Charles Maldarelli

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

Stability of symmetric and unsymmetric thin liquid films to short and long wavelength perturbations

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

Nonlinear interfacial stability of core‐annular film flows

The stability of thin (< 100 nm) symmetrical and unsymmetrical membranes, assimilated with viscous liquids, to short and long wavelength perturbations is investigated. The asymmetry is due to the two different viscous phases surrounding the membrane and to the different interfacial tensions on the two faces of the membrane. The cell membrane is a case for which the present treatment is of significance. The dynamics of the membrane to perturbations is described by the Navier-Stokes equation modified with a body force which accounts for the fact that the range of the interaction forces is larger than the thickness of the film. The body force is computed assuming pair-wise additivity and accounting for the deformation of the interface produced by the perturbation. General dispersion equations are derived, and these equations describe the squeezing and stretching modes of perturbations. The growth coefficient is expressed as a function of the wavelength for various ratios of the viscosities of the two surrounding phases and various values of the two interfacial tensions. In the limiting cases of interfacial tension ratio equal to unity and wavelength large compared to the thickness of the film results of the previous investigators are obtained. For the symmetrical case expressions are derived for the critical and dominant wavelength of the squeezing and stretching modes. An application of the results to a cell membrane shows that the growth of the instability is dominated in this case by the stretching mode since the time scale of growth of the perturbations is four orders of magnitude less than that in the squeezing mode. For unsymmetrical systems the effect of differences between the interfacial tensions on the two faces on the ratio of the amplitudes of perturbations on the two faces is investigated. The results show in what manner differences in interfacial tension convey amplified or damped messages across the membrane.

Remobilizing surfactant retarded fluid particle interfaces. I. Stress‐free conditions at the interfaces of micellar solutions of surfactants with fast sorption kinetics

In this paper the weakly nonlinear stability of two‐phase core‐annular film flows in the limit of small film thickness and in the presence of both viscosity stratification and interfacial tension is examined. Rational asymptotic expansions are used to derive some novel nonlinear evolution equations for the interface between the phases. The novel feature of the equations is that they include a coupling between core and film dynamics thus enabling a study of its effect on the nonlinear evolution of the interface. The nonlinear interfacial evolution is governed by modified Kuramoto–Sivashinsky equations in the cases of slow and moderate flow [the former also developed by Frenkel, Sixth Symposium on Energy Engineering Sciences (Argonne Lab. Pub. CONF‐8805106, 1988), p.100, using different asymptotic methods], which include new nonlocal terms that reflect core dynamics. These equations are solved numerically for given initial conditions and a range of parameters. Some interesting behavior results, such as tran...

Measurement of the kinetic rate constants for the adsorption of superspreading trisiloxanes to an air/aqueous interface and the relevance of these measurements to the mechanism of superspreading

Surfactant molecules adsorb onto the interfaces of moving fluid particles and are convected to regions in which the surface flow converges. Accumulation of surfactant in these regions creates interfacial tension gradients that retard the surface flow. In this study it is argued theoretically and demonstrated experimentally that fluid movement on the surface of a drop or bubble can remain unhindered in the presence of a single adsorbed surfactant if, relative to the convective rate of transport of adsorbed surfactant along the surface, desorption is fast, and the bulk concentration is high enough so that diffusion away from the particle is fast. For this circumstance, a uniform surface concentration of surfactant is maintained, and no gradients in surface tension arise to retard the surface velocity. The fluid particle flow behaves as it would in the absence of surfactant save that it has a reduced, uniform surface tension. The remobilization of surfactant‐laden interfaces of fluid particles is demonstrate...

The size of stagnant caps of bulk soluble surfactant on the interfaces of translating fluid droplets

