<html> <head> <TITLE>Harris Wong's research</TITLE> </head> <BODY bgcolor="#ffffff"> <H2>Summary of Research Work</H2> <H3>Surface Tension Dominated Phenomena</H3> In the world of submicrons to millimeters, capillary forces usually dominate over other forces. Given a fluid or solid object of size L, capillary forces are in general proportional to L, whereas surface and body forces vary as L<sup>2</sup> and L<sup>3</sup>, respectively. Thus, as L decreases, surface tension becomes increasingly important. It is therefore not surprising that many natural phenomena and industrial processes are governed by capillarity. Below are four areas I have made contributions to. <OL> <LI><H4> Thin Solid Films </H4> <UL> <A NAME="line"></A> <LI> "Capillary instabilities in solid thin films: lines," <u>J. Applied Physics</u> <b>79</b>, 7604-7611 (1996) by M.S. McCallum, <A HREF="http://www.matsci.nwu.edu/faculty/pwv.html">P. W. Voorhees</A>, <a href="http://www.esam.nwu.edu/people/professors/miksis/index.html"> M. J. Miksis</a>, <a href="http://www.esam.nwu.edu/people/professors/davis/index.html"> S. H. Davis</a>, and H. Wong. <br> <b>Abstract</b><br> The linear morphological instability of a line of film on a substrate has been examined for contact angles between 0 and 180&#730. The base state of the line is an infinitely long cylinder with cross-sectional shape a segment of a circle. We assume that mass flows by diffusion along the film surface and that local equilibrium holds. We find that for non-zero contact angles there is a finite range of perturbation wavenumbers in the axial direction which correspond to instability and will potentially lead to agglomeration of the line of film. All unstable perturbations are of the varicose (sausage) type. The presence of the substrate is stabilizing; the range of unstable wavelengths is always less than that of a freely-suspended circular cylinder and decreases to zero width at zero contact angle. The maximum growth rate of the instability varies strongly with the contact angle and approaches zero as the contact angle approaches zero. Our results agree qualitatively with the experimentally observed wavelength and growth rates of the instability. <br> <p> <A NAME="wedge"></A> <LI> "Capillarity driven motion of solid film wedges," <u>Acta Materialia</u> <b>45</b>, 2477-2484 (1997) by H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis. <br> <b>Abstract</b><br> A solid film freshly deposited on a substrate may form a non-equilibrium contact angle with the substrate, and will evolve. This morphological evolution near the contact line is investigated by studying the motion of a solid wedge on a substrate. The contact angle of the wedge changes at time t = 0 from the wedge angle α to the equilibrium contact angle β, and its effects spread into the wedge via capillarity driven surface diffusion. The film profiles at different times are found to be self-similar, with the length scale increasing as t<sup>1/4</sup>. The self-similar profile is determined numerically by a shooting method for α and β between 0 and 180&#730. In general, we find that the film remains a wedge when α = β. For α < β, the film retracts, whereas for α > β, the film extends. For α = 90&#730, the results describe the growth of grain-boundary grooves for arbitrary dihedral angles. For β = 90&#730, the solution also applies to a free-standing wedge, and the thin-wedge profiles agree qualitatively with those observed in transmission electron microscope specimens.<br> <A href="fortran.html">FORTRAN program.</A> <br> <p> <A NAME="pinch off"></A> <LI> "Universal pinch off of rods by capillarity driven surface diffusion," <u>Scripta Materialia</u> <b>39</b>, 55-60 (1998) by H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis. <br> <b>Conclusions</b><br> We have derived local similarity solutions for the axisymmetric pinch off of rods when the morphological evolution is by capillarity-driven surface diffusion. These local solutions describe the approach to and departure from the topological singularity where a rod pinches into two separate bodies. During pinching, the self-similar surface profile far away from the neck approaches two opposing cones with a unique half-cone angle of 46.04&#730. It is thus likely that all rods must pinch off with this cone angle. This assertion is supported by