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Cite this: Lab Chip, 2013, 13, 1121

Fundamentals of inertial focusing in microchannels

Received 13th November 2012,Accepted 3rd January 2013

DOI: 10.1039/c2lc41248a

www.rsc.org/loc

Jian Zhou and Ian Papautsky*

Inertial microfluidics has been attracting considerable interest in recent years due to immensely promising

applications in cell biology. Despite the intense attention, the primary focus has been on development of

inertial microfluidic devices with less emphasis paid to elucidation of the inertial focusing mechanics. The

incomplete understanding, and sometimes confusing experimental results that indicate a different

number of focusing positions in square or rectangular microchannels under similar flow conditions, have

led to poor guidelines and difficulties in design of inertial microfluidic systems. In this work, we describe

and experimentally validate a two-stage model inertial focusing in microchannels. Our analysis and

experimental results show that not only the well-accepted shear-induced and wall-induced lift forces act

on particles within flow causing equilibration near microchannel sidewalls, but the rotation-induced lift

force influences the position of these equilibria. In addition, for the first time, we experimentally measure

lift coefficients, which previously could only be obtained from numerical simulations. More importantly,

insights offered by our two-stage model of inertial focusing are broadly applicable to cross-sectional

geometries beyond rectangular. With elucidation of the equilibration mechanism, we envision better

guidelines for the inertial microfluidics community, ultimately leading to improved performance and

broader acceptance of the inertial microfluidic devices in a wide range of applications, from filtration to

cell separations.

Introduction

Inertial microfluidics has been attracting considerable interestin recent years due to the promising applications ranging fromfiltration1–4 to separation5–10 to cytometry of cells.11–13 As apassive technique, it manipulates cells and particles inmicrochannels without an externally applied field, andcombines the benefits of a passive approach with extremelyhigh throughput. In this phenomenon, cells and particlesmigrate across streamlines and order deterministically atequilibrium positions near channel walls. This behavior is dueto the inertial forces, which are typically neglected in themicrofluidic low Reynolds number flows. In straight channels,inertial migration is believed to be caused by balance of liftforces arising from the curvature of the velocity profile (theshear-induced lift) and the interaction between particle andthe channel wall (the wall-induced lift) as illustrated in Fig. 1a.In square microchannels, the inertial migration of cells orparticles leads to focusing in four equilibrium positionscenters at the faces of the channels (Fig. 1b).2,3,14–16

Despite the recent interest, the fundamentals of particleordering in microfluidic systems remain to be elusive.17

Indeed, the majority of work to date has focused on devicesand applications. The limited number of studies discussing

the fundamental mechanisms have in fact introduced anumber of inconsistencies. For example, in square channels,different particle behaviors have been described2,3,16,18 andnot only four but also eight equilibrium positions have beenreported by several groups.2,3,14,15 Even further migration intotwo equilibrium positions has been recently observed inrectangular microchannel.18 These observations cannot beexplained from the current understanding of inertial focusing(i.e. balance of two lift forces), which dictates that once particleequilibrates its position should be stable. This incompleteunderstanding of the mechanism for particle migration inmicrochannels stems in part from the difficulties in investi-gating forces that act on particles. Even on the macroscle, themajority of investigations reported to date are numerical innature. This absence of underlying principles can lead todifficulties in design of devices and impede their develop-ment.

In this work, we experimentally investigate forces acting onneutrally-buoyant particles flowing through a microchannel.We re-examine the forces that are responsible for particleequilibration and for the first time experimentally measure thelift forces that cause particle equilibration in rectangularmicrochannels. Previous work by Chun and Ladd14 proposedand showed numerically that particle migration in squarechannels occurs in two stages. Herein, we experimentallyconfirm that particle migration occurs in two stages, eachdominated by a different lift force balance, and show that the

BioMicroSystems Lab, School of Electronic and Computing Systems, University of

Cincinnati, Cincinnati, OH 45221 E-mail: [email protected];

Fax: +1 (513) 556-7326; Tel: +1 (513) 556-2347

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same mechanism is applicable to rectangular microchannels.This work will improve the understanding of the underlyingmechanisms of particle inertial migration and will offer auseful guide in the development of inertial microfluidicsystems.

