Experimental and numerical investigation of capillary flow in SU8 and PDMS microchannels with integrated pillars

Microfluidic channels with integrated pillars are fabricated on SU8 and PDMS substrates to understand the capillary flow. Microscope in conjunction with high-speed camera is used to capture the meniscus front movement through these channels for ethanol and isopropyl alcohol, respectively. In parallel, numerical simulations are conducted, using volume of fluid method, to predict the capillary flow through the microchannels with different pillar diameter to height ratio, ranging from 2.19 to 8.75 and pillar diameter to pitch ratio, ranging from 1.44 to 2.6. The pillar size (diameter, pitch and height) and the physical properties of the fluid (surface tension and viscosity) are found to have significant influence on the capillary phenomena in the microchannel. The meniscus displacement is non-uniform due to the presence of pillars and the non-uniformity in meniscus displacement is observed to increase with decrease in pitch to diameter ratio. The surface area to volume ratio is observed to play major roles in the velocity of the capillary meniscus of the devices. The filling speed is observed to change more dramatically under different pillar heights upto 120 μm and the change is slow with further increase in the pillar height. The details pertaining to the fluid distribution (meniscus front shapes) are obtained from the numerical results as well as from experiments. Numerical predictions for meniscus front shapes agree well with the experimental observations for both SU8 and PDMS microchannels. It is observed that the filling time obtained experimentally matches very well with the simulated filling time. The presence of pillars creates uniform meniscus front in the microchannel for both ethanol and isopropyl alcohol. Generalized plots in terms of dimensionless variables are also presented to predict the performance parameters for the design of these microfluidic devices. The flow is observed to have a very low Capillary number, which signifies the relative importance of surface tension to viscous effects in the present study.     

A study of capillary flow from a pendant droplet

Surface tension driven capillary flow from a pendant droplet into a horizontal glass capillary is investigated in this paper. Effect of the droplet surface on dynamic behavior of such capillary flow is examined and compared with surface tension driven capillary flow from an infinite reservoir. In the experiment, capillaries of 300–700 μm in diameter were used with glycerol–DI water mixture solutions having viscosities ranging from 80 to 934 mPa s. It is observed that compared to the capillary flow from an infinite reservoir, the capillary flow from a droplet exhibits higher rates of meniscus displacement. This is due to an additional driving force resulted from change in droplet surface area (or curvature). The two main parameters influencing the flow are the dimensionless droplet geometry parameter (k) and the dynamic contact angle (θ _D). The molecular kinetics theory of Blake and De Coninck’s model [Adv Colloid Interface Sci 96(1–3):21–36, 2002] is used to interpret the dynamic contact angle. This theory considers a molecular friction coefficient (ζ) at the liquid front flowing over a solid surface. Moreover, three models are proposed to describe the shape of the pendant droplet during capillary action. It is found that the egg-shaped model provides a more realistic model to compute the shape of the pendant droplet deformed during the capillary action. Thus the predictions by the egg-shaped model are in good agreement with the experimental data.     

Simulation of liquid micro-jet in free expanding high-speed co-flowing gas streams

We present the development of an experimentally validated computational fluid dynamics model for liquid micro jets. Such jets are produced by focusing hydrodynamic momentum from a co-flowing sheath of gas on a liquid stream in a nozzle. The numerical model based on laminar two-phase, Newtonian, compressible Navier–Stokes equations is solved with finite volume method, where the phase interface is treated by the volume of fluid approach. A mixture model of the two-phase system is solved in axisymmetry using?~?300,000 finite volumes, while ensuring mesh independence with the finite volumes of the size 0.25 µm in the vicinity of the jet and drops. The numerical model is evaluated by comparing jet diameters and jet lengths obtained experimentally and from scaling analysis. They are not affected by the strong temperature and viscosity changes in the focusing gas while expanding at nozzle outlet. A range of gas and liquid-operating parameters is investigated numerically to understand their influence on the jet performance. The study is performed for gas and liquid Reynolds numbers in the range 17–1222 and 110–215, and Weber numbers in the range 3–320, respectively. A reasonably good agreement between experimental and scaling results is found for the range of operating parameters never tackled before. This study provides a basis for further computational designs as well as adjustments of the operating conditions for specific liquids and gases.     