several numerical simulations. After pinch off, the smoothening of the cone tip is again self-similar. The results obtained here for rods also apply to the pinch off of cylindrical pore channels. <br> <p> <A NAME="multilayer"></A> <LI> "Self-similar growth of a compound layer in thin-film binary diffusion couples," <u>Acta Materialia</u> <b>48</b>, 1371-1382 (2000) by H. Zhang and H. Wong. <br> <b>Abstract</b><br> The diffusion controlled growth of a compound phase A<sub>n</sub>B between two thin films of material A and B is studied with the nonlinear Kirkendall effect included. This growth process is important in electronic materials processing and in synthesis of high-temperature materials using multilayer films. Previous models of the growth rate do not solve the diffusion equation, and thus do not fully utilize the predictive capability. This paper describes a self-similar transformation that reduces the nonlinear, time-dependent diffusion equation with two free boundaries into a nonlinear ordinary differential equation, which is solved numerically by a shooting method. It is found that the intrinsic diffusion coefficients of A and B in A<sub>n</sub>B can be determined from the positions of the interfaces without using the concentration profile. This provides a simpler method for measuring intrinsic diffusion coefficients. An asymptotic solution valid for small concentration gradients is derived and agrees with the numerical results. <br> <p> <A NAME="step film"></A> <LI> "Periodic mass shedding of a retracting solid film step," <u>Acta Materialia</u> <b>48</b>, 1719-1728 (2000) by H. Wong, P. W. Voorhees, M. J. Miksis, and S. H. Davis. <br> <b>Abstract</b><br> A semi-infinite, uniform film on a substrate tends to contract from the edge to reduce the surface energy of the system. This work studies the two-dimensional retraction of such a film step, assuming that the film evolves by capillarity-driven surface diffusion. It is found that the retracting film edge forms a thickened ridge followed by a valley. The valley sinks with time and eventually touches the substrate. The ridge then detaches from the film. The new film edge retracts to form another ridge accompanied again by a valley, and the mass shedding cycle is repeated. This periodic mass shedding is simulated numerically for contact angle α between 30 to 180&#730. For smaller α, a small-slope late-time asymptotic solution is found that agrees with the numerical solution for α = 30&#730. Thus, the complete range of α is covered. The long-time retraction speed and the distance traveled per cycle agree quantitatively with experiments. <br> <p> <A NAME="delta-function"></A> <LI> "A δ-function model of facets," <u>Surface Science</u> <b>487</b>, L529-L533 (2001) by T. Xin and H. Wong. <br> <b>Abstract</b><br> Facets or planar surfaces appear often on crystalline solids, and need to be accurately modeled in studying surface evolution. Previous models of facets prescribe the surface free energy γ as a function of crystallographic orientation θ. However, when the anisotropic is strong, the reduced surface energy γ + d<sup>2</sup>γ /dθ<sup>2</sup> can become negative, which may induce ill-posedness in surface evolution problems. Here, we prescribe the reduced surface energy instead of the surface energy. This approach allows arbitrarily strong anisotropy, but avoids the ill-posedness. Further, a facet is represented by the Dirac delta function. This is the first time that the delta function is used to model facets. The new approach is demonstrated by modeling two-dimensional equilibrium square crystals. <br> <p> <A NAME="coupled1"></A> <LI> "Coupled grooving and migration of inclined grain boundaries: Regime I," <u>Acta Materialia</u> <b>50</b>, 1983-1994 (2002) by H. Zhang and H. Wong. <br> <b>Abstract</b><br> Grain-boundary migration controls the growth and shrinkage of crystalline grains and is important in materials synthesis and processing. A grain boundary ending at a free surface forms a groove at the tip, which affects its migration. This coupled grooving and migration is studied for an initially straight, inclined grain boundary intercepting a horizontal free surface. The groove deepens by surface diffusion. Previous work on a groove migrating at constant speed suggests that the grain boundary is pinned if the inclination angle