher precision than the traditional planar elements. This theoretical framework is illustrated with numerical examples that provide further physical insight into the role of the surface curvature in scattering problems.
Q. Sun, E. Klaseboer, A. J. Yuffa, D. Y. C. Chan
Robust Field-Only Surface Integral Equations: Scattering from a Dielectric Body
Journal of the Optical Society of America 37, 284-293 (2020). pdf
A robust and eﬃcient ﬁeld-only nonsingular surface integral method to solve Maxwell’s equations for the components of the electric ﬁeld on the surface of a dielectric scatterer is introduced. In this method, both the vector Helmholtz equation and the divergence-free constraint are satisﬁed inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric ﬁeld across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic ﬁelds are expressed in terms of the electric ﬁeld across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric ﬁeld inside and outside the scatterer is solved by a fully desingularized surface integral method. Comparing with the most popular surface integral methods based on the Stratton–Chu formulation or the PMCHWT formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents and the use of complicated vector basis functions can be avoided altogether. Also, our method is not aﬀected by numerical issues such as the zero frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric ﬁeld and the magnetic ﬁeld, our theoretical framework can also be used to solve for the magnetic ﬁeld.
J. Oshitani, S. Hayashi, D. Y. C. Chan
Order from Chaos: Dynamics of density segregation in continuously aerated granular systems
Advanced Powder Technology 31, 843-847 (2020) DOI , pdf, supp
Under continual disturbance such as vibration, tumbling, flow or aeration, granular or powder systems can display solid or fluid like behavior. Using a well-mixed system of same size (0.2 mm) non-cohesive glass beads and iron powder, we show that gentle aeration can completely segregate the components thereby reducing the entropy of mixing to create near total order from an initially chaotic mixture. We quantify the time dependence of the segregation process and identify two dynamic pathways that dominate depending on the intensity of the aeration. Such findings can facilitate the search for energy efficient methods to process granular systems in pharmaceutical, mining and waste recovery industries.
I. U. Vakarelski, F. Yang, Y. S. Tian, E. Q. Li, D. Y. C. Chan, S. T. Thoroddsen
Mobile-surface bubbles and droplets coalesce faster but bounce stronger
Enhancing the hydrodynamic interfacial mobility of bubbles and droplets in multiphase systems is expected to reduce the characteristic coalescence times and hence affect the stability of gas or liquid emulsions that are of wide industrial and biological importance. However, by comparing the controlled collision of bubbles or water droplets with mobile and immobile liquid interfaces that have been designed to have the same interfacial tension in a pure fluorocarbon liquid, we demonstrated that collisions involving mobile surfaces result in a significantly stronger series of rebounds prior to the rapid coalescence event. The stronger rebound is explained by the lower viscous dissipation during collisions involving mobile surfaces. We present direct numerical simulations confirming that the observed rebound is enhanced with increased surface mobility. Good quantitative agreement between simulation and experiments substantiates the prediction that, in contrast to the intuitive expectation of surface mobility decreasing colloidal stability, there is a dynamic regime in which mobile surface droplets will bounce apart, whereas immobile surfaces droplets coalesce. These novel observations require a reassessment of the role of the surfaces mobility and open new avenues for controlling of the dynamics stability of gas or liquid emulsion systems relevant to a wide range of processes from microfluidics and pharmaceuticals to food and crude oil processing.
E. Klaseboer, F. D. E. Charlet, B.-C. Khoo, Q. Sun, D. Y. C. Chan
Eliminating the ﬁctitious frequency problem in BEM solutions of the external Helmholtz equation
Engineering Analysis with Boundary Elements 109, 106-116 (2019). pdf
The problem of the ﬁctitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Green’s function is replaced by a modiﬁed Green’s function, it is shown that these ﬁctitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these ﬁctitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.
A. S. Jayaraman, E. Klaseboer, D. Y. C. Chan
The unusual ﬂuid dynamics of particle electrophoresis
The classical problem of the electrophoretic motion of a spherical particle has been treated theoretically by Overbeek in his 1941 PhD thesis and almost 40 years later by O’Brien & White. Although both approaches used identical assumptions, the details are quite diﬀerent. Overbeek solved for the pressure, velocity ﬁelds as well as the electrostatic potential whereas O’Brien & White obtained the electrophoretic mobility without the need to consider the pressure and velocity explicitly. In this paper, we establish the equivalence of these two approaches which allow us to show that the tangential component of the ﬂuid velocity has a maximum near the surface of the particle and outside the double layer, the velocity decays as 1/r³, where r is the distance from the sphere, instead of 1/r in normal Stokes ﬂow. Associated with this behavior is that of an irrotational outer ﬂow ﬁeld. This is consistent with the fact that a sphere moving with a constant electrophoretic velocity experiences zero net force. A study of the forces on the particle also provides a physical explanation of the independence of the electrophoretic mobility on the electrostatic boundary conditions or dielectric permittivity of the particle. These results are important in situations where inter-particle interaction is considered, for instance, in electrokinetic deposition.
A. J. Yuffa, J. Markkanen, Q. Sun, E. Klaseboer, D. Y. C. Chan
A New Perspective on an Old Problem: Scattering by a Perfect Electric Conductor
We explore several different singular surface integral equation formulations. These formulations are speciﬁcally designed for time-harmonic scattering by a perfect electric conductor and are obtained by choosing non-conventional boundary unknowns.
E. Klaseboer, Q. Sun, D. Y. C. Chan
Dynamic linear elasticity using a Helmholtz decomposition and boundary elements
The displacement field for three-dimensional (3D) dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which an analytical solution has been derived. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the in- verse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the physical displacement field, remains finite in the zero frequency limit.
I. U. Vakarelski, R. Manica, E. Q. Li, E. S. Basheva, D. Y. C. Chan, S. T. Thoroddsen
Coalescence dynamics of mobile and immobile fluid interfaces
By using high speed camera imaging we examine the free rise and coalescence of small air-bubbles (100 to 1300 μm) with a liquid interface. A perfluorocarbon liquid, PP11 is used as a model liquid to investigate coalescence dynamics between fully-mobile and immobile deformable interfaces. The mobility of the bubble surface was determined by measuring the terminal rise velocity of small bubbles rising at Reynolds numbers approaching zero (Re < 0.1) and the mobility of free PP11 surface by measuring the deceleration kinetics of the small bubble toward the interface. Induction or film drainage times of a bubble at the mobile PP11-air surface were found to be more than two orders of magnitude shorter compared to the case of bubble and an immobile PP11-water interface. A theoretical model is used to illustrate the effect of hydrodynamics and interfacial mobility on the induction time.
E. Klaseboer, Q. Sun, D. Y. C. Chan
A field only integral equation method for time domain scattering of electromagnetic pulses
A combination of a non-singular boundary integral method to solve directly for the electromagnetic field components in the frequency domain and Fourier transform is used to describe the space-time behavior in electromagnetic problems. The approach is stable for wavelengths both small and large relative to characteristic length scales. Amplitudes and phases of field values can be obtained accurately on or near material boundaries. Local field enhancement effects due to multiple scattering of interest to applications in microphotonics are demonstrated.
J. D. Berry, I. U. Vakarelski, D. Y. C. Chan, S. T. Thoroddsen
Navier slip model of drag reduction by Leidenfrost vapour layers
Recent experiments found that a hot solid sphere that is able to sustain a stable Leidenfrost vapor layer in a liquid exhibits significant drag reduction during free fall. The variation of the drag coefficient with Reynolds number deviates substantially from the characteristic drag crisis behavior at high Reynolds numbers. Measurements based on liquids of different viscosities show that onset of the drag crisis depends on the viscosity ratio of the vapor to the liquid. Here we attempt to characterize the complexity of the Leidenfrost vapor layer with respect to its variable thickness and possible vapor circulation within, in terms of the Navier slip model that is defined by a slip length. Such a model can facilitate tangential flow and thereby alter the behaviour of the boundary layer. Direct numerical and large eddy simulations of flow past a sphere at moderate to high Reynolds numbers (10^2 ≤ Re ≤ 4 × 10^4) are employed to quantify comparisons with experimental results, including the drag coefficient and the form of the downstream wake on the sphere. This provides a simple one parameter characterization of the drag reduction phenomenon due to a stable vapor layer that envelops a solid body.