Physics of inertial migration

Inertial migration of particles was originally discovered incylindrical pipes in 1960s, when Segre and Silberberg observedthat randomly dispersed y1 mm diameter particles formed anannulus in a 1 cm diameter pipe.19,20 With advent ofmicrofluidics, observations of the same phenomenon wereconfirmed in microchannels in the recent years.17,21,22

Particles flowing in a microchannel are subjected to two

dominant forces: the viscous drag (FD), which entrainsparticles along streamlines, and the inertial lift force (FL) thatleads to migration across the streamlines. Matas et al.discussed two components of inertial lift that together act toyield an equilibrium position between the channel wall andthe centerline.16,21,22 A wall-induced lift force Fw that acts upthe velocity gradient away from the wall toward the channelcenterline, and a shear-induced lift force Fs that acts down thevelocity gradient toward the channel walls (Fig. 1a). It is thenet lift force, a balance of the shear-induced and wall-inducedlift forces, that is responsible for particle migration into anannulus y0.2D (diameter of the pipe) away from the wall.

Early theoretical investigations into inertial migration ofparticles have identified that for particles of diameter a in achannel of hydraulic diameter Dh, the net lift force scales as FL

3 a4.23 To arrive to a theoretical prediction, Asmolov23

introduced a non-dimensional lift coefficient (CL) to relate FL

to its dependent variables, such that

FL = CLG2ra4 (1)

where r is fluid density and G is the shear rate. The liftcoefficient is a function of the Reynolds number (Re) and hasbeen shown to decrease with increasing Re.24,25 Consideringthat the shear rate is a function of average flow velocity and Uf

and channel dimension (G = 2Uf/Dh),23 the net lift force can beexpressed as

FL~4rCLUf

2a4

3pmDh2

(2)

The lateral migration velocity of particles (UL) can then becalculated by balancing the inertial lift with Stokes drag (FD =3pmaUL) to arrive to an expression.2,3

UL~4rCLUf

2a3

3pmDh2

(3)

Thus, particles migrate a lateral distance toward equili-brium position that is proportional of the flow velocity and thedownstream position.

Recent work by Di Carlo et al.16 has shown that lift forcescaling is dependent on the particle position in the channel,suggesting that disparate fluid dynamic effects act to createthe inertial lift equilibrium positions. Motion of particles nearthe microchannel centerline is dominated by the shear-induced lift Fs due to the vorticity v around particlesurface.24,26,27 The direction of this force is toward channelwalls, determined by the cross-product of the vorticity and therelative velocity Ur (Fs = v 6 Ur). We note here that the relativeparticle velocity Ur is the difference between the particlevelocity and flow velocity, as particles lag behind theflow;21,28,29 the direction of Ur is opposite to the direction ofUf (if we assume as simplified 2-D model; the more complex3-D case is discussed by Loth and Dorgan26). Early work withnumerical models has shown that Fs is strongly dependent on

Fig. 1 Inertial focusing in rectangular microchannels. (a) Two lift forcesorthogonal to the flow direction act to equilibrate microparticles near wall. Theshear-induced lift force Fs is directed down the shear gradient and drivesparticles toward channel walls. The wall-induced lift force Fw directs particlesaway from the walls and drives particles toward the channel centerline. Thebalance of these two lift forces causes particles to equilibrate. (b) In squarechannels, at moderate Re randomly distributed particles focus into fourequilibrium positions at the wall centers. (c) Fluorescent images illustratingmigration of 20 mm diameter particles toward microchannel center at Re = 30(100 mm wide 6 27 mm high cross-section). (d) In a low aspect ratiomicrochannel at moderate Re, microparticles first migrate from the channel bulktoward equilibrium positions near walls under the influence of the shear-induced lift force Fs and the wall-induced lift force Fw. Then, particles migrateparallel to channel walls into wall-centered equilibrium positions under theinfluence of the rotation-induced lift force FV.