is small. We find that the grain boundary is never pinned. The coupled motion can be separated into two time regimes. In Regime I, both the groove and grain-boundary profiles grow with time following similarity laws. The groove profile is symmetric about the groove root which turns the grain boundary tip vertically. This bending drives the migration. The self-similar profiles are shown to be linearly stable, and they grow continuously into Regime II. <br> <p> <A NAME="coupled2"></A> <LI> "Coupled grooving and migration of inclined grain boundaries: Regime II," <u>Acta Materialia</u> <b>50</b>, 1995-2012 (2002) by H. Zhang and H. Wong. <br> <b>Abstract</b><br> This work studies the coupled grooving and migration of an initially straight, inclined grain boundary ending at a horizontal free surface with an inclination angle β << 1. The coupled motion is separated into two time regimes. In Regime I, the grain boundary turns vertically at the groove root. In Regime II, the turning relaxes following two different paths depending on σ/β, where σ is the supplementary dihedral angle. For β > σ/6, the groove root positions (x<sub>0</sub>, y<sub>0</sub>) ~ (t<sup>1/2</sup>, t<sup>1/6</sup>) as time t -> &#8734, whereas for β < σ/6, (x<sub>0</sub>, y<sub>0</sub>) ~ (t<sup>1/4</sup>, t<sup>1/4</sup>) as t -> &#8734. These results come from asymptotic expansions and agree with a finite-difference solution of the coupled equations. They show that the grain boundary is never pinned. The asymptotic solutions also apply to the Sun-Bauer method of measuring mobility, and predict grain-boundary profiles that agree better with experiments. <br> <p> <A NAME="multiple layers"></A> <LI> "Self-similar growth of multiple compound layers in binary diffusion couples with application to the multi-foil method," <u>Acta Materialia</u> <b>50</b>, 3831-3844 (2002) by W. Kan and H. Wong. <br> <b>Abstract</b><br> The diffusion-controlled growth of multiple compound phases is studied with the nonlinear Kirkendall effect included. Previous work [Zhang & Wong, 2000] has analyzed the growth of one compound layer. In that analysis, the nonlinear, time-dependent diffusion equation with two free boundaries is reduced by a self-similar transformation into an ordinary differential equation, which is then solved numerically by a shooting method. This work extends that analysis to arbitrary N layers. The method of extracting intrinsic diffusion coefficients from only the positions of interfaces is expanded to two layers. In addition, the asymptotic solution valid for small concentration gradients is applied to the "multi-foil" method of measuring intrinsic diffusion coefficients and yields an analytic expression for the displacement curve. It is found that two vertex positions of the triangular displacement curve are sufficient to calculate the intrinsic diffusion coefficients. <br> <p> <A NAME="migrating grooves"></A> <LI> "A model of migrating grain-boundary grooves with application to two mobility-measurement methods," <u>Acta Materialia</u> <b>50</b>, 5155-5169 (2002) by D. Min and H. Wong. <br> <b>Abstract</b><br> Grain-boundary migration controls grain growth and is important in materials processing and synthesis. The mobility of grain boundaries is usually measured by the "quarter-loop" and Sun-Bauer methods. In these methods, a grain boundary migrates and its tip position along a free surface is recorded to infer the mobility. At the tip, a groove develops to reduce the combined surface energy. The groove is small and adjusts quickly. Thus, in both methods, the groove can be treated at each instant as migrating at constant speed. We study this quasi-steady groove formed via surface diffusion, and find that the groove turns the grain boundary (by angle θ) away from being perpendicular to the free surface. We add this tilting effect into both measurement methods by solving the migrating grain-boundary profiles for arbitrary θ. Computed profiles agree well with two Sun-Bauer experiments in which θ = 18 and 30&#730. <br> <p> <A NAME="faceted grooves"></A> <LI> "Grain-boundary grooving by surface diffusion with strong surface energy anisotropy," <u>Acta Materialia</u> <b>51</b>, 2305-2317 (2003) by T. Xin and H. Wong. <br> <b>Abstract</b><br> A vertical grain boundary intercepting a horizontal free surface forms a groove to reduce