Designing for chaos: applications of chaotic advection at the microscale

Design and High-Resolution Analysis of an Efficient Periodic Split-and-Recombination Microfluidic Mixer

We developed a highly efficient passive mixing device based on a split-and-recombine (SAR) configuration. This micromixer was constructed by simply bonding two identical microfluidic periodical open-trench patterns face to face. The structure parameters of periodical units were optimized through numerical simulation to facilitate the mixing efficiency. Despite the simplicity in design and fabrication, it provided rapid mixing performance in both experiment and simulation conditions. To better illustrate the mixing mechanism, we developed a novel scheme to achieve high-resolution confocal imaging of serial channel cross-sections to accurately characterize the mixing details and performance after each SAR cycle. Using fluorescent IgG as an indicator, nearly complete mixing was achieved using only four SAR cycles in an aqueous solution within a device's length of less than 10 mm for fluids with a Péclet number up to 8.7 × 104. Trajectory analysis revealed that each SAR cycle transforms the input fluids using three synergetic effects: rotation, combination, and stretching to increase the interfaces exponentially. Furthermore, we identified that the pressure gradients in the parallel plane of the curved channel induced vertical convection, which is believed to be the driving force underlying these effects to accelerate the mixing process.

Chaotic advection and mixing by a pair of microrotors in a circular domain

In this work we study chaotic mixing induced by point microrotors in a bounded two-dimensional Stokes flow. The dynamics of the pair of rotors, modeled as rotlets, are non-Hamiltonian in the bounded domain and produce chaotic advection of fluid tracers in subsets of the domain. A complete parametric investigation of the fluid mixing as a function of the initial locations of the rotlets is performed based on pseudophase portraits. The mixing of fluid tracers as a function of relative positions of microrotors is studied using finite-time entropy and locational entropy. The finite-time locational entropy is used to identify regions of the fluid that produce good versus poor mixing, and this is visualized by the stretching and folding of blobs of tracer particles. Unlike the case of the classic blinking vortex dynamics, the velocity field of the flow modeled using rotlets inside a circular boundary is smooth in time and satisfies the no-slip boundary condition. This makes the considered model a more realistic case for studies of mixing in microfluidic devices using magnetic-actuated microspheres.

Computational Modeling and Analysis of Fluid Structure Interaction in Micromixers with Deformable Baffle

A passive micromixer with obstacles in the form of deformable baffles is examined numerically. The model deploys an Arbitrary Lagrangian-Eulerian framework with Fluid-structure interaction coupled with a diffusion–advection model. Numerical analysis is carried out in the Reynolds number [Re] range of 0.01≤Re≤300. The objective of the present study is to enhance mixing between two component flow streams in a microchannel encompassing a deformable baffle. In the present work, the baffle deforms only due to the dynamic force of fluids. No external forces are applied. To exemplify the effectiveness of the present design, water and a suspension of curcumin drug loaded nanoparticles are taken as two fluids. Mixing index based on the variance of the local concentration of the suspension is employed to appraise the mixing performance of the micromixer. The introduction of the deformable baffle in a micromixer proliferates the mixing performance with minimal pressure drop over the tested Reynolds number range.

Using heteroclinic orbits to quantify topological entropy in fluid flows

Topological approaches to mixing are important tools to understand chaotic fluid flows, ranging from oceanic transport to the design of micro-mixers. Typically, topological entropy, the exponential growth rate of material lines, is used to quantify topological mixing. Computing topological entropy from the direct stretching rate is computationally expensive and sheds little light on the source of the mixing. Earlier approaches emphasized that topological entropy could be viewed as generated by the braiding of virtual, or "ghost," rods stirring the fluid in a periodic manner. Here, we demonstrate that topological entropy can also be viewed as generated by the braiding of ghost rods following heteroclinic orbits instead. We use the machinery of homotopic lobe dynamics, which extracts symbolic dynamics from finite-length pieces of stable and unstable manifolds attached to fixed points of the fluid flow. As an example, we focus on the topological entropy of a bounded, chaotic, two-dimensional, double-vortex cavity flow. Over a certain parameter range, the topological entropy is primarily due to the braiding of a period-three orbit. However, this orbit does not explain the topological entropy for parameter values where it does not exist, nor does it explain the excess of topological entropy for the entire range of its existence. We show that braiding by heteroclinic orbits provides an accurate computation of topological entropy when the period-three orbit does not exist, and that it provides an explanation for some of the excess topological entropy when the period-three orbit does exist. Furthermore, the computation of symbolic dynamics using heteroclinic orbits has been automated and can be used to compute topological entropy for a general 2D fluid flow.

