Volume 17, Issue 10, October 2005
 SPECIAL TOPIC: TRANSPORT PHENOMENA IN MICRO AND NANODEVICES



An extended volumeoffluid method for micro flows with shortrange interactions between fluid interfaces
View Description Hide DescriptionA formulation is developed allowing to take into account the interaction of fluid interfaces in computational methods for freesurface flow. The necessity for such an extension of the standard computational approaches is traced back to a large number of experimental and theoretical results showing that in the presence of surfactants, fluid interfaces often interact in such a way that coalescence of bubbles or droplets is slowed down or suppressed. The strategy pursued in this paper to incorporate such effects relies on a Debyescreened scalar field which develops steep gradients in the vicinity of an interface. The force density acting in a gap between two interfaces is computed using a function of the local field values and a cut off to eliminate those regions where no interaction takes place. It is shown that in such a way a powerlaw behavior of the force as a function of separation between the interfaces can be reproduced. The model is implemented into a standard volumeoffluid scheme and it is exemplified that, compared to conventional approaches, completely different scenarios for micro flows of droplets can be reproduced. The important qualitative difference is that coalescence is avoided, so that the formation and transport of multibubble∕multidroplet arrangements can be studied.

Electron vortices in semiconductors devices^{a)}
View Description Hide DescriptionThe hydrodynamic model of electron transport in semiconductors is analyzed and, in analogy with vortices in fluid mechanics, the curl of electron velocity is defined as electron vorticity, and the transportequation for the electron vorticity is derived. Aside from the classical hydrodynamic sources of vorticity, collision terms in the continuity and momentum equations are identified as sources and sinks of electron vorticity. Similar to threedimensional fluid flows there is a vortex stretching term in the vorticityequation. This term could be responsible for the possible cascade of electron kinetic energy to small scales and formation of chaotic turbulent electron transport regimes. A scale analysis of the electron vorticityequation is performed and the relative order of magnitude of each sources of vorticity is found. This analysis and the calculation of electron meanfreepath due to electron–electron and electron–phonon scatterings characterize a transport regime with significant electron vorticity effects. Furthermore, conditions for observation of electron vortices in semiconductor devices are predicted.

Ultrasoundmediated destruction of contrast microbubbles used for medical imaging and drug delivery
View Description Hide DescriptionMicronsize bubbles encapsulated by a stabilizing layer of surfaceactive materials are used in medicalultrasound imaging and drug delivery. Their destruction stimulated by ultrasoundin vivo plays a critical role in both applications. We investigate the destruction process of microbubbles in a commercially available contrast agent by measuring the attenuation of ultrasound through it. The measurement is performed with singlecycle bursts from an unfocused transducer (with a center frequency of ) for varying pressure amplitudes at 50, 100, and pulse repetition frequencies (PRF) with duty cycles 0.001%, 0.002%, and 0.004%, respectively. At low excitation, the attenuation is found to increase with time. With increased excitation level, the attenuation level decreases with time, indicating destruction of microbubbles. There is a critical pressure amplitude for all three PRFs, below which there is no significant bubble destruction. Above the critical pressure amplitudes the rate of destruction depends on excitation levels. But at highpressure amplitudes the destruction becomes independent of excitation pressure amplitude. The results are interpreted to identify two different mechanisms of bubble destruction by its signature in attenuation, namely, slow dissolution by diffusion and catastrophic shell rupture. The different modes are discussed in detail with their implications in medical applications.

Effect of the surface charge on ion transport through nanoslits
View Description Hide DescriptionA description of ion transport through geometrically defined nanoslits is presented. It is characterized by the effective surface charge density and was obtained by impedance spectroscopy measurements of electrolytes with different physicochemical properties. The fluid channels were fabricated in a Pyrex–Pyrex field assisted bonding process with an intermediate layer of amorphous silicon. The height of the nanoslits was defined by the thickness of the amorphous silicon layer. Two microfluidic channels, containing electrodes for the characterization of the nanoslits, maintained fresh liquid on both sides of the nanoapertures. By changing the KCl concentration of the electrolyte, a conductance plateau (in log–log scale) was observed due to the dominance of the effective surface charge density, resulting in an excess of mobile counterions in the nanoslits at low salt concentrations. The effective surface charge density of the Pyrex nanoslits could be modified by changing the of the solution. It was verified that at higher values the nanoslit conductance increased. Fieldeffect experiments allowed changing the effective surface charge density as well. The polarity of the external voltage could be chosen such that the effective surface charge density was increased or decreased, resulting in a higher or lower nanoslit conductance. This regulation of ionic flow can be exploited for the fabrication of nanofluidic devices.