1122 | Lab Chip, 2013, 13, 1121–1132 This journal is � The Royal Society of Chemistry 2013

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particle diameter and scales as Fs 3 a2.26,30 As particlesapproach microchannel walls, a wall-induced lift force Fw

arises due to proximity to walls.25–27,31–33 Direction of thisforce can also be described by as a cross-product of thevorticity and the relative particle velocity vectors Fw = v 6 Ur,as both lift forces act orthogonal to channel walls. Numericalmodels by Zeng et al.24,27 show that vorticity near walls is inthe direction opposite to that of the shear induced lift force,causing the wall-induced lift force to act away from walls(Fig. 1d). Williams et al. showed that the wall-induced lift forceFw exhibits even stronger dependence on particle size(compared to the shear-induced lift) and the distance to walld, scaling as Fw 3 a3/d.34

While the balance of the two lift forces can successfullyexplain particle focusing in a round duct, square andrectangular channels present a more complex situation dueto radial asymmetry. The expressions above could be stillapplied to estimate the FL and UL by defining hydraulicdiameter Dh = 2WH/(W + H) for a channel W wide and H high.Since the shear-induced lift causes particles to migrate awayfrom the channel center, down the shear gradient toward thechannel wall, one would expect particles to equilibrate alongthe perimeter of the channel (including corners) in order toachieve force balance. However, work in microfluidic channelshas identified four distinct focusing positions centered at eachface in square microchannels, as illustrated in Fig. 1b.2,3,16,35

The absence of particles in corners suggests that additionallateral migration effects take place near channel walls thatcause particle migration toward wall centers.

Indeed, the early work by Saffman36 proposed a rotation-induced lift force FV. This Saffman lift force plays animportant role near channel walls, but was considerednegligible compared with the shear-induced force whileparticles are far away from the channel wall.27,30,31,33,36

Recent numerical simulations by Dorgan et al.37 haveconfirmed that in most flows particles are more likely toexperience a shearing behavior. Consequently, most descrip-tions of inertial flows do not consider this lift force.

While negligible away from the wall, we believe that therotation-induced lift is significant at the channel wall and canhelp explain the asymmetric equilibrium positions in rectan-gular microchannels. Cherukat and Mclaughlin33 have shownthat the effect of rotation is very small but becomes importantwhen the shear is large and particle is close to wall. Due to theparabolic nature of the Poiseuille flow, the shear rate increasesdramatically near the wall, which satisfies Cherukat andMcLaughlin’s conclusion and lead us to believe that therotation-induced lift is critical. Rubinow and Keller,38 andmore recently Loth and Dorgan,26 proposed that the rotation-induced lift scales with particle diameter as FV 3 a3.Analogous to the other lift forces, the direction of this forceis determined by the cross-product of the rotation and therelative particle velocity vectors FV = V 6 Ur as illustrated inFig. 1d.26 Since the rotation is due to shear28,29 and theneutrally buoyant particles always lag behind flow velocity inPoiseuille flow,21,28,29 this rotation-induced lift acts against the

flow velocity gradient and is directed toward the center face ofthe channel wall. In this case, particles near the channel wallwould further migrate to the center of the wall, which is inagreement with the experimental observations in square andrectangular channels. Indeed, our results in this work alsosuggest that particle spinning is dominant in case of a wallregion where shear and wall forces cancel each other.

From the introduction of rotation-induced lift force, a morecomplex model of inertial migration at finite Re emerges,illustrated in Fig. 1d. In rectangular channels, particles firstmigrate from the channel bulk toward equilibrium positionsnear walls. Then, particles migrate parallel to channel wallsinto wall-centered equilibrium positions. Particles in thecorner can migrate along either of the two nearby wallsdepending on the local shear rate,33 which suggests that bothchannel aspect ratio (AR = height/width) and Re can modify thefocusing positions in rectangular channel since both of themcan alter the magnitude of local shear rate close to walls. Morespecifically, if the AR y 1 or at high Re, the four equilibriumpositions emerge even in rectangular channels.6,18

Nevertheless, in low aspect ratio channels (AR % 1), particleswill focus into two equilibrium positions centered at the topand bottom walls, as demonstrated in Fig. 1c. Similar behaviorcan be observed for high aspect ratio channels (AR & 1), withtwo equilibrium positions developing at the center of side-walls.