I. U. Vakarelski, E. Klaseboer, A. Jetly, M. M. Mansoor, A. A. Aguirre-Pablo, D. Y. C. Chan, S. T. Thoroddsen
Self-determined shapes and velocities of giant zero-drag gas cavities
Minimizing the retarding force on a solid moving in liquid is the canonical problem in the quest for energy saving by
friction and drag reduction. For an ideal object that cannot sustain any shear stress on its surface, theory predicts that
drag force will fall to zero as its speed becomes large. However, experimental verification of this prediction has been
challenging. We report the construction of a class of self-determined streamlined structures with this free-slip surface,
made up of a teardrop-shaped giant gas cavity that completely encloses a metal sphere. This stable gas cavity is
formed around the sphere as it plunges at a sufficiently high speed into the liquid in a deep tank, provided that
the sphere is either heated initially to above the Leidenfrost temperature of the liquid or rendered superhydrophobic
in water at room temperature. These sphere-in-cavity structures have residual drag coefficients that are typically less 1/10 those of solid objects of the same dimensions, which indicates that they experienced very small drag forces. The self-determined shapes of the gas cavities are shown to be consistent with the Bernoulli equation of potential flow applied on the cavity surface. The cavity fall velocity is not arbitrary but is uniquely predicted by the sphere density and cavity volume, so larger cavities have higher characteristic velocities.
E. Klaseboer, S. Sepehrirahnama, D. Y. C. Chan
Space-time domain solutions of the wave equation by a non-singular boundary integral method and Fourier transform
The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the fast Fourier transform is used to obtain the full space-time solution of the wave equation. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.
Q. Sun, E. Klaseboer, D. Y. C. Chan
A Robust Multi-Scale Field-Only Formulation of Electromagnetic Scattering
We present a boundary integral formulation of electromagnetic scattering by homogeneous bodies that are characterized by linear constitutive equations in the frequency domain. By working with the Cartesian components of the electric, E and magnetic, H fields and with the scalar functions (r.E) and (r.H), the problem can be cast as having to solve a set of scalar Helmholtz equations for the field components that are coupled by the usual electromagnetic boundary conditions at material boundaries. This facilitates a direct solution for the surface values of E and H rather than having to work with surface currents or surface charge densities as intermediate quantities in existing methods. Consequently, our formulation is free of the well-known numerical instability that occurs in the zero frequency or long wavelength limit in traditional surface integral solutions of Maxwell’s equations and our numerical results converges uniformly to the static results in the long wavelength limit. Furthermore, we use a formulation of the scalar Helmholtz equation that is expressed as classically convergent integrals and does not require the evaluation of principal value integrals or any knowledge of the solid angle. Therefore, standard quadrature and higher order surface elements can readily be used to improve numerical precision for the same number of degrees of freedom. In addition, near and far field values can be calculated with equal precision and multiscale problems in which the scatterers possess characteristic length scales that are both large and small relative to the wavelength can be easily accommodated. From this we obtain results for the scattering and transmission of electromagnetic waves at dielectric boundaries that are valid for any ratio of the local surface curvature to the wave number. This is a generalisation of the familiar Fresnel formula and Snell’s law, valid at planar dielectric boundaries, for the scattering and transmission of electromagnetic waves at surfaces of arbitrary curvature. Implementation details are illustrated with scattering by multiple perfect electric conductors as well as dielectric bodies with complex geometries and composition.
E. Klaseboer, Q. Sun, D. Y. C. Chan
Non-singular field-only surface integral equations for electromagnetic scattering
IEEE Transactions on Antennas and Propagation 65, 972-977 (2017). pdf
A boundary integral formulation of electromagnetics that involves only the components of E and H is derived without the use of surface currents that appear in the classical PMCHWT theory. The kernels of the boundary integral equations for E and H are non-singular so that all field quantities at the surface can be determined to high precision and geometries with closely spaced surfaces present no numerical difficulties. Higher order quadratic elements can readily be used to represent the surfaces so that the surface integrals can be calculated to higher numerical precision than using planar elements for the same numbers of degrees of freedom.
I. U. Vakarelski, J. D. Berry, D. Y. C. Chan, S. T. Thoroddsen
Leidenfrost vapor layers reduce drag without the crisis in high viscosity liquids
The drag coefficient, C_D of a solid sphere moving in fluid is known to be only a function of the Reynolds number, Re and diminishes rapidly at the drag crisis around Re ∼ 3 × 10^5. A Leidenfrost vapor layer on a hot sphere surface can trigger the onset of the drag crisis at lower Re. By using a range of high viscosity perfluorocarbon liquids, we show that the drag reduction effect, can occur over a wide range of Re, from as low as ∼ 600 to 10^5. The Navier slip model with a viscosity dependent slip length can fit the observed drag reduction and wake shape.
R. Manica, E. Klaseboer, D. Y. C. Chan
The hydrodynamics of bubble rise and impact with solid surfaces
A bubble smaller than 1 mm in radius rises along a straight path in water and attains a constant speed due to the balance between buoyancy and drag force. Depending on the purity of the system, within the two extreme limits of tangentially immobile or mobile boundary conditions at the air-water interface considerably different terminal speeds are possible. When such a bubble impacts on a horizontal solid surface and bounces, interesting physics can be observed. We study this physical phenomenon in terms of forces, which can be of colloidal, inertial, elastic, surface tension and viscous origins. Recent advances in highspeed photography allow for the observation of phenomena on the millisecond scale. Simultaneous use of such cameras to visualize both rise/deformation and the dynamics of the thin film drainage through interferometry are now possible. These experiments confirm that the drainage process obeys lubrication theory for the spectrum of micrometre to millimetre-sized bubbles that are covered in this review. We aim to bridge the colloidal perspective at low Reynolds numbers where surface forces are important to high Reynolds number fluid dynamics where the effect of the surrounding flow becomes important. A model that combines a force balance with lubrication theory allows for the quantitative comparison with experimental data under different conditions without any fitting parameter.
Q. Sun, E. Klaseboer, D. Y. C. Chan
A robust and accurate formulation of molecular and colloidal electrostatics
Journal of Chemical Physics 145, 054106 [1-12] (2016). pdf
This paper presents a re-formulation of the boundary integral method for the Debye- Hu ̈ckel model of molecular and colloidal electrostatics that removes the mathematical singularities that have to date been accepted as an intrinsic part of the conventional boundary integral equation method. The essence of the present boundary regular- ized integral equation formulation (BRIEF) consists of subtracting a known solution from the conventional boundary integral method in such a way as to cancel out the singularities associated with the Green’s function. This approach better reflects the non-singular physical behavior of the systems on boundaries with the benefits of (i) the surface integrals can be evaluated accurately using quadrature without any need to devise special numerical integration procedures, (ii) being able to use quadratic or spline function surface elements to represent the surface more accurately and the variation of the functions within each element is represented to a consistent level of precision by appropriate interpolation functions, (iii) being able to calculate electric fields, even at boundaries, accurately and directly from the potential without having to solve hypersingular integral equations and this imparts high precision in calcu- lating the Maxwell stress tensor and consequently, intermolecular or colloidal forces, (iv) a reliable way to handle geometric configurations in which different parts of the boundary can be very close together without being affected by numerical instabili- ties, therefore potentials, fields and forces between surfaces can be found accurately at surface separations down to near contact, and (v) having the simplicity of a for- mulation that does not require complex algorithms to handle singularities will result in significant savings in coding effort and in the reduction of opportunities for coding errors. These advantages are illustrated using examples drawn from molecular and colloidal electrostatics.