Droplet formation and ejection from a micromachined ultrasonic droplet generator: Visualization and scaling
View Description Hide DescriptionVisualization and scaling of dropondemand and continuousjet fluid atomization of water are presented to elucidate the fluid physics of the ejection process and characterize the modes of operation of a novel micromachined ultrasonicdropletgenerator. The device comprises a fluid reservoir that is formed between a bulk ceramic piezoelectric transducer and an array of liquid horn structures wet etched into (100) silicon. At resonance, the transducergenerates a standing ultrasonicpressurewave within the cavity and the wave is focused at the tip of the nozzle by the horn structure. Device operation has been demonstrated by water droplet ejection from orifices at multiple resonant frequencies between 1 and . The intimate interactions between focused ultrasonicpressurewaves and capillary waves formed at the liquid–air interface located at the nozzle tip are found to govern the ejection dynamics, leading to different ejection modalities ranging from individual droplets to continuous jet. Specifically, we report the results of highresolution stroboscopic optical imaging of the liquid–air interface evolution during acoustic pumping to elucidate the role of capillary waves in the dropletformation and ejection process. A basic understanding of the governing physics gained through careful visualization and scaling forms the basis for development of improved theoretical models for the dropletformation and ejection processes by accounting for key fluid mechanical features of the phenomena.

Characterization of surface roughness effects on pressure drop in singlephase flow in minichannels
View Description Hide DescriptionRoughness features on the walls of a channel wall affect the pressure drop of a fluid flowing through that channel. This roughness effect can be described by (i) flow area constriction and (ii) increase in the wall shear stress. Replotting the Moody’s friction factor chart with the constricted flow diameter results in a simplified plot and yields a single asymptotic value of friction factor for relative roughness values of in the fully developed turbulent region. After reviewing the literature, three new roughness parameters are proposed (maximum profile peak height , mean spacing of profile irregularities , and floor distance to mean line ). Three additional parameters are presented to consider the localized hydraulic diameter variation (maximum, minimum, and average) in future work. The roughness is then defined as . This definition yields the same value of roughness as obtained from the sandgrain roughness [H. Darcy, Recherches Experimentales Relatives au Mouvement de L’Eau dans les Tuyaux (MalletBachelier, Paris, France, 1857); J. T. Fanning, A Practical Treatise on Hydraulic and Water Supply Engineering (Van Nostrand, New York, 1877, revised ed. 1886); J. Nikuradse, “Laws of flow in rough pipes” [“Stromungsgesetze in Rauen Rohren,” VDIForschungsheft 361 (1933)]; Beilage zu “Forschung auf dem Gebiete des Ingenieurwesens,” Ausgabe B Band 4, English translation NACA Tech. Mem. 1292 (1937)]. Specific experiments are conducted using parallel sawtooth ridge elements, placed normal to the flow direction, in aligned and offset configurations in a wide rectangular channel with variable gap (resulting hydraulic diameters of with Reynolds numbers ranging from 200 to 7200 for air and 200 to 5700 for water). The use of constricted flow diameter extends the applicability of the laminar friction factor equations to relative roughness values (sawtooth height) up to 14%. In the turbulent region, the aligned and offset roughness arrangements yield different results indicating a need to further characterize the surface features. The laminar to turbulent transition is also seen to occur at lower Reynolds numbers with an increase in the relative roughness.

Flow of gaseous mixtures through rectangular microchannels driven by pressure, temperature, and concentration gradients
View Description Hide DescriptionThe flow of binary gaseous mixtures through rectangular microchannels due to small pressure, temperature, and molar concentration gradients over the whole range of the Knudsen number is studied. The solution is based on a mesoscale approach, formally described by two coupled kinetic equations, subject to diffuse scattering boundary conditions. The model proposed by McCormack substitutes the complicated collision term and the resulting kinetic equations are solved by an accelerated version of the discrete velocity method. Typical results are presented for the flow rates and the heat fluxes of two different binary mixtures (Ne–Ar and He–Xe) with various molar concentrations, in twodimensional microchannels of different aspect (height to width) ratios. The formulation is very efficient and can be used instead of the classical method of solving the Navier–Stokes equations with slip boundary conditions, which is restricted by the hydrodynamic regime. Moreover, the present formulation is a good alternative to the direct simulation Monte Carlo method, which often becomes computationally inefficient.

