aDepartment of Bioengineering, University of California, 420 Westwood Plaza,
5121 Engineering V, P.O. Box 951600, Los Angeles, CA, 90095, USA.
E-mail: [email protected]; Fax: +1 310 794 5956; Tel: +1 310 983 3235bCalifornia NanoSystems Institute, 570 Westwood Plaza, Building 114,
Los Angeles, CA, 90095, USAc Graduate School of Nanoscience and Technology, Korea Advanced Institute of
Science and Technology, N5-2322 291 Daehak-Ro, YuseongGu, Daejeon, South Korea
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4lc00128a‡ Current address: Illumina Inc., 25861 Industrial Blvd, Hayward, CA 94545, USA.
Cite this: Lab Chip, 2014, 14, 2739
Received 29th January 2014,Accepted 27th May 2014
DOI: 10.1039/c4lc00128a
www.rsc.org/loc
Inertial microfluidic physics†
Hamed Amini,‡ab Wonhee Leec and Dino Di Carlo*ab
Microfluidics has experienced massive growth in the past two decades, and especially with advances in
rapid prototyping researchers have explored a multitude of channel structures, fluid and particle
mixtures, and integration with electrical and optical systems towards solving problems in healthcare,
biological and chemical analysis, materials synthesis, and other emerging areas that can benefit from the
scale, automation, or the unique physics of these systems. Inertial microfluidics, which relies on the
unconventional use of fluid inertia in microfluidic platforms, is one of the emerging fields that make use
of unique physical phenomena that are accessible in microscale patterned channels. Channel shapes that
focus, concentrate, order, separate, transfer, and mix particles and fluids have been demonstrated,
however physical underpinnings guiding these channel designs have been limited and much of the
development has been based on experimentally-derived intuition. Here we aim to provide a deeper
understanding of mechanisms and underlying physics in these systems which can lead to more effective
and reliable designs with less iteration. To place the inertial effects into context we also discuss related
fluid-induced forces present in particulate flows including forces due to non-Newtonian fluids, particle
asymmetry, and particle deformability. We then highlight the inverse situation and describe the effect of
the suspended particles acting on the fluid in a channel flow. Finally, we discuss the importance of
structured channels, i.e. channels with boundary conditions that vary in the streamwise direction, and
their potential as a means to achieve unprecedented three-dimensional control over fluid and particles in
microchannels. Ultimately, we hope that an improved fundamental and quantitative understanding of
inertial fluid dynamic effects can lead to unprecedented capabilities to program fluid and particle flow
towards automation of biomedicine, materials synthesis, and chemical process control.
Introduction
The belief that all practically achievable and useful flows inmicrofluidic systems operate not only in the laminar flowregime but also in a Stokes flow regime (Re → 0, Re = ρUH/μwhere U is the average velocity, H is the channel dimensionand ρ and μ are fluid density and dynamic viscosity,respectively) was widespread in the microfluidic communityuntil recent years.1 Following this belief, the inertia of the fluidis ignored inmost microfluidic platforms and contributions offluidmomentum are omitted from the Navier–Stokes equationsresulting in linear, and thus time-reversible, equations of
motion for Newtonian fluids. However, the importance ofintermediate range flow (~1 < Re < ~100) was emphasizedrecently2–4 in which nonlinear and irreversible motions areobserved for fluid and particles in microchannels. Thisregime, in which both the inertia and the viscosity of thefluid are finite, still lies within the realm of laminar flow(Re ≪ 2300) which provides a deterministic nature and thuscontrollability of fluid and particles within. The velocitydifference across the particle or obstacle length scale (andtherefore velocity gradient) is especially important in theemergence of inertial effects as it impacts the inertial liftforce via the lift coefficient magnitude or secondary flowmagnitude respectively, which will be discussed in followingsections. It should be noted that the necessity for largevelocity gradients to manipulate particles and fluids in inertialmicrofluidic platforms has likely prevented observation ofsimilar behaviors in larger channels filled with fluids withviscosity similar to water at room temperature and pressure.We show this with an example: in a typical microchannelwith a diameter of 100 μm and water as a working fluid(ρ ~ 1000 kg m−3 and μ ~ 0.001 Pa s) a flow rate of 150 μL min−1
results in a maximum velocity of ~0.375 m s−1, which isachieved at the channel center only 50 μm away from
the wall, and corresponds to Re ~ 25 (a common set of easilyachievable conditions in inertial microfluidic systems). Ifone were to scale up all the elements to achieve a similarvelocity gradient in a channel that is only 1 cm in diameter(i.e. 100× wider) the required maximum velocity would be~37.5 m s−1, which corresponds to Re ~ 250 000! Even thoughsuch a flow rate is achievable (since the pressure to drivesuch a flow actually decreases due to the increased channeldimension), the flow would be highly chaotic and turbulent(Re ≫ 2300) not allowing the precise quantitative controlachievable with laminar flow.
In this paper we aim to revisit the fluid physics in suchmicroscale systems operating in a laminar but finiteRe regime and demonstrate how integration of the physicalphenomena occurring in this scale introduces a quantitativeframework to passively control and manipulate fluid andparticles in channels, similar to the framework semiconduc-tor physics provided for controlling the flow of electrons thatrevolutionized electronics.
We first discuss straight microchannels and introduce thedifferent mechanisms that can be used to cause lateralmigration and control of particles due to fluid inertia, fluidviscoelasticity, particle shape and particle deformability. Wealso review the inter-particle interactions in these systemsand discuss how particles can interestingly be used as“dynamic/moving structures” to manipulate fluid in straightmicrochannels. We then argue how channels can be modifiedto control the fluid and particles flowing inside. Astraight channel with no boundary irregularities can onlybe used to manipulate particles, but not fluid. However,“structured” channels (i.e. channels with irregular geometryin a streamwise direction) can be used to manipulate bothparticles and fluid. Any deviation from a straight channel canbe considered a structured channel, for instance the presenceof curvature in a channel, or introduction of pillars andobstacles in the channel or grooves on the walls of the channel.We discuss how any geometrical deviation can inducenet secondary flows (i.e. net recirculation perpendicular tothe main flow direction) and present the different typesof fluid/particle manipulation that can be achieved byengineering channel irregularities and structures. We discussthe importance of inertia in each of these cases and show howthe two regimes, namely Stokes and inertial flow, can beused to achieve unprecedented control over fluid and particlesin a completely passive manner. Throughout, we emphasizethe differences between inertial and Stokes flow andconnect the reversibility associated with Stokes flow to the“mirror symmetry time reversal” theorem,5 which leads tothe conclusions that with fore-aft asymmetries even Stokesflow can cause net twisting of fluid streamlines while inertialflow can inherently lead to such flow deformations even insymmetric geometries. In combination with introducing thegoverning fluid physics for each of these systems we alsointroduce simplified “rules of thumb” (Tutorial Boxes 1–5) thatwe hope will aid novices in the design of inertial microfluidicsystems.
2740 | Lab Chip, 2014, 14, 2739–2761
i. Particles in a straight channel
A straight channel with a square/rectangular cross-section isthe simplest form of conduit that can be used inmicrofluidics (although the concepts introduced in this sectioncan be generalized to other cross-sections and geometry scalesas well). Introduction of different forms of nonlinearity intothese simple systems can produce lift forces (i.e. forcesdirected perpendicular to the main flow) that can be used tomanipulate flowing particles.
a) Inertial migration and focusing
Dominant inertial lift forces. In order to achieve spatialcontrol of particles in microchannels, a mechanism of lateralmigration is required; a manipulation tool that can be usedto introduce order into an otherwise randomly distributed setof particles in a flow. Such a behavior was first reported bySegre and Silberberg6 where they observed that randomlydistributed millimeter-sized particles migrated laterally tofocus on an annulus with a radius ~0.6 times the radius fromthe center of a one-centimeter-diameter pipe. The lateralmotion was unexpected at the time and proved the existenceof some form of lift force acting on these particles. Additionally,since the particles reach a stable dynamic equilibrium at theannulus, there are possibly two (or more) types of opposinglift forces acting on the particles in the Poiseuille flow. Latertheoretical analyses suggested7–9 that there are two opposingforces that dominate in this situation for neutrally buoyantparticles: 1) the wall-induced lift force (or wall effect liftforce), due to the interaction between the particle and theadjacent wall, which directs the particle away from the walland 2) the shear gradient induced lift force, due to the curva-ture of the velocity profile, which directs the particle awayfrom the channel center (Fig. 1A). Note that the finite andwall-directed gradient in the shear rate (second derivative ofvelocity) in Poiseuille flow (which is due to its natural para-bolic shape) plays an important role leading to stabledynamic equilibrium positions.
Weaker inertial lift forces. Additional lift forces also arisewhen the particle leads, lags, or rotates in the fluid, but theseforces play a dominant role only in special circumstances.Saffman showed that a particle lagging/leading the fluid inPoiseuille flow experiences a channel center/wall-directed liftwhich is an inertial effect (slip-shear). Note that as a resultof the presence of the walls, particles tend to lag theundisturbed flow velocity at the particle centerline in generalin Poiseuille flow, however, these slip-shear effects scale withan order of magnitude higher a/H (ratio of particle diameter,a, to channel dimension, H).8,10 Hence, this type of lift isnot generally applicable to inertial focusing systems givensmall a/H ≪ 1. However, in special circumstances, such asan additional force acting on a particle (e.g. electrical, gravi-tational, or magnetic), slip-shear lift can begin to dominateparticle behavior.11 In another study, Rubinow and Kellerdescribed that the flow about a spinning sphere moving in
Fig. 1 Inertial lift in straight channels. (A) Two lift forces act on a particle at finite Re (Rp > ~1) in Poiseuille flow, (i) a wall effect lift that pushes aparticle away from the wall, and (ii) a shear gradient lift which pushes a particle away from the channel center. The balance between the twoforces defines the equilibrium position for the particle. This phenomenon is called inertial focusing.3 (B) In square channels, inertial focusingcreates four symmetric equilibrium positions. The force map acting on a particle in a channel cross-section obtained by finite element methodsolutions yields an equilibrium position approximately at ~0.6 of the distance from center to channel wall.14 (C) If the channel cross-section isrectangular, there are usually two preferred equilibrium positions. This can be explained by the force distribution across the channel, which almostalways leads a randomly distributed particle to the stable equilibrium positions near the wider channel faces.16 (D) Increasing Re causes thefocusing position to shift slightly away from the channel center. This presumably also leads to stabilization of the two previously unstableequilibrium positions in rectangular channels. (E) Based on the theory proposed by Ho and Leal8 the direction of the shear gradient induced liftdepends on the shear rate and its gradient. For Poiseuille flow with a parabolic velocity profile, this force always points away from the channelcenter. However, in an arbitrary velocity profile with an inflection point, the direction of the force could change across the channel.