Mixing in microfluidic devices and enhancement methods

Mixing in microfluidic devices presents a challenge due to laminar flows in microchannels, which result from low Reynolds numbers determined by the channel's hydraulic diameter, flow velocity, and solution's kinetic viscosity. To address this challenge, novel methods of mixing enhancement within microfluidic devices have been explored for a variety of applications. Passive mixing methods have been created, including those using ridges or slanted wells within the microchannels, as well as their variations with improved performance by varying geometry and patterns, by changing the properties of channel surfaces, and by optimization via simulations. In addition, active mixing methods including microstirrers, acoustic mixers, and flow pulsation have been investigated and integrated into microfluidic devices to enhance mixing in a more controllable manner. In general, passive mixers are easy to integrate, but difficult to control externally by users after fabrication. Active mixers usually take efforts to integrate within a device and they require external components (e.g. power sources) to operate. However, they can be controlled by users to a certain degree for tuned mixing. In this article, we provide a general overview of a number of passive and active mixers, discuss their advantages and disadvantages, and make suggestions on choosing a mixing method for a specific need as well as advocate possible integration of key elements of passive and active mixers to harness the advantages of both types.

Flows and mixing in channels with misaligned superhydrophobic walls

Aligned superhydrophobic surfaces with the same texture orientation reduce drag in the channel and generate secondary flows transverse to the direction of the applied pressure gradient. Here we show that a transverse shear can be easily generated by using superhydrophobic channels with misaligned textured surfaces. We propose a general theoretical approach to quantify this transverse flow by introducing the concept of an effective shear tensor. To illustrate its use, we present approximate theoretical solutions and Dissipative Particle Dynamics simulations for striped superhydrophobic channels. Our results demonstrate that the transverse shear leads to complex flow patterns, which provide a new mechanism of a passive vertical mixing at the scale of a texture period. Depending on the value of Reynolds number two different scenarios occur. At relatively low Reynolds number the flow represents a transverse shear superimposed with two corotating vortices. For larger Reynolds number these vortices become isolated, by suppressing fluid transport in the transverse direction.

Topological chaos, braiding and bifurcation of almost-cyclic sets

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.

Microfluidic mixing using contactless dielectrophoresis

The first experimental evidence of mixing enhancement in a microfluidic system using contactless dielectrophoresis (cDEP) is presented in this work. Pressure-driven flow of deionized water containing 0.5 μm beads was mixed in various chamber geometries by imposing a dielectrophoresis (DEP) force on the beads. In cDEP the electrodes are not in direct contact with the fluid sample but are instead capacitively coupled to the mixing chamber through thin dielectric barriers, which eliminates many of the problems encountered with standard DEP. Four system designs with rectangular and circular mixing chambers were fabricated in PDMS. Mixing tests were conducted for flow rates from 0.005 to 1 mL/h subject to an alternating current signal range of 0-300 V at 100-600 kHz. When the time scales of the bulk fluid motion and the DEP motion were commensurate, rapid mixing was observed. The rectangular mixing chambers were found to be more efficient than the circular chambers. This approach shows potential for mixing low diffusivity biological samples, which is a very challenging problem in laminar flows at small scales.

Topological chaos and periodic braiding of almost-cyclic sets

In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that "stir" the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.