Spreading behavior of an impacting drop on a structured rough surface
View Description Hide DescriptionThe spreading of water drops impinging on structuredrough surfaces is studied experimentally. The rough surfaces are specially prepared with a regular pattern of surface asperities. The arrangement of the squareshaped surface asperities creates channellike grooves on the surface. A video microscope along with a controlled light exposure system is used to construct the image sequences of the spreading process. The images are digitally analyzed to measure the temporal variation of the spreading drop diameter . Results are obtained for three rough surfaces with varying asperity heights in the range of and for different impact drop conditions with Weber number We in the range of 35–225. The results on the temporal variation of show that, on the structuredrough surfaces, the spreading occurs simultaneously both inside and above the texture pattern of the surfaces. For a given surface geometry, the volume of liquid flowing inside the grooves of the surface increases with increasing We. Consequently, the values of measured inside the texture pattern are larger than those measured above the texture pattern, and their difference increases with increasing We. The arrangement of the surface asperities influences the spreading pattern of an impacting drop spreading axisymmetrically. For the texture geometry used in the present study, the spreading pattern resembles a regular rhombus shape for the impact of low We drops and becomes complex at high We. The spreading distances, measured both inside and above the texture pattern of the structuredrough surfaces, are nearer to the measurements recorded on the smooth surface if the asperity height of the rough surface is smaller than the thickness of the spreading liquidlamella; however, the surface asperities influence the spreading pattern drastically and create a liquid splash.

The usefulness of higherorder constitutive relations for describing the Knudsen layer
View Description Hide DescriptionThe Knudsen layer is an important rarefaction phenomenon in gas flows in and around microdevices. Its accurate and efficient modeling is of critical importance in the design of such systems and in predicting their performance. In this paper we investigate the potential that higherorder continuum equations may have to model the Knudsen layer, and compare their predictions to highaccuracy DSMC (direct simulation Monte Carlo) data, as well as a standard result from kinetic theory. We find that, for a benchmark case, the most common higherorder continuum equation sets (Grad’s 13 moment, Burnett, and superBurnett equations) cannot capture the Knudsen layer. Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure. To make quantitative comparisons, we obtain additional boundary conditions (needed for unique solutions to the higherorder equations) from kinetic theory. However, we find the quantitative agreement with kinetic theory and DSMC data is only slight.

Molecular number flux detection using oxygen sensitive luminophore
View Description Hide DescriptionExperimental analyses of thermofluid phenomena with a high Knudsen number, related to lowdensity gas flows or nanotechnologies, need the measurement techniques based on atoms or molecules, such as emission and absorption of photons. Because the principle of the pressure sensitive paint (PSP) technique is based on oxygen quenching of luminescence, the technique has the capability to be applied to high Knudsen number flows such as microflows and lowdensity gas flows. In this study, to inspect the feasibility of PSP for measurement of pressure on a solid surface in high Knudsen number flows, fundamental properties of three types of PSP [palladium tetrakis (pentafluorophenyl) porphyrin,palladium octaethylporphyrin (PdOEP), and platinum tetrakis (pentafluorophenyl) porphyrin bound by poly[1–(trimethylsilyl)1–propyne] (poly(TMSP))] are examined especially in the range of pressure below 130 Pa (about 1 Torr). As an application of PSP to high Knudsen number flows, we measure the pressure distribution on a jetimpinging solid surface using with very high sensitivity. Moreover, the “pressure” distribution obtained by the PSP is compared with the distribution of the molecular number flux onto the solid surface to investigate the feasibility of number flux measurement by PSP.

Oscillatory sheardriven gas flows in the transition and freemolecularflow regimes
View Description Hide DescriptionWe investigate oscillatory sheardriven gas flows in the transition and freemolecularflow regimes. Analytical results valid through slip flow and the early transition regime are obtained using a recently proposed, rigorous secondorder slip model with no adjustable coefficients. Analytical solution of the collisionless Boltzmann equation provides a description of the high Knudsen number limit including the bounded shear layers present in the limit of high oscillation frequency. These layers are analogous to the Stokes layers observed in the limit, but contrary to the latter, they exhibit a nonconstant wave speed as demonstrated by Park, Bahukudumbi, and Beskok in Phys. Fluids.16, 317 (2004). All theoretical results are validated by direct Monte Carlo simulations. We find that the secondorder slip results are in good agreement with direct simulation Monte Carlo (DSMC) solutions up to ; in some cases these results continue to provide useful approximations to quantities of engineering interest, such as the shear stress, well beyond . The collisionless theory provides, in general, a good description of DSMC results for , while in the high frequency limit the agreement is very good for Knundsen numbers as low as .