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an unbounded and stationary fluid exerts a lift force on it(slip-spin),12 originating from rotational differences betweenthe sphere and underlying flow.8 Should this lift be appliedto a particle in Poiseuille flow the particle would be directedtowards the channel center. Following the analysis of Hoand Leal, slip-spin effects contribute minimally compared towall-effect and shear-gradient lift in Poiseuille flow, scalingwith three orders higher a/H8,12 given that the particle rota-tion will generally match the undisturbed rotation in thefluid. The origins and directions of action for weaker anddominant lift are tabulated in Table S1.†
Inertial lift scaling. The scaling of inertial lift forces hasrecently become more clearly elucidated to depend on thefinite size of particles. Following Hood, Lee and Roper, forsmall particles (a/H ≪ 1) lift scales as FL∝ρU2a4/H2 but shiftsto scale less strongly with particle diameter as the particlesize becomes more comparable to the channel dimension(a common occurrence for many microfluidic flows).13 Fora finite-size particle (a/H between 0.05 and 0.2), at Re of20 and 80, finite element method simulations yielded anet lift acting on the particle scaling as FL∝ρU2a3/H nearthe channel center and as FL∝ρU2a6/H4 near the channelwall. The dimensionless lift coefficient fL remains a weakfunction of Re and strongly depends on position within thechannel.14
Focusing in square and rectangular channels. As a resultof these lift forces, for a particle Reynolds number, Rp
of order of equal or larger than 1 (where Rp = Re(a/H)2 =ρUa2/μH) the randomly dispersed particles in a squaremicrochannel inertially focus to four symmetric equilibriumpositions near the center of each channel wall face and, inagreement with Segre and Silberberg's observation, ~0.6(H/2)away from the channel center (Fig. 1B) as confirmedexperimentally.2,15 The finite element solutions also providea clear map of the lift imposed on the particle in thesechannels which is in agreement with experimental andtheoretical work (Fig. 1B).
Interestingly, when particles are introduced into achannel with rectangular cross-section (without the four-foldsymmetry of a square) the number of favored dynamic equi-librium positions is decreased to two, located near the widerfaces of the channel (Fig. 1C). Numerical solutions show thatthe force field present in the channel results in a set of stableequilibrium positions and saddle points in the channel.16 Tounderstand how this can lead to two points of dominantequilibrium, we can think of releasing a particle at a pointin a quadrant of the channel and assessing the probability ofit approaching either of the two reachable equilibriumpositions (i.e. on the long and the short face). In such across-section (Fig. 1C) the shear gradient lift force, which is
dominant anywhere away from the walls, is stronger in thez-direction but much weaker in the y-direction (due to theblunted velocity profile in the y-direction). Therefore, a parti-cle that is randomly released in this channel has a greaterchance of being pushed away from channel's y-centerline(blue line). Consequently, the particle will approach the longchannel face (green line) in the z-direction first and then dueto dominant wall-effect lift is directed towards the centerline(red line) in the y-direction until it settles at its stableequilibrium position (Fig. 1C). Along the channel face (greenline), wall-effect lift dominates likely because of the lowerfluid velocity – and local fluid inertia – in this region, asdescribed further below. Overall, for a moderate Re the liftforces in this cross-section lead to limited particle paths thatarrive at the other equilibrium positions at the short channelfaces. Significant biasing to two focusing positions can beobserved for even slight deviations from square geometries(e.g. aspect ratio of 3/2).
Effect of Re on focusing positions. At higher Re (up to~150) the equilibrium positions in square and rectangularchannels tend to shift slightly towards the walls (Fig. 1D).This also can be explained by taking the two competing liftforces into account. It is clear that both the wall effect liftand the shear gradient lift increase with increasing velocityor Re. However, the increase in shear gradient lift is relativelylarger for a given increase in Re.7,16 Therefore, increasing Recauses the shear gradient lift to become more dominant and,consequently, the particle equilibrium positions shift closerto the wall. A similar behavior occurs in rectangular channels,in which the two previously unstable positions on the shortfaces now collect additional particles, as the wall effect becomesless dominant in directing particles away along the long face,leading to an increase in the total number of focusing positionsto four (Fig. 1D).17,18 Therefore, the distribution of lift forceswithin the channel is altered such that the competing y-directedshear gradient lift force can direct particles into the previouslyunstable/unreachable equilibrium position. This was foundto occur at above Rp ~ 4.5–6 experimentally.17,18 Above Rp ~ 8,equilibrium positions were observed to shift again but nowtowards streamlines closer to the center of the channel.18
We propose a similar argument can provide intuitionabout the local relative importance of wall-effect and shear-gradient lift within a channel in which the cross-sectionalflow shape remains the same. The local Re, i.e. local relativeimportance of inertial to viscous effects, can be greatlyvariable in a quarter of the channel (Fig. 1C) due to the largevelocity differences across a channel. Our conjecture is thatdue to relatively higher velocities near the channel center(blue line), shear gradient lift is dominant in this region,thus directing particles towards the wall in the y-direction.On the other hand, flow is slower near the walls (green line)due to the local no-slip condition. Therefore, wall effect liftdominates directing the particles back towards the channelcenter (negative y-direction).
Effect of a/H on focusing positions. Particle size can alsoaffect the focusing position. For a/H ≪ 1 the focusing
2742 | Lab Chip, 2014, 14, 2739–2761
position approaches ~0.6(H/2) as in the original studiesof Segre and Silberberg. But both experimental andnumerical results indicate that the equilibrium position ofparticles shift towards the channel center as a/H approaches1.14,17,19 This is largely due to the fact that steric effectsbecome more important as a/H increases, especially above~0.4. For instance, in the limiting case when a/H = 1 theparticle is geometrically forced to pass through the channelcenterline.
Particle concentration effects. It has also been shown thatboth the location and the number of focusing positionsdepend on the number of particles per unit length along thechannel,20 such that at high length fractions, i.e. φ > ~75%(φ, length fraction, defined as the fraction of particlediameters per channel length)3 multiple streams are observedacross the channel. At these increased concentrations, ifthe particles are to occupy two to four focused streams theyare forced to stay in close proximity to each other. In thiscase, there will be a large increase in particle–particlehydrodynamic interactions which have been shown to beimportant once particles are within a short distance of oneanother (< ~10a).21 As a result, a portion of the particles arepushed out of the lift-specified focusing streams and formnew nearby focusing streams. These particle–particle inter-actions act as one limit on achieving precision focusing ofall particles in a channel.
Building physical intuition for inertial lift. Despitemultiple works discussing the lift forces on particles inchannel flow,7,8,22 there is no simple intuitive explanation forthe physical origins of inertial lift. In their theoretical workon the lift forces exerted on infinitely small particles, Lealand Ho8 conclude that the lateral force mainly originatesfrom the shear field acting on the sphere and the induceddisturbance flow (i.e. flow difference due to the presence of aparticle in the stream) rather than the presence of wall-induced lag velocity (Saffman lift/slip-shear) or slip-spin, inagreement with our previous description. They decomposenet lift into “wall-effect” which is the interaction of the dis-turbance stresslet and its wall reflection interacting with thebulk shear, and “shear-gradient lift” which is the interactionbetween the stresslet and the curvature of the bulk velocityprofile. A stresslet, with its characteristic orthogonal “flow-in–flow-out” quadrapolar pattern, can be shown to be relatedto the disturbance around a particle in a flow with finite rateof strain. Rigid particles induce stresslets in such flowsbecause they do not deform appreciably with the underlyingflow. To maintain their shape the particle must apply a forcefield back on the flow that resists the underlying flow defor-mation, resulting in a stresslet.23 Furthermore, they concludethat the direction of the net lateral force depends on thedirection of both the shear rate and its gradient, and arisesfrom the interaction of the local disturbance velocity fieldwith both the shear rate and its gradient. For instance, ina Poiseuille flow in a rectangular channel the shear rateis always positive and its gradient negative as an inherentcharacteristic of the shape of the velocity profile (Fig. 1E).
Therefore, the shear gradient induced lift is always directedtowards the wall. However, if the suspension was to flowthrough a channel with a more complex cross-sectionalshape, such as the one shown in Fig. 1E (bottom), the veloc-ity profile correspondingly becomes more complex and evenpossesses an inflection point. In this case, the direction ofshear rate and shear rate gradient could alter across thechannel width which, based on Leal and Ho's analysis, leadsto shear gradient lift forces that change signs across thechannel. Numerical predictions and experimental resultswith additional channel shapes will likely clarify the impor-tance of this effect, and have the potential to open up awhole new set of design tools for particle manipulation andcontrol based on velocity field sculpting.
Length required for focusing. Considering shear-gradientlift alone we have previously presented an expression basedon the cross-streamline motion of finite-sized particles3
that proscribes a channel length required for particles toreach lateral equilibrium positions (Lf) in rectangular
channels L hU a ffm L
2
2, where Um is the maximum channel
velocity (~1.5U, the mean channel velocity). The average fL isabout 0.02–0.05 for channel aspect ratios (height/width)between 2 and 0.5. For instance, let us consider focusing of10 μm particles in a 40 × 60 μm2 rectangular channel withwater as working fluid, flowing at 150 μL min−1. In this caseRp ~ 3.3 which is not high enough to give rise to four stableequilibrium positions. Therefore, particles focus along the
Fig. 2 Applications of inertial microfluidics. (A) Massive parallelization of pafocusing, which enables extremely high-throughput flow cytometry.17 (B) Kblood cells is achieved by controlling the separation distance to focus largequilibrium positions in low aspect ratio channels is used for particle excha
longer face, i.e. along w = 60 μm. A simple calculation showsUm ~ 1.56 m s−1 and h/w = 0.66. Therefore, assuming fL ~ 0.04,the channel length required for particle focusing is Lf ~ 2 cm.Entry length required for focusing has also been exploredexperimentally using holographic imaging of particles,24
obtaining general agreement with this basic equation. Also inagreement with our discussions above, Choi et al.24 found thatmigration down the shear-gradient, for which Lf is calculated,preceded wall-induced migration to the center of channelfaces, suggesting that to describe complete focusing furtherwork to develop a more accurate approximation for Lf willbe required.