R. Manica, E. Klaseboer, D. Y. C. Chan
The impact and bounce of air bubbles at a flat fluid interface
The rise and impact of bubbles with an initially flat but deformable liquid-air interface in ultraclean liquid systems is modelled by taking into account the buoyancy force, hydrodynamic drag, inertial added mass effect and drainage of the thin film between the bubble and the interface. The bubble-surface interaction is analyzed using lubrication theory that allows for both bubble and surface deformations under a balance of normal stresses and surface tension as well as the long-ranged nature of the deformation along the interface. The quantitative result for the collision and bounce is sensitive to the impact velocity of the rising bubble. This velocity is controlled by the combined effects of interfacial tension via the Young-Laplace equation and hydrodynamic stress on the surface that determine the deformation of the bubble. The drag
force that arise from the hydrodynamic stress in turn depends on the hydrodynamic boundary condition on the bubble surface and its shape. These interrelated factors are accounted for in a consistent manner. The model can predict the rise velocity and shape of millimeter-size bubbles in ultra-clean water, in two silicone oils of different density and viscosity and in ethanol without any adjustable parameters. The collision and bounce of such bubbles with a flat water/air, silicone oil/air and ethanol/air interface can then be predicted with excellent agreement when compared to experimental
J. Oshitani, T. Sasaki, T. Tsuji, K. Higashida, D. Y. C. Chan
Anomalous sinking of spheres due to local fluidization of apparently fixed powder beds
The sinking of an intruder sphere into a powder bed in the apparently fixed bed regime exhibits complex behavior in the sinking rate and the final depth when the sphere density is close to the powder bed density. Evidence is adduced that the intruder sphere locally fluidizes the apparently fixed powder bed, allowing the formation of voids and percolation bubbles that facilitates spheres to sink slower but deeper than expected. By adjusting the air injection rate and the sphere-to- powder bed density ratio this phenomenon provides the basis of a sensitive large particle separation mechanism.
D. Y. C. Chan
Electric double layer interaction between dissimilar charge-conserved conducting plates
Small metallic particles used in forming nano-structured to impart novel optical, catalytic or tribo-rheological can be modeled as conducting particles with equipotential surfaces that carry a net surface charge. The value of the surface potential will vary with the separation between interacting particles and in the absence of charge transfer or electrochemical reactions across the particle surface, the total charge of each particle must also remain constant. These two physical conditions require the electrostatic boundary condition for metallic nano-particles to satisfy an equipotential whole-of-particle charge conservation constraint that has not been studied previously. This constraint gives rise to a global charge conserved constant potential (CCCP) boundary condition that results in multi-body effects in the electric double layer interaction that are either absent or are very small in the familiar constant potential or constant charge or surface electrochemical equilibrium condition.
I. U. Vakarelski, D. Y. C. Chan, S. T. Thoroddsen
Drag moderation by the melting of ice surface in contact with water
We report measurements of the effects of the melting ice surface on the hydrodynamic drag of ice-shell-metal-core spheres free falling in water at Reynolds number, Re ~ 4 x 10^4 to 4 x 10^6 and demonstrate that the melting surface induces the early onset of the drag crisis, thus reducing the hydrodynamic drag by more than 50%. Direct visualization of the flow pattern demonstrates the key role of surface melting. Our observations support the hypothesis that the drag reduction is due to the disturbance of the viscous boundary layer by the melting ice surface.
R. Manica, E. Klaseboer, D. Y. C. Chan
A force balance model for bubble rise, impact and bounce from solid surfaces
A force balance model for the rise and impact of air bubbles in a liquid against rigid horizontal surfaces that takes into account effects of buoyancy and hydrodynamic drag forces, bubble deformation, inertia of the fluid via an added mass force, and a film force between the bubble and the rigid surface is proposed. Numerical solution of the governing equations for the position and velocity of the center of mass of the bubbles is compared against experimental data taken with ultraclean water. The boundary condition at the air-water interface is taken to be stress free, which is consistent for bubbles in clean water systems. Features that are compared include bubble terminal velocity, bubbles accelerating from rest to terminal speed, and bubbles impacting and bouncing off different solid surfaces for bubbles that have already or are yet to attain terminal speed. Excellent agreement between theory and experiments indicates that the forces included in the model constitute the main physical ingredients to describe the bouncing phenomenon.
J. D. Berry, M. J. Neeson, R. R. Dagastine, D. Y. C. Chan, R. F. Tabor
Measurement of surface and interfacial tension using pendant drop tensiometry
Pendant drop tensiometry offers a simple and elegant solution to determining surface and interfacial tension - a central parameter in many colloidal systems including emulsions, foams and wetting phenomena. The technique involves the acquisition of a silhouette of an axisymmetric fluid droplet, and iterative fitting of the Young-Laplace equation that balances gravitational deformation of the drop with the restorative interfacial tension. Since the advent of high-quality digital cameras and desktop computers, this process has been automated with high speed and precision. However, despite its beguiling simplicity, there are complications and limitations that accompany pendant drop tensiometry connected with both Bond number (the balance between interfacial tension and gravitational forces) and drop volume. Here, we discuss the process involved with going from a captured experimental image to a fitted interfacial tension value, highlighting pertinent features and limitations along the way. We introduce a new parameter, the Worthington number, Wo, to characterise the measurement precision. A fully functional, open-source acquisition and fitting software is provided to enable the reader to test and develop the technique further.
Q. Sun, E. Klaseboer, B. C. Khoo, D. Y. C. Chan
Boundary regularized integral equation formulation of Stokes flow
This paper shows that the Stokes equation with prescribed boundary conditions can be cast as a boundary regularized integral equation that is completely free of singularities that exist in the traditional formulation. The singularities are removed by subtracting a related auxiliary flow field, w, that can be constructed from one of many known fundamental solutions of the Stokes equation without the introduction of additional cutoff parameters. The numerical implementation of this boundary regularized integral equation formulation affords considerable savings in coding effort with improved numerical accuracy, in particular, the evaluation of field variables near boundaries. The high accuracy of this formulation does not deteriorate for problems where parts of the boundaries may almost be in contact.
Q. Sun, E. Klaseboer, B. C. Khoo, D. Y. C. Chan
Boundary regularised integral equation formulation of the Helmholtz equation in acoustics
Royal Society Open Science 2, 140520-140529 (2015). pdf
A boundary integral solution of the Helmholtz equation is developed in which the solid angle and singular behavior on the boundaries of radiating or scattering bodies in acoustic wave problems are removed analytically. This facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution is assured without the need to solve hypersingular integrals.
C. Shi, X. Cui, L. Xie, Q. Liu, D. Y.C. Chan, J. N. Israelachvili, H. Zeng
Measuring Forces and Spatiotemporal Evolution of Thin Water Films between an Air Bubble and Solid Surfaces of Different Hydrophobicity
A combination of atomic force microscopy (AFM) and reflection interference contrast microscopy (RICM) was used to measure simultaneously the interaction force and the spatiotemporal evolution of the thin water film between a bubble in water and mica surfaces with varying hydrophobicity. Stable films, supported by the repulsive van der Waals-Casimir-Lifshitz force, were always observed between an air bubble and hydrophilic mica surface (water contact angle, \theta_w < 5 deg) whereas bubble attachment occurred on hydrophobized mica surfaces. A theoretical model, based on the Reynolds lubrication theory and augmented Young-Laplace equation including the effects of disjoining pressure, provided excellent agreement with experiment results, indicating the essential physics involved in the interaction between air bubble and solid surfaces can be elucidated. A hydrophobic interaction free energy per unit area of the form: W(h) = - \gamma [1 - cos(\theta_w)] exp(-h/D_H) can be used to quantify the asymmetric interaction between the bubble and the hydrophobized solid substrate at separation, h, with \gamma being the surface tension of water. For the water contact angle in the range 45 deg < \theta_w < 90 deg, the decay length D_H varied in the range 0.8 nm to 1.0 nm. This study quantified the hydrophobic interaction in asymmetric system between air bubble and hydrophobic surfaces, and provides a feasible method for synchronous measurements of the interaction forces with nN resolution and the drainage dynamics of thin films down to nm thickness.