Liquids: The holy grail of microfluidic modeling
View Description Hide DescriptionTraditional fluid mechanics edifies the indifference between liquid and gas flows as long as certain similarity parameters—most prominently the Reynolds number—are matched. This may or may not be the case for flows in nanodevices or microdevices. The customary continuum, Navier–Stokes modeling is ordinarily applicable for both air and water flowing in macrodevices. Even for common fluids such as air or water, such modeling is bound to fail at sufficiently small scales, but the onset for such failure is different for the two forms of matter. Moreover, when the noslip, quasiequilibrium Navier–Stokes system is no longer applicable, the alternative modeling schemes are different for gases and liquids. For dilute gases, statistical methods are applied and the Boltzmann equation is the cornerstone of such approaches. For liquidflows, the dense nature of the matter precludes the use of the kinetic theory of gases, and numerically intensive molecular dynamics simulations are the only alternative rooted in first principles. The present paper discusses the above issues as well as outlines physical phenomena unique to liquidflows in minute devices.

Current and current fluctuations in quantum shuttles
View Description Hide DescriptionWe review the properties of electron shuttles, i.e., nanoelectromechanical devices that transport electrons one by one by utilizing a combination of electronic and mechanical degrees of freedom. We focus on the extreme quantum limit, where the mechanical motion is quantized. We introduce the main theoretical tools needed for the analysis, e.g., generalized master equations and Wigner functions, and we outline the methods how the resulting large numerical problems can be handled. Illustrative results are given for current, noise, and full counting statistics for a number of model systems. Throughout the review we focus on the physics behind the various approximations, and some simple examples are given to illustrate the theoretical concepts. We also comment on the experimental situation.

Transient micromixing: Examples of laminar and chaotic stirring
View Description Hide DescriptionThe efficiency of a micromixing device may be quantified by the time taken for a given initial state of separated fluids to reach a desired level of homogenization. In the physically relevant case of high Peclet number the accurate prediction of the mixing time is a challenging problem, even in simple twodimensional flows within bounded domains. In this paper a closedform solution for the time dependence of mixing in an annular micromixer is derived and verified by numerical simulation. The mixing time is found to scale with Peclet number as a power law, but the powerlaw exponent depends on the level of homogeneity desired in the final state. Numerical simulation of a recent model of chaotic mixing reveals a vortexlike stirring effect in quasiperiodic islands of the Poincaré map of the flow, which strongly influences the mixing time. This stirring effect is identified with an exponential decrease in solute variance on an intermediate time scale, being subdominant to the asymptotic longtime decay, but sensitive to the initial loading of fluids in the mixer. The subdominant decay rate is calculated to scale with Peclet number as the square root of the dominant decay rate.

The impact of subcontinuum gas conduction on topography measurement sensitivity using heated atomic force microscope cantilevers
View Description Hide DescriptionNanometerscale topographical imaging using heated atomic force microscope(AFM) cantilevers, referred to here as thermal sensing AFM (TSAFM), is a promising technology for high resolution topographical imaging of nanostructured surfaces. Heated AFM cantilevers were developed for highdensity data storage, where the heated cantilever tip can form and detect 20 nm indents made in a thermoplastic polymer. The scan height of the cantilever heater platform is typically near 500 nm, but could be made much smaller to improve reading sensitivity. Under atmospheric conditions the continuum models used in previous studies to model the gas phase heat transfer are invalid for the smallest operating heights. The present study uses a molecular model of subcontinuum heat transfer coupled with a finite difference simulation to predict the behavior of a TSAFM system. A direct simulation Monte Carlo model and a kinetic theory based macromodel are separately developed and used to model subcontinuum gas conduction. For the working gas (argon) the simple macromodel is found to be accurate and is used to predict cantilever operation. This systemslevel modeling approach for TSAFM operation can aid data interpretation and seeks to improve microcantilever design.