All-hexahedral mesh generation via inside-out advancing front based on harmonic fields

The generation of hexahedral meshes is an open problem that has undergone significant research. This paper deals with a novel inside-out advancing front method to generate unstructured all-hexahedral meshes for given volumes. Two orthogonal harmonic fields, principal and radial harmonic fields, are generated to guide the inside-out advancing front process based on a few user interactions. Starting from an initial hexahedral mesh inside the given volume, we advance the boundary quadrilateral mesh along the streamlines of radial field and construct layers of hexahedral elements. To ensure high quality and uniform size of the hexahedral mesh, quadrilateral elements are decomposed in such a way that no non-hexahedral element is produced. For complex volume with branch structures, we segment the complex volume into simple sub-volumes that are suitable for our method. Experimental results show that our method generates high quality all-hexahedral meshes for the given volumes.     

The flow structure inside a microfabricated inkjet printhead

A micrometer resolution particle image velocimetry system has been adapted to measure instantaneous velocity fields in an inkjet printhead. The technique uses 700-nm-diameter fluorescent flow tracing particles, a pulsed Nd:YAG laser, an epi-fluorescent microscope, and a cooled interline transfer charge-coupled device camera to record images of flow tracing particles at two known instances in time. Instantaneous velocity vector fields are obtained with spatial resolutions of 5-10 μm and temporal resolutions of 2-5 μs. The relationship between instantaneous velocity fields is compared to instantaneous shapes of the meniscus. The flow in the nozzle is highly unsteady and characterized by a maximum velocity of 8 ms^-1, Reynolds numbers of Re=500, and accelerations of up to 70 000 times gravity (i.e., 70 000 g). Since the flow field is periodic for each ejection cycle, the instantaneous measurements can be phased averaged to determine the evolution of the average flow field. The ejection cycle period is 500 μs, and consists of four primary phases: infusion, inversion, ejection, and relaxation. During infusion, the actuator plate is deflected downward creating a low pressure that draws fluid into the inkjet cavity through the orifice and pulls the meniscus into the cavity through the nozzle. The meniscus grows, begins to decrease in size, and then deforms in shape, becoming inverted for approximately 6 μs. The meniscus exits the cavity through the nozzle during droplet ejection. During relaxation, the flow undergoes viscously-damped oscillations, and reaches equilibrium before the next ejection cycle begins     

A single channel capillary microviscometer

We have developed a microviscometer analyzing the fluid dynamics in a single channel glass microfluidic chip with a closed end. The device is able to test sample volumes of a few microliters by inserting one drop in the inlet. The fluid enters the channel driven by capillary pressure and an optical sensor registers the motion. The equation that describes the fluid dynamics is function of the channel geometry, atmospheric pressure, fluid viscosity, and capillary pressure. Knowing the first two, the last parameters can be obtained as fitting parameters from the meniscus position as a function of time plot. We have successfully tested Newtonian fluids with different viscosities and capillary pressure.     

High throughput micro droplet generator array controlled by two-dimensional dynamic virtual walls

A chamber-free two-dimensional-array micro droplet generator has been realized by precise time-delayed control of micro bubble arrays as virtual chamber walls. Droplets can be ejected out by the bubbles around the ejection site in specific configuration of excitation, thus replacing physical chamber walls for pressure preservation. The micro droplet generator array was fabricated by heater lithography and direct nozzle formation on a laminated SU-8 dry film without any solid chamber wall among heaters. The nozzle density of this compact droplet generator can be five to ten times higher than that of commercial inkjet printheads in one-dimensional formats. The volume and initial speed of the generated droplets was 3.6–5.7 pL and 14–15 m/s, respectively, meeting the standard of commercial printheads. The micro droplet generator is free of satellite droplets due to the precise meniscus control. The analyzed data shows the meniscus undergoes a “push–pull–push” progress which effectively cuts the liquid column short. The refilling time of the innovative micro droplet generator was estimated to be 0.296 μs from the simplified chamber model, and it was one-tenth of the commercial printheads. In addition, the frequency response was estimated to be higher than 20 kHz by observing the meniscus fluctuation condition. Finally, a 3 × 5 heater array was used to generate two droplets simultaneously, which shows that the crosstalk problem can be eliminated by precise time-delayed control. An interlacing operation was also proposed to address the large array control algorithm. To summarize, a 330-dpi monolithic micro droplet generator prototype has been proposed for high speed and large 2D format printing.     