the combined surface energy of the system. The groove grows with time and is commonly used for measuring surface diffusion coefficients. This work studies grooving by capillarity-driven surface diffusion with strong surface energy anisotropy and finds that faceted grooves still grow with time t as t<sup>1/4</sup>. However, an anisotropic groove can be smooth if the groove surface does not cross a facet orientation. The groove has the same shape as the corresponding isotropic groove, but the growth rate is reduced by a factor that depends on the degree of anisotropy. This reduction induces an error in the surface diffusion coefficient if the isotropic model is applied to a smooth, but anisotropic groove. We show how to correct for this error. <br> <p> <A NAME="spike-function"></A> <LI> "A spike-function model of facets," <u>Materials Science and Engineering A</u> <b>364</b>, 287-295 (2004) by T. Xin and H. Wong. <br> <b>Abstract</b><br> Facets appear often on crystalline solids and affect surface evolution. Previous models of facets prescribe the surface energy γ as a function of crystallographic orientation θ. However, when the anisotropy is strong, the reduced surface energy γ + d<sup>2</sup>γ/dθ<sup>2</sup> can become negative, which may make surface-evolution problems ill-posed. Recently, a new model has been proposed in which the reduced surface energy is prescribed instead of the surface energy [Surf. Sci. 487 (2001) L529]. In this approach, a facet is represented by the Dirac delta function, and the anisotropy can be arbitrarily strong, but it does not induce ill-posedness. The delta-function model is illustrated in this work by an octagonal crystal with unequal sides. To facilitate numerical simulations, the delta function is replaced by a spike function with width ε. A square crystal is simulated to show the effect of varying ε. We also demonstrate that, given a measured equilibrium crystal shape, the new model can calculate the surface energy if the temperature is at or above the roughening temperature. <br> <p> <A NAME="fingering"></A> <LI> "Fingering instability of a retracting solid film edge," <u>J. Applied Phys.</u> <b>97</b>, 043515 (2005) by W. Kan and H. Wong. <br> <b>Abstract</b><br> A thin gold film under annealing on a silica substrate can develop &Prime;fingers&Prime; at the perimeter of the film. The perimeter retracts to leave behind longer fingers, which eventually pinch off to reduce the surface energy of the system. New fingers then form at the film edge and the process continues until the entire film disintegrates. To maintain the structure integrity of annealed thin films, this fingering instability must be understood. The retraction of a straight film edge via capillarity-driven surface diffusion has been analyzed in two dimensions by Wong et al [Acta mater. 48, 1719 (2000)]. They found that a retracting film is thickened at the edge followed by a valley before the film thickness becomes uniform. We study the three-dimensional linear stability of this two-dimensional film profile and find one unstable mode of perturbation. The growth rate of the perturbation is determined as a function of the wavelength of the perturbation and the speed of the receding edge. The results show that a straight film edge becomes wavy when perturbed. The wavelength λ<sub>m</sub> of the fastest growing perturbation agrees with the distance between adjacent fingers observed in a gold-film experiment. Fingers can also form during annealing at the retracting edges of cracks in sapphire, and our predicted λ<sub>m</sub> compares well with the measured finger spacing. <br> <p> </UL> <LI><H4>Thin Liquid Films</H4> <UL> <A NAME="tear"></A> <LI>"Deposition and thinning of the human tear film," <u> J. Colloid and Interface Sci.</u> <b>184</b>, 44-51 (1996) by H. Wong, I. Fatt, and <a href="http://www.cchem.berkeley.edu/~cjrgrp/">C. J. Radke</a>.