Applications of inertial focusing. The simplicity of thephenomenon suggests its ability to impact many applications.For instance, since there are no extra force fields or devicesrequired (only the channel itself is needed for alignment ofparticles), this method can be easily parallelized by branchingoff many channels from a single inlet and arraying them inclose proximity. This leads to extremely high throughputparticle focusing (Fig. 2A) which has been used for high-throughput sheath-less flow cytometry.17,18,25 This maybe particularly useful in reducing the size and cost ofoperation of flow cytometers, which normally require a largebulk of aqueous sheath fluid to pinch cell streams prior tooptical interrogation. Remaining challenges to overcome tospeed adoption for this application include focusing of cellsized particles in channels with larger channel dimensions26
(currently precise focusing requires channels with dimen-sions approximately less than 5–7 times particle diameters).
Lab Chip, 2014, 14, 2739–2761 | 2743
rticle focusing channels is possible due to the passive nature of inertialinetic size-based separation of smaller pathogenic bacteria from largerer blood cells only.27 (C) Lateral migration of particles to reach stable
Achieving this would require less frequent replacement ofthe focusing chip and lower pressure pumping systems.Going beyond the use of straight rectangular channels, asdiscussed below, can aid in this direction.
Inertial focusing in straight channels has also been usedto separate particles or pathogenic bacteria cells from dilutedblood based on differential migration rates (kinetic separa-tion, Fig. 2B).27–29 In this simple design, the larger bloodcells or particles become focused in a shorter channel lengthfollowing the aforementioned equation for Lf, while thesmaller bacteria or particles are less affected by the inertiallift forces and remain dispersed throughout the channel (dueto their smaller sizes). These approaches also all rely on thereduced number of equilibrium positions in high-aspectratio rectangular channels which enable two outer collectionoutlets. In the work from Mach et al. a gradual expansion ofthe channel width is also used such that the focused cellsare then moved to a position closer to the side walls (whilemaintaining their focused state), leaving a large portion ofthe channel cross-section free of blood cells. The focusedstreams are then captured with side outlets while theremaining fluid that contains most of the bacteria is cap-tured separately. An advantage of separations in straightchannel designs is the ability to radially parallelize the sys-tem which has been shown to process up to 240 mL h−1
with a throughput of 400 million cells min−1 and with>80% bacteria removal from blood after two passes.27 Aswith focusing, inter-particle interactions at high blood cellconcentrations can prevent precise focusing and reduce theaccuracy of separation, such that separations using inertiallift forces operate best in semi-dilute solutions with lengthfractions below ~50–60%.
Lateral inertial migration across laminar co-flowingstreams has also been used to achieve fast fluid exchangearound cells and particles.16 This technique relies on theintelligent use of the presence of two (rather than four) stableequilibrium positions in low aspect ratio channels. Here, theparticle suspension is co-flowed with an exchange solutionthat has a higher flow rate (~2-fold) into a low aspect ratiochannel (i.e. h/w smaller than 1 with w in the directionorthogonal to the line of contact of streams). Once in thismain channel, particles start to migrate towards new stableequilibrium positions in the center of the new channel,which due to the asymmetry in the rates of the co-flows islocated along streamlines occupied by the exchange solution(Fig. 2C). Consequently, with an appropriate outlet designthe particles can be captured in the exchange solutionwith high purity, high yield and at a high rate (which is ageneral characteristic of inertial microfluidic platforms).The high rate necessary for operation is not ideal forexchanging solution around one or a few cells from arare sample, which would require much more delicateinstrumentation, but is ideal for applications requiring milli-second solution exchange around hundreds to thousands ofcells or particles per second, with the capability of down-stream inline analysis.
2744 | Lab Chip, 2014, 14, 2739–2761
High throughput is a primary advantage of inertialmicrofluidic platforms which makes them compatible withother high-speed platforms. For instance, in a recent develop-ment, cancer and blood cells inertially focused in straight chan-nels were analyzed by a novel line-scan based high-speedimaging technology (serial time-encoded amplified microscopy– STEAM) and achieved nearly 100 000 particle s−1 real-timeimage-based screening (compared to the state of the art~1000 particle s−1).30 In this system, particles, including bloodcells and budding yeast, are first ordered and focused in ainertial microfluidic channel fabricated in thermosettingpolyester (TPE),31 to prevent fluctuations in channel dimen-sions in softer polymer channels that affect the optical signalquality, and achieve higher pressure flows without device fail-ure. The typically high operating flow rates of inertial micro-fluidics platforms leads to very high pressure drops in thesechannels. In practice, the inability of Polydimethylsiloxane(PDMS)-based devices, which are the gold standard for rapidand cheap device fabrication, to withhold such high pressuredrops acts as one of the limiting factors, especially in proto-typing and research development. Therefore, the develop-ment of stiff material that can withstand higher pressuredrops for rapid prototyping of microfluidic devices is impor-tant and could push the limits on both research and applica-tion fronts.31
Inertial microfluidics can also be integrated with othermicrofluidic schemes to offer more comprehensive solutions.For instance, Toner et al. combined different effects in theirdevice where, first, small red blood cells were separated fromblood using size-based deterministic lateral displacement,32
followed by inertial focusing of remaining cells which are fur-ther distinguished downstream using magnetophoresis.33
Although inertial focusing is well-suited for dilute solu-tions a useful frontier is investigating focusing and separa-tion in concentrated particulate solutions to address thechallenge of focusing and separation of cell populations pres-ent in whole blood. Development of novel techniques andevaluation approaches that are better compatible with highcell concentrations has aided in this endeavor. For instance,conventional imaging techniques used for studying inertialfocusing – including high-speed bright-field imaging andlong exposure fluorescence imaging – are generally limited incomplex fluids such as whole blood due to interference fromthe large numbers of red blood cells. In a recent study, Toneret al. applied particle trajectory analysis (PTA) as a means toobserve inertial migration of particles in dilute and wholeblood. In this technique, fluorescently labeled particles aresuspended in the biological fluid, and images taken at multi-ple vertical positions in the channel, using extremely short(~10 ns) but intense pulses, are used to find in-focus particlesand determine their diameter and two-dimensional spatialcoordinates in the channel cross-section. This led to un-covering a radical shift in focusing behavior of PC-3 prostatecancer cells in whole blood.34 Tanaka et al. also studied migra-tion of cancer cells in concentrated blood cell suspensionsup to 40% hematocrit, but results suggested minimal or
abnormal migration, as determined by reduced separationpurity downstream, until hematocrit was decreased to about10%.35
An understanding of inertial focusing of rigid sphericalparticles in straight rectangular channels provides a solidfoundation for applications in focusing, separation, enrich-ment, and ordering of cells and particles. Our current under-standing is summarized in Table S2,† including what hasbeen proven analytically, numerically, and experimentally. Amore complete understanding of migration behavior for real-world particles and applications requires additional knowl-edge of how fluid properties, particle shape and deformability,and channel structures contribute to overall motion.
b) Non-Newtonian fluids
Lateral migration in non-Newtonian fluid. In addition tofinite-Re flow conditions, non-Newtonian fluids have longbeen known to cause lateral migration of single particles in apressure-driven channel flow. The focusing positions for par-ticles in a non-Newtonian flow depend on the rheologicalproperties of the fluid. We will briefly touch upon the visco-elastic lift force that can be used (or must be taken intoaccount for some fluids) to manipulate particles in inertialmicrofluidic systems.
Particles migrate typically towards the center or the wall ofa pressure driven channel flow, depending on rheologicalproperties of fluid (i.e. shear thinning or shear thickening)and the channel blockage ratio (α = a/H).36,37 Ho and Lealpredicted that a non-uniform normal stress distribution in asecond-order fluid results in lateral particle migration.38
A second order fluid is defined to have stresses in the fluidthat are not linearly dependent on the shear rate – as in aNewtonian fluid. Particles migrate in the direction of decreas-ing absolute shear rate, which is towards the center of thechannel for a Poiseuille flow.
The non-dimensional Weissenberg number (Wi �
where � is the shear rate and λ is the fluid relaxation time)
is often used to characterize the viscoelasticity of a fluid.
Tutorial Box 1 – Practical design rules and considerations for ichannels
Due to the complexity of inertial microfluidic physics and interplay of multplatform could be challenging for the beginner. However, here we provide simBoxes. These estimates are meant to provide the reader with a reasonableexperimental design affect particle and fluid behavior. Note that the accuraexperimentally, yet the proposed rules serve as a decent starting point.
The channel length required for particles to reach lateral equilibrium posi
maximum channel velocity (~1.5U, the mean channel velocity). The averag
2 and 0.5. Consequently, in order to achieve inertial focusing over a length of L
conversion of the maximum velocity to the average velocity in the flow through
For a rectangular channel, average shear rate becomes
Uh/2
and Wi can be expressed as Wi 2 2
2
Uh
Qwh
where Q is
the flow rate. The Elasticity number, El is the ratio betweenWi and Re, and indicates the relative strength of elasticforces to inertial effects. For a rectangular channel
El WiRe
h wh w 2 .
As both Wi and Re are proportional to the flow rate, Elbecomes independent of the flow rate. This assumptionwould be valid if the viscosity is constant. However, in manycases viscosity is not constant due to shear thinning andthickening.
In recent years, experimental and numerical studies havedemonstrated the ability to apply viscoelasticity-inducedforces for particle focusing and separation.39–44 These forceshave also been combined with inertial lift forces to enablehigher throughput manipulation of particles.42–44
Applications of elasto-inertial flows. In a circular tube,(e.g. glass capillary43) particles in viscoelastic flow migratetoward the center of the tube for a wide range of Wi and α,these parameters can be tuned to achieve a bi-stable conditionwhere particles migrate both towards the channel center andthe walls depending on the initial position of the particle.In this study, Re was sufficiently small that inertial effectscould be ignored, which also makes this technique moresuitable for applications requiring small volume operations.With typical microfluidic channels of rectangular cross-section,elastic force focuses particles to the center and the four cornersinstead.42,44 The addition of inertial lift forces that focusparticles near the walls (Fig. 3A) can lead to a reduction in thenumber of focusing positions. By controlling flow rate, inertial liftforces and viscoelastic lift forces can be carefully balanced toachieve an Elasticity number that yields a single line of focusingalong the channel center-line (Fig. 3A and B).42 Single-line focus-ing without external fields or sheath fluid has advantages inapplication areas such asminiaturization of flow cytometers. Theviscoelastic flows that yield single-line focusing have Re ~ O(1).
