Liquid flow through microchannels with grooved walls under wetting and superhydrophobic conditions

Recent developments in superhydrophobic surfaces have enabled significant reduction in the frictional drag for liquid flow through microchannels. There is an apparent risk when using such surfaces, however, that under some conditions the liquid meniscus may destabilize and, consequently, the liquid will wet the entire patterned surface. This paper presents analytical and experimental results that compare the laminar flow dynamics through microchannels with superhydrophobic walls featuring ribs and cavities oriented both parallel and transverse to the direction of flow under both wetting and non-wetting conditions. The results show the reduction in the total frictional resistance is much greater in channels when the liquid phase does not enter the cavity regions. Further, it is demonstrated that the wetting and non-wetting cavity results represent limiting cases between which the experimental data lie. Generalized expressions enabling prediction of the classical friction factor-Reynolds number product as a function of the relevant governing dimensionless parameters are also presented for both the superhydrophobic and wetting states. Experimental results are presented for a range of parameters in the laminar flow regime.     

Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves

Superhydrophobic surfaces have been demonstrated to be capable of reducing fluid resistance in micro- and nanofluidic applications. The objective of this paper is to present analytical solutions for the Stokes flow through microchannels employing superhydrophobic surfaces with alternating micro-grooves and ribs. Results are presented for both cases where the micro-grooves are aligned parallel and perpendicular to the flow direction. The effects of patterning the grooves on one or both channel walls are also analyzed. The reduction in fluid resistance has been quantified in terms of a dimensionless effective slip length, which is found to increase monotonically with the shear-free fraction and the periodic extent of each groove–rib combination normalized by the channel half-height. Asymptotic relationships have been derived for the normalized effective slip length corresponding to large and small limiting values of the shear-free fraction and the normalized groove–rib period. A detailed comparison has been made between transverse and longitudinal grooves, patterned on one or both channel walls, to assess their effectiveness in terms of enhancing the effective slip length. These comparisons have been carried out for small and large limiting values, as well as finite values of the shear-free fraction and normalized groove–rib period. Results for the normalized effective slip length corresponding to transverse and longitudinal grooves are further applied to model the Stokes flow through microchannels employing superhydrophobic surfaces containing a periodic array of micro-grooves inclined at an angle to the direction of the applied pressure gradient. Results are presented for the normalized effective slip lengths parallel to the direction of the applied pressure gradient and the normalized cross flow rate perpendicular to the direction of the applied pressure gradient.     

Three-dimensional hydrodynamic focusing in polydimethylsiloxane (PDMS) microchannels

This paper presents a generalization of the hydrodynamic focusing technique to three-dimensions. Three-dimensional (3-D) hydrodynamic focusing offers the advantages of precision positioning of molecules in both vertical and lateral dimensions and minimizing the interaction of the sample fluid with the surfaces of the channel walls. In an ideal approach, 3-D hydrodynamic focusing could be achieved by completely surrounding the sample flow by a cylindrical sheath flow that constrains the sample flow to the center of the channel in both the lateral and the vertical dimensions. We present here design and simulation, 3-D fabrication, and experimental results from a piecewise approximation to such a cylindrical flow. Two-dimensional (2-D) and 3-D hydrodynamic focusing chips were fabricated using micromolding methods with polydimethylsiloxane (PDMS). Three-dimensional hydrodynamic focusing chips were fabricated using the "membrane sandwich" method. Laser scanning confocal microscopy was used to study the hydrodynamic focusing experiments performed in the 2-D and 3-D chips with Rhodamine 6G solution as the sample fluid and water as the sheath fluid.     

Effects of interface curvature on Poiseuille flow through microchannels and microtubes containing superhydrophobic surfaces with transverse grooves and ribs

This paper presents numerical results pertaining to the effects of interface curvature on the effective slip behavior of Poiseuille flow through microchannels and microtubes containing superhydrophobic surfaces with transverse ribs and grooves. The effects of interface curvature are systematically investigated for different normalized channel heights or tube diameters, shear-free fractions, and flow Reynolds numbers. The numerical results show that in the low Reynolds number Stokes flow regime, when the channel height or tube diameter (normalized using the groove–rib spacing) is sufficiently large, the critical interface protrusion angle at which the effective slip length becomes zero is θ _c ≈ 62°–65°, which is independent of the shear-free fraction, flow geometry (channel and tube), and flow driving mechanism. As the normalized channel height or tube diameter is reduced, for a given shear-free fraction, the critical interface protrusion angle θ _c decreases. As inertial effects become increasingly dominant corresponding to an increase in Reynolds number, the effective slip length decreases, with the tube flow exhibiting a more pronounced reduction than the channel flow. In addition, for the same corresponding values of shear-free fraction, normalized groove–rib spacing, and interface protrusion angle, longitudinal grooves are found to be consistently superior to transverse grooves in terms of effective slip performance.     