Super-spreading trisiloxane surfactants are a class of amphiphiles which consist of nonpolar trisiloxane headgroups ((CH3)3-Si-O)2-Si(CH3)(CH2)3-) and polar parts composed of between four and eight ethylene oxides (ethoxylates, -OCH2CH2-). Millimeter-sized aqueous drops of trisiloxane solutions at concentrations well above the critical aggregate concentration spread rapidly on very hydrophobic surfaces, completely wetting out at equilibrium. The wetting out can be understood as a consequence of the ability of the trisiloxanes at the advancing perimeter of the drop to adsorb at the air/aqueous and aqueous/hydrophobic solid interfaces and to reduce considerably the tensions of these interfaces, creating a positive spreading coefficient. The rapid spreading can be due to maintaining a positive spreading coefficient at the perimeter as the drop spreads. However, the air/aqueous and solid/aqueous interfaces at the perimeter are depleted of surfactant by interfacial expansion as the drop spreads. The spreading coefficient can remain positive if the rate of surfactant adsorption onto the solid and fluid surfaces from the spreading aqueous film at the perimeter exceeds the diluting effect due to the area expansion. This task is made more difficult by the fact that the reservoir of surfactant in the film is continually depleted by adsorption to the expanding interfaces. If the adsorption cannot keep pace with the area expansion at the perimeter, and the surface concentrations become reduced at the contact line, a negative spreading coefficient which retards the drop movement can develop. In this case, however, a Marangoni mechanism can account for the rapid spreading if the surface concentrations at the drop apex are assumed to remain high compared to the perimeter so that the drop is pulled out by the higher tension at the perimeter than at the apex. To maintain a high apex concentration, surfactant adsorption must exceed the rate of interfacial dilation at the apex due to the outward flow. This is conceivable because, unlike that at the contact line, the surfactant reservoir in the liquid at the drop center is not continually depleted by adsorption onto an expanding solid surface. In an effort to understand the rapid spreading, we measure the kinetic rate constants for adsorption of unaggregated trisiloxane surfactant from the sublayer to the air/aqueous surface. The kinetic rate of adsorption, computed assuming the bulk concentration of monomer to be uniform and undepleted, represents the fastest that surfactant monomer can adsorb onto the air/aqueous surface in the absence of direct adsorption of aggregates. The kinetic constants are obtained by measuring the dynamic tension relaxation as trisiloxanes adsorb onto a clean pendant bubble interface. We find that the rate of kinetic adsorption is only of the same order as the area expansion rates observed in superspreading, and therefore the unaggregated flux cannot maintain very high surface concentrations at the air/aqueous interface, either at the apex or at the perimeter. Hence in order to maintain either a positive spreading coefficient or a Marangoni gradient, the surfactant adsorptive flux needs to be augmented, and the direct adsorption of aggregates (which in the case of the trisiloxanes are bilayers and vesicles) is suggested as one possibility.

Temporal instability of compound threads and jets

Abstract When a spherical fluid droplet translates through a liquid medium containing surfactants with a desorption rate much slower than the convective rate, surfactant collects at the trailing pole in a stagnant cap of angle φ which reduces the terminal velocity. An exact solution for the terminal velocity, at low Reynolds numbers, can be obtained in terms of φ when a linear gaseous surface constitutive equation is used. The present analysis shows that the linear equation of state is only valid for large Marangoni numbers (> 10). The extension of this equation to regimes of small Marangoni numbers significantly underpredicts the stagnant cap angle and, consequently, yields an erroenously larger terminal velocity. The nonlinear Frumkin constitutive equation, on the other hand, provides more accurate computation of larger cap angles and limits the viscous compression of the monolayer in the cap to its maximum possible packing density.

Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles

Compound threads and jets consist of a core liquid surrounded by an annulus of a second immiscible liquid. Capillary forces derived from axisymmetric disturbances in the circumferential curvatures of the two interfaces destabilize cylindrical base states of compound threads and jets (with inner and outer radii R 1 and aR 1 respectively). The capillary instability causes breakup into drops; the presence of the annular phase allows both the annular- and core-phase properties to influence the drop size. Of technological interest is breakup where the core snaps first, and then the annulus. This results in compound drops. With jets, this pattern can form composite particles, or if the annular fluid is evaporatively removed, single drops whose size is modulated by both fluids. This paper is a study of the linear temporal instability of compound threads and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formation. The temporal dispersion equation is solved numerically for non-dimensional annular thicknesses a of order one, and analytically for thin annuli ( a – 1 = e [Lt ] 1) by asymptotic expansion in e. There are two temporally growing modes: a stretching mode, unstable for wavelengths greater than the undisturbed inner circumference 2π R 1 , in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelengths greater than 2π aR 1 , which grows exactly out of phase. Growth rates are always real, indicating that in jetting configurations disturbances convect downstream with the base velocity. For order-one thicknesses, the growth rate of the stretching mode is higher for the entire range of system parameters examined. The drop size scales with the wavenumber of the maximally growing wave ( k max ). We find that for the dominant stretching mode and a = 2, variations from 0.1 to 10 in the ratios of the annulus to core viscosity, or the tension of the outer surface to that of the inner interface, can result in changes in k max by a factor of approximately 2. However, for these changes in the system ratios, the growth rate ( s max ) and the ratio of the amplitude of the outer to the inner interface ( A max ) for the fastest growing wave only change marginally, with A max near one. The system appears most sensitive to the ratio of the density of the annulus to the core fluid. For a variation between 0.1 and 10, k max again changes by a factor of 2, but A max and s max vary more significantly with large amplitude ratios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave ( A max ) indicates whether the film or core will break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for low values of the density ratio, our results indicate that most system conditions promote compound drop formation. For thin annuli, the growth rate disparity between modes becomes even greater. In the limit e → 0, the squeezing growth rate is roughly proportional to e 2 while the stretching mode growth rate is roughly proportional to e 0 and asymptotes to a single jet with radius R 1 and tension equal to the sum of the two tensions. Thus, in this limit the growth rate and k max are independent of the film density and viscosity. The amplitude ratio of the stretching mode becomes equal to one for all wavenumbers; so thin films break as compound drops. Our results compare favourably with previously published measurements on unstable waves in compound jets.