Applications of micromixing technology

This review presents an application of micromixer technologies, which have driven a number of critical research trends over the past few decades, particularly for chemical and biological fields. Micromixer technologies in this review are categorized according to their applications: (1) chemical applications, including chemical synthesis, polymerization, and extraction; (2) biological applications, including DNA analysis, biological screening enzyme assays, protein folding; and (3) detection/analysis of chemical or biochemical content combined with NMR, FTIR, or Raman spectroscopies. In the chemical application, crystallization, extraction, polymerization, and organic synthesis have been reported, not only for laboratory studies, but also for industrial applications. Microscale techniques are used in chemical synthesis to develop microreactors. In clinical medicine and biological studies, microfluidic systems have been widely applied to the identification of biochemical products, diagnosis, drug discovery, and investigation of disease symptoms. The biological and biochemical applications also include enzyme assays, biological screening assays, cell lysis, protein folding, and biological analytical assays. Nondestructive analytical/detection methods have yielded a number of benefits to chemical and biochemical processes. In this chapter, we introduce analytical methods those are frequently integrated into micromixing technologies, such as NMR, FT-IR, and Raman spectroscopies. From the study of micromixers, we discovered that the Re number and mixing time depends on the specific application, and we clustered micromixers in various applications according to the Re number and mixing performance (mixing time). We expect that this clustering will be helpful in designing of micromixers for specific applications.

Improving alternate flow mixing by obstacles located along a micro-channel

An essential requirement for any practical fully integrated lab-on-a-chip device is the ability to mix two or more fluids thoroughly and efficiently, i.e., in a reasonable amount of time. This paper presents a way to improve mixing in microfluidic systems combining alternate flows with obstacles using passive mixers. Numerical simulations show that the layers of high and low solute concentration, created by the alternate flow, are split into smaller chunks of fluid, due to the obstacles inserted in the mixing channel, increasing the contact area between high and low concentration regions and decreasing the critical mixing length. This improvement can lead to shorter mixing channels and to low-cost mixers fabricated by planar lithographic technology.

Micro-Particle Image Velocimetry (microPIV): recent developments, applications, and guidelines

In this review we discuss the state of the art of the optical whole-field velocity measurement technique micro-scale Particle Image Velocimetry (microPIV). microPIV is a useful tool for fundamental research of microfluidics as well as for the detailed characterization and optimization of microfluidic applications in life science, lab-on-a-chip, biomedical research, micro chemical engineering, analytical chemistry and other related fields of research. An in depth description of the microPIV method is presented and compared to other flow visualization and measurement methods. An overview of the most relevant applications is given on the topics of near-wall flow, electrokinetic flow, biological flow, mixing, two-phase flow, turbulence transition and complex fluid dynamic problems. Current trends and applications are critically reviewed. Guidelines for the implementation and application are also discussed.

Reciprocating flow-based centrifugal microfluidics mixer

Proper mixing of reagents is of paramount importance for an efficient chemical reaction. While on a large scale there are many good solutions for quantitative mixing of reagents, as of today, efficient and inexpensive fluid mixing in the nanoliter and microliter volume range is still a challenge. Complete, i.e., quantitative mixing is of special importance in any small-scale analytical application because the scarcity of analytes and the low volume of the reagents demand efficient utilization of all available reaction components. In this paper we demonstrate the design and fabrication of a novel centrifugal force-based unit for fast mixing of fluids in the nanoliter to microliter volume range. The device consists of a number of chambers (including two loading chambers, one pressure chamber, and one mixing chamber) that are connected through a network of microchannels, and is made by bonding a slab of polydimethylsiloxane (PDMS) to a glass slide. The PDMS slab was cast using a SU-8 master mold fabricated by a two-level photolithography process. This microfluidic mixer exploits centrifugal force and pneumatic pressure to reciprocate the flow of fluid samples in order to minimize the amount of sample and the time of mixing. The process of mixing was monitored by utilizing the planar laser induced fluorescence (PLIF) technique. A time series of high resolution images of the mixing chamber were analyzed for the spatial distribution of light intensities as the two fluids (suspension of red fluorescent particles and water) mixed. Histograms of the fluorescent emissions within the mixing chamber during different stages of the mixing process were created to quantify the level of mixing of the mixing fluids. The results suggest that quantitative mixing was achieved in less than 3 min. This device can be employed as a stand alone mixing unit or may be integrated into a disk-based microfluidic system where, in addition to mixing, several other sample preparation steps may be included.