Fluid flows in microchannels with cavities

Pressure-driven gas and liquid flows through microchannels with cavities have been studied using both experimental measurements and numerical computations. Several microchannels with cavities varying in shape, number and dimensions have been fabricated. One set of microdevices, integrated with sensors on a silicon wafer, is used for flow rate and pressure distribution measurements in gas flows. Another set of microdevices, fabricated using glass-to-silicon wafer bonding, is utilized for visualization of liquid flow patterns. The cavity effect on the flow in the microchannel is found to be very small, with the mass flow rate increasing slightly with increasing number of cavities. The flow pattern in the cavity depends on two control parameters; it is fully attached only if both the reduced Reynolds number and the cavity number are small. A flow regime map has been constructed, where the critical values for the transition from attached to separated flow are determined. The numerical computations reveal another control parameter, the cavity aspect ratio. The flow in the cavity is similar only if all three control parameters are the same. Finally, the vorticity distribution and related circulation in the cavity are analyzed. [1546].     

Geometric Entropies of Mixing (EOM)

Trigonometric and trigonometric-algebraic entropies are introduced and are given an axiomatic characterization. Regularity increases the entropy and the maximal entropy is shown to result when a regular n-gon is inscribed in a circle. A regular n-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy, which occurs in the asymptotic infinite n-limit. The EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.     

Electroosmotic flow in microchannels with prismatic elements

Fundamental understanding of liquid flow through microchannels with 3D prismatic elements is important to the design and operation of lab-on-a-chip devices. In this paper, we studied experimentally and theoretically the electroosmotic flow (EOF) in slit microchannels with rectangular 3D prismatic elements fabricated on the bottom channel wall. The average electroosmotic velocity measured by the current-monitoring technique was found lower than that in a smooth microchannel. This velocity reduction becomes larger in microchannel with larger but less number of the prisms even though the space taken by the prisms are identical. The velocity distribution and streamlines on two typical horizontal planes in the microchannel are measured and visualized by a particle-based technique. These experimental observations are in good agreement with the numerical simulation. The comparison of streamlines near the prisms in the pressure-driven flow with that in the EOF showed that the EOF was more sensitive to the local geometry.     

Seasonal patterns in leaf reflectance red-edge characteristics

Abstract

Leaves from ten tree species, including three conifers from a woodlot in southern Ontario were sampled at weekly intervals for a period of 150 days spanning the phenological events in deciduous trees of leaf development and expansion (flushing), leaf maturity and leaf senescence. The highly diverse seasonal red-edge reflection patterns were studied collectively and individually both from the perspective of long-term trends and relatively short-term or oscillating trends. The phenological events characteristic of deciduous trees were most effectively described using the red-edge position (λ_pg) and the chlorophyll-well position (λ_og ) derived from the inverted Gaussian model. Moreover, these parameters appeared consistent with what is known about the seasonal turnover of leaf chlorophyll and with other parameters R₅₅₀ or R_og which more specifically quantify leaf chlorophyll. By these terms of reference chlorophyll declines considerably earlier in the season than the onset of other physiological or structural changes normally associated with senescence. Both parameters λ_PR and λ_og and the real red-edge position (λ_pr) conjointly provide specific information about each species and differentiate clearly deciduous from coniferous growth forms. Individual species variation in these and other variables was more ‘short term’ than ‘long term’, providing support for the perception that the long-term trends as visualized by λ_pg and λ_og are species-specific and can probably be correlated with ecological parameters such as tolerance.

Short-term or oscillating variations were observed in all species most specifically in the shoulder reflectance (R_gr), the red-edge parameter (λ_pr ) and, to a lesser extent, in the inverted Gaussian model parameters, the wavelength at minimum reflectance (λ_og) and the average red-edge position (λ_pg). The generally synchronous behaviour of R_sr between all species, and of reflectance and spectral parameters within specific species, suggest the oscillations were not attributable to sampling error. Instead they appeared to be associated with rainfall and temperature events. Furthermore, systematic uncoupling of short-term variations between the spectral and the reflectance parameters observed in other species suggest different casual factors for spectral and reflectance parameters of the red edge.     