Full transient response of Taylor cones to a step change in electric field

We studied experimentally the complete transient response of Taylor cones subject to a step change in external electric field with the goal of finding optimal conditions to reduce the overall response time and achieve the highest possible switching bandwidth. The transient behavior of electrified menisci is of interest for many applications that would benefit from active control of on/off switching of the electrospray, such as femtoliter droplet-on-demand or novel fuel injectors in next generation internal combustion engines. We first investigated the transient behavior of ethanol, a typical solvent for droplet-on-demand. We then expanded the study to fuels such as JP-8 and E-30 biogas, a biofuel with 30% ethanol (vol.). The system response is a multi-stage process that can last from ~100?μs to ~100?ms. Potential bottleneck stages include liquid accumulation, meniscus oscillation, and cone relaxation, depending on the experimental conditions. A typical full response time is ~1?ms, and the shortest transient process observed is ~400?μs. For a given liquid, nozzle outer diameter (OD) and applied voltage are the two most important parameters to influence the full response time. Onset or near-onset voltage for the establishment of the cone jet often leads to a large number of oscillation cycles and should be avoided. Changes in conductivity and viscosity by less than a factor of 10 have negligible effects on the transient process. Using JP-8 or E-30 biogas, 90?μm OD nozzle with extractor, and flow rate of 0.4?mL/h, we can routinely achieve bandwidth of 1?kHz, corresponding to a full response time of 1?ms, after which quasi-monodispersed droplets of ~10?μm are generated. Adaptation of an inviscid model of a charged oscillating droplet to the oscillating meniscus satisfactorily explains several key phenomena observed in our experiments, such as the full response time and the overshoot of the meniscus height.     

Capillary filling with the effect of pneumatic pressure of trapped air

This article presents an investigation into the effects of pneumatic pressure of trapped air on the dynamics of capillary filling. Controlled experiments were carried out in horizontal closed-end capillaries with diameters of 200–700 μm. Glycerol–DI water mixture solutions having viscosities ranging from 8 to 80 mPa s were used as the filling liquids. The pneumatic air backpressure is built up as a result of the air compressed at the closed end of the capillary. A model is presented based on the conventional theory of capillary filling (i.e., Washburn’s equation) with consideration of the effect of air backpressure force on the advancing meniscus. The molecular kinetics theory of Blake and De Coninck’s model (Adv Colloid Interface Sci 96:21–36, 2002) is also incorporated in the model to account for the dependence of dynamic contact angle on wetting velocity. The model predictions agree reasonably well with the experimental data. It is observed that due to the presence of air backpressure, the smaller the capillary diameter, the longer the length that the liquid fills the capillary, regardless of the liquid viscosity. It is also shown that the increased pneumatic air backpressure reduces the equilibrium contact angle (θ ⁰). A relation is then proposed among liquid penetration, capillary length and radius, and contact angle. In addition, a dimensionless analysis is performed on experimental data, and the power law dependence of dimensionless meniscus position on dimensionless time is obtained.     

Water diffusion in polymer nano-films measured with microcantilevers

We present a method to measure the absorption of water molecules from the liquid and the vapour phase into polymer nano-films and the diffusion inside these films. Film thickness can be down to 45 nm. To demonstrate the possibilities of this method we use polymer films that are deposited on the upper side of a silicon cantilever by plasma polymerization of norbornene. When a microdrop of water is deposited onto the initially straight cantilever, the drop causes the cantilever to bend while it evaporates. Evaporation of such small water drops usually takes less than a second. An upwards bending is due to capillary forces and a downwards bending is due to the diffusion of water into the polymer film – and the consequent volume expansion (swelling) of the film. The magnitude of the capillary forces and the extent of swelling continuously change during drop evaporation. When drop evaporation is over the cantilever returns to its initial straight position. We simulate the time dependent bending with a numerical model that qualitatively agrees with the experiment. From the time dependence of cantilever bending we are able to determine the diffusion coefficient of water in the thin polymer film.     

Dynamic meniscus models for MEMS elements

The dynamics of a liquid meniscus bridge between solid surfaces were analyzed based on the continuum lubrication theory assuming a small vibration of the spacing. The geometry of the meniscus considered in this study was the finite meniscus ring. The following two meniscus models were considered: (1) the fixed boundary position of the meniscus and variable contact angle (VCA) model and (2) the fixed contact angle and variable boundary position (VBP) model. The time-dependent Reynolds equation was solved under the boundary condition considering the Laplace pressure, assuming that the mass of the liquid in the meniscus is conserved. It was found by linearization that the pressures and the load-carrying capacities of both models have three terms, i.e., a time-dependent squeeze term due to the viscosity of the liquid, a spring term due to the dynamic Laplace pressure and a static meniscus force term. The comparisons between these models and experimental results were also presented.     