Biofabrication: a 21st century manufacturing paradigm

Biofabrication can be defined as the production of complex living and non-living biological products from raw materials such as living cells, molecules, extracellular matrices, and biomaterials. Cell and developmental biology, biomaterials science, and mechanical engineering are the main disciplines contributing to the emergence of biofabrication technology. The industrial potential of biofabrication technology is far beyond the traditional medically oriented tissue engineering and organ printing and, in the short term, it is essential for developing potentially highly predictive human cell- and tissue-based technologies for drug discovery, drug toxicity, environmental toxicology assays, and complex in vitro models of human development and diseases. In the long term, biofabrication can also contribute to the development of novel biotechnologies for sustainable energy production in the future biofuel industry and dramatically transform traditional animal-based agriculture by inventing 'animal-free' food, leather, and fur products. Thus, the broad spectrum of potential applications and rapidly growing arsenal of biofabrication methods strongly suggests that biofabrication can become a dominant technological platform and new paradigm for 21st century manufacturing. The main objectives of this review are defining biofabrication, outlining the most essential disciplines critical for emergence of this field, analysis of the evolving arsenal of biofabrication technologies and their potential practical applications, as well as a discussion of the common challenges being faced by biofabrication technologies, and the necessary conditions for the development of a global biofabrication research community and commercially successful biofabrication industry.

Design and evaluation of a passive alcove-based microfluidic mixer

A novel passive microfluidic silicon mixer has been designed, optimized and fabricated. The architecture of the mixer consists of a simple "T" junction, made up by a 20 microm wide by 82 microm deep channel, followed by three repeats of an alcove, each with a triangular obstruction, arranged in a zigzag fashion. Numerical simulations were employed to optimize the geometry, particularly the dimensions of the alcoves, the relative orientation and the spacing between them, and the degree of intrusion associated with them. The simulation results demonstrate that chaotic flow due to recirculation within the alcoves results in transverse velocity that promotes effective fluid mixing. The microfluidic mixer with the simulation-optimized geometry was fabricated with photolithographic techniques and characterized by optical imaging, fluorescence, and Raman microscope spectroscopy. At a sample flow rate of 20 microL/s, the mixer exhibits a short mixing deadtime of approximately 22 micros and a high mixing efficiency under both low and high viscosity conditions. The alcove-based microfluidic silicon mixer offers unique advantages for its short deadtime and slow sample consumption rate. In addition, it provides a valuable component for laboratory-on-a-chip applications for its ease of development into multiple networks for massively parallel analytical processes.

Something old, something new

Some thoughts on the development of theoretical fluid dynamics in this century and some reflections on the developments of the past century were provided as an introduction to a broad-ranging conference. A written synopsis of these remarks is provided here.

A practical guide to the staggered herringbone mixer

An analytical model of mixing in the staggered herringbone mixer (SHM) was derived to estimate mixing parameters and provide practical expressions to guide mixer design and operation for a wide range of possible solutes and flow conditions. Mixing in microfluidic systems has historically been characterized by the mixing of a specific solute system or by the redistribution of flow streams; this approach does not give any insight into the ideal operational parameters of the mixer with an arbitrary real system. For Stokes-flow mixers, mixing can be computed from a relationship between solute diffusivity, flow rate, and mixer length. Confocal microscopy and computational fluid dynamics (CFD) modeling were used to directly determine the extent of mixing for several solutes in the staggered herringbone mixer over a range of Reynolds numbers (Re) and Péclet numbers (Pe); the results were used to develop and evaluate an analytical model of its behavior. Mixing was found to be a function of only Pe and downstream position in the mixer. Required mixer length was proportional to log(Pe); this analytical model matched well with the confocal data and CFD model for Pe<5 x 10(4), at which point the experiments reached the limit of resolution. For particular solutes, required length and mixing time depend upon Re and diffusivity. This analytical model is applicable to other solute systems, and possibly to other embodiments of the mixer, to enable optimal design, operation, and estimation of performance.

Tailored mixing inside a translating droplet

Tailored mixing inside individual droplets could be useful to ensure that reactions within microscopic discrete fluid volumes, which are used as microreactors in "digital microfluidic" applications, take place in a controlled fashion. In this paper we consider a translating spherical liquid drop to which we impose a time periodic rigid-body rotation. Such a rotation not only induces mixing via chaotic advection, which operates through the stretching and folding of material lines, but also offers the possibility of tuning the mixing by controlling the location and size of the mixing region. Tuned mixing is achieved by judiciously adjusting the amplitude and frequency of the rotation, which are determined by using a resonance condition and following the evolution of adiabatic invariants. As the size of the mixing region is increased, complete mixing within the drop is obtained.