Thin film evaporation in microchannels with interfacial slip

We investigate the role of interfacial slip on evaporation of a thin liquid film in a microfluidic channel. The effective slip mechanism is attributed to the formation of a depleted layer adhering to the substrate–fluid interface, either in a continuum or in a rarefied gas regime, as a consequence of intricate hydrophobic interactions in the narrow confinement. We appeal to the fundamental principles of conservation in relating the evaporation mechanisms with fluid flow and heat transfer over interfacial scales. We obtain semi-analytical solutions of the pertinent governing equations, with coupled heat and mass transfer boundary conditions at the liquid–vapor interface. We observe that a general consequence of interfacial slip is to elongate the liquid film, thereby leading to a film thickening effect. Thicker liquid films, in turn, result in lower heat transfer rates from the wall to liquid film, and consequently lower mass transfer rates from the liquid film to the vapor phase. Nevertheless, the total mass of evaporation (or equivalently, the net heat transfer) turns out to be higher in case of interfacial slip due to the longer film length. We also develop significant physical insights on the implications of the relative thickness of the depleted layer with reference to characteristic length scales of the microfluidic channel on the evaporation process, under combined influences of the capillary pressure, disjoining pressure, and the driving temperature differential for the interfacial transport.     

Entropies of Mixing (EOM)and the Lorenz Order

Polynomial nonadditive, or pseudo-additive (PAE), entropies are related to the Shannon entropy in that both are derived from two classes of parent distributions of extreme-value theory, the Pareto and power distributions.The third class is the exponential distribution, corresponding to the Shannon entropy, to which the other two tend as their shape parameters increase without limit. These entropies all belong to a single class of entropies referred to as EOM. EOM is defined as the normalized difference between the dual of the Lorentz function and the Lorenz function. Sufficient conditions for majorization involve finding a separable, Schur-concave function, like the EOM, which increases as the distribution becomes more uniform or less spread out. Lorenz ordering has been associated to the degree in which the Lorenz curve is bent. This criterion is valid for tail distributions, and fails in the case where the distribution is limited on the right. EOM provide criteria for inequality in the Lorenz ordering sense: In the Pareto case,an increase in the shape parameter implies a decrease in inequality and the EOM decreases, whereas for the power distribution an increase in the shape parameter corresponds to an increase in inequality leading to an increase in the EOM. An analogy is drawn between Gauss' invariant distribution for the probability of the fractional part of a continued fraction and the area criterion in Lorenz ordering, analogous to the Gini index criterion. The tendency to approach the invariant distribution, as the number of partial quotients increases without limit, is shown to be analogous to the tendency to approach the invariant area, as the shape parameters increase without limit.     

Optimal replica placement in hierarchical Data Grids with locality assurance

In this paper, we address three issues concerning data replica placement in hierarchical Data Grids that can be presented as tree structures. The first is how to ensure load balance among replicas. To achieve this, we propose a placement algorithm that finds the optimal locations for replicas so that their workload is balanced. The second issue is how to minimize the number of replicas. To solve this problem, we propose an algorithm that determines the minimum number of replicas required when the maximum workload capacity of each replica server is known. Finally, we address the issue of service quality by proposing a new model in which each request must be given a quality-of-service guarantee. We describe new algorithms that ensure both workload balance and quality of service simultaneously.     

Investigations into mixing of fluids in microchannels with lateral obstructions

This work presents a study of a passive micromixer with lateral obstructions along a microchannel. The mixing process is simulated by solving the continuity, momentum and diffusion equations. The mixing performance is quantified in terms of a parameter called ‘mixing efficiency’. A comparison of mixing efficiencies with and without obstructions clearly indicates the benefit of having obstructions along the microchannel. The numerical model was validated by comparing simulation results with experimental results for a micromixer. An extensive parametric study was carried out to investigate the influence of the geometrical and operational parameters in terms of the mixing efficiency and pressure drop, which are two important criteria for the design of micromixers. A very interesting observation reveals that there exists a critical Reynolds number (Re _cr  ~ 100) below which the mixing process is diffusion dominated and thus the mixing efficiency is reduced with increase in Re and above which the mixing process is advection dominated and mixing efficiency increases with increase in Re. Microchannels with symmetric and staggered protrusion arrangements were studied and compared. The mixing performance of the staggered arrangement was comparable with that of symmetric arrangement but the pressure drop was lower in the case of staggered arrangements making it more suitable.     