M. Shahalami, L. Wang, C. Wu, J. H. Masliyah, Z. Xu, D. Y. C. Chan
Measurement and modeling on hydrodynamic forces and deformations of an air bubble approaching a solid sphere in aqueous solutions with the integrated thin film drainage apparatus (ITFDA)
Advances in Colloid and Interface Science 217, 31-42 (2015). pdf
The interaction between bubbles and solid surfaces is central to a broad range of industrial and biological processes. Various experimental techniques have been developed to measure the interactions of bubbles approaching solids in a liquid. A main challenge is to accurately and reliably control the relative motion over a wide range of hydrodynamic conditions and at the same time to determine the interaction forces, bubble-solid separation and bubble deformation. The existing experimental methods are able to focus only on one of the aspects of this problem, mostly for bubbles and particles with characteristic dimensions either below 100 μm or above 1 cm. As a result, either the interfacial deformations are measured directly with the forces being inferred from a model, or the forces are measured directly with the deformations to be deduced from the theory. The recently developed integrated thin film drainage apparatus (ITFDA) filled the gap of intermediate bubble/particle size ranges that are commonly encountered in mineral and oil recovery applications. Equipped with side-view digital cameras along with bimorph as force sensor and speaker diaphragm as the driver for bubble to approach solid sphere, the ITFDA has the capacity to measure simultaneously and independently the forces and interfacial deformations as a bubble approaches a solid sphere in a liquid. Coupled with the thin liquid film drainage modeling, the ITFDA measurement allows elucidating the critical role of surface tension, fluid viscosity and bubble approach velocity in determining bubble deformation (profile) and hydrodynamic forces. Here we compare the available methods of studying bubble-solid interactions and demonstrate unique features and advantages of the ITFDA for measuring both forces and bubble deformations in systems of Reynolds numbers as high as 10. The consistency and accuracy of such measurement are tested against the well established Stokes-Reynolds-Young-Laplace model. The potential to use the design principles of the ITFDA for fundamental and developmental research is demonstrated.
M. J. Neeson, D. Y. C. Chan, R. F. Tabor
Compound pendant drop tensiometry for surface tension measurement at zero Bond number
A widely used method to determine the interfacial tension between fluids is to quantify the pendant drop shape that is determined by gravity and interfacial tension forces. Failure of this method for small drops or small fluid density differences is a critical limitation in microfluidic applications and when only small fluid samples are available. By adding a small spherical particle to the interface to apply an axisymmetric deformation, both the particle density and the interfacial tension can be simultaneously and precisely determined, providing an accurate and elegant solution to a long-standing problem.
E. Klaseboer, R. Manica, D. Y. C. Chan
Universal behavior of the initial stage of drop impact
During the early stages of the impact of a drop on a solid surface, pressure builds up in the intervening thin lubricating air layer and deforms the drop. The extent of the characteristic deformation is determined by the competition between capillary, gravitational and inertial forces that has been encapsulated in a simple analytic scaling law. For millimetric drops, variations of the observed deformation with impact velocity, V , exhibits a universal maximum at Weber number, W e = 1. The deformation scales as V^(1/2) at the low velocity capillary regime and as V^−(1/2) at the high velocity inertia regime, in excellent agreement with a variety of experimental systems.
C. Shi, D. Y. C. Chan, Q. Liu, H. Zeng
Probing Hydrophobic Interaction between Air Bubble and Partially Hydrophobic Surfaces Using Atomic Force Microscopy
Journal of Physical Chemistry C 118, 25000-25008 (2014). pdf, supp
The hydrophobic interaction plays an essential role in various natural phenomena and industrial processes. Previous studies on the hydrophobic interaction focused mainly on the interaction between hydrophobic solid surfaces for which the effective range of hydrophobic attraction was reported to vary from ~10 nm to >1 μm. Here, we report studies of the interaction between an air bubble in water used as a probe attached to the cantilever of an atomic force microscope and partially hydrophobized mica surfaces. No bubble attachment was observed for bare hydrophilic mica but attachment behaviors and attraction with an exponential decay length of 0.8-1.0 nm were observed between air bubble and partially hydrophobized mica as characterized by a water contact angle on the mica surface that varied from 45° to 85°. Our results demonstrate the important roles of the additional attraction at partially hydrophobized surfaces and hydrodynamic conditions in bubble attachment to substrate surfaces, and provide new insights into the basic understanding of this interaction mechanism in various applications such as mineral flotation.
M. J. Neeson, R. R. Dagastine, D. Y. C. Chan, R. F. Tabor
Evaporation of a capillary bridge between a particle and a surface
The liquid bridge that forms between a particle and a flat surface, and the dynamics of its evaporation are pertinent to a range of physical processes including paint and ink deposition, spray drying, evaporative lithography and the flow and processing of powders. Here, using time-lapse photography, we investigate the evaporative dynamics of a sessile liquid bridge between a particle and a planar substrate. Different wetting characteristics of the particle and substrate are explored, as well as the effects of contact line pinning and stick-slip boundary conditions. A theoretical framework is developed to quantify and analyse the experimental observations. For the size range of particles and drops used in this study, gravity is by far the smallest force in the system when compared to the surface tension and capillary interactions that are present, but in certain circumstances it dictates the key evolution stages of the geometry of the particle-drop-substrate systems. Analysis of evaporation dynamics and capillary forces indicate that at low Bond numbers, surface tension forces dominate and provide unique opportunities for the control of particles on surfaces.
E. Klaseboer, R. Manica, M. H. W. Hendrix, C.-D. Ohl, D. Y. C. Chan
A force balance model for the motion, impact and bounce of bubbles
A force balance model has been developed to predict the terminal velocity of a sub-millimetric bubble as its rises in water under buoyancy. The dynamics of repeated collisions and rebounds of the bubble against a horizontal solid surface is modeled quantitatively by including forces due to buoyancy, added mass, drag and hydrodynamic lubrication – the last arises from the drainage of water trapped in the thin film between the solid surface and the surface of the deformable bubble. The result is a self-contained, parameter-free model that is capable of giving quantitative agreement with measured trajectories and observed collisions and rebounds against a solid surface as well as the spatio-temporal evolution of the thin film during collision as measured by interferometry.
R. F. Tabor, F.Grieser, R. R. Dagastine, D. Y. C. Chan
The hydrophobic force: measurements and methods
Physical Chemistry Chemical Physics (Perspective) 16, 18065-18075 (2014). pdf
The hydrophobic force describes the attraction between water-hating molecules (and surfaces) that draws them together, causing aggregation, phase separation, protein folding and many other inherent physical phenomena. Attempts have been made to isolate the range and magnitude of this interaction between extended surfaces for more than four decades, with wildly varying results. In this perspective, we critically analyse the application of common force-measuring techniques to the hydrophobic force conundrum. In doing so, we highlight possible interferences to these measurements and provide physical rationalisation where possible. By analysing the most recent measurements, new approaches to establishing the form of this force become apparent, and we suggest potential future directions to further refine our understanding of this fundamental physical force.
E. Q. Li, I. U. Vakarelski, D. Y. C. Chan, S. T. Thoroddsen
Stabilization of Thin Liquid Films by Repulsive van der Waals Force
Using high-speed video recording of bubble rise experiments we study the stability of thin liquid films trapped between a rising bubble and a surfactant-free liquid-liquid meniscus interface. Using different combinations of non-polar oils and water that are all immiscible, we investigate the extent to which film stability can be predicted by attractive and repulsive van der Waals (vdW) interactions that are indicated by the relative magnitude of the refractive indices of the liquid combinations e.g. water (refractive index, n = 1.33), perfluorohexane (n = 1.23) and tetradecane (n = 1.43). We show that when the film-forming phase was oil (perfluorohexane or tetradecane) the stability of the film could always be predicted from the sign of the vdW interaction with a repulsive vdW force resulting in a stable film and an attractive vdW force resulting in film rupture. However, if aqueous electrolyte is the film-forming bulk phase between the rising air bubble and the upper oil phase, the film always ruptured even when a repulsive vdW interaction was predicted. We interpret these results as supporting the hypothesis that a short-ranged hydrophobic attraction determines the stability of the thin water film formed between an air phase and a non-polar oil phase.
R. Manica, M. H. W. Hendrix, R. Gupta, E. Klaseboer, C.-D. Ohl, D. Y. C. Chan
Modelling bubble rise and interaction with a glass surface
Applied Mathematical Modelling 38, 4249-4261 (2014). pdf
A theoretical model has been developed to analyse experimental data of millimetre-size bubbles rising under gravity in water and colliding with a horizontal glass plate. Based on lubrication theory, it can account for bubble deformations on the micrometre scale in the interaction zone with quantitative precision. Experimental data were obtained using synchronised high-speed cameras to visualise the bubble rise and bounce from the side and interferometry was used to deduce the position and time-dependent thickness of the thin water film trapped between the deformed bubble and the glass plate. Our experiments were performed using deionised water with trace amount of impurities typical of most industrial applications so that the bubble surface would behave as a tangentially immobile interface. This result was confirmed with theoretical comparisons of bubble rise velocity and thin film drainage rates during impact. Quantitative comparisons between experimental data and lubrication theory, which is accurate if the film Reynolds number is smaller than unity, are presented side by side for two bubbles of different size to quantify and discuss similarities and differences. To the knowledge of the authors, this is the first time that impacting bubbles with such high velocities have been modelled with such quantitative precision.
