Rupture of thin stagnant films on a solid surface due to random thermal and mechanical perturbations

A generalized formalism for the rupture of a nondraining thin film on a solid support due to imposed random thermal and mechanical perturbations, modeled as a Gaussian white noise, is presented. The evolution of amplitude of perturbation is described by a stochastic differential equation. The average film rupture time is the average time for the amplitude of perturbation to equal to the film thickness and is calculated by employing a first passage time analysis for different amplitudes of imposed perturbations, wavenumbers, film thickness, van der Waals and electrostatic interactions and surface tensions. The results indicate the existence of an optimum wavenumber at which the rupture time is minimum. A critical film thickness is identified based on the sign of the disjoining pressure gradient, below which the film is unstable in that the rupture time is very small. The calculated values of rupture time as well as the optimum wavenumber in the present analysis agree well with the results of linear stability analysis for immobile as well as completely mobile gas-liquid film interfaces. For stable films, the rupture time is found to increase dramatically with film thickness near the critical film thickness. As expected, the average rupture time was found to be higher for smaller amplitudes of imposed perturbations, larger surface potentials, larger surface tensions and smaller Hamaker constants.     

Effect of interfacial mobility on rupture of thin stagnant films on a solid surface due to random mechanical perturbations

Previous analysis of Narsimhan [G. Narsimhan, J. Colloid Interface Sci. 287 (2005) 624-633] for the evaluation of rupture of a nondraining thin film on a solid support due to imposed random mechanical perturbations modeled as a Gaussian white noise has been extended for partially mobile gas-liquid interfaces. The average rupture time of film is evaluated by first passage time analysis (as the mean time for the amplitude of perturbation to become equal to film thickness). The interfacial mobility is accounted for through surface viscosity as well as Marangoni effect. The mean rupture time for partially mobile gas-liquid interface, as characterized by two dimensionless groups, dimensionless surface viscosity and Marangoni number, lies between the two extreme limits for fully mobile and immobile films. The critical wavenumber for minimum rupture time is shown to be insensitive to interfacial mobility. However, the critical dimensionless surface viscosity and critical Marangoni number at which the behavior of thin film deviates from that of fully mobile film and the behavior approaches that of fully immobile film are smaller for higher Hamaker constants, smaller film thickness and smaller surface potentials.     

The effect of interfacial viscosity on the droplet dynamics under flow field

The effects of interfacial viscosity on the droplet dynamics in simple shear flow and planar hyperbolic flow are investigated by numerical simulation with diffuse interface model. The change of interfacial viscosity results in an apparent slip of interfacial velocity. Interfacial viscosity has been found to have different influence on droplet deformation and coalescence. Smaller interfacial viscosity can stabilize droplet shape in flow field, while larger interfacial viscosity will increase droplet deformation, or even make droplet breakup faster. Different behavior is found in droplet coalescence, where smaller interfacial viscosity speeds up film drainage and droplet coalescence, but larger interfacial viscosity postpones the film drainage process. This is due to the change of film shape from flat‐like for smaller interfacial viscosity to dimple‐like for larger interfacial viscosity. The film drainage time still scales as Ca⁰ at smaller capillary number (Ca), and Ca^1.5 at higher capillary number when the interfacial viscosity changes. The interfacial viscosity only affects the transition between these limiting scaling relationships. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1505–1514, 2008     

Instability of thin liquid films between membranes

The conditions for instability of the thin liquid film between two plane-parallel membranes were derived taking into account the influence of the membrane tension, the membrane bending elasticity, the film viscosity and the disjoining pressure. It was shown that the liquid film could be unstable if the negative (attractive) disjoining pressure is predominant. The characteristic timeτ _m of growth of perturbation due to thermal or other fluctuations of the membrane shape increases with increasing the film viscosity, the membrane tension and the membrane bending elasticity, and decreasing the film thickness and the negative disjoining pressure. It is of the order of 10^?2÷10³ sec. When the membranes approach each other at certain value of the average film thicknessh _cr called critical, the fastest growing perturbations lead to formation of a liquid film with smaller (or zero) thickness. It was found that the critical thickness increases with increasing the negative disjoining pressure and the membrane area and decreasing the membrane tension and the bending elasticity having typical values of the order of 10^?6÷10^?5 cm. The case of a membrane approaching a solid plane was also considered. Excluding the small differences in numerical coefficients the results are similar to the case of two identical membranes.     