Lab Chip, 2014, 14, 2739–2761 | 2745
nertial microfluidic systems: inertial focusing in straight
itude of physical parameters, design of an effective inertial microfluidicplified rules and general considerations for these designs in five Tutorialidea of how and to what extent different physical parameters in theircy of these assumptions have not been rigorously and universally tested
tions (Lf) in rectangular channels is L hU a ffm L
2
2, where Um is the
e fL is about 0.02–0.05 for channel aspect ratios (height/width) between
Fig. 3 Elasto-inertial flows. (A) While inertial effects create four focusing positions near the center of channel walls (left), elastic properties ofnon-Newtonian fluids are shown to cause focusing near the corners as well as the center of the channel (middle). The combined effect ofthe two, called elasto-inertial flow (i.e. presence of inertial effects in non-Newtonian fluids), can lead to single stream focusing of particles (right).42
(B) Inertial and viscoelastic lift forces are dependent on particle size and the combined force is predicted to have a quadratic dependence ofmigration velocity on particle size. This effect has been used for kinetic size-based separation of microparticles.44
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Considering inertial focusing (without viscoelastic effects)applied to flow cytometry applications have operated at Re ~ O(10–100),45,46 viscoelastic focusing provides 1–2 orders ofmagni-tude slower flow rates. Increases in flow rate will result in higherRe, which in turn requires higher relaxation timematerials or vis-cosity to balance inertial lift forces and viscoelastic forces. In prac-tice, high viscosity fluid limits the flow rate due to high pressurefor driving flow, however, using very dilute DNA (0.0005 (w/v)%)solutions that possess a longer relaxation time47 may beapplied for particle focusing over a wide range of flow rates.The λ-DNA that was used in Kang et al. has a relaxation time of140 ms, while poly(ethyleneoxide) (PEO) that is water-solublesynthetic polymer with similar contour length with λ-DNA
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(16.4 μm) has a relaxation time of 0.7 ms. Viscoelasticfocusing can also be applied to separate particles of differentsizes using the fact that lift forces (inertial and viscoelastic)are dependent on particle size39,44 leading to a predictedquadratic dependence of migration velocity on particle size.Small particles (2.4 μm) are relatively less focused and thelarger particles are tightly focused at the center of the channel.Similarly, 1 μm and 5 μm particles were separated by intro-ducing particles (Fig. 3B)44 in a side channel and relying ondifferences in the migration rate towards the center of thechannel (a kinetic separation48).
Effect of fluid viscosity on throughput. High flow rate(typically >10 μL min−1) is one important feature of inertial
microfluidic systems but is more challenging with viscoelasticfocusing systems. Addition of polymers to utilize viscoelasticlift forces leads to an increase in fluid viscosity and the highviscosity limits maximum flow rates and potential applicationsof the manipulation scheme in areas that can benefit fromhigh-throughput. The change in viscosity of shear-thinningfluids and shear-thickening fluids will also affect themaximumthroughput. Shear-thinning and -thickening fluids aredefined as fluids having a viscosity that decrease or increasein magnitude with increasing shear stress respectively. Use ofshear thinning fluids such as blood and typical polymer solu-tions will result in lower fluidic resistance at higher flow ratewhich could help to increase maximum throughput. For thesame reason, use of shear thickening fluid will be disadvanta-geous for high throughput applications. On the contrary,these viscoelastic focusing approaches may be better suitedto deal with small samples, as the sample would not bewasted during the start-up time required to achieve the focus-ing flow rate. Microchannels that can withstand high pres-sure31 may also be used to allow higher maximum flow rateswith non-Newtonian fluids. Experimental and theoreticalstudies of non-Newtonian fluids in microchannels areexpected to attract more attention over the coming years to(i) develop a systematic understanding of viscoelastic effectsacting on particles for fluids that can be viscoelastic in manydifferent ways (with a range of non-Newtonian constitutiveequations), and (ii) engineer structured or shaped channels49
that can introduce additional forces to achieve further sepa-ration and focusing control.
c) Shaped particles (with or without fluid inertia)
Non-spherical particles in real systems. A sphere is theideal shape for a particle in many cases and studies owing toits ultimate symmetry and lack of singularities, which lead tomathematical simplicity. Therefore, in a large portion of thework describing particulate flows (theoretical, numerical andexperimental), a sphere has been used as a simplified case,an approximation. However, in a large portion of realsystems non-spherical particles exist. For instance, a humanred blood cell (RBC) adopts a discoid shape in the body, farfrom being a sphere. Shape represents an important factor tospecifically identify a bioparticle,50 including bacteria, viralparticles, budding yeast, and marine micro-organisms. Syn-thesized barcoded particles represent another example of par-ticles that would be helpful to operate on andmanipulate.51–53 Therefore, it is important to improve ourunderstanding of systems that contain shaped particles andthe corresponding lift forces and dynamics of these particles.One of the classic works in this area was the investigation ofthe motion of ellipsoidal particles in unbounded shear flowin a Stokes flow regime by Jeffery54 which led to the discoveryof the so-called “Jeffery orbit” in which a particle rotates in aclosed repeating cycle around any of an infinite set of axesthat depend on its initial orientation.55 Here we discuss howparticle shape and fluid inertia combined can affect particle
motion, focusing and separation dynamics in microchannelsystems.
Mirror symmetry time reversal theorem. An interestingpoint about shaped particles is that they can be subject to liftforces even in the absence of inertia (i.e. Stokes flow) whichcan be intuitively described via an exception to Bretherton's“mirror symmetry time reversal” theorem (MSTR theorem).5
Simply put, the theorem indicates that if a sufficientlysymmetric particle moves in a uni-directional viscous shearflow without inertia and we momentarily reverse time, theresulting configuration (in terms of the flow field, pressurefield, forces, etc.) is mirror symmetric to another configura-tion of the particle that is rotated, at some angle, around therotational axis defined by its Jeffery orbit. We will definewhat a “sufficiently symmetric” particle entails in the follow-ing discussion. This is due to the linearity of the Stokes equa-tions, and a full proof can be found in Bretherton's paper.5
The MSTR theorem, can also apply to a channel flow inwhich a velocity gradient is also present leading to particlerotation.56
For a rigid spherical particle, the MSTR theorem providesa proof that there can be no lift forces acting on the particlein Stokes flow for a simple channel (Fig. 4A). If we assume
there is a net lift on the particle (�FL ), then due to the linear-
ity of the equations of motion there is an opposite directed
force (� �F FL L ) if we reverse the time. However, the initial
and time-reserved configurations are exactly the same inevery way (due to the fore-aft symmetry of the system). There-
fore the two forces need to be equal (� �F FL L ) which is only
satisfied for�FL 0 Hence, there can be no lift on a spherical
particle in Stokes flow through a channel.If the particle has an ellipsoidal shape, the scenario is
slightly different (Fig. 4B): an ellipsoidal particle follows anorbit similar to a Jeffery orbit (Fig. 4B.i). If time is reversed,the direction of the instantaneous lift force is also reversed(Fig. 4B.ii). The configuration that is mirror symmetric(Fig. 4B.iii), however, is in fact a rotation of the original con-figuration. Therefore, instantaneous lift can possibly existduring the particle's rotational cycle. However, the lift forceat any point has a complete counterpart during the cycle withopposite directed lift, thus there can be no net lift on the par-ticle during a full orbit. This result requires that the particleis sufficiently symmetric, that is, the particle is mirror sym-metric to a rotation around its rotational axis. Classes ofshapes that can satisfy this condition include disks, cylin-ders, rectangular prisms, triangular prisms, and cones.
For a rotationally asymmetric particle, the MSTR theoremcannot be applied (Fig. 4C) and a net lift force is possible fora rotating particle. For example if we have an h-shaped parti-cle in the channel (i) and we reverse time (ii), the mirror-symmetric configuration is impossible to achieve by anydegree of rotation. Therefore, the MSTR theorem does notapply to such asymmetric particles, and cannot be used to
Fig. 4 Lift forces on shaped particles in straight channels. A, B and C are in the frame of reference of the channel. (A) The mirror symmetry timereversal (MSTR) theorem teaches that there is no lift possible on a spherical particle in Stokes flow. (B) The MSTR theorem also applies forellipsoidal particles that are rotationally symmetric. In this case, while instantaneous lift is possible, no net lift can be induced over a completerotation of the particle. (C) However, the MSTR theorem does not apply to classes of asymmetric shaped particles such as an h-shaped particle,since the mirror symmetry of the time-reversed configuration cannot be achieved through rotation of the original particle. Therefore, a net lift isallowed in this case. (D) Lift forces and the focusing behavior of various shapes of rigid particles such as symmetric disks (ii), cylinders (iii) andasymmetric disks (iv) have been experimentally studied.59 (E) These studies reveal that most shaped particles follow the focusing behavior of aspherical particle of the equivalent rotational diameter, with the exception of the rotationally asymmetric h-shaped particle.59 (F) Ellipsoidalparticles with higher aspect ratio focus closer to the center line compared to spheres of the same volume. The dependence of focusing positionon the shape and aspect ratio of particles has been used for shape-based separation of budding yeasts.57
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eliminate the possibility of a net lift force acting on the parti-cle throughout its rotational cycle, even in Stokes flow.
Clearly, the rotational axis of the particle is important indefining the lift forces and subsequent motion of a particle,and this preferred rotational axis can be set due to the pres-ence of fluid inertia. In Stokes flow we know that particlescan occupy an infinite set of rotational axes that depend oninitial conditions. This situation changes due to the presenceof fluid inertia.57,58 The addition of inertia and channel con-finement leads to the precession of particle orbits to a stablerotational axis in the parabolic flow.