I. U. Vakarelski, D. Y. C. Chan, S. T. Thoroddsen
Leidenfrost vapour layer moderation of drag crisis and trajectories of superhydrophobic and hydrophilic spheres falling in water
We investigate the dynamic effects of a Leidenfrost vapour layer sustained on the surface of heated steel spheres during free fall in water. We find that a stable vapour layer sustained on textured superhydrophobic surface spheres falling through 95C water can reduce the hydrodynamic drag by up to 75% and stabilize the sphere trajectory for Reynolds number between 10^4-10^6, spanning the drag crisis in the absence of the vapour layer. For hydrophilic spheres under the same conditions, the transition to drag reduction and trajectory stability occurs abruptly at a temperature different from the static Leidenfrost point. The observed drag reduction effects are attributed to the disruption of the viscous boundary layer by the vapour layer whose thickness depends on the water temperature. Both the drag reduction and the trajectory stabilization effects are expected to have significant implications for development of the sustainable vapour layers based technologies.
Q. Sun, E. Klaseboer, B. C. Khoo, D. Y. C. Chan
A robust and non-singular formulation of the boundary integral method for the potential problem
Engineering Analysis with Boundary Elements 43, 117-123 (2014). pdf
A general formulation of the boundary integral method is presented for the Laplace equation applicable for potential flow whereby the singularities are removed analytically. Special consideration is given to problems involving 3D semi-infinite domains and problems with axial symmetry. The approach also provides a numerically robust way to evaluate the potential at points near boundaries. This non-singular approach affords con- siderable simplification to the numerical implementation and has general applicability in the study of potential flow problems in fluid dynamics. Examples are given to illustrate the application of our approach.
C. Wu, D. Y. C. Chan, R. F. Tabor
A simple and accurate method for calculation of the structure factor of interacting charged spheres
Journal of Colloid and Interface Science 426, 80-82 (2014). pdf | supp
Calculation of the structure factor of a system of interacting charged spheres based on the Ginoza solution of the Ornstein-Zernike equation has been developed and implemented on a stand- alone spreadsheet. This facilitates direct interactive numerical and graphical comparisons between experimental structure factors with the pioneering theoretical model of Hayter-Penfold that uses the Hansen-Hayter renormalisation correction. The method is used to fit example experimental structure factors obtained from the small-angle neutron scattering of a well-characterised charged micelle system, demonstrating that this implementation, available in the Supplementary Information, gives identical results to the Hayter-Penfold-Hansen approach for the structure factor, S(q) and provides direct access to the pair correlation function, g(r). Additionally, the intermediate calculations and outputs can be readily accessed and modified within the familiar spreadsheet environment, along with information on the normalisation procedure.
R. F. Tabor, C. Wu, F. Grieser, R. R. Dagastine, D. Y. C. Chan
Measurement of the hydrophobic force in a soft matter system
Journal of Physical Chemistry Letters 4, 3872-3877 (2013). pdf | supp
The hydrophobic attraction describes the familiar tendency for nonpolar molecules and surfaces to agglomerate in water, controlled by the re-organisation of intervening water molecules to minimize disruption to their hydrogen bonding network. Measurements of the attraction between chemically-hydrophobised solid surfaces have reported ranges varying from tens to hundreds of nanometers, all attributed to hydrophobic forces. Here, by studying the interaction between two hydrophobic oils drops in water under well-controlled conditions where all known surface forces are suppressed, we observe only a strong, short-ranged attraction with an exponential decay length of 0.30 +/- 0.03 nm - comparable to molecular correlations of water molecules. This attraction is implicated in a range of fundamental phenomena from self-assembled monolayer formation to the action of membrane proteins and non-stick surface coatings.
I. U. Vakarelski, D. Y. C. Chan, J. O. Marston, S. T. Thoroddsen
Dynamic Air Layer on Textured Superhydrophobic Surfaces
We provide an experimental demonstration that a novel macroscopic, dynamic continuous air layer or plastron can be sustained indefinitely on textured superhydrophobic surfaces in air-superhydrophobic surfaces. We show that such plastron can be sustained on the surface of a centimeter size superhydrophobic sphere immersed in heated water and variations of its dynamic behavior with air saturation of the water can be regulated by rapid changes of the water temperature. The simple experimental set-up allows for quantification of the air flux into the plastron and identification of the air transport model of the plastron growth. Both the observed growth dynamics of such plastrons and of millimeter-sized air bubbles seeded on hydrophilic surface under identical air-supersaturated solution condition are consistent with the predictions of a well-mixed gas transport model.
R. Manica, M. H. W. Hendrix, E. Klaseboer, C.-D. Ohl, D. Y. C. Chan
Effects of hydrodynamic film boundary conditions on bubble-wall impact at high Reynolds number
We report high speed video studies of the impact of mm-size rising bubbles onto a glass surface in deionized water. The bubble terminal velocities (Reynolds number ∼ 0.01) are consistent with tangentially immobile bubble-water interfaces, implying interfacial impurities. Two clearly different types of bounce behavior were observed. They are consistent with either a tangentially immobile or stress-free (mobile) air-water interface of the thin water film trapped between the bubble and the glass surface. Drainage and dimpling of the μm-thick water film agree quantitatively with lubrication theory.
C. Wu, H. S. Leese, D. Mattia, R. R. Dagastine, D. Y. C. Chan, R. F. Tabor
A study of fluid and transport properties of porous anodic aluminium membranes by dynamic atomic force microscopy
Recent work on carbon nanotubes (CNT) has focused on their potential application in water treatment due to their predicted and observed enhanced flow rates. Recent work on the lesser known porous anodic alumina membranes (PAAMs) has also shown flow enhancement, albeit only a fraction of what has been observed in CNTs. Despite their potential applications, little research has been conducted on PAAMs hydrodynamic properties and in this paper we present experimental results and theoretical models that explore the fluid flow behavior around and through these membranes. The experiments were conducted using an atomic force microscope (AFM) that pushed a solid silica particle against PAAMs that were characterized with different pore diameters. Furthermore the PAAMs were classified as either 'closed' or 'open' with the latter allowing fluid to pass through. The theoretical model developed to describe the experimental data incorporates Derjaguin-Landau-Verwey-Overbeek (DLVO) effects, cantilever drag and hydrodynamic forces. By using the slip boundary condition for the hydrodynamic forces, we were able to fit the model to experimental findings and also demonstrate that the difference between 'closed' and 'open' PAAMs was negligible. The slip lengths did not correspond to any physical feature of the PAAMs but our model does provide a simple yet effective means of describing the hydrodynamics of not only PAAMs but for membranes in general.
R. F. Tabor, C. Wu, F. Grieser, D. Y. C. Chan, R. R. Dagastine
Non-linear and cyclical collisions between drops and bubbles: using AFM to understand droplet interactions in micro-scale flows
Understanding the mechanics and outcome of droplet and bubble collisions is central to a range of processes from emulsion stability to mineral flotation. The atomic force microscope has been shown to be sensitive and accurate in measuring the forces during such interactions; in combination with a suitable model framework, a powerful tool is obtained for understanding surface forces and droplet stability in dynamic systems. Here we demonstrate for the first time that this process is not limited to linear motion, and that accelerating, decelerating and cyclical droplet velocities can be used to explore the collisions between droplets and bubbles in ways that much more closely mimic real systems. In particular, the motion of droplets experiencing oscillating pumping pressures is explored, providing insight into fluid handling for microfluidics. By modelling a range of processes in which drops collide and deform, and sometimes coalesce, the validity of the theoretical model – which accounts for deformation, surface forces and dynamic lubrication – is demonstrated. Further, it is shown how this model can be used as a predictive tool to determine whether a given droplet collision will be stable or coalescent.