Model for drop coalescence in a locally isotropic turbulent flow field

The proposed model views drop coalescence in a turbulent flow field as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet due to the drainage of the intervening film of continuous phase under the action of colloidal (van der Waals and electrostatic) and random turbulent forces. The turbulent flow field was assumed to be locally isotropic. A first-passage-time analysis was employed for the random process of intervening continuous-phase film thickness between the two drops of a doublet in order to evaluate the first two moments of coalescence-time distribution of the doublet. The average drop coalescence time of the doublet was dependent on the barrier for coalescence due to the net repulsive force (net effect of colloidal repulsive and turbulent attractive forces). The predicted average drop coalescence time was found to be smaller for larger turbulent energy dissipation rates, smaller surface potentials, larger drop sizes, larger ionic strengths, and larger drop size ratios of unequal-sized drop pairs. The predicted average drop coalescence time was found to decrease whenever the ratio of average turbulent force to repulsive force barrier became larger. The calculated coalescence-time distribution was broader, with a higher standard deviation, at lower energy dissipation rates, higher surface potentials, smaller drop sizes, and smaller size ratios of unequal drop pairs. The model predictions of average coalescence-rate constants for tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate (SDS) in a high-pressure homogenizer agreed fairly well with the inferred experimental values as reported by Narsimhan and Goel (J. Colloid Interface Sci. 238 (2001) 420-432) at different homogenizer pressures and SDS concentrations.     

Linear stability of a draining film squeezed between two approaching droplets

In the present paper we analyze the effect of infinitesimal non-axisymmetric perturbations in determining the critical gap thickness at which a draining, finite radius thin-film becomes unstable. The film is part of the suspending fluid trapped between two approaching deformable drops under the action of a flow field. We carry out a linear stability analysis in the context of a quasi-static approximation where the rate of growth of the disturbances is assumed to be much faster than the rate of film drainage. An analytical solution is derived for the model in the special case of a uniformly thick film, for two types of perturbation: fixed-end and free-end. It is shown, for this special case, when the hydrodynamic force pushing the drops together from the external flow is constant, that the four most unstable disturbances are of the free-end kind, associated with the lowest frequency modes of azimuthal variation in the film thickness. Higher modes are stabilized by surface tension. Our analysis also shows that adopting the unretarded form of the van der Waals disjoining pressure yields results similar to the analysis when electromagnetic retardation effects are included in the calculation. A second case is analyzed where the film is also of uniform thickness but its lateral extent and the gap thickness are both time-dependent. This case was included to extend the predictions to glancing drop-collisions where the external hydrodynamic force is time-dependent. We find that there is a maximum capillary number below which the film becomes unstable, and that there is range of angles in the trajectory where the film becomes unstable, but that outside this range the film is stable.     

The Dynamics of Marangoni-Driven Local Film Drainage between Two Drops

A study of Marangoni-driven local continuous film drainage between two drops induced by an initially nonuniform interfacial distribution of insoluble surfactant is reported. Using the lubrication approximation, a coupled system of fourth-order nonlinear partial differential equations was derived to describe the spatio-temporal evolution of the continuous film thickness and surfactant interfacial concentration. Numerical solutions of these governing equations were obtained using the Numerical Method of Lines with appropriate initial and boundary conditions. A full parametric study was undertaken to explore the effect of the viscosity ratio, background surfactant concentration, the surface Péclet number, and van der Waals interaction forces on the dynamics of the draining film for the case where surfactant is present in trace amounts. Marangoni stresses were found to cause large deformations in the liquid film: Thickening of the film at the surfactant leading edge was accompanied by rapid and severe thinning far upstream. Under certain conditions, this severe thinning leads directly to film rupture due to the influence of van der Waals forces. Time scales for rupture, promoted by Marangoni-driven local film drainage were compared with those associated with the dimpling effect, which accompanies the approach of two drops, and implications of the results of this study on drop coalescence are discussed. Copyright 2001 Academic Press.     