In summary, the inertial migration of particles in micro-channels follows the balance of the lift forces and occurs intwo stages. Both shear and wall induced lift forces are due tothe vorticity around particle surface, dominating particlemigration toward channel walls (stage I). Once the initialequilibrium is reached, near channel walls particle motion isdominated by the rotation-induced lift force. As a result,particles migrate to the center points of walls (stage II). Thisnew model of inertial focusing is generally applicable torectangular microchannels of any aspect ratio at finite Re, andcan be used to aid design of inertial microfluidic systems. Inthe subsequent sections, we discuss our experimental valida-tion of this two-stage model and for the first time measure theassociated lift coefficients.

Experimental approach

The lift forces are characterized by their associated liftcoefficients, with the sign of coefficients representing direc-tion of the lift force. For particles flowing in a microchannel,the sign of the lift coefficient due to the balance of the shear-induced and wall-induced lift forces is negative, as particlesmigrate away the channel centerline and orthogonal to thechannel wall.3,23 Conversely, the lift coefficient due torotational is positive, which implies lift up the velocity profileand parallel to channel wall.30 Herein, the two coefficients aredenoted as CL

2 and CL+ and the corresponding lift forces are

FL2 for Fs (since Fs and Fw act against each other) and FL

+ forFV respectively.


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The channel length L required to fully focus particles can bewritten as2,3

L~UmLm

UL~

3pmDh2Lm

2rUf CLa3(4)

where Um is the maximum flow velocity (Um = 2Uf) and Lm isthe particle migration distance. Thus, the lift coefficient isgiven as

CL~3pmDh

2

2rUf a3|

Lm

L(5)

Therefore, as long as we are able to determine the particlemigration distance Lm and the focusing length L, the liftcoefficient can be obtained.

In this work, we use microchannels with rectangular cross-section for experimental determination of the lift coefficientsbecause of the two equilibrium positions that emerge atcomplete focusing. In a low aspect ratio channel, randomly-distributed particles will rapidly migrate and equilibrate nearthe top and bottom walls under the influence of negative liftFL

2, and thus the migration distance is half of the channelheight length (Lm

2 = H/2). This initial stage I focusing can beobserved experimentally using a high aspect ratio channel(essentially rotating channel by 90u), an approach we success-fully used in the past.2,3 In stage II, particles migrate towardthe center of the top and bottom walls under the influence ofthe positive lift FL

+ and the transportation length is approxi-mately half of the width (Lm

+ = W/2). The stage II focusing inlow aspect ratio channels can be observed directly with aninverted microscope. The same approach can be extended tohigh aspect ratio channels, by appropriately switching the Hand W. Experimental determination of the downstreamfocusing length can then be used in calculation of theappropriate lift coefficients using eqn (5) (i.e. CL

2 from L2,and CL

+ from L+). Since lift coefficients are a function ofparticle position within the flow, the measurements in thiswork are averaged along the migration direction. In thesubsequent sections we discuss experimental measurementsof both positive and negative lift coefficients and theirdependence on the flow parameters and microchannelgeometry.

Determination of the negative liftcoefficient

Particles in high AR channels focus near sidewalls as long asthe ratio of particle size to channel hydraulic diameter ismaintained a/Dh . 0.07.2,3 We first used small particles (a/Dh =0.11) which exhibit longer focusing length to minimizemeasurement errors. Fig. 2a shows a suspension of particlesat the inlet (L = 0 mm), spanning the entire channel width. At L= 9 mm downstream, particles order into two streams nearsidewalls, generating a particle-free region in the center. Thisis in agreement with our previous work.2,3 To determine the

focusing length, we measured fluorescent intensity acrosschannel width at successive downstream positions.