<br> <b>Abstract</b><br> The exposed part of the eyeball is covered by a tear film, which is vital for the proper function of the eye. The film thickness has been measured to be roughly 10 μ. However, how a tear film of this thickness is generated has not been clearly explained. We propose that the tear film is deposited analogous to a coating process by the rising meniscus of the upper lid during a blink. A coating model is formulated that not only predicts correctly the film thickness, but also captures the post-blink lipid spreading commonly observed in experiments. A deposited tear film thins rapidly near the tear meniscus surrounding the film. Numerical simulation of this thinning reveals that the minimum film height obeys a power law. When the minimum height reaches the effective range of dewetting intermolecular forces, the film ruptures. The thinning time therefore defines a breakup time, and the thinning law shows explicitly how this breakup time is related to tear viscosity, surface tension, meniscus radius, and initial and final film thicknesses. The calculated breakup time agrees with those observed experimentally. <br> <p> <A NAME="disjoining"></A> <LI>"A slope-dependent disjoining pressure for non-zero contact angles," <u>J. Fluid Mechanics</u> <b>506</b>, 157-185 (2004) by Q. Wu and H. Wong <br> <b>Abstract</b><br> A thin liquid film experiences additional intermolecular forces when the film thickness h is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by disjoining pressure Π. Since Π dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, Π = Π(h) because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope h<sub>x</sub> leads to Π = Π(h, h<sub>x</sub>) that allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates Π to an excess interaction energy E = E(h, h<sub>x</sub>). By considering a fluid wedge on a solid substrate, E(h, h<sub>x</sub>) is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit <br> Π = B(α<sup>4</sup> - h<sub>x</sub><sup>4</sup> + 2hh<sub>x</sub><sup>2</sup>h<sub>xx</sub>)/h<sup>3</sup> <br> where α is the contact angle and B is a material constant. The term containing the curvature h<sub>xx</sub> is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. Evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that h<sub>xx</sub> = 0 at the contact line is sufficient to specify the contact angle. <p> </UL> <p> <LI><H4>Two-Phase Flow in Porous Media</H4> <UL> <A NAME="bubble film"></A> <LI>"The motion of long bubbles in polygonal capillaries: I. Thin films," <u>J. Fluid Mechanics</u> <b>292</b>, 71-94 (1995) by H. Wong, C.J. Radke, and <a href="http://www.me.berkeley.edu/faculty/morris/index.html"> S. Morris</a>.<br> <b>Abstract</b><br> Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure-velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The "piston" is reluctant to move because of a large drag exerted by the capillary side walls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments. <br> Part I of this work studies the fluid films deposited on capillary walls in the limit Ca -> 0 (Ca = μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part II (Wong et al., 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 << x << Ca<sup>-1</sup>, the film profile is frozen with a thickness of order Ca<sup>2/3</sup> at the center and order Ca at the sides. For x ~ Ca<sup>-1</sup>, surface tension rearranges the film at the center into a parabolic shape while the film at the sides thins to order Ca<sup>4/3</sup>. For x >> Ca<sup>-1</sup>, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ~ Ca<sup>-5/3</sup>, the height of the parabola is order Ca<sup>2/3</sup>. Finally, for x >> Ca<sup>-5/3</sup>, the height decreases as Ca<sup>1/4</sup>x<sup>-1/4</sup>. <p> <A NAME="bubble drag"></A> <LI>"The motion of long bubbles in polygonal capillaries: II. Drag, fluid pressure, and fluid flow," <u>J. Fluid Mechanics</u> <b>292</b>, 95-110 (1995) by H. Wong, C.J. Radke, and S. Morris.<br> <b>Abstract</b><br> This work determines the pressure-velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part I (Wong et al., 1995), we find that the drag scales as Ca<sup>2/3</sup> in the limit Ca -> 0 (Ca = μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca<sup>2/3</sup>. The proportionality constant for six different polygonal capillaries is roughly the same and is about 1/3 that for the circular capillary. <br> The liquid in a polygonal capillary flows by pushing the bubble (plug flow) or by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure-velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa<sup>2</sup>/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Q<sub>c</sub>, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Q<sub>c</sub> << 10<sup>-6</sup>. For Q<sub>c</sub> << Q << 1, the plug flow dominates, and the gradient in liquid pressure varies with Q<sup>2/3</sup>. For Q << Q<sub>c</sub>, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media. </UL> <p> <LI><H4>Dynamic Surface Tension</H4> <UL> <A NAME="clean bubble"></A> <LI>"Experiment and theory on the low Reynolds number expansion and contraction of a bubble pinned at a submerged tube tip," <u>J. Fluid Mechanics</u> <b>356</b>, 93-124 (1998) by H. Wong, D. Rumschitzki, and C. Maldarelli.