Stability of draining plane-parallel films containing surfactants

The stability of partially mobile draining thin liquid films with respect to axisymmetric fluctuations was studied. The material properties of the interfaces (Gibbs elasticity, surface and bulk diffusions) were taken into account. When studying the long wave stability of films, the coupling between the drainage and perturbation flows was considered and the lubrication approximation was applied. Two types of wave modes were examined: radially-bounded and unbounded waves. The difference between the thickness of loss of stability, h(st), the transitional thickness, h(tr), at which the critical wave causing rupture becomes unstable, and the critical thickness, h(cr), when the film ruptures, is demonstrated. Both the linear and the non-linear theories give h(st) > h(tr) > h(cr). The numerical results show that the interfacial mobility does not significantly influence the thickness of the draining film rupture. The interfacial tension and the disjoining pressure are the major factors controlling the critical thickness. The available experimental data for critical thicknesses of foam and emulsion films show excellent agreement with the theoretical predictions. The important role of the electromagnetic retardation term in the van der Waals interaction is demonstrated. Other published theories of the film stability are discussed.     

Uniform liquid flow around porous spherical capsule

The problem of a liquid flow that is uniform at infinity around a spherical porous capsule is solved. The flow in a porous layer is described by the Brinkman equation assuming that the viscosity of the Brinkman medium differs from the viscosity of the liquid flowing around. The tangential stress jump condition is imposed on the porous medium-liquid interface. Velocity and pressure distributions are determined and the hydrodynamic force applied to the capsule is calculated.     

Droplet size effects on film drainage between droplet and substrate

When a droplet approaches a solid surface, the thin liquid film between the droplet and the surface drains until an instability forms and then ruptures. In this study, we utilize microfluidics to investigate the effects of film thickness on the time to film rupture for water droplets in a flowing continuous phase of silicone oil deposited on solid poly(dimethylsiloxane) (PDMS) surfaces. The water droplets ranged in size from millimeters to micrometers, resulting in estimated values of the film thickness at rupture ranging from 600 nm down to 6 nm. The Stefan-Reynolds equation is used to model film drainage beneath both millimeter- and micrometer-scale droplets. For millimeter-scale droplets, the experimental and analytical film rupture times agree well, whereas large differences are observed for micrometer-scale droplets. We speculate that the differences in the micrometer-scale data result from the increases in the local thin film viscosity due to confinement-induced molecular structure changes in the silicone oil. A modified Stefan-Reynolds equation is used to account for the increased thin film viscosity of the micrometer-scale droplet drainage case.     

Flow around a capsule with a fractal core

The problem of a liquid flow that is uniform at infinity around a capsule containing a fractal aggregate is solved. The flow in the porous layer of the capsule is described by the Brinkman equation, assuming that the viscosity of the Brinkman medium differs from the viscosity of the pure liquid. The tangential stress jump condition is imposed on the liquid-porous medium interface. Velocity and pressure distributions are found, and the hydrodynamic force applied to the capsule is calculated.     

Dynamics of thin liquid films on surfaces with a time-periodic wettability

The dynamics of thin liquid films on surfaces whose wettability changes in a time-periodic manner are examined in this work. A nonlinear evolution equation based on the lubrication approximation is used to describe the film height, and attractions due to van der Waals forces are incorporated. Film wettability is varied through an imposed sinusoidal modulation of the Hamaker constant. A linear stability analysis predicts that if the mean Hamaker constant is negative, disturbances at the film surface will eventually decay regardless of the amplitude and frequency of the oscillation. However, numerical solution of the evolution equation shows that the film can rupture at a given frequency if the amplitude is sufficiently large. The associated characteristic wavelength can be predicted from results for constant-wettability surfaces if an appropriate effective Hamaker constant is used. For positive mean Hamaker constants, film rupture can be accelerated, delayed, or prevented depending on how the Hamaker constant changes early in the oscillation cycle. The effects of spatial gradients in wettability are also considered, and it is found that oscillation can delay but not prevent rupture. Inclusion of short-range repulsive forces leads to the formation of droplet-like structures separated by ultra-thin films, but this can be prevented by sufficiently large and slow oscillations of the Hamaker constant. The results of this work may find use in applications that make use of surfaces whose wettability can be controlled by external stimuli.     