Progressive entrainment of particles as they flow down-stream is shown in Fig. 2b. The intensity of the middle regiondecreases, suggesting depletion of particles and their migra-tion away from the center. The intensity near sidewalls grows,forming two peaks, due to migration of particles towardchannel sidewalls. Quality of particle focusing can be judgedby measuring full width at half maximum (FWHM) of theseintensity peaks. Progressive reduction in the FWHM indicatesprogressively tighter focusing. The downstream position wheremigration toward the channel wall stops and FWHM valuestabilizes is the focusing length. We should note here thatparticle migration along the vertical sidewall does not affectthe peak width in this case. Plotting focusing quality data atmultiple Re as a function of downstream length (Fig. 2c)reveals that complete focusing can be achieved as indicated byvalues approaching particle diameter. Increasing Re leads tolonger channel length needed for particle focusing, suggestinglift coefficient is not constant. Indeed, similar observationshave been reported by others on the macroscale.23,27

At first glance, Fig. 2c appears to be counterintuitive sincelarger the Re (Uf), the higher the lift force acting on theparticle, and hence the shorter channel downstream length forinertial migration. However, the lift coefficient CL in eqn (2) isalso a function of Uf, and thus the relationship between thefocusing length and flow velocity (Re) is more complex.Previous numerical results24,25,27 have shown CL to decreasewith increasing Re, leading to an optimal flow Re that offersthe minimum focusing length. Close examination of Fig. 2cshows that particle migration is not uniform, as FWHM firstdecreases slowly, followed by a sharp drop indicating devel-opment of two peaks near sidewalls. This can be partiallyattributed to the parabolic nature of the velocity profile whichresults in a shear rate lowest in the channel center. Since shearforce is dependent on the second power of shear rate, asexpressed in eqn (1), the lateral migration velocity acceleratesin a quadratic manner which leads to fast migration. Asparticles approach channel walls, the migration velocityreduces exponentially due to the logarithmic increase of thedrag force, and the wall induced lift force counteracting theshear induced lift force.25,39

Further experiments were performed to explore effects ofparticle size which is supposed to modify the focusing length.Four particle sizes, ranging from 7.32 to 20 mm in diameter,were introduced into the same channel at identical flowconditions and their focusing progress is shown in Fig. 2d. Asexpected, large particles migrate much faster due to the largerlift force according to eqn (1). Thus, they require shorterchannel length for focusing. Decreasing particle diameter,however, does not lead to drastic increase in focusing lengthbut rather acts in a more linear way with respect to reciprocalof particle size.

Presenting focusing length as a function of Re illustratesthe non-linear behavior and validates our hypothesis of a non-constant lift coefficient (Fig. 3a). The separation between


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adjacent curves representing particles of different sizeincreases linearly as particle size reduces. Small particles(e.g., 7.32 mm diameter) are more easily affected by the flow Reas the curve is less flat than that of the larger (e.g., 20 mmdiameter) particles. All curves appear to be parabolic in nature,suggesting existence of optimal flow conditions.

Calculating the negative lift coefficient (CL2) based on the

focusing length (eqn (5)) and plotting it as a function of Re

(Fig. 3b) leads to exponentially-decaying curves. Similar trendand magnitude have been reported by others based numericalmodels.25 Presenting our experimental results as a function ofRe20.5 (Fig. 3b inset) shows a strong linear relationship (R2 .

0.99). This is in agreement with the recent proposition by Lothand Dorgan26 that Saffman lift is dependent on Re20.5. Thesame relationship may also be inferred from the earlieranalysis by Asmolov,23 which further validates our results. This

Fig. 3 Focusing length (a) and corresponding negative lift coefficient (b) as a function of Re. The inset shows well-defined linear curves for each particle diameterwhen related to Re20.5.

Fig. 2 Focusing of microparticles in a high aspect ratio microchannel (50 mm 6 100 mm). (a) Fluorescent images at four downstream positions illustrating ordering of7.32 mm diameter microparticles at Re = 120. Dotted lines represent an approximate position of channel walls. (b) Fluorescent intensity line scans at four downstreampositions (7.32 mm diameter particles). (c) FWHM as a function of downstream length for various Re (7.32 mm diameter particles). (d) FWHM as a function ofdownstream position for four particle sizes at Re = 50. In all experiments, 0.025%. volume fraction was used.


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is the first experimental determination of CL2 and it confirms

the speculation and numerical models recently proposed byothers.