Boundarylayer exchange by bubble: A novel method for generating transient nanofluidic layers
View Description Hide DescriptionUnstirred layers (i.e., Nernst boundary layers) occur on every dynamic solidliquid interface, constituting a diffusion barrier, since the velocity of a moving liquid approaches zero at the surface (no slip). If a macromoleculesurface reaction rate is higher than the diffusion rate, the Nernst layer is solute depleted and the reaction rate becomes masstransport limited. The thickness of a Nernst boundary layer generally lies between 5 and . In an evanescent wave rheometer, measuring fibrinogen adsorption to fused silica, we made the fundamental observation that an air bubble preceding the sample through the flow cell abolishes the masstransport limitation of the Nernst diffusion layer. Instead exponential kinetics are found. Experimental and simulation studies strongly indicate that these results are due to the elimination of the Nernst diffusion layer and its replacement by a dynamic nanofluidic layer maximally thick. It is suggested that the air bubble leads to a transient boundarylayer separation into a novel nanoboundary layer on the surface and the bulk fluid velocity profile separated by a vortex sheet with an estimated lifetime of . A bubbleinduced boundarylayer exchange from the Nernst to the nanoboundary layer and back is obtained, giving sufficient time for the measurement of unbiased exponential surface kinetics. Noteworthy is that the nanolayer can exist at all and displays properties such as (i) a long persistence and resistance to dissipation by the bulk liquid (boundarylayerexchangehysteresis) and (ii) a lack of solute depletion in spite of boundarylayer separation. The boundarylayerexchange by bubble (BLEB) method therefore appears ideal for enhancing the rates of all types of diffusionlimited macromolecular reactions on surfaces with contact angles between 0° and 90° and only appears limited by slippage due to nanobubbles or an air gap beneath the nanofluidic layer on very hydrophobic surfaces. The possibility of producing nanoboundary layers without any nanostructuring or nanomachining should also be useful for fundamental physical studies in nanofluidics.
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 ARTICLES

 Interfacial Flows

Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects
View Description Hide DescriptionThe weak viscous oscillations of a bubble are examined, in response to an elongation that perturbs the initial spherical shape at equilibrium. The flow field in the surrounding liquid is split in a rotational and an irrotational part. The latter satisfies the Laplacian and can be obtained via an integral equation. A hybrid boundaryfinite element method is used in order to solve for the velocity potential and shape deformation of axisymmetric bubbles. Weak viscous effects are included in the computations by retaining firstorder viscous terms in the normal stress boundary condition and satisfying the tangential stress balance. An extensive set of simulations was carried out until the bubble either returned to its initial spherical shape, or broke up. For a relatively small initial elongation the bubble returned to its initial spherical state regardless of the size of the Ohnesorge number; . For larger initial elongations there is a threshold value in above which the bubble eventually breaks up giving rise to a “donut” shaped larger bubble and a tiny satellite bubble occupying the region near the center of the original bubble. The latter is formed as the round ends of the liquid jets that approach each other from opposite sides along the axis of symmetry, coalesce. The size of the satellite bubble decreased as the initial elongation or increased. This pattern persisted for a range of large initial deformations with a decreasing threshold value of the as the initial deformation increased. As its equilibrium radius increases the bubble becomes more susceptible to the above collapse mode. The effect of initial bubble overpressure was also examined and it was seen that small initial overpressures, for the range of initial bubble deformations that was investigated, translate the threshold of to larger values while at the same time increasing the size of the satellite bubble.

Thermocapillary instability of coreannular flows
View Description Hide DescriptionThermocapillary instability of a coreannular flow is asymptotically examined in the thin annulus limit. Two sets of scalings are established to study the interplays between base flows,interfacial tension, and thermocapillary effects. For each scaling case, an interfacial evolution equation is derived for describing the leading order stability of the system. Both linear and weakly nonlinear stabilities are examined. When the core fluid is warmer (cooler) than the wall, thermocapillarity linearly stabilizes (destabilizes) the system, and hence suppresses (promotes) the capillary instability. For a moderate thermocapillary force and a strong capillary force, the linear instability can be arrested within the weakly nonlinear regime. For a weak thermocapillary force and a moderately strong interfacial tension, the weakly nonlinear evolution is governed by a modified KuramotoSivashinsky equation. The influence of thermocapillarity on the route to chaos is discussed.

Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity
View Description Hide DescriptionA theoretical analysis of the thermal effects on the dynamics of a thin nonuniform film of a nonvolatile incompressible viscous fluid on a heated rotating disk has been considered and the effects of temperaturedependent viscosity and surface tension have been analyzed. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory. It has been observed that the infinitesimal disturbances decay for small wave numbers and are transiently stable for large wave numbers, for both zero and nonzero values of Biot number.