Inertial focusing of shaped particles. In recent works,Hur et al. and Masaeli et al. used straight microchannels(Fig. 4D.i) to experimentally study the focusing behavior ofmultiple types of particles such as symmetric disks (Fig. 4D.ii),cylinders (Fig. 4D.iii), ellipsoids (Fig. 4F), and asymmetricdisks (Fig. 4D.iv).57,59 Above a critical Reynolds number(Re ~ 50) particles predominantly adopted preferred
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equilibrium positions and “tumbling” rotational axes. Forparticles symmetric about this rotational axis focusingposition was found to be independent of cross-sectionalshape and mainly depends on rotational diameter (Dmax).During the tumbling motion, when the major axis rotatesto be perpendicular to the wall, wall effect lift increasessubstantially due to the closer distance. As a result, theintegrated wall effect (over time) on these particles is higherthan that of spheres of the same volume. Interestingly, onlythe highly asymmetric h-shaped disk did not follow thistrend and focused closer to the walls than expected for itsDmax, in agreement with the MSTR theorem allowing anadditional lift force acting on this rotationally asymmetricparticle (Fig. 4E). Interestingly, if complete particle rotationis prevented (due to channel confinement or other forces)lift acting on shaped particles can be employed to directparticles to locations in the channel cross-section in a shape-dependent manner.59,60
Shape-based differential focusing has potential applica-tions, and has already been used to sort and separate bud-ding yeast at the high rate of 1500 cells s−1, an order ofmagnitude improvement from previous methods.57
Fig. 5 Deformable particles. (A) Three models for deformable particlesinclude solid elastic particles, deformable droplets (with fluid interiorand a slipping interface) and deformable capsules (with fluid interiorbut an elastic solid boundary). (B) Deformability of a particle inPoiseuille flow generally leads to an additional lift force which directsthe particle away from the wall. Balance of these forces creates afocusing position closer to the channel center, compared to rigid solidspherical particles.61 (C) Experimental data shows that, in general,increasing deformability corresponds with a focusing position closer tothe center (for λd from ∞ to ~4.6). There is a shift in the trend whendrop viscosity approaches the underlying fluid viscosity (λd < ~4.6).61
d) Deformable particles
Models for particle deformability. In many studies ofparticulate systems, solid rigid particles are used as simplecomponents to study system behavior. However, bioparticlessuch as cells and vesicles, or two-phase emulsion droplets arenot rigid, but deformable. Importantly, deformability of parti-cles can create nonlinearity in these systems which can leadto additional lift forces on these particles61 (Table S1†).
Three types of models are generally used to approximatecomplex bioparticles or more simple emulsions when study-ing their motion in flow. Models approximating a deformableparticle in a shear flow are (i) an elastic solid particle, (ii) adeformable drop, and (iii) a deformable capsule (Fig. 5A). (i)An elastic solid particle:62 the whole particle is considered asa single solid body which can deform with a particular elasticconstant. In this case, for the flowing particle to retain asteady-state shape it needs to constantly deform or “tank-tread” as it is rotating under the shear flow. (ii) A deformabledrop:61,63–66 the particle is assumed to be a drop with fluidon the inside and a boundary that can slip under the drivingflow. In this case, recirculation of fluid can occur inside thedroplet due to the outer flow and the elasticity of the dropletis defined by its surface tension. (iii) A deformablecapsule:67–69 the inner fluid is covered by a deformable butsolid shell with a specified elastic constant. The solid shellundergoes a tank-treading motion to maintain a steady-statedeformed shape while fluid recirculation can occur insidethe capsule.
Effect of particle deformability on lateral migration. Thesestudies suggest that in a shear or Poiseuille flow, regardlessof the approach to approximate deformability, there is acomponent of force perpendicular to the free-stream direc-tion that causes the particle to migrate across undisturbedstreamlines. This lateral migration is argued to be due to theshape-change of the particle and can occur even in theabsence of inertia.62 This links with our discussion of shapedparticles in which instantaneous lift can act on a particle inStokes flow (Fig. 4B). The difference in this case is that thedeformable particle in many cases does not tumble to occupymirror symmetric configurations but rather can perform atank-treading motion such that it maintains a steady-stateshape as it flows downstream. Therefore, in contrast with arigid elongated rotational symmetric particle, a deformableparticle can have a net lift leading to its lateral migration.
In addition to lift caused by the deformed shape, lateralmigration of deformable particles is reported to also arisefrom nonlinearities caused by the matching of velocities andstresses at the deformable particle/droplet interface.70 In gen-eral this means that the magnitude of lateral drift is relatedagain to the deformed shape of the particle or gradients in
ratio, λd = μd/μ where μd is the dynamic viscosity of fluidinside the droplet, is another important parameter in definingthe shape of the deformed drop. By measuring the shiftingbalance between deformability-induced lift and inertial lift,Hur et al. experimentally demonstrated that deformability-induced lift increased with increasing drop deformation.Droplets shifted towards the channel center as λd decreasesfrom 970 to 4.6 (i.e. more deformable particle). Counter-intuitively, as the viscosity ratio decreased below λd < 4.6,the focusing position shifted back towards the wall(Fig. 5C), in agreement with theoretical predictions70 thatmay also implicate strong internal circulations within adrop affecting lift. A recent study by Stan et al.66 also sug-gests that interfacial effects (e.g. gradients in surface ten-sion that develop in the shear field) yield lift forces thatcan play a dominant role especially in flows of bubbles ordroplets. In fact, lift arising from surface tension gradientsis starting to be actively controlled and used to steer drop-lets.71 Similar to droplets, numerical studies on deformablecapsules in inertial flows69 suggest that particle equilibriumposition depends on the shell compliance and particleswith more flexible shells move closer to the channel center.
Scaling of deformability-induced lift. As Ca increases andRp decreases deformability-induced lift begins to dominatemotion of drops and bubbles in microchannels. Chan andLeal give the conditions λd < 1 and λd > 10 for deformability-induced lift force being directed towards center of the chan-nel. Their analytical study suggests that when a drop or bub-ble is not too close to the walls (e.g. the separation betweenthe drop and the wall is larger than a) the deformability-
induced lift force scales as F Ua aH
dH
fL,d dCa
3
,
where d is the distance between drop and center of the chan-nel.72 Experimental measurements of inertial lift force inmicrochannels by Di Carlo et al.14 have also been used toextract an equation for inertial force near the center of thechannel.66 For d/H < 0.1 the negative sign indicates that thisforce is directed towards the walls and fits
F R Ua dHpL,inertial
center
10 . When both drop deformation and
inertial forces are small (for Ca < 0.01 and Rp < 0.01), Stanet al.66 suggest an empirical formula for net lift force on bub-
ble or droplet as F C Ua aH
dHL,empirical L
3
, where CL is an
empirical factor which has to be determined experimentallyfor a given pair of carrier liquid and drop or bubble fluid.However, for order-of-magnitude estimates CL ~30 can beused. This formula is applicable for drops or bubbles that are(nearly) spherical (since Ca < 0.01) and positioned near thechannel center (d < ~0.15H). A summary of our currentunderstanding concerning deformable particles is providedin Table S2.†
In Poiseuille flow, the direction of deformability-inducedlift force is towards the center of the channel under most
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operating conditions, owing to the parabolic velocityprofile.62 This lift force is attributed as a factor responsiblefor the cell-free layer in channel flow of red blood cells evenwith negligible fluid inertia.63 In inertial flow, deformability-induced lift leads to equilibrium positions closer to thechannel center for deformable particles in comparison withrigid particles61 (Fig. 5B). If the force balance is calibratedprecisely this effect can be used to measure the mechanicalproperties of the suspended cells as it correlates with equilib-rium position in a microchannel.61
Furthermore, particle deformability can also affect thefocusing dynamics and the velocity of lateral migration ininertial flows. Center-directed deformability-induced lift com-bined with wall-directed shear-gradient lift leads to an overallwall-directed lift that is lower in magnitude. This would leadto slower migration to equilibrium positions which wasreported in recent studies that found that the migration ofcancer cells (which are deformable) required a longer chan-nel length than for rigid beads of the same size.35
ii. Effects of particles on flow
The two main elements of particle-laden fluidic systems arethe fluid and the particles with several possible interactionsbetween them. There have been multiple studies on the effectof the fluid on particles, and we also discussed the interac-tions between particles. The other important interaction isthe effect of particles on the fluid flow which we discuss here.In general, particles significantly perturb the fluid aroundthem and cannot be considered passive components of achannel system.
a) Reversing streamlines
Reversing streamlines are one of the distinguishing featuresof the flow field around a freely-rotating particle placed in ashear flow with finite Reynolds number. By reversing stream-lines we refer to fluid that approaches and then moves awayfrom a particle in its reference frame (Fig. 6). Normally onewould expect fluid to be diverted around a rigid particle andreturn to a similar location upon passing the particle andcontinue in the same direction as shown in Fig. 6A, however,under conditions described in this section the fluid canchange flow direction near the particle and form reversingstreamlines as shown in Fig. 6B. The reversing streamlinescan lead to repulsive particle–particle interactions.21,73,74
Acrivos et al. showed that streamlines near a cylinder in asimple 2D shear flow differs in the case of inertial flow andStokes flow.75 The region of closed streamlines diminisheswith increasing Reynolds number and reversing streamlinesappear due to fluid inertia (Fig. 6A, B). Later they experimen-tally confirmed the presence of reversing streamlines arounda cylinder and a sphere.76 Similar reversing streamline forma-tion for finite Reynolds number flows were also found inlater studies.74,77–79 Fluid inertia leading to reversing stream-lines also resulted in breaking down of closed stream linessuch that fluids around the particle spiral outwards
Fig. 6 Effect of particles on flow. (A) There will be only closed streamlines near a particle in a simple shear flow with no confinement (i.e. noreversing streamlines) under Stokes flow conditions.78 (B) However, the addition of fluid inertia to the system (without confinement, unboundedflow) leads to creation of reversing streamlines and saddle points near the particle.78 (C) Channel confinement, without fluid inertia, also leads tothe similar creation of reversing streamlines. Interestingly, the reversing flow fore and aft of the particle are somewhat mirrored on the other halfof the channel.21 (D) Real systems usually include both inertia and confinement, which inevitably create the reversing streamlines as well.(E) Presence of reversing streamlines has been experimentally demonstrated for very low Re as well as finite Re flows.21 (F) The reversingstreamlines act as a repulsive mechanism between particle pairs and contribute to their inertial ordering.21 (G) Inertially focused particles createstrong net secondary flows in the channel, as they rotate and flow downstream.56 (H) The presence of particle-induced convection has also beendemonstrated experimentally with applications for fluid mixing.56 (I) Type, direction and strength of the induced secondary flows by the particlehighly depend on lateral position of the particle. Therefore, inertial focusing plays an important role in the output of the induced secondary flow.56
(Fig. 6A, B). The spiraling streamlines are expected toenhance mass and heat transfer.78
Zurita–Gotor et al. suggested another mechanism ofreversing streamline formation that does not require fluidinertia.73 Numerical studies in a confined geometry revealedthat the presence of a wall near a particle leads to breakingof closed streamlines and generation of open and reversingstreamlines. Reflection of the stresslet flow at the near wallleads to this disturbance and lateral forces that can act onanother nearby particle. In Stokes flow, this results in twoparticles moving along streamlines with swappingtrajectories. Note that the stresslet disturbance flow decayswith 1/r2 such that this is expected to be a short rangeinteraction.