Arrays of high-aspect ratio microchannels for high-throughput isolation of circulating tumor cells (CTCs)

Microsystem-based technologies are providing new opportunities in the area of in vitro diagnostics due to their ability to provide process automation enabling point-of-care operation. As an example, microsystems used for the isolation and analysis of circulating tumor cells (CTCs) from complex, heterogeneous samples in an automated fashion with improved recoveries and selectivity are providing new opportunities for this important biomarker. Unfortunately, many of the existing microfluidic systems lack the throughput capabilities and/or are too expensive to manufacture to warrant their widespread use in clinical testing scenarios. Here, we describe a disposable, all-polymer, microfluidic system for the high-throughput (HT) isolation of CTCs directly from whole blood inputs. The device employs an array of high aspect ratio (HAR), parallel, sinusoidal microchannels (25 × 150 μm; W × D; AR = 6.0) with walls covalently decorated with anti-EpCAM antibodies to provide affinity-based isolation of CTCs. Channel width, which is similar to an average CTC diameter (10–20 μm), plays a critical role in maximizing the probability of cell/wall interactions and allows for achieving high CTC recovery. The extended channel depth allows for increased throughput at the optimized flow velocity (2 mm/s in a microchannel); maximizes cell recovery, and prevents clogging of the microfluidic channels during blood processing. Fluidic addressing of the microchannel array with a minimal device footprint is provided by large cross-sectional area feed and exit channels poised orthogonal to the network of the sinusoidal capillary channels (so-called Z-geometry). Computational modeling was used to confirm uniform addressing of the channels in the isolation bed. Devices with various numbers of parallel microchannels ranging from 50 to 320 have been successfully constructed. Cyclic olefin copolymer (COC) was chosen as the substrate material due to its superior properties during UV-activation of the HAR microchannels surfaces prior to antibody attachment. Operation of the HT-CTC device has been validated by isolation of CTCs directly from blood secured from patients with metastatic prostate cancer. High CTC sample purities (low number of contaminating white blood cells) allowed for direct lysis and molecular profiling of isolated CTCs.     

Inertial focusing of microparticles in curvilinear microchannels with different curvature angles

Inertial microfluidics has become one of the emerging topics due to potential applications such as particle separation, particle enrichment, rapid detection and diagnosis of circulating tumor cells. To realize its integration to such applications, underlying physics should be well understood. This study focuses on particle dynamics in curvilinear channels with different curvature angles (280°, 230°, and 180°) and different channel heights (90, 75, and 60 µm) where the advantages of hydrodynamic forces were exploited. We presented the cruciality of the three-dimensional particle position with respect to inertial lift forces and Dean drag force by examining the focusing behavior of 20 µm (large), 15 µm (medium) and 10 µm (small) fluorescent polystyrene microparticles for a wide range of flow rates (400–2700 µL/min) and corresponding channel Reynolds numbers. Migration of the particles in lateral direction and their equilibrium positions were investigated in detail. In addition, in the light of our findings, we described two different regions: transition region, where the inner wall becomes the outer wall and vice versa, and the outlet region. The maximum distance between the tight particle stream of 20 and 15 µm particles was obtained in the 90 high channel with curvature angle of 280° at Reynolds number of 144 in the transition region (intersection of the turns), which was the optimum condition/configuration for focusing.     

Dispersion and deposition of nanoparticles in microchannels with arrays of obstacles

Air pollutants are among the hazardous materials for human health. Therefore, many scientists are interested in removing particles from the carrier gas. In this study, flow of air and airborne particles through the virtual multi-fibrous filters that consist of different fiber cross-sectional shapes and arrangements is simulated where particle deposition and filtration performance are studied. Regular and irregular arrangements of fibers with the circular, elliptical, and equilateral triangular cross sections have been considered. Effects of important parameters such as solid volume fraction, internal structure, and filter thickness on particle collection efficiency and pressure drop are investigated, and the best internal structure of the multi-fiber filters for the collection of particles is obtained. Results indicate that an increase in the ratio of the vertical distances between fibers to fiber diameter and the solid volume fraction would result in a dramatic reduction of filtration. Collection efficiency of multi-fiber filters is validated with the available numerical studies.     

Analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with slip-dependent zeta potential

The present study is an analysis of pressure-driven electrokinetic flows in hydrophobic microchannels with emphasis on the slip effects under coupling of interfacial electric and fluid slippage phenomena. Commonly used linear model with slip-independent zeta potential and the nonlinear model at limiting (high-K) condition with slip-dependent zeta potential are solved analytically. Then, numerical solutions of the electrokinetic flow model with zeta potential varying with slip length are analyzed. Different from the general notion of “the more hydrophobic the channel wall, the higher the flowrate,” the results with slip-independent and slip-dependent zeta potentials both disclose that flowrate becomes insensitive to the wall hydrophobicity or fluid slippage at sufficiently large slip lengths. Boundary slip not only assists fluid motion but also enhances counter-ions transport in EDL and, thus, results in strong streaming potential as well as electrokinetic retardation. With slip-dependent zeta potential considered, flowrate varies non-monotonically with increasing slip length due to competition of the favorable and adverse effects with more complicated interactions. The influence of the slip on the electrokinetic flow is eventually nullified at large slip lengths for balance of the counter effects, and the flowrate becomes insensitive to further hydrophobicity of the microchannel. The occurrence of maximum, minimum, and insensitivity on the flowrate-slip curves can be premature at a higher zeta potential and/or larger electrokinetic separation distance.     

Comprehensive model of electrokinetic flow and migration in microchannels with conductivity gradients

A comprehensive model of electrokinetic flow and transport of electrolytes in microchannels with conductivity gradients is developed. The electrical potential is modeled by a combination of an electrostatic and an electrodynamic approach. The fluid dynamics are described by the Navier–Stokes equations, extended by an electrical force term. The chemistry of the system is represented by source terms in the mass transport equations, derived from an equilibrium approach. Moreover, the interaction between ionic species concentration and physicochemical properties of the microchannel substrate (i.e. zeta-potential) is taken into consideration by an empirical approach. Approximate analytical solutions for all variables are found which are valid within the electrical double layer. By using the method of matched asymptotic expansions, these solutions provide boundary conditions for the numerical simulation of the bulk liquid. The models are implemented in a Finite-Element-Code. As an example, simulations of an electrophoretic injection/separation process in a micro-electrophoresis device are performed. The results of the simulations show the strong coupling between the involved physicochemical phenomena. Simulations with a constant and a concentration-depend zeta-potential clarify the importance of a proper modeling of the physicochemical substrate characteristics.     

Measurement of pressure drop and flow resistance in microchannels with integrated micropillars

In the present study, we investigate single phase fluid flow through microchannels with integrated micropillars to calculate the pressure drop and flow resistance. The microchannels, which contain micropillars arranged in square and staggered arrangement, are fabricated in silicon substrate using standard photolithography and deep reactive ion etching (DRIE) techniques. The DRIE technique results in precise and accurate fabrication with smooth and vertical wall profiles. Pressure drop measurements are performed on microchannels with integrated micropillars under creeping flow regime over a range of water flow rates from 50 to 600 μl/min. It is observed that the pressure drop varies linearly with increasing flow rates. Flow resistance ( $\Updelta P/Q$ ) is calculated using the pressure drop values and is found to be decreasing as the Darcy number ( $\sqrt{K/h^2}$ ) increases. In general, the square arrangement of pillars offers higher resistance to flow than their staggered counterparts. It is observed that the existing theoretical models fail to accurately predict the permeability of the microchannel with integrated micro-pillars, particularly for cases where the micropillars have smooth and accurate geometric conformity, as obtained in the microfabricated structures used in the present study.     

Experimental investigations of liquid flow in rib-patterned microchannels with different surface wettability

The effects of rib-patterned surfaces and surface wettability on liquid flow in microchannels were experimentally investigated in this study. Microchannels were fabricated on single-crystal silicon wafers by photolithographic and wet-etching techniques. Rib structures were patterned in the silicon microchannel, and the surface was chemically treated by trichlorosilane to create hydrophobic condition. Experiments with water as the working fluid were performed with these microchannels over a wide range of Reynolds numbers between 110 and 1914. The results for the rib-patterned microchannels showed that the friction factor with the hydraulic diameter based on the rib-to-upper-wall height was lower than that predicted from incompressible theory with the same height. The friction factor-Reynolds number products for the hydrophobic condition increased as Reynolds number increased in the laminar flow regime. The experimental results were also compared with the predictive expressions from the literature, and it was found that the experimental data for the small rib/cavity geometry was in good agreement with those in the literature.