Liquid recirculation in microfluidic channels by the interplay of capillary and centrifugal forces

We demonstrate a technique to recirculate liquids in a microfluidic channel by alternating predominance of centrifugal and capillary forces to rapidly bring the entire volume of a liquid sample to within one diffusion length, δ, of the surface, even for sample volumes hundreds of times the product of δ and the geometric device area. This is accomplished by repetitive, random sampling of an on-disc sample reservoir to form a thin fluid layer of thickness δ in a microchannel, maintaining contact for the diffusion time, then rapidly exchanging the fluid layer for a fresh aliquot by disc rotation and stoppage. With this technique, liquid volumes of microlitres to millilitres can be handled in many sizes of microfluidic channels, provided the channel wall with greatest surface area is hydrophilic. We present a theoretical model describing the balance of centrifugal and capillary forces in the device and validate the model experimentally.     

Nozzle size effects on the nanoelectrospraying of Au nanocolloid in a fully voltage-controlled form

Nozzle size effects on the nanoelectrospraying of Au nanocolloid were investigated in a pure voltage controlled form. The nanoelectrospraying system was set up in a nozzle-to-plate geometry and the spray current was monitored to evaluate the performance of the nozzles during the atomization of Au colloid. Current–voltage characteristic shows that nanoelectrospray properties strongly depend on the nozzle size. In the spraying of Au nanocolloid using nozzles sized from 4 to 30 μm in diameter, two well-distinguished regimes of pulsation and cone-jet were observed with an increasing of voltage. The onset voltages for entering both modes were found to increase with the nozzle size. In the pulsation mode, pulsating frequency ranged from several tens kHz up to hundred kHz was witnessed and current with low oscillation amplitude was detected in the case of small nozzles, indicating the rate of ejected charges decreases with a fine nozzle. Meanwhile, the DC equivalent resistance derived from the best-fit model was determined to be in the range of 0.89–3.16 GΩ. In cone-jet region the equivalent resistance, representing an electrical equivalence of the spray gap, was measured to be between 0.49 and 1.41 GΩ. The relatively low resistance at high field implies better efficiency of spray in cone-jet mode compared with in pulsating mode. In both modes the equivalent resistances fall with increasing nozzle size, reflecting large nozzles easier process high volume ejections.     

A novel technique for producing metallic microjets and microdrops

A new technique for producing steady metallic jets is proposed. It allows the production of supercritical jets with Weber numbers well below unity, which entails important technological advantages over existing techniques. The metallic liquid is injected through a micrometer converging nozzle located inside a gas stream. Both the liquid jet and the coflowing gas current cross an orifice located in front of the nozzle. The gas stream stabilizes the jet by sweeping away the capillary waves growing on the free surface. In this way, one can steadily produce microjets with a kinetic energy much lower than the interfacial energy, a possibility that has been predicted theoretically (Gañán-Calvo in Phys Rev E 78:026304, 2008). Experiments were conducted with mercury to assess the performance of the new technique. The experimental results agreed remarkably well with the predictions calculated from the convective/absolute instability transition of the jet. The jet breakup mechanism did not correspond to classical Rayleigh instability, but to the growth of surface waves over a capillary column which ends at a fixed location. The results were compared with those obtained with the well-established flow focusing method to show that the new technique considerably favors the jet’s stability.     

Behavior of microdroplets in diffuser/nozzle structures

This paper investigates the behavior of microdroplets flowing in microchannels with a series of diffuser/nozzle structures. Depending on the imposed flow direction, the serial structures can act either as a series of diffusers or nozzles. Different serial diffuser/nozzle microchannels with opening angles ranging from 15° to 45° were considered. A 2D numerical model was employed to study the dynamics of the microdroplet during its passage through the diffuser/nozzle structures. The deformation of the microdroplet was captured using a level set method. On the experimental front, test devices were fabricated in polydimethylsiloxane using soft lithography. T-junctions for droplet formation, diffuser/nozzle structures and pressure ports were integrated in a single device. Mineral oil with 2% w/w surfactant span 80 and de-ionized water with fluorescent worked as the carrier phase and the dispersed phase, respectively. The deformation of the water droplet and the corresponding pressure drop across the diffuser/nozzle structures were measured in both diffuser and nozzle configurations at a fixed flow rate ratio between oil and water of 30. The results show a linear relationship between the pressure drop and the flow rate. Furthermore, the rectification effect was observed in all tested devices. The pressure drop in the diffuser configuration is higher than that of the nozzle configuration. Finally, the pressure measured results with droplet and without droplet were analyzed and compared.     