Reversing streamlines can also be found in inertial micro-fluidic systems in which both fluid inertia and confiningwalls are present.21 Microfluidic channels typically have rect-angular cross-sections and the reversing streamlines arealmost identical to those in a simple shear flow, except thesymmetry in the vertical direction is broken due to the para-bolic base flow (Fig. 6C, D). As in linear shear flow, reversingstreamlines appear in a channel flow for Stokes flow condi-tions. This formation of reversing streamlines at both finite-Reynolds numbers and zero-Reynolds number was confirmedwith numerical simulation and experiments in microchannelflows (Fig. 6C, D, E).
The reversing streamlines act to create repulsive particle–particle interactions in the process of inertial ordering(Fig. 6F). These particle-induced reversing streamlines initi-ate the self-assembly of ordered structures by pushing parti-cles in transverse directions (vertical direction inschematics, Fig. 6F), which directs particle centers ontostreamlines with opposite flow directions. The subsequentparticle pair's motion resembles harmonic oscillations withdamping. Pair dynamics that lead to self-assembled stateshave several steps. First, particles are directed vertically dueto reversing streamlines and move away from each other.Then, inertial lift forces direct particles back towards thefocusing position. The particles approach but the interac-tions repeat if particles overshoot the focusing position.Note this self-assembled particle system is not in equilib-rium, therefore inter-particle spacing can be tuned with
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Tutorial Box 2 – Practical design rules and considerations for i
While individual particles migrate towards focusing positions, particle induceparticle interactions decay rapidly (~1/r2), thus the effect is local and more easrandom particle positions lead to faster particles catching up to slower particinteractions before they migrate to the focusing positions even at low concentrainterparticle spacing (inertial ordering). The interparticle spacing has been showParticles are forced to rearrange in the expansion channel due to inter-particleparticle spacings become larger due to mass conservation. In principle, the ratshould be an inverse of the ratio of channel cross-section area. However, pardepending on local particle concentration, expansion channel length and Rethere is a lack of void spaces that neighboring particles can occupy. Note thesize by the following formula: ds = a(2 − φ)/φ if particles are focused in a singleor completeness, of the inter-particle spacing to fill voids.
simple expansion–contraction channels.21 Since the interac-tion strength of the reversing streamlines increases rapidlyas particles approach each other in the expansion, particlesare pushed off the focusing position again and find a newself-assembled state which is further expanded upon reduc-tion of the channel cross-section downstream. The capabil-ity to achieve inter-particle spacing control, in addition toinertial focusing, allows particle manipulation in all direc-tions, which can be useful for applications such as flowcytometry and controlling cell numbers for tissueprinting.80
In this section, we focused on inertial microfluidic phys-ics and the effects of reversing streamlines in such systems,however, reversing streamlines appear in a broader range ofscenarios in which there is Stokes flow with confinement,and particle–particle interactions due to reversing stream-lines should be widely considered for any two-phase flows inmicrofluidic channels. Table S2† summarizes our currentunderstanding of these particle-induced disturbances andthe need for additional numerical and analytical under-standing especially for how they contribute to particleordering.
b) Particle-induced convection
Particles flowing in confined channels with finite inertia notonly produce reversing disturbance flows that occupylocations up and downstream of the particle but also inducenet secondary flows in the channel cross-section. This effect,termed “particle-induced convection,” results in the creationof a set of net flows that span across the channel from afocused particle and recirculate around the top and bottomof the channel to conserve mass56 (Fig. 6E). The net second-ary flow, which is not present in a confined particulateflow in Stokes flow, is mostly created by flow differencesupstream of the particle for inertial flow. The intensity ofthis mode of lateral fluid transport scales strongly with par-ticle size (~a3), exhibits an inertial scaling with flow velocity(~U2) and linearly scales with fluid density. Numerical studyof the physically constrained system suggests that bothtranslational and rotational components of particle motionaffect the magnitude of the net secondary flow: increasing
d disturbance flows can influence motion of other particles. These inter-ily noticeable in flows with high particle concentrations. However, initiallyles, which means that a large portion of particles experience inter-particletion. Inter-particle interactions in inertial microfluidic systems lead to regularn to be tunable with a simple expansion–contraction channel geometry.21
interactions. Then when the particles enter the contraction channel inter-io of inter-particle spacing before and after an expansion–contraction channelticles in expansion channels may not rearrange with sufficient regularity: high particle concentration (e.g. φ > ~25%) can limit tunability becausefinal inter-particle spacing, ds is related to the length fraction and particleline. Repeating contraction–expansion structure can improve the regularity,
Tutorial Box 3 – Practical design rules and considerations for inertial microfluidic systems: particle affecting flow
Particles are observed to considerably affect and disturb the flow around themselves via creation of net secondary flows, an effect referred to as “particle-induced convection”. Particle size, flow conditions and particle concentration are important parameter in the creation of particle-inducedconvection. At a/H > 0.1 the particle is large enough to create strong net secondary flows and disturb the flow field. Flow conditions (including flow rate,channel geometry and particle size) leading to Rp > ~2 lead to significant particle-induced convection. As each particle acts as a dynamic convection site,the effect becomes more prevalent at higher length fractions, e.g. φ > 35% to achieve considerable convection over a few centimeters in a straight channel.This effect is cumulative over channel length, so the overall disturbance of the main flow depends on the distance over which particles have affected it. Forinstance, for 10 μm cells in a 100 × 100 μm2 channel, flowing ~4 × 106 cells mL−1 (i.e. φ = 40%) at 1.5 mL min−1 (i.e. Rp = 2.5) leads to cross-stream motionthat spans the channel ~4 cm downstream.
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with increasing drag force on the particle (i.e. increasing slipof the particle relative to the fluid velocity), and also withdecreasing torque (i.e. increasing particle rotation rate), withthe former having a more significant effect. It should benoted however that these are difficult conditions to physi-cally achieve unless significant body forces act on the parti-cle and for the majority of cases particle motion is passivelyadjusted by the system such that the particle flow force- andtorque-free.
Particle-induced convection can have significant impacton the flow field at Rp > ~2. The behavior is the result of anintricate interplay between geometry and flow conditions –
the direction and magnitude of the flow depend heavily uponfluid inertia, the location of the particle in the channel, parti-cle rotation rate and the extent to which the particle lags orleads the underlying channel flow. As shown in Fig. 6I, if thelateral position of the particle is slightly shifted the second-ary flows created alter drastically, to the point of changingdirection. Consistent with the importance of velocity gradi-ents discussed elsewhere, the weakest secondary flow and areversal in direction are observed when particle is closest tothe center which corresponds with the smallest local velocitydifferences across the particle.
Experimental results confirm that particles occupying iner-tial focusing equilibrium positions have uniform flows thateach act as a dynamic site for mass convection while flowingdownstream and constructively add to effectively transferfluid across channels (Fig. 6H). Consequently, a high concen-tration of particles in the channel, e.g. φ > ~35%, whichequates to more convection sites can also contribute to thedominance of the effect. Interestingly, the particles remainfocused and are not disturbed by the self-induced secondaryflow.56 Therefore, the unique lateral positioning of the parti-cles due to inertial focusing plays an important role in creat-ing a strong directed secondary flow. As a result, at Rp > ~2the particle-induced convection dominates the diffusioncompletely as a means of lateral fluid transport, especially athigh enough concentrations of particles, φ > ~35%. Thismechanism has been used for applications such as simplefluid switching and mixing around beads and cells. Otherpotential applications include enhanced transport in micro-cooling systems and bead surface functionalization. Similarto Dean flow, this secondary flow accumulates along thechannel, so one needs to characterize and time it appropri-ately (via geometrical design, flow conditions, etc.) to achieve
the desired transport rather than rotating the fluid aroundthe channel fully.
iii. Effects of microstructures on flowand particles – flow programming
Thus far, we have focused on how different forms of nonline-arities or particle asymmetry in straight channel flows lead tolift forces and unique flow disturbances. In this section, wediscuss the effects of channel geometric changes on the flowand how these changes of boundary conditions induce arange of secondary flows that can be utilized to control fluidand particles. The deviations from a straight channel can beimposed in different forms such as through channel curva-ture, grooves on channel walls or obstacles within the chan-nel. Theoretical analysis of the perturbations introduced bychannel geometry is generally difficult due to the complex setof varying boundary conditions imposed on the already com-plex and nonlinear Navier–Stokes equations. However, weprovide some insight into the physics of these systems, aswell as present their applications.
