Harris Wong

On the surfactant mass balance at a deforming fluid interface

The amount of surfactants ~surface active agents! adsorbed onto a fluid interface affects its surface tension. Thus the distribution of surfactants must be determined to find the jump in the normal and tangential stresses across the interface. Scriven ~see also Aris, Slattery, and Edwards et al.! uses differential geometry to derive the correct surface balance equation for an arbitrary surface coordinate system. Also invoking differential geometry, Waxman develops a correct form in ~‘‘fixed’’! surface coordinates that advance only normal to the surface. To arrive at this balance without appealing to differential geometry, Stone presents a simple physical derivation which leads to a form of the mass balance which is easy to solve numerically. Unfortunately, Stone’s derivation leaves the nature of the unsteady time derivative ambiguous. Here we follow the spirit set in Stone to derive geometrically the surface balance in a way that keeps the nature of the time derivative explicit. We verify that in Stone’s form the time derivative must hold the fixed coordinates constant, as the numerical implementation of this form of the mass balance actually do. We also derive a new form valid in an arbitrary surface coordinate system. Consider a fixed point A on a fluid surface with local normal n as in Fig. 1. We locate any two perpendicular planes which intersect along n. The intersection of each of these planes with the surface near the point A define curves whose unit tangents are t1 and t2 . By construction ]t1/]s152~1/R1!n and ]t2/]s252~1/R2!n, where ds1 and ds2 are differential arcs and R1~.0! and R2~.0! are the radii of curvature of the curves. Geometrically, these differential arcs are ds15R1df1 and ds25R2df2 , where df1 and df2 are the differential angles in the figure, and ]t1/]f152n and ]t2/]f252n. Thus in this locally orthogonal system, the components of the surface metric tensor aab are: Aa115R1 , a1250, and Aa225R2 and the diagonal elements simply act as scale factors. These arcs define a patch of area dA5Aa11Aa22df1df25Aadf1df2 where a is the determinant of the metric tensor. The diagonal components of the curvature tensor bab are defined by @]ta /]fa#–n 5 baa /Aaaa ~no sum on a!; so b1152R1 and b2252R2 . The curvatures are negative because as drawn in Fig. 1 both arcs are concave down with respect to the normal. If U is the instantaneous material velocity vector at the fixed point, its components along

Periodic mass shedding of a retracting solid film step

n,t1 ,t2% are U5Us(1)t11Us(2)t21Wn, where W is the normal component and Us(1) and Us(2) are the physical components tangent to the surface. The fixed point advances along the normal ~n! as shown in the Fig. 1 a distance WDt so that the patch perimeters have lengths (R11WDt)df1 and (R21WDt)df2 at the time t1Dt; thus the change in area of the patch is WDt(R11R2)df1df2 and the per unit area per unit time rate of change is

Fingering instability of a retracting solid film edge

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 {alpha} between 30 and 180{degree}. For smaller {alpha}, a small-slope late-time solution is found that agrees with the numerical solution for {alpha} = 30{degree}. Thus, the complete range of {alpha} is covered. The long-time retraction speed and the distance traveled per cycle agree quantitatively with experiments.

A slope-dependent disjoining pressure for non-zero contact angles

A thin gold film under annealing on a silica substrate can develop “fingers” 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 str...

Grain-boundary grooving by surface diffusion with strong surface energy anisotropy

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 the 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 x leads to Π = Π(h, h x ), which 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 x ). By considering a fluid wedge on a solid substrate, E(h, h x ) is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit Π = B h 3 (α 4 - h 4 x + 2hh 2 xh xx ), where a is the contact angle and B is a material constant. The term containing the curvature h xx 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. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that h xx = 0 at the contact line is sufficient to specify the contact angle.

Coupled grooving and migration of inclined grain boundaries: Regime I

Abstract 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 1/4 . 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.

Fluid flow and heat transfer in a dual-wet micro heat pipe

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.

Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip

Micro heat pipes have been used to cool micro electronic devices, but their heat transfer coefficients are low compared with those of conventional heat pipes. In this work, a dual-wet pipe is proposed as a model to study heat transfer in micro heat pipes. The dual-wet pipe has a long and narrow cavity of rectangular cross-section. The bottom-half of the horizontal pipe is made of a wetting material, and the top-half of a non-wetting material. A wetting liquid fills the bottom half of the cavity, while its vapour fills the rest. This configuration ensures that the liquid–vapour interface is pinned at the contact line. As one end of the pipe is heated, the liquid evaporates and increases the vapour pressure. The higher pressure drives the vapour to the cold end where the vapour condenses and releases the latent heat. The condensate moves along the bottom half of the pipe back to the hot end to complete the cycle. We solve the steady-flow problem assuming a small imposed temperature difference between the two ends of the pipe. This leads to skew-symmetric fluid flow and temperature distribution along the pipe so that we only need to focus on the evaporative half of the pipe. Since the pipe is slender, the axial flow gradients are much smaller than the cross-stream gradients. Thus, we can treat the evaporative flow in a cross-sectional plane as two-dimensional. This evaporative motion is governed by two dimensionless parameters: an evaporation number E defined as the ratio of the evaporative heat flux at the interface to the conductive heat flux in the liquid, and a Marangoni number M . The motion is solved in the limit E →∞ and M →∞. It is found that evaporation occurs mainly near the contact line in a small region of size E −1 W , where W is the half-width of the pipe. The non-dimensional evaporation rate Q * ~ E −1 ln E as determined by matched asymptotic expansions. We use this result to derive analytical solutions for the temperature distribution T p and vapour and liquid flows along the pipe. The solutions depend on three dimensionless parameters: the heat-pipe number H , which is the ratio of heat transfer by vapour flow to that by conduction in the pipe wall and liquid, the ratio R of viscous resistance of vapour flow to interfacial evaporation resistance, and the aspect ratio S . If HR ≫1, a thermal boundary layer appears near the pipe end, the width of which scales as ( HR ) −1/2 L , where L is the half-length of the pipe. A similar boundary layer exists at the cold end. Outside the boundary layers, T p varies linearly with a gradual slope. Thus, these regions correspond to the evaporative, adiabatic and condensing regions commonly observed in conventional heat pipes. This is the first time that the distinct regions have been captured by a single solution, without prior assumptions of their existence. If HR ~ 1 or less, then T p is linear almost everywhere. This is the case found in most micro-heat-pipe experiments. Our analysis of the dual-wet pipe provides an explanation for the comparatively low effective thermal conductivity in micro heat pipes, and points to ways of improving their heat transfer capabilities.

Grain-boundary grooving by surface diffusion with asymmetric and strongly anisotropic surface energies

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 of the capillary number Ca →0 with the Reynolds number Re = o ( Ca −1 , non-zero Gibbs elasticity ( G ), and arbitrary Bond number ( Bo ). ( Ca =μ Q / a 2 σ c , where μ is the liquid viscosity, a is the tube radius, and σ c is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M →∞, where M = G / Ca 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.

A model of migrating grain-boundary grooves with application to two mobility-measurement methods

Grain-boundary migration controls grain growth, which is important in material processing and synthesis. When a vertical grain boundary ends at a horizontal free surface, a groove forms at the tip to reduce the combined grain-boundary and surface energies. The groove affects the migration of the grain boundary, and its effect must be understood. This work studies grain-boundary grooving by capillarity-driven surface diffusion with asymmetric and strongly anisotropic surface energies. The surface energies are described by the delta-function facet model that holds for temperatures above the roughening temperature of the bicrystal. Since the asymmetric anisotropy does not introduce a length scale, the asymmetric groove still grows with time t as t1∕4. Thus, the nonlinear partial differential equation that governs grooving is reduced by a self-similar transformation to an ordinary differential equation, which is then solved numerically by shooting methods. We vary systematically the crystallographic orientati...