Q. Sun, E. Klaseboer, B. C. Khoo, D. Y. C. Chan
Stokesian dynamics of pill-shaped Janus particles with stick and slip boundary conditions
We study the forces and torques experienced by pill-shaped Janus particles of different aspect ratios where half of the surface obeys the no-slip boundary condition and the other half obeys the Navier slip condition of varying slip lengths. Using a recently developed boundary integral formulation whereby the traditional singular behaviour of this approach is removed analytically, we quantify the strength of the forces and torques experienced by such particles in a uniform flow field in the Stokes regime. Depending on the aspect ratio and the slip length, the force transverse to the flow direction can change sign. This is a novel property unique to the Janus nature of the particles.
Interfacial nanobubbles (INBs) on a solid surface in contact with water have drawn widespread research interest. Although several theoretical models have been proposed to explain their apparent long lifetimes, the underlying mechanism still remains in dispute. In this work, the morphological evolution of INBs was examined in air-equilibrated and partially-degassed water with the use of Atomic Force Microscopy (AFM). Our results show that (1) INBs shrank in the partially-degassed water while they slightly grew in the air-equilibrated water, (2) the three-phase boundary of the INBs was pinned during the morphological evolution of the INBs. Our analyses show that (1) the lifetime of INBs was sensitive to the saturation level of dissolved gases in the surrounding water, especially when the concentration of dissolved gases was close to the saturation, (2) the pinning of the three-phase boundary could significantly slow down the kinetics of both the growth and the shrinkage of the INBs. We developed one-dimensional version of the Epstein-Plesset model of gas diffusion to account for the effect of the pinning.
D. Wang, L. Pevzner, C. Li, K. Peneva, C. Y. Li, D. Y. C. Chan, K Muellen, M. Mezger, K. Koynov, H.-J. Butt
Layer with reduced viscosity at water-oil interfaces probed by fluorescence correlation spectroscopy
The two-dimensional diffusion of isolated molecular tracers at the water–n-alkane interface was studied with fluorescence correlation spectroscopy. The interfacial diffusion coefficients of larger tracers with a hydrodynamic radius of 4.0 nm agreed well with the values calculated from the macroscopic viscosities of the two bulk phases. However, for small molecule tracers with hydrodynamic radii of only 1.0 and 0.6 nm, notable deviations were observed, indicating the existence of an interfacial region with reduced effective viscosity and increased mobility.
R. F. Tabor, C. Wu, F. Grieser, D. Y. C. Chan, R. R. Dagastine
Non-linear and cyclical collisions between drops and bubbles: Using AFM to understand droplet interactions in micro-scale flows
One of the challenges that remains in the field of microfluidics is to develop a scalable understanding of multi-phase flows in channels, based on quantitative theoretical descriptions that account for both surface chemistry and hydrodynamic interactions for deformable droplets and bubbles. In particular, it is desirable to have control over events such as stability and coalescence of droplets within devices, and to understand the effects of different characteristic pumping methods on the behaviour of drops in channels. Here, we employ the atomic force microscope (AFM) as a tool to analyse collisions between pairs of bubbles and droplets of appropriate size to microfluidic devices, determining the conditions under which coalescence can occur for flows encountered under different commonly-used pumping methods. Additionally, it is shown that a theoretical model based on Reynolds lubrication theory and accounting for surface forces and local deformation of drops can accurately predict the coalescence behaviour seen during such interactions. We use this model to predict the coalescence behaviour of binary systems from flow characteristics and surface forces, providing a map of stability for interacting drops, and suggest methods by which closer modelling of on-chip drop behaviour can be achieved.
E. Klaseboer, R. Manica, D. Y. C. Chan
Rising and bouncing bubnbles
9th International Conference on CFD, CSIRO, Melbourne (2012). pdf
It is well known that the Boundary Element Method (BEM) based on potential flow theory is an ideal model to simulate high Reynolds number flows. However, for rising bubbles viscous effects can no longer be neglected since drag forces will ultimately have to compensate the buoyancy force for a terminal velocity to set in. We present a method to include viscous effects in the boundary element method for high Reynolds numbers, based on a model by Joseph and Wang (2004). The model takes into account surface tension effects and was used to predict the terminal velocity and the shape of millimetre sized bubbles in a pure system (Duineveld, 1995; Klaseboer et al. 2011) with good agreement.
Subsequently, this model is used to predict the bouncing of a bubble against a surface. A thin film between the bubble and the (horizontal) boundary builds up a pressure that pushes the bubble back. A comparison with experimental data (Zawala et al. 2007) shows excellent agreement, including shape oscillations that occur just after the rebound. For bigger bubbles, several bounces are observed before the bubble settles at the solid boundary.
The current framework leads to rapid simulation times as compared to other numerical methods, yet containing all the essential physics of the problem and could lead to a better understanding of problems involving bubbles in many industrial applications in terms of bouncing and coalescence behavior.
M. J. Neeson, R. F. Tabor, F. Grieser, R. R. Dagastine, D. Y. C. Chan
Compound drops arise from the contact of three immiscible fluids and can assume various geometric forms based on the inter- facial chemistry of the phases involved. Here we present a study of a new class of compound drops that is sessile on a solid surface. The possible geometries are demonstrated experimentally with appropriate fluid combinations and accounted for with a quantitative theoretical description. Although such systems are broadly controlled by relative interfacial energies, subtleties such as the Van der Waals force and effects of micro-gravity, despite drop sizes being well below the capillary length, come into play to determine the equilibrium state that is achieved. The drying of a compound sessile drop is measured experimentally and the process revealed a novel transition between different characteristic configurations of compound sessile drops. Such drops may prove useful as the first step towards development of functional surfaces in applications such as soft optics, photonics and surface encapsulation.
I. U. Vakarelski, N. A. Patankar, J. O. Marston, D. Y. C. Chan, S. T. Thoroddsen
Stabilization of Leidenfrost Vapour Layer by Textured Superhydrophobic Surfaces
In 1756 Leidenfrost observed that water drops skittered on a sufficiently hot skillet due to levitation by an evaporative vapor film. The stability of such films, only above a critical temperature, is a cornerstone effect in boiling. Low thermal conductivity of the vapour layer inhibit heat transfer in boiling operations, whereas their collapse in cooling applications can result in vapor explosions that are particularly detrimental, for instance in nuclear plants. Vapor films can also reduce liquid-solid drag, similar in principle to how a liquid film lubricates the motion of ice skaters. However, the ability to fabricate surfaces that can control vapor film stability remains elusive. Here we show that vapor film collapse does not occur at textured superhydrophobic surfaces. This result fundamentally alters what has been observed for more than two centuries. Whereas at smooth hydrophobic surfaces, the vapor film does collapse, albeit at a lower critical temperature, and switches explosively to nucleate boiling, in contrast, at textured superhydrophobic surfaces, the vapor layer gradually relaxes until the surface is completely cooled, without exhibiting nucleate boiling. This result demonstrates that topological texture on superhydrophobic materials is critical in stabilizing the vapor layer and thus in controlling phase change at surfaces. The concept can potentially be applied to control other phase transitions like ice or frost formation, leading to novel applications to regulate phase behavior using textures as well as to the design of novel low drag surfaces at which the vapor phase is stabilized in the grooves of textures without heating.
S. Cui, R. Manica, R. F. Tabor, D. Y. C. Chan
Interpreting atomic force microscopy measurements with a non-linear parametric estimation
Review of Scientific Instruments 83, 103702 (2012). pdf
A non-linear parameter estimation method is developed to extract the separation-dependent surface force and cantilever spring constant from AFM data taken at various speeds between a silica colloidal probe and plate in aqueous solution. The approach provides an unbiased estimate of the separation-dependent colloidal force and an independent measurement of the in situ cantilever spring constant. In combination with static force measurements, this approach can be used to quantify the extent of hydrodynamic slip.
M. H. W. Hendrix, R. Manica, E. Klaseboer, D. Y. C. Chan, C.-D. Ohl
Spatiotemporal evolution of thin liquid films during impact of water bubbles on glass on a micrometer to nanometer scale
Physical Review Letters 108, 247803 (2012). pdf | supp | Movie
Collisions between millimeter-size bubbles in water against a glass plate are studied using high- speed video. Bubble trajectory and shape are tracked simultaneously with laser interferometry between the glass and bubble surfaces that monitors spatial-temporal evolution of the trapped water film. Initial bubble bounces and the final attachment of the bubble to the surface have been quantified. While the global Reynolds number is large (~10^2), the film Reynolds number remains small and permits analysis with lubrication theory with tangentially immobile boundary condition at the air-water interface. Accurate predictions of dimple formation and subsequent film drainage are obtained.