a) Curving channels – Dean flow
Secondary flow in curving channels. Dean flow – thesecondary flow induced in curving channels is a well-knowninertial effect which is due to the momentum mismatch offluid parcels at different cross-sections of a channel as theseparcels pass around a curve (Fig. 7A). In this case, fluid par-cels near the center flow faster and have a larger inertiadirected outwards compared to the fluid parcels near thewalls. In order to conserve mass, the fluid near the top andbottom walls move inwards, resulting in a set of secondaryflows. The magnitude and qualitative features of the second-ary flow is characterized by the dimensionless Dean numberDe = Re(H/2R)0.5, where R is the radius of channel curvature.The strength of the secondary flow depends both on thestrength of the underlying downstream flow (Re), as well asthe geometrical ratio H/2R which, intuitively, means that afaster turn in the channel (i.e. smaller radius of curvature R)leads to a stronger Dean flow. As an inertial effect the sec-ondary flow velocity is expected to scale with downstreamvelocity squared as UD ~ De2μ/(ρH)1. Berger et al. also notedthat increases in Dean number are associated with changesin shape of the secondary flow, with centers of the symmetric
Fig. 7 Dean flow and its applications. (A) Dean flow, an inertially-induced secondary flow, is created due to the velocity mismatch of fluid parcelsin a curving channel.85 (B) Dean flow was employed to increase the interfacial surface between two fluid streams to enhance microscale mixing.4
Two major classes of curving channels are generally employed in inertial microfluidic platforms: spiraling channels (C)85 and alternatingasymmetric curving channels (D).2 (E) Spiraling channels have been used for differential focusing and separation of particles.89 (F) Alternatingchannels have been employed for accurate positioning of cells for deformability cytometry.97
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vortices moving towards the outer wall and development ofboundary layers and additional vortices within the cross-section with increasing De.81
Applications of Dean flow. Traditionally, secondary flowsin microfluidic systems have been utilized to achievemixing. Ugaz et al. employed Dean flow to control theinterface of two fluids and perform fast mixing via a split-and-recombine arrangement (Fig. 7B).4 The ability to controlfluid streams with Dean flow has also been used for 3Dhydrodynamic focusing,82 and for creating a variable iner-tially modulated focal length lens.83 More recently, there hasbeen an increasing interest in using curving channels tocontrol particles in microchannels, as the developed second-ary flow enhances the lateral motion of particles across thechannel and alters inertial focusing equilibrium positions byimposing a drag force proportional to the Dean flow velocity,UD (Fig. 7A).2,84,85
The two major roles channel curvature plays in inertialmicrofluidic platforms are: (i) it allows for mixing of the
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flow and particles to sample equilibrium positions faster.Gossett et al. demonstrated that at the sufficiently highDean number (De ~ 17 in their example) distance requiredfor particle focusing is nearly 5 times shorter compared tothe case of a straight channel of the same cross-sectionaldimensions.86 (ii) It allows for unique equilibrium positionsfor particles with different properties due to Rf, the lift todean drag force ratio (FL/FD) varying with particle dimensionsand locations across the channel. Dean drag follows theunderlying secondary velocity field, directed outwards nearthe channel center, and inwards near the wall (Table S1†).Details of how system parameters can affect the equilibriumposition in curved channels can be found in previous work.86,87
Spiral and asymmetric curving channels. The two majorclasses of curving channels used to enhance inertial focusingare spiraling channels, with one direction of curvature(Fig. 7C),85,88–96 and sigmoidal channels with alternatingcurvature (Fig. 7D).2,86,97 A comprehensive set of studies ofinertial focusing in spiral microchannels of varying widths in
Tutorial Box 4 – Practical design rules and considerations for inertial microfluidic systems: inertial focusing in curvingchannels
Because curved microchannels easily perturb flowing particles from unstable positions, they can be used to enhance the speed of lateral particle migrationto stable equilibria. This, however, requires considerable secondary flow in the curved channel. For instance for similar flow conditions, when Re or Deare low (22 or 4), focusing length was reported to be the same. However, at increased Re or De (89 or 17), focusing length of the curving channel(with repeating features) was nearly 5 times shorter than the required length on the straight channel. One important practical advantage of shorterfocusing distance is decreased fluidic resistance, which directly affects the upper limit of flow rate, thus throughput, in a channel. Therefore, the length
required for focusing in curving channels can be obtained by multiplying the straight channel result by a factor such that L f hU a ffm L
2
2 wheref = 0.2–1 depending on the curving channel design.
Furthermore, the ratio of inertial lift and Dean drag (Rf) must be taken into consideration. If this ratio is too small, Dean drag can be so strong asto lead to mixing and disrupt particle focusing. As a result, the following inequality is proposed to be satisfied when designing these systems:
R a RHf 2 0 08
2
3 ~ .
Evidently, if Rf ≫ 1 inertial lift is dominant and the user is not benefiting from the effect of Dean flow to achieve shorter focusing length.
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which Re, De, and a/H were independently controlled hasprovided many of the missing pieces to the overall inertialfocusing parameter space.85,87 This work has highlighted thetransitions in focusing behavior as a function of increasingchannel curvature ratio for Re up to 400, in which top andbottom particle equilibrium positions in a low aspect ratiochannel (here aspect ratio is height/width, where width is inthe direction of channel radius of curvature) shift inwards,yield a single stable equilibrium position and then moveback outwards as the balance between lift and drag changes(Fig. 7A). As the velocity profile becomes flattened for widerchannels, the shear gradient lift in this wide directiondecreases (Fig. 7A.i), as discussed previously. Simultaneously,the Dean drag on the particle increases with channel sizeand becomes stronger in wider channels (Fig. 7A.iii).Therefore, at a critical channel width, Dean drag will becomethe dominant mechanism for lateral migration of theparticles (Fig. 7C). This, as mentioned above, modifies thefocusing behavior of the particle by shortening focusinglength and altering the equilibrium position. Recent workfurther modified the balance between inertial lift and Deandrag within the cross-section of a channel by using a non-rectangular slanted geometry. This had an unexpected effectof creating particle size-dependent equilibrium positions thatswitch quickly with increases in Re allowing larger separationin space within the channel cross-section, which is useful toimprove separation accuracy.95
Note that unlike for spiral channels which approach asteady-state Dean flow, incorporation of alternating curvaturemay prevent the secondary flow from reaching steady stateand leads to more complex inertial focusing behavior as afunction of Reynolds number.85,86 In using the alternatingcurvature design it is important to note that opposing chan-nel segments should be asymmetric, such that the curvature-induced secondary flows will not act counter to each otherand reduce the overall effect of the Dean drag or mixingaction (Fig. 7D).
Spiral and asymmetric curving channels each have uniquesituations to switch they are better suited. Dean flow inspiraling channels can be employed for differential focusingand separation of microparticles (Fig. 7E)88,94,95 which hasbeen used for high-throughput cell cycle synchronization89
and isolation of circulating tumor cells.91 Spiral channelshave achieved higher performance separations than asym-metric curving channels.98 Spiral channels have also beenused for ordering and deterministic cell-in-droplet encapsula-tion.90 Alternating curvature channels were initially intro-duced to reduce the number of focused streams fromstraight channels2 and have also been found to enhance iner-tial focusing (tighter focusing at shorter downstream dis-tances) compared to straight channels.86 These designs areeasier to layout as a segment within a microfluidic circuitbecause they occupy the same footprint as a straight channeland can be easily parallelized in a radial or array layout. Forexample an asymmetric focusing geometry has been insertedinline and used to achieve accurate localization of cells inmicrochannels to enable uniform hydrodynamic stretchingin order to characterize single cells at high rates using“deformability cytometry” (Fig. 7F).97
Density mismatch effect (inertia of particles). Anothermechanism that can lead to cross-stream migration in particu-late systems is the density mismatch between the particle andfluid. If the fluid suddenly changes direction due to changingchannel curvature or structures, the particle may continue fora time on a straight trajectory due to its inertia. For example,particles with larger density than the fluid will experience acentrifugal force directed radially outward in its frame of refer-ence when the channel curves. Because of this relatively sim-ple physical mechanism, particle migration in spiralingchannels can often be understood mistakenly as an effect ofcentrifugal force due to density mismatch rather than Deanflow. The Stokes number, Stk = (2ρp + ρ)a2ΔU/36μdc (where ρpis density of the particle, ΔU is the velocity change and dc isthe length scale of the change), is a dimensionless parameter
that compares the flow timescale – a distance divided by veloc-ity difference – to the particle acceleration time scale, andprovides a measure of the fidelity of the particle's ability tofollow fluid streamlines which is especially important inparticle image velocimetry (PIV). However, this number is usu-ally quite small (Stk < 0.01) in microfluidic flows and eveninertial microfluidic flows (unless there is an extremely largedensity mismatch or abrupt changes in velocity). As a result,fluid inertia, particle shape, or particle deformability mostoften dominate effects from particle inertia.2
b) Grooves and herringbones on channel walls
Addition of grooves and herringbones onto the wall surfacesof straight microchannels has also been known as a simplemeans for inducing strong secondary flows which was intro-duced in the seminal work of Stroock et al. (Fig. 8A).99 Thesesecondary flows may be used instead of Dean flow to modifyinertial focusing behavior,19 however, most of such systemshave been operated in a low-Re flow regime, where inertia istypically negligible. In Stokes flow, in order for the phenome-non to be effective, the grooves/herringbones must lack fore-aft symmetry otherwise, based on the MSTR theorem, thereshould be no net deformation to the fluid flow as it passesthe groove which explains the need for slanted structures inthese systems.
Applications in which herringbone-induced flow deforma-tions are used provide a good starting point for identifyingapplications of inertia-induced secondary flows and flowdeformations. The herringbone structure has been used tocreate hydrodynamically focused sheath flow100 to performflow cytometry101 (Fig. 8B). Numerical simulations also sug-gest that these systems can be used for size-based sorting ofmicroparticles in combination with inertial lift forces.19 Inthis scheme, particles of different sizes are inertially focusednear the top and bottom walls in a low aspect ratio channel,where smaller particles have equilibrium positions closer tochannel walls. Periodical ridges on top and bottom walls gen-erate two symmetrical secondary flows. Due to their differentinitial z-positions by inertial focusing, larger and smaller par-ticles are forced to move in opposing y-directions by the sec-ondary flows, resulting in their distinct lateral locations. Stott
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Fig. 8 Herringbone structures. (A) Herringbones are known to induce stroinduce chaotic mixing in microchannels.99 (B) This phenomenon can be usefocused sheath flow for flow cytometry.101 (C) The secondary flow createdinteraction and capture cells on surfaces.102
et al. have also employed the microvortices created with thesestructures to mix blood, allow cells to better sample andinteract with surfaces, and effectively isolate circulatingtumor cells.102 However, similar to most other secondary flowinducing mechanisms, this technique has most widely beenutilized to achieve fast mixing.99,103
c) Pillars and other structures
With the addition of fluid inertia, even symmetric structuresin channels – like circular pillars – can create net secondaryflows. Lauga et al. previously showed that flow in a channelwith varying cross-section is three-dimensional even in Stokesflow,104 however, the secondary flows are completely reversedand there is no net deformation when the channel cross-section returns to its original shape symmetrically down-stream (Fig. 9A). With finite inertia symmetric changes inchannel geometry, such as the presence of a circular pillar,yield a net deformation of fluid streamlines instead.105 Inbrief, pillars inside a channel can induce a net secondary flowwhich causes the fluid to inertially deform around theseobstacles and not return to its original configurationdownstream.