The linear, hydrodynamic stability of an interfacially perturbed, transversely isotropic, thin, planar viscoelastic film: I. General formulation and a derivation of the dispersion equation

The buoyant motion of a bubble rising through a continuous liquid phase can be retarded by the adsorption onto the bubble surface of surfactant dissolved in the liquid phase. The reason for this retardation is that adsorbed surfactant is swept to the trailing pole of the bubble where it accumulates and lowers the surface tension relative to the front end. The difference in tension creates a Marangoni force which opposes the surface flow, rigidifies the interface and increases the drag coefficient. Surfactant molecules adsorb onto the bubble surface by diffusing from the bulk to the sublayer of liquid adjoining the surface, and kinetically adsorbing from the sublayer onto the surface. The surface surfactant distribution which defines the Marangoni force is determined by the rate of kinetic adsorption and bulk diffusion relative to the rate of surface convection. In the limit in which the rate of either kinetic or diffusive transport of surfactant to the bubble surface is slow relative to surface convection and surface diffusion is also slow, surfactant collects in a stagnant cap at the back end of the bubble while the front end is stress free and mobile. The size of the cap and correspondingly the drag coefficient increases with the bulk concentration of surfactant until the cap covers the entire surface and the drag coefficient is that of a bubble with a completely tangentially immobile surface. Previous theoretical research on the stagnant cap regime has not studied in detail the competing roles of bulk diffusion and kinetic adsorption in determining the size of the stagnant cap angle, and there have been only a few studies which have attempted to quantitatively correlate simulations with measurements. This paper provides a more complete theoretical study of and a validating set of experiments on the stagnant cap regime. We solve numerically for the cap angle and drag coefficient as a function of the bulk concentration of surfactant for a spherical bubble rising steadily with inertia in a Newtonian fluid, including both bulk diffusion and kinetic adsorption. For the case of diffusion-limited transport (infinite adsorption kinetics), we show clearly that very small bulk concentrations can immobilize the entire surface, and we calculate the critical concentrations which immobilize the surface as a function of the surfactant parameters. We demonstrate that the effect of kinetics is to reduce the cap angle (hence reduce the drag coefficient) for a given bulk concentration of surfactant. We also present experimental results on the drag of a bubble rising in a glycerol–water mixture, as a function of the dissolved concentration of a polyethoxylated non-ionic surfactant whose bulk diffusion coefficient and a lower bound on the kinetic rate constants have been obtained separately by measuring the reduction in dynamic tension as surfactant adsorbs onto a clean interface. For low concentrations of surfactant, the experiments measure drag coefficients which are intermediate between the drag coefficient of a bubble whose surface is tangentially mobile and one whose surface is completely immobilized. Using the separately obtained value for the diffusion coefficient of the polyethoxylate, we undertake simulations which provide, upon comparison with the measured drag coefficients, a tighter bound on the kinetic rate constants than were otherwise obtained using dynamic surface tension measurement, and this suggests a new method for the measurement of kinetic rate constants.

Diffusiophoretic self-propulsion of colloids driven by a surface reaction: The sub-micron particle regime for exponential and van der Waals interactions

Abstract A dispersion equation which describes the linear, hydrodynamic stability of an interfacially perturbed, thin (O(10–100 nm)), planar, uncharged, transversely isotropic, viscoelastic film bounded by electrolytic Newtonian fluids is developed for the case in which the film interfaces are tangentially immobile. The linear viscoelastic rheology of the film is described by a Boltzmann superposition in which the stress relaxation tensor is formulated by utilizing Kelvin models. The influence of the electrical interactions of the film system on the linear dynamics is derived explicitly by integrating the normal mode electrostatic field equations. An investigation of the adjoint properties of the normal mode mechanical field relations indicates that for a certain class of films, (i) the principle of exchange of stabilities is valid, (ii) instability is nonoscillatory, and (iii) oscillatory states decay. A simplified dispersion equation for a symmetric film system is deduced, and it is shown that this equation describes squeezing and stretching eigenmodes.