<br> <b>Abstract</b><br> The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re << 1. Bubble shape, gas pressure, surface velocities, and detached bubble volume are determined by a boundary integral method for various Bond (Bo = ρga<sup>2</sup>/σ) and capillary (Ca = μQ/σa<sup>2</sup>) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration. <br> Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01 - 100) and Bo (0.01 - 0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01 < Ca < 100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca << 1, and asymptote as Q<sup>3/4</sup> for Ca >> 1, in agreement with analytic predictions. In the limit Ca -> 0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable. <br> Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare excellently with numerical solutions. It is observed that a bubble contracting at high Ca snaps off. <p> <A NAME="insol bubble"></A> <LI>"Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip," <u>J. Fluid Mechanics</u> <b>379</b>, 279-302 (1999) by H. Wong, D. Rumschitzki, and C. Maldarelli.<br> <b>Abstract</b><br> This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit the capillary number Ca -> 0 with the Reynolds number Re = o(Ca<sup>-1</sup>), nonzero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca = ρQ/σ a<sup>2</sup>, where ρ is the liquid viscosity, a is the tube radius, and σ is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M -> &#8734, where M is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble area changes. To maintain a uniform distribution at all times, a tangential flow is induced, the magnitude of which is more than twice that in the clean case. This is in contrast to the surface immobilizing effect of surfactant on an isolated translating bubble. These conclusions are confirmed by a boundary integral solution of Stokes flow valid for arbitrary Ca, G, and Bo. The uniformity in surfactant distribution validates the first assumption in the bubble and drop methods, but the enhanced tangential flow contradicts the second. <p> <A NAME="oscillating tensiometry"></A> <LI>"Oscillating drop/bubble tensiometry: effect of viscous forces on the measurement of interfacial tension," <u> J. Colloid and Interface Sci.</u> <b>282</b>, 128-132 (2005) by E.M. Freer, H. Wong, and C. J. Radke.<br> <b>Abstract</b><br> The oscillating drop/bubble technique is increasingly popular for measuring the interfacial dilatational properties of surfactant/polymerladen fluid/fluid interfaces. A caveat of this technique, however, is that viscous forces are important at higher oscillation frequencies or fluid viscosities; these can affect determination of the interfacial tension. Here, we experimentally quantify the effect of viscous forces on the interfacial-tension measurement by oscillating 100 and 200 cSt poly(dimethylsiloxane) (PDMS) droplets in water at small amplitudes and frequencies ranging between 0.01 and 1 Hz. Due to viscous forces, the measured interfacial tension oscillates sinusoidally with the same frequency as the oscillation of the drop volume. The tension oscillation precedes that of the drop volume, and the amplitude varies linearly with Capillary number, Ca = Δμ ωΔV/γa<sup>2</sup>, where Δμ = μ<sub>D</sub> - μ is the difference between the bulk Newtonian viscosities of the drop and surrounding continuous fluid, ω is the oscillation frequency of the drop, ΔV is the amplitude of volume oscillation, γ is the equilibrium interfacial tension between the PDMS drop and water, and a is the radius of the capillary. A simplified model of a freely suspended spherical oscillating-drop well explains these observations. Viscous forces distort the drop shape at Ca > 0.002, although this criterion is apparatus dependent. <p> </UL> </OL> <hr> <a href="http://me.lsu.edu/~hwong"> Return to Harris Wong's Home Page</a> <hr> </BODY> </html>