Constitutive equations for the flow behavior of entangled polymeric systems: application to star polymers

A semimicroscopic derivation is presented of equations of motion for the density and the flow velocity of concentrated systems of entangled polymers. The essential ingredient is the transient force that results from perturbations of overlapping polymers due to flow. A Smoluchowski equation is derived that includes these transient forces. From this, an equation of motion for the polymer number density is obtained, in which body forces couple the evolution of the polymer density to the local velocity field. Using a semimicroscopic Ansatz for the dynamics of the number of entanglements between overlapping polymers, and for the perturbations of the pair-correlation function due to flow, body forces are calculated for nonuniform systems where the density as well as the shear rate varies with position. Explicit expressions are derived for the shear viscosity and normal forces, as well as for nonlocal contributions to the body force, such as the shear-curvature viscosity. A contribution to the equation of motion for the density is found that describes mass transport due to spatial variation of the shear rate. The two coupled equations of motion for the density and flow velocity predict flow instabilities that will be discussed in more detail in a forthcoming publication.     

A model for the stability of films stabilized by randomly packed spherical particles

Particle stabilized thin films occur in a range of industrial applications where their properties affect the efficiency of the process concerned. However, due to their dynamic and unstable nature they are difficult to observe experimentally. As such, a tractable way of gaining insight into the fundamental aspects of this complicated system is to use computer simulations of particles at interfaces. This paper presents modeling results of the effect of nonuniform packing of spherical particles on the stability of thin liquid films. Surface Evolver was used to model cells containing up to 20 particles, randomly packed in a thin liquid film. The capillary pressure required to rupture the film for a specific combination of particle arrangement, packing density, and contact angle was identified. The data from the periodic, randomly packed models has been used to find a relationship between particle packing density, contact angle, and critical capillary pressure which is refined to a simple equation that depends on the film loading and contact angle of the particles it contains. The critical capillary pressure for film rupture obeys the same trends observed for particles in regular 2D and 3D packing arrangements. The absolute values of P*(crit), however, are consistently lower than those for regular packing. This is due to the irregular arrangement of the particles, which allows for larger areas of free film to exist, lowering the critical capillary pressure required to rupture the film.     

Decay of standing foams: drainage,coalescence and collapse

A summary of recent theoretical work on the decay of foams is presented. In a series of papers, we have proposed models for the drainage, coalescence and collapse of foams with time. Each of our papers dealt with a different aspect of foam decay and involved several assumptions. The fundamental equations, the assumptions involved and the results obtained are discussed in detail and presented within a unified framework.Film drainage is modeled using the Reynolds equation for flow between parallel circular disks and film rupture is assumed to occur when the film thickness falls below a certain critical thickness which corresponds to the maximum disjoining pressure. Fluid flow in the Plateau border channels is modeled using a Hagen-Poiseuille type flow in ducts with triangular cross-section.The foam is assumed to be composed of pentagonal dodecahedral bubbles and global conservation equations for the liquid, the gas and the surfactant are solved to obtain information about the state of the decaying foam as a function of time. Homogeneous foams produced by mixing and foams produced by bubbling (pneumatic foams) are considered. It is shown that a draining foam eventually arrives at a mechanical equilibrium when the opposing forces due to gravity and the Plateau-border suction gradient balance each other. The properties of the foam in this equilibrium state can be predicted from the surfactant and salt concentration in the foaming solution, the density of the liquid and the bubble radius.For homogeneous foams, it is possible to have conditions under which there is no drainage of liquid from the foam. There are three possible scenarios at equilibrium: separation of a single phase (separation of the continuous phase liquid by drainage or separation of the dispersed phase gas via collapse), separation of both phases (drainage and collapse occurs) or no phase separation (neither drainage nor collapse occurs). It is shown that the phase behavior depends on a single dimensionless group which is a measure of the relative magnitudes of the gravitational and capillary forces. A generalized phase diagram is presented which can be used to determine the phase behavior.For pneumatic foams, the effects of various system parameters such as the superficial gas velocity, the bubble size and the surfactant and salt concentrations on the rate of foam collapse and the evolution of liquid fraction profile are discussed. The steady state height attained by pneumatic foams when collapse occurs during generation is also evaluated.Bubble coalescence is assumed to occur due to the non-uniformity in the sizes of the films which constitute the faces of the polyhedral bubbles. This leads to a non-uniformity of film-drainage rates and hence of film thicknesses within any volume element in the foam. Smaller films drain faster and rupture earlier, causing the bubbles containing them to coalesce. This leads to a bubble size distribution in the foam, with the bubbles being larger in regions where greater coalescence has occurred.The formation of very stable Newton black films at high salt and surfactant concentrations is also explained.     