R. Manica, E. Klaseboer, D. Y. C. Chan
Modelo e Simulacao Numerica de Interacoes Envolvendo Bolhas e Gotas
Tendencias em Matematica Aplicada e Computacional 13, 121-132 (2012). pdf
Formulamos um modelo teórico para o estudo de escoamento em filmes finos envolvendo superfícies deformáveis (por exemplo bolhas e gotas) que estão interagindo a baixas velocidades. Assumimos que a condição de contorno na in- terface entre os fluidos é de não-deslizamento tangencial. As equações evolutivas resultantes constituem um sistema algébrico-diferencial na qual a posição da fron- teira avança e deforma ao mesmo tempo e depende da solução global do sistema. O foco principal do trabalho é na derivação do modelo e nos detalhes da imple- mentação numérica. As equações são resolvidas usando uma rotina do Matlab e os resultados numéricos são comparados com dados experimentais da literatura que foram produzidos por pesquisadores em differentes laboratórios e usando diversas técnicas, comprovando que o modelo é adequado para uma variedade de sistemas.
O. Manor, T. Chau, G. W. Stevens, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Polymeric stabilized emulsions: steric effects and deformation in soft systems
Polymeric stabilizers are used in a broad range
of processes and products, from pharmaceuticals and engine
lubricants to formulated foods and shampoos. In rigid
particulate systems, the stabilization mechanism is attributed
to the repulsive force that arises from the compression of the
polymer coating or “steric brush” on the interacting particles.
This mechanism has dictated polymer design and selection for
more than thirty years. Here we show, through direct
measurement of the repulsive interactions between immobi-
lized drops with adsorbed polymers layers in aqueous electrolyte solutions, that the interaction is a result of both steric stabilization and drop deformation. Drops driven together at slow collision speeds, where hydrodynamic drainage effects are negligible, show a strong dependence on drop deformation instead of brush compression. When drops are driven together at higher collision speeds where hydrodynamic drainage affects the interaction force, simple continuum modeling suggests that the film drainage is sensitive to flow through the polymer brush. These data suggest, for drop sizes where drop deformation is appreciable, that the stability of emulsion drops is less sensitive to the molecular weight or size of the adsorbed polymer layer than for rigid particulate systems.
H. Lockie, R. Manica, R. F. Tabor, G. W. Stevens, F. Grieser, D. Y. C. Chan, R. R. Dagastine
Anomalous Pull-Off Forces Between Surfactant Free Emulsion Drops in Different Aqueous Electrolytes
A systematic study of collisions between
surfactant-free organic drops in aqueous electrolyte solutions
reveals the threshold at which continuum models provide a
complete description of thin-film interactions. For collision
velocities above ∼1 μm/s, continuum models of hydro-
dynamics and surface forces provide a complete description of
the interaction, despite the absence of surfactant. This includes
accurate prediction of coalescence at high salt concentration
(500 mM). In electrolyte solutions at intermediate salt
concentration (50 mM), drop−drop collisions at lower
velocity (<1 μm) or extended time of forced drop−drop interaction exhibit a strong pull-off force of systematically varying magnitude. The observations have implications on the effects of ion-specificity and time-dependence in drop−drop interactions where kinetic stability is marginal.
R. F. Tabor, F. Grieser, R. R. Dagastine, D. Y. C. Chan
Measurement and analysis of forces in bubble and droplet systems using AFM
The use of atomic force microscopy to measure and understand the interactions between deformable colloids – particularly bubbles and drops – has grown to prominence over the last decade. Insight into surface and structural forces, hydrodynamic drainage and coalescence events has been obtained, aiding in the understanding of emulsions, foams and other soft matter systems. This article provides information on experimental techniques and considerations unique to performing such measurements. The theoretical modelling frameworks which have proven crucial to quantitative analysis are presented briefly, along with a summary of the most significant results from drop and bubble AFM measurements. The advantages and limitations of such measurements are noted in the context of other experimental force measurement techniques.
E. Klaseboer, Q. Sun, D. Y. C. Chan
Non-singular boundary integral methods for fluid mechanics applications
Journal of Fluid Mechanics 696, 468-478 (2012). pdf
A formulation of the boundary integral method for solving partial differential equations has been developed whereby the usual weakly singular integral and the Cauchy principal value integral can be removed analytically. The broad applicability of the approach is illustrated with a number of problems of practical interest to fluid and continuum mechanics including the solution of the Laplace equation for potential flow, the Helmholtz equation as well as the equations for Stokes flow and linear elasticity.
C. Browne, R. F. Tabor, D. Y. C. Chan, R. R. Dagastine, M. Ashokkumar, F. Grieser
Bubble Coalescence during Acoustic Cavitation in Aqueous Electrolyte Solutions
Bubble coalescence behavior in aqueous electro-
lyte (MgSO4, NaCl, KCl, HCl, H2SO4) solutions exposed to an
ultrasound field (213 kHz) has been examined. The extent of
coalescence was found to be dependent on electrolyte type and
concentration, and could be directly linked to the amount of
solubilized gas (He, Ar, air) in solution for the conditions used.
No evidence of specific ion effects in acoustic bubble coalescence was found. The results have been compared with several
previous coalescence studies on bubbles in aqueous electrolyte
and aliphatic alcohol solutions in the absence of an ultrasound field. It is concluded that the impedance of bubble coalescence by electrolytes observed in a number of studies is the result of dynamic processes involving several key steps. First, ions (or more likely, ion-pairs) are required to adsorb at the gas/solution interface, a process that takes longer than 0.5 ms and probably fractions of a second. At a sufficient interfacial loading (estimated to be less than 12% monolayer coverage) of the adsorbed species, the hydrodynamic boundary condition at the bubble/solution interface switches from tangentially mobile (with zero shear stress) to tangentially immobile, commensurate with that of a solidliquid interface. This condition is the result of spatially nonuniform coverage of the surface by solute molecules and the ensuing generation of surface tension gradients. This change reduces the film drainage rate between interacting bubbles, thereby reducing the relative rate of bubble coalescence. We have identified this point of immobilization of tangential interfacial fluid flow with the “critical transition concentration” that has been widely observed for electrolytes and nonelectrolytes. We also present arguments to support the speculation that in aqueous electrolyte solutions the adsorbed surface species responsible for the immobilization of the interface is an ion-pair complex.
R. F. Tabor, H. Lockie, D. Y. C. Chan, F. Grieser, I. Grillo, K. J. Mutch, R. R. Dagastine
Structural forces in soft matter systems: unique flocculation pathways between deformable droplets
Oscillatory structural forces caused by colloidal additives including micelles, microemulsion droplets and particles were explored between rigid and deformable interfaces using direct force measurements with the atomic force microscope. The observed oscillations from rigid surfaces become distorted when confinement occurs between deformable interfaces, giving rise to a force hysteresis between approaching and separating interfaces. Small-angle neutron scattering was used to determine the bulk structure of the colloidal additives, as a basis for comparison with their behaviour when confined in thin films. It is seen that confinement itself does not appear to significantly alter the structure of the colloidal additive when compared to the bulk; however, at small separations, interactions with the confining interfaces may become important. The combined approach uncovered an unique flocculation pathway that is available to deformable emulsion droplets, and that the strength of this flocculation can be tuned by changes in the size and concentration of the structuring colloid, the emulsion droplet size, and the ionic strength of the solution.
R. F. Tabor, C. Wu, H. Lockie, R. Manica, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Homo- and hetero-interactions between air bubbles and oil droplets measured by atomic force microscopy
The atomic force microscope was used to analyse the interactions between bubbles and oil droplets in surfactant-free aqueous solutions. Both homo- (bubble-bubble and drop-drop) and hetero- (bubble-drop) interactions were examined to elucidate the role of colloidal and hydrodynamic forces which, together with interfacial deformations dictate the stability in these systems. It is shown that electrical double-layer forces can be rendered attractive within a small pH range, and that the Van der Waals force can be switched from attractive to repulsive by material choice and ionic strength through salt effects on the so-called ‘zero-frequency’ term of the Lifshitz theory. By measuring interaction events between bubbles and drops at higher velocities, it is seen that deformation of the bodies and lubrication in the film generated between them can be predicted with a continuum hydrodynamic theory. These results suggest that solution pH and droplet material choice can be used to enhance or inhibit coalescence in such multi-component and multi-phase systems, and this may prove useful in controlling the behaviour of systems in microfluidics, as well as dispersion and formulation science.