Analysis and experiment of capillary valves for microfluidics on a rotating disk

This paper presents an analytical expression of the pressure barrier in a capillary-burst valve for flow regulation in centrifugal microfluidics. The analysis considers variations of the interfacial energies at the meniscus of three-dimensional (3D) configuration in a rectangular microchannel with a sudden expansion in cross-section. We derive a simple expression that predicts the critical burst pressure or rotational speed to overcome the capillary valve. Experiments were carried out for capillary valves that were integrated with microchannels on a rotating disk having various cross-sectional dimensions (300 and 400 μm in width and 80–600 μm in depth) and wedge angles (30°–100°) of sudden expansion. The flow visualization of the meniscus development across the capillary valve supports the assumptions made for the present analysis. The measurements of burst rotational speeds for the capillary valves are in good agreement with the predictions by the simple expression except that those with a larger channel width and wider wedge angles are nearly 10% lower than the predicted values.     

Aggregate growth and breakup in particulate suspension flow through a micro-nozzle

A computational study is reported on the growth of aggregates in flow of a particulate suspension through a micro-nozzle. The study employs a soft-sphere discrete element method (DEM) with van der Waals adhesion force between the particles in two-dimensional, incompressible channel flow. A new computational approach for particle transport in complex domains is developed which uses a background Cartesian grid for efficient flow field interpolation at the particle locations, together with a level-set method to represent the nozzle boundaries in the particle computation. Three mechanisms for the growth or breakup of particulate aggregates in the micro-nozzle are examined: (1) enhanced particle collision due to lateral compression as fluid elements pass through the nozzle, (2) stretching of aggregates due to axial stretching of fluid elements, and (3) collision and intermittent adhesion of particles to the nozzle wall. The first of these mechanisms leads to aggregate growth, and the second to aggregate breakup. The wall collision and adhesion mechanism can enhance either aggregate growth or breakup, but it is found in most cases to be a primary agent in the breakup of incident aggregates as part of the aggregate attaches to the nozzle wall and is torn from the remainder of the aggregate due to the high shear near the walls. Simplified models for these processes are developed and used to interpret the trends observed in the DEM simulations. The effects of particle adhesion parameter, particle size and density, particle concentration, and nozzle geometry are examined. It is found that passage of a particulate suspension through a nozzle can lead to either a substantial decrease in aggregate size or a modest increase under different conditions, depending in part on the size of the incident aggregates.     

Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from \(10^{-4}\) to 1 and inner-to-outer viscosity ratios, \(\lambda\), from 0, i.e. a bubble, to high-viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers (\(Ca<10^{-3}\)) agrees well with Bretherton’s scaling for bubbles as long as \(\lambda <1\). For larger viscosity ratios, the film thickness increases monotonically, before saturating for \(\lambda>10^3\) to a value \(2^{2/3}\) times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous and Quéré (Phys Fluids 12(10):2367–2371, 2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for \(Ca>10^{-3}\).     

Asymmetric surface roughness measurements and meniscus modeling of polysilicon surface micromachined flaps

Polycrystalline silicon (polysilicon) films are primary structural materials for microelectromechanical systems (MEMS). Due to relatively high compliance, large surface-to-volume ratio, and small separation distances, micromachined polysilicon structures are susceptible to surface forces which can result in adhesive failures. Since these forces depend on surface properties especially surface roughness, three types of microhinged flaps were fabricated to characterize their roughness and adhesive meniscus properties. The flaps enabled access to both the top and bottom surfaces of the structural polysilicon layers. Roughness measurements using an atomic force microscope revealed that MEMS surfaces primarily exhibit non-Gaussian surface height distributions, and for the release procedures studied, the bottom surface of the structural layers was significantly smoother and prone to higher adhesion compared to the top surface. A non-symmetric surface roughness model using the Pearson system of frequency curves was coupled with a capillary meniscus adhesion model to analyze the effects of surface roughness parameters (root-mean-square, skewness, and kurtosis), relative humidity, and surface contact angle on the interfacial adhesion energy. Using the measured roughness properties of the flaps, four different surface pairs were simulated and compared to investigate their effects on capillary adhesion. It was found that since the base polysilicon layer (poly0) was rougher than the base silicon nitride and the structural layer on poly0 was also rougher than that on silicon nitride, depositing MEMS devices on poly0 layer rather than directly on silicon nitride will reduce the adhesion energy.