Pressure–voltage oscillations as a diagnostic tool for PEFC cathodes

In this work, pressure-induced voltage oscillations are explored as a novel diagnostic tool for PEFC cathodes. In this method, a small signal oscillation is imposed on the cathode outlet pressure. As a response to this pressure perturbation, the fuel cell voltage exhibits oscillations with the same frequency. The amplitude ratio and phase difference between the voltage and pressure oscillations embody diagnostic information about the operating conditions and processes in the PEFC cathode.     

Critical capillary pressure for destruction of single foam films and foam: effect of foam film size

Foams and single foam films stabilised by ionic and amphiphile polymer surfactants are studied with foam pressure drop technique (FPDT) and thin liquid film-pressure balance technique (TLF-PBT). A pressure is reached at which the single foam films rupture and the foams destruct very fast (avalanche-like). For film rupture we named this pressure—critical capillary pressure of film rupture, P_cr,film while for foam destruction, we introduced a new parameter—critical capillary pressure of foam destruction, P_cr,foam. The surfactant kind and foam film type (common thin, common black and Newton black) affect the values of both parameters. When below 20 kPa, P_cr,film and P_cr,foam are close by value, when over 20 kPa, there is a significant difference between them. The P_cr,film versus film size and P_cr,foam versus foam dispersity dependences, indicate that the film size and foam dispersity strongly affects the critical capillary pressure values. Film size distribution histograms reveal that a foam always contains films that are of a larger than the most probable size. They rupture at lower pressures, does initiating the destruction of the whole foam, which can be an explanation why higher than 20 kPa there is a difference between P_cr,film and P_cr,foam values. This parameter, P_cr,foam is considered of significant with respect to foam stability and could find use in industry.     

FAIMS operation for realistic gas flow profile and asymmetric waveforms including electronic noise and ripple

The use of field asymmetric waveform ion mobility spectrometry (FAIMS) has rapidly grown with the advent of commercial FAIMS systems coupled to mass spectrometry. However, many fundamental aspects of FAIMS remain obscure, hindering its technological improvement and expansion of analytical utility. Recently, we developed a comprehensive numerical simulation approach to FAIMS that can handle any device geometry and operating conditions. The formalism was originally set up in one dimension for a uniform gas flow and limited to ideal asymmetric voltage waveforms. Here we extend the model to account for a realistic gas flow velocity distribution in the analytical gap, axial ion diffusion, and waveform imperfections (e.g., noise and ripple). The nonuniformity of the gas flow velocity profile has only a minor effect, slightly improving resolution. Waveform perturbations are significant even at very low levels, in some cases approximately 0.01% of the nominal voltage. These perturbations always improve resolution and decrease sensitivity, a trade-off controllable by variation of noise or ripple amplitude. This trade-off is physically inferior to that obtained by adjusting the gap width and/or asymmetric waveform frequency. However, the disadvantage is negligible when the perturbation period is much shorter than the residence time in FAIMS, and ripple adjustment appears to offer a practical method for modifying FAIMS resolution.     

Transient instability in case II diffusion

A well-known model of one-dimensional Case II diffusion is reformulated in two dimensions. This 2-D model is used to study the stability of 1-D planar Case II diffusion to small spatial perturbations. An asymptotic solution based on the assumption of small perturbations and a small driving force is developed. This analysis reveals that while 1-D planar diffusion is indeed asymptotically stable to small spatial perturbations, it may exhibit a transient instability. That is, although any small perturbation is damped out over sufficiently long times, the amplitude of any perturbation initially grows with time. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 2941–2947, 1998     

Rupture of thin liquid films induced by impinging air-jets

Thin liquid films on partially wetting substrates are subjected to laminar axisymmetric air-jets impinging at normal incidence. We measured the time at which film rupture occurs and dewetting commences as a function of diameter and Reynolds number of the air-jet. We developed numerical models for the air flow as well as the height evolution of the thin liquid film. The experimental results were compared with numerical simulations based on the lubrication approximation and a phenomenological expression for the disjoining pressure. We achieved quantitative agreement for the rupture times. We found that the film thickness profiles were highly sensitive to the presence of minute quantities of surface-active contaminants.