Magnetically driven colloidal microstirrer

Paramagnetic colloidal particles dispersed in water and deposited above magnetic bubble domains of a uniaxial ferrimagnetic garnet film are used as microscopic stirrer when subjected to external rotating magnetic fields. The hydrodynamic flow field above the stirrer is detected by tracking of nonmagnetic microspheres. The vorticity of the flow falls off inversely proportionally to the distance from the bubble center and is proportional to the field frequency. The device provides complete control over the mixing capability. This alternative method of active mixing might be used for microfluidics applications where mechanical stirring cannot be achieved easily with other machinery parts.

An ultrashort mixing length micromixer: the shear superposition micromixer

We report for the first time a laminar high-performance continuous micromixing process of two fluids over a length of 200 microns in under 10 milliseconds achieved by an optimization of the control parameters amplitude and frequency in the mixing device denoted as 'Shear Superposition Micromixer'. We improve mixing time by approximately 5 orders of magnitude over diffusion-limited mixing. The data indicate that rapid mixing is a result of the combined action of Taylor-Aris dispersion in the main and secondary microchannels and unsteady vortex motion that occurs at finite Reynolds number, which occurs above a threshold amplitude and frequency. The mixing performance is quantified using micron-resolution particle image velocimetry (micro-PIV) and computational fluid dynamics (CFD) simulations.

Mixing with bubbles: a practical technology for use with portable microfluidic devices

This paper demonstrates a methodology for micromixing that is sufficiently simple that it can be used in portable microfluidic devices. It illustrates the use of the micromixer by incorporating it into an elementary, portable microfluidic system that includes sample introduction, sample filtration, and valving. This system has the following characteristics: (i) it is powered with a single hand-operated source of vacuum, (ii) it allows samples to be loaded easily by depositing them into prefabricated wells, (iii) the samples are filtered in situ to prevent clogging of the microchannels, (iv) the structure of the channels ensure mixing of the laminar streams by interaction with bubbles of gas introduced into the channels, (v) the device is prepared in a single-step soft-lithographic process, and (vi) the device can be prepared to be resistant to the adsorption of proteins, and can be used with or without surface-active agents.

Chaotic mixer improves microarray hybridization

Hybridization is an important aspect of microarray experimental design which influences array signal levels and the repeatability of data within an array and across different arrays. Current methods typically require 24h and use target inefficiently. In these studies, we compare hybridization signals obtained in conventional static hybridization, which depends on diffusional target delivery, with signals obtained in a dynamic hybridization chamber, which employs a fluid mixer based on chaotic advection theory to deliver targets across a conventional glass slide array. Microarrays were printed with a pattern of 102 identical probe spots containing a 65-mer oligonucleotide capture probe. Hybridization of a 725-bp fluorescently labeled target was used to measure average target hybridization levels, local signal-to-noise ratios, and array hybridization uniformity. Dynamic hybridization for 1h with 1 or 10ng of target DNA increased hybridization signal intensities approximately threefold over a 24-h static hybridization. Similarly, a 10- or 60-min dynamic hybridization of 10ng of target DNA increased hybridization signal intensities fourfold over a 24h static hybridization. In time course studies, static hybridization reached a maximum within 8 to 12h using either 1 or 10ng of target. In time course studies using the dynamic hybridization chamber, hybridization using 1ng of target increased to a maximum at 4h and that using 10ng of target did not vary over the time points tested. In comparison to static hybridization, dynamic hybridization reduced the signal-to-noise ratios threefold and reduced spot-to-spot variation twofold. Therefore, we conclude that dynamic hybridization based on a chaotic mixer design improves both the speed of hybridization and the maximum level of hybridization while increasing signal-to-noise ratios and reducing spot-to-spot variation.

Introduction: mixing in microfluidics

In this paper we briefly review the main issues associated with mixing at the microscale and introduce the papers comprising the Theme Issue.