I. U. Vakarelski, J. O. Marston, D. Y. C. Chan,
S. T. Thoroddsen
Drag Reduction by Leidenfrost Vapor Layers
Physical Review Letters 106, 214501 (2011). pdf | supp
We demonstrate and quantify a highly effective drag reduction technique that exploits the Leidenfrost effect to create a continuous and robust lubricating vapor layer on the surface of a heated solid sphere moving in a liquid. Using high-speed video, we show that such vapor layers can reduce the hydrodynamic drag by over 85%. These results appear to approach the ultimate limit of drag reduction possible by different methods based on gas-layer lubrication and can stimulate the development of related energy saving technologies.
R. F. Tabor, H. Lockie, D. Mair, R. Manica, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Combined AFM-Confocal Microscopy of Oil Droplets: Absolute Separations and Forces in Nano-Films
Quantitative interpretation of the dynamic forces between micrometer-sized deformable droplets and bubbles has previously been limited by the lack of an independent measurement of their absolute separation. Here, we use in situ confocal fluorescence microscopy to directly image the position and separation of oil droplets in an atomic force microscopy experiment. Comparison with predicted force vs. separation behavior to describe the interplay of force and deformation showed excellent agreement with continuum hydrodynamic lubrication theory in aqueous films less than 30 nm thick. The combination of force measurement and 3D visualization of geometric separation and surface deformation is applicable to interactions between other deformable bodies.
R. F. Tabor, A. J. Morfa, F. Grieser, D. Y. C. Chan, R. R. Dagastine
The effect of gold oxide in measurements of colloidal force
Atomic force microscopy, contact-angle, and spectroscopic ellipsometry measurements were employed to investigate the presence and properties of gold oxide on the surface of gold metal. It was found that, in agreement with available literature, unoxidized gold surfaces were hydrophobic, whereas oxidation rendered the surface highly hydrophilic. The oxide could be removed with ethanol or base but appeared to be stable over long periods in water or salt solutions between pH 3 and 7. After oxidation, the oxide layer thickness, determined using ellipsometry, was consistent with an approximate monolayer of AuO bonds at the gold surface. The presence of gold oxide was found to alter significantly the electrical double-layer characteristics of the gold surface below pH 6 and may explain the apparent inconsistencies in observed force behavior where gold is employed as well as aiding in design of future microfluidic systems which incorporate gold as a coating.
R. F. Tabor, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Oscillating structural forces arise when nanoscale colloids are confined at high concentration between two approaching surfaces. As layers of colloid are squeezed out, changes in osmotic pressure cause alternating regions of repulsion and attraction. Here, we provide direct measurements of such oscillatory structural forces between the soft interfaces of two emulsion droplets. Quantitative comparison indicates that the deformable nature of droplets allows them to act as far more sensitive probes than solid spheres. In addition, the responsive nature of soft surfaces can give rise to unexpected behaviors not encountered in rigid systems including reversible aggregation/flocculation for emulsion droplets and, potentially, spatial ordering within concentrated emulsion phases.
H. Lockie, R. Manica, G. W. Stevens, F. Grieser, D. Y. C. Chan, R. R. Dagastine
Precision AFM measurements of dynamic interactions between deformable drops in aqueous surfactant and surfactant-free solutions
The atomic force microscope (AFM) has provided
unprecedented opportunities to study velocity-dependent inter-
actions between deformable drops and bubbles under a range of
solution conditions. The challenge is to design an experimental
system that enables accurate force spectroscopy of the interac-
tion between deformable drops and thus the extraction of
accurate quantitative information about the physically important
force-separation relation. This step requires very precise control
and knowledge of the interfacial properties of the interacting
drops, the drive conditions of the force-sensing cantilever, the
disposition of the interacting drops on the substrate and on the
cantilever, and transducer calibrations of the instrument in order
to quantify the effects of approach velocities and interfacial deformation. This article examines and quantifies in detail all experimental conditions that are necessary to facilitate accurate processing of dynamic force spectroscopy data from the AFM using the well-defined system of tetradecane drops in aqueous solutions under surfactant and surfactant-free conditions over a range of force magnitudes that has not been attained before. The ability of drops to deform and increase the effective area of interaction instead of decreasing the distance of closest approach when disjoining pressure exceeds the Laplace pressure means that the DLVO paradigm of colloidal stability as being determined by a balance of kinetic energy against the height of the primary maximum is no longer valid. The range of interfacially active species present in alkane-aqueous systems investigated provides insight into the applicability of the tangentially immobile boundary condition in colloidal interactions.
R. F. Tabor, R. Manica, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Repulsive Van der Waals forces in soft matter: why bubbles don't stick
Measurements of nonequilibrium hydrodynamic interactions between bubbles and solid surfaces in water provide direct evidence that repulsive van der Waals forces of quantum origin control the behavior of liquid films on solids in air. In addition to being the simplest and most universal 3-phase system, the deformable air-water interface greatly enhances the sensitivity of force measurements compared with rigid systems. The strength of the repulsive interaction, controlled by the choice of solid, is sufficient to prevent coalescence (sticking) on separation due to hydrodynamic interactions.
R. F. Tabor, D. Y. C. Chan, F. Grieser, R. R. Dagastine
Anomalous stability of carbon dioxide in pH-controlled bubble coalescence
Angewandte Chemie Int Ed 50, 3454-3456 (2011). pdf
We explore bubble coalescence as a function of pH and gas type, demonstrating that CO2 has a suprising and vital role, by comparing pure CO2 bubbles with air (which has CO2 as a minor component), argon, and nitrogen (pure, inert gases).
D. Y. C. Chan, E. Klaseboer, R. Manica
Theory of non-equilibrium force measurements involving deformable drops and bubbles
Advances in Colloid and Interface Science 165, 70 (2011). pdf
Over the past decade, direct force measurements using the Atomic Force Microscope (AFM) have been extended to study non-equilibrium interactions. Perhaps the more scientifically interesting and technically challenging of such studies involved deformable drops and bubbles in relative motion. The scientific interest stems from the rich complexity that arises from the combination of separation dependent surface forces such as Van der Waals, electrical double layer and steric interactions with velocity dependent forces from hydrodynamic interactions. Moreover the effects of these forces also depend on the deformations of the surfaces of the drops and bubbles that alter local conditions on the nanometer scale, with deformations that can extend over micrometers. Because of incompressibility, effects of such deformations are strongly influenced by small changes of the sizes of the drops and bubbles that may be in the millimeter range. Our focus is on interactions between emulsion drops and bubbles at around 100 μm size range. At the typical velocities in dynamic force measurements with the AFM which span the range of Brownian velocities of such emulsions, the ratio of hydrodynamic force to surface tension force, as characterized by the capillary number, is ~10−6 or smaller, which poses challenges to modeling using direct numerical simulations. However, the qualitative and quantitative features of the dynamic forces between interacting drops and bubbles are sensitive to the detailed space and time-dependent deformations. It is this dynamic coupling between forces and deformations that requires a detailed quantitative theoretical framework to help interpret experimental measurements. Theories that do not treat forces and deformations in a consistent way simply will not have much predictive power. The technical challenges of undertaking force measurements are substantial. These range from generating drop and bubble of the appropriate size range to controlling the physicochemical environment to finding the optimal and quantifiable way to place and secure the drops and bubbles in the AFM to make reproducible measurements. It is perhaps no surprise that it is only recently that direct measurements of non-equilibrium forces between two drops or two bubbles colliding in a controlled manner have been possible. This review covers the development of a consistent theory to describe non-equilibrium force measurements involving deformable drops and bubbles. Predictions of this model are also tested on dynamic film drainage experiments involving deformable drops and bubbles that use very different techniques to the AFM to demonstrate that it is capable of providing accurate quantitative predictions of both dynamic forces and dynamic deformations. In the low capillary number regime of interest, we observe that the dynamic behavior of all experimental results reviewed here are consistent with the tangentially immobile hydrodynamic boundary condition at liquid–liquid or liquid–gas interfaces. The most likely explanation for this observation is the presence of trace amounts of surface-active species that are responsible for arresting interfacial flow.