We demonstrated, with numerical simulations and experi-ments, that the pressure gradient created by the presence of apillar irreversibly pushes fluid parcels laterally which lead tocreation of a set of secondary flows. This behavior has featuresin common with Dean flow. However, in contrast with Deanflow, here one can use lateral position of the pillar to tune thelateral location of the secondary flows. Furthermore, whileDean flow presents a steady-state behavior along the curva-ture, this phenomenon creates a net motion with additionalcomplexities. Importantly, four dominant modes have beenidentified for this phenomenon based on the number anddirection of the secondary flows (Fig. 9B) which is determinedbased on flow conditions and channel geometry. For instance,Mode 1 occurs in low aspect ratio channels and Re ~ <40. Forthis mode, the fluid parcels near the channel center deformoutwards away from the pillar while, to conserve mass, thefluid parcels near the top and bottom of the channel moveback towards the pillar center, creating a symmetric set of netflow deformations inside the channel (Fig. 9A).
ng secondary flows in a Stokes flow regime, and have been utilized tod to manipulate the fluid, for instance for creation of hydrodynamicallyby these structures can also be used to manipulate particles to increase
Fig. 9 Inertial flow deformation near pillars.105 (A) Centrally located cylindrical pillars in a straight channel lead to irreversible deformation of fluidstreams in inertial flow. This creates a set of net secondary flows inside the microchannel. (B) There are four dominant modes of operation for thisinertially induced secondary flow, depending on channel geometry and flow conditions. (C) Lateral position of the secondary flow depends on thelateral position of the pillar. (D) Sequences of pillars can be used for sculpting of fluid streams to create engineered shapes within the channelcross-section. (E) The secondary flows can also be combined with inertial focusing to manipulate particles and fluids, for applications such as fluidexchange.
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If the pillar has a circular cross-section and is located inthe middle of the channel, dimensional analysis suggeststhat three non-dimensional groups are needed to fullydescribe the system. These independent non-dimensionalgroups were chosen as D/w, h/w and Re, where D is the pillardiameter. We observed inertial flow deformation to increasein magnitude as Re increases when the pillar diameter iscomparable with channel width, and magnitude alsoincreases strongly with D/w. Besides increasing magnitude,the overall topology of the secondary flows depends on thesegeometric and flow parameters. Four dominant modes ofoperation occur for different combinations of these non-dimensional groups, which are categorized based on thenumber of net secondary flows they induce in a quadrant ofthe channel and the directions of these flows (Fig. 9B). Wefurther showed that by simply having the pillar at a differentlocation across the channel, the location of the net secondaryflows can be tuned (Fig. 9C). For instance, as shown inFig. 9C, if we have two half-pillars on the sides of the channel(instead of a full pillar on the center) the net secondary flowsshift a half-width to induce a total flow that is opposite to theflow for a centrally-located pillar.105
The local ability to deform flow along with the ability tosequentially apply these deformations leads to an ability toperform more arbitrary manipulations on fluids. It is possi-ble to obtain an accurate cross-sectional transformation func-tion for a given pillar (size, shape and position) in a specificchannel (width and height) and with given flow conditions(fluid properties and flow velocity). If the pillars in a channelare separated by a large-enough spacing, such that their flow
fields are independent, their singular transformation func-tions can then be sequentially combined to generate a fastand accurate solution to the total flow deformation inducedin a flow (i.e. total lateral displacement of each fluid parcel)by a specific series of pillars. The ability to use this approachto sculpt fluid flow has been demonstrated in our study(Fig. 9D) where we were able to create different classes ofnon-trivial cross-sectional shapes from a simple sheath flow.A plethora of applications for this newly introduced paradigmto sculpt flows is anticipated. For instance, the combinationof this effect with inertial focusing has been used for solutionexchange around microparticles (Fig. 9E). Other applicationsinclude stream splitting, particle size-based separation andfluid mixing,105 while applications such as fiber-genera-tion,106 3D particle fabrication and gradient generation arenatural next steps.
In general, these studies suggest that the presence of anytype of irregularity in the channel could lead to net secondaryflows which could be used to control fluid or particles. Theeffects of the varying channel cross-section also are apparentin sudden expansion–contraction channels leading to second-ary flows that act in addition to inertial lift.107–109 In themulti-orifice flow fractionation (MOFF) technology, inertialfocusing is combined with the presence of multiple suddenexpansion–contraction regions that lead to hydrodynamiccross-stream motion of particles.107 Jung et al. further devel-oped multistage MOFF devices for continuous size-based sep-aration of spherical particles110 and used the techniquecombined with dielectrophoresis to separate rare breast can-cer cells from blood.111 Han et al. combined inertial focusing
Tutorial Box 5 – Practical design rules and considerations for inertial microfluidic systems: structures affecting flow
Structures, such as pillars, or even minor channel expansions or contractions (i.e. pillars on the side walls or weirs on the top or bottom surfaces), can beused to manipulate the flow. It is useful to think of these systems in terms of non-dimensional groups, mainly: Re, h/w (channel aspect ratio) and D/w (nor-malized pillar dimension), The two major concepts to think about when designing these systems are: (i) what mode of secondary flows are being inducedby the pillars, and (ii) how strong the secondary flows are.
There are four major modes of secondary flows reported when utilizing cylindrical pillars in straight channels. The phase diagram provides an estimate ofthe expected mode of operation for a certain set of parameters in the system. Modes 3 and 4 mostly occur in tall channels, which are typically moredifficult to prototype. It is also worth noting that modes 2 and 4 have two sets of secondary flows that make these modes more complicated.
The magnitude of the secondary flows is highly dependent on the pillar size. A larger D/w (i.e. a larger pillar) leads to a stronger secondary flow, with a ruleof thumb of D/w > 0.3 to achieve motion that would span the channel with less than 10–15 pillars. Also, Re > 10 is recommended to achievesignificant lateral displacement per pillar. For instance, in a 50 × 200 μm2 channel (i.e. h/w = 0.25) operating in mode 1 (i.e. pushing fluid away from thepillar at the channel center and back towards the pillar near the walls) with 100 μm pillars (i.e. D/w = 0.5) deformation that spans the channel widthfollowing ~15 pillars can be observed atQ> 150 μLmin−1, corresponding to Re> ~20.
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with similar structured channels in which the secondary flowalong with steric pinching of larger cells allowed separationof larger rare cells from blood cells.108 Operating in Mode 1structures introduced in these works would lead to a net cen-ter directed secondary flow that is useful in mixing and cen-tering larger cancer cells at inertial lift equilibrium positionsat the channel centerline. A limitation of this approach, likeother inertial focusing-based approaches, is that high cellconcentrations prevent accurate separations. Herringbonestructures also offer a unique ability to deterministicallymanipulate flow and shape fluid cross-section, especially atlow-Re and Stokes flow which is starting to be explored andapplied.100
iv. Conclusions and perspectives
In this work we reviewed a variety of mechanisms by whichsystem nonlinearity (mainly inertia) and asymmetry in chan-nel or particle geometry can be used to control particles andflows in microfluidic platforms. Importantly, based on accu-mulating evidence we conclude that generation of lift or sec-ondary flows is not the exception, but the rule for any
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perturbation in channel geometry or particle propertiesbeyond the simplest cases especially in the inertial flowregime (see Table S1† and S2 for summaries). In other words,any type of irregularity will give rise to some form of lift orsecondary flow. Although most of the applications of suchsecondary flows in microsystems to date have focused on“mixing” and creating chaos, we suggest that due to theirdeterministic nature in laminar flow these techniques can beexploited to move in the opposite direction: to induce orderinto an otherwise random system by utilizing flows transverseto downstream fluid streamlines to offer a strong tool forcontrollably positioning fluid parcels and particles insidemicrochannels. The ability to make the most of these effectswill require libraries of pre-simulated structured channels foraccurate and fast numerical prediction of total transforma-tions. We have already demonstrated such a capability forpillar-induced flow deformations112 and similar strategies areapplicable for any structured channel such as herringbonesand curving channels.100 We anticipate that the complimen-tary use of these different types of structures should enableunprecedented 3D control of a fluid stream and engineeringits shape inside channels.
Inertial microfluidics is still a nascent field despite theincreasing number of research articles published eachyear. Even though a basic tool kit of operations that can beperformed on particles and fluids exists, the underlyingnonlinear fluid physics is extremely rich with a multitudeof fundamental directions that are still unexplored. Forinstance, the physics of how different modes of operationare created in pillar-induced secondary flows is yet to beunderstood, predictions for how combinations of Deandrag and lift forces interact within a channel cross sectionrequire painstaking experiments, and unexpected changes inmodes of inertial focusing in straight channels alone asRe increases require improved physical intuition. Channelirregularities and changes in boundary conditions that createflow deformations offer a vast number of other areas tobe explored.
Emerging applications of this field are being discovered ata fast pace and are only recently finding their way to translatefrom academic research settings into real-world and commer-cial devices. Examples include the “CTC-iChip” from HarvardMedical School and Massachusetts General Hospital which isen route to be commercialized presumably by Johnson &Johnson,33 and uses inertial focusing to sort rare CTCs fromwhole blood at 107 cell s−1. Another example is themechanophenotyping platform developed at UCLA and beingcommercialized by CytoVale, which employs inertial focusingalongside deformability cytometry to offer a new class ofbiomarkers. By measuring a range of biophysical cell markersat rates above 2000 cells s−1, CytoVale could offer a high-throughput, low-costmethod of detecting disease.113 ClearbridgeBioMedics, a spin-off from National University of Singapore,is also using inertial focusing in spiral channels to separatelarger cancer cells from other blood components in theirClearCell FX system.91 Still a better fundamental understand-ing of all these systems will be necessary to enable the inte-gration into more sophisticated and useful platforms towardsa range of novel applications in biomedicine, industrial filtra-tion, and beyond.
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