Smaller particles exhibit a larger negative lift, suggesting aninverse relationship. The negative lift coefficient for particlesat multiple Re in Fig. 4a presents the experimental data as afunction of a22. Again, low Re exhibits higher CL

2, which isconsistent with our previous observations. From these results,we conclude that CL

2 is proportional to a22 (CL2 3 a22). If

this relationship is used in eqn (2), it yields FL2 3 a2, which is

corresponds to Saffman lift or shear-induced lift.26,30 Thisagreement with numerical results also indicates thatAsmolov’s equation for parabolic flow in macroscale channelsis applicable in on the microscale.

We next investigated the effects of channel dimensions onthe negative lift coefficient. First we fixed the channel aspectratio (AR y 2), but scaled the cross-section and measuredfocusing length for various particle suspensions. From thesemeasurements we calculated CL

2 for each channel (Fig. 4b). Asexpected, the coefficient increased with increasing channelsize. Comparison of the slopes shows that CL

2 scalesapproximately with cross-sectional area. Additional experi-ments which fixed channel width showed that the liftcoefficient varies with channel width (CL

2 3 W2) which isthe smaller dimension, as data in Fig. 4c show. Plotting datafor varying heights (Fig. 4d) shows that the two channels,which have similar width, are approximately the same. From

these results, we conclude that the negative lift coefficient isdominated by the smaller dimension in rectangular channel.This actually has been implied previously by us2,3 and byothers12 when channel width was used instead of hydraulicdiameter as the characteristic dimension of a high AR channel.The predominant effect of width here is related to thecoincidence of migration direction and the higher shear ratealong the small dimension.

Overall, our experimental results show that in high aspectratio microchannels negative lift coefficient is stronglydependent on the particle diameter and channel width, butis only weakly dependent on the flow Re. These experimentstherefore lead us to an expression indicating that the negativelift coefficient scales as

CL{!

W 2

a2ffiffiffiffiffiffi

Rep ,HwW (6)

In concert with eqn (4), this expression permits accurateprediction of channel length for particle lateral migration.According to this expression, we are able to obtain the negativecoefficient in other rectangular channels at various conditionsin terms of particle size and flow rate. As a result, it enables anumber of potential applications taking advantage of theshear force dominated migration, such as guidance indesigning or improving performance of filtration microfluidic

Fig. 4 (a) Negative lift coefficient exhibits inverse square dependence on the particle size (a). (b) Negative lift coefficient as a function of Re at fixed aspect ratio (7.32mm diameter particles). (c) Negative lift coefficient as a function of Re for microchannels with height fixed at H = 50 mm (7.32 mm diameter particles). (d) Negative liftcoefficient as a function of Re for microchannels with width fixed at W = 50 mm (7.32 mm diameter particles).


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systems in high AR channels or accurate evaluation of effectsof shear force acting on cells which are susceptible to shearrate. Next we examine the positive lift coefficient.

Determination of the positive lift coefficient

Particle rotation-induced lift force (positive lift) has not beendescribed in microchannels, yet it is associated with migrationbehavior of particles.14–17 Unlike the negative lift that directsparticles along the velocity gradient, the positive lift guidesparticle migration against the gradient. It is thereforeresponsible for particle migration along microchannel peri-meter toward sidewall centers. The result is the fourequilibrium positions near sidewall centers in a squarechannel or two stable positions along the middle of the longersidewalls in a rectangular channel.

To investigate the effect of positive lift, we performedexperiments in low aspect ratio (AR , 1) microchannels. As inthe experiments described above, we measured fluorescentintensity across microchannel width at successive downstreampositions. The results for 7.32 mm diameter particles (Fig. 5a)show three streams – one clearly-visible, broad stream in thecenter and two weaker streams near sidewalls, whichdisappear after sufficient downstream length. This develop-ment of three streams is a new phenomenon, previously not

observed in low aspect ratio channel. In fact, based on ourprevious work, we expected a single broad band spanningnearly the entire width of the microchannel as we havereported recently.2,3

This new phenomenon may be explained by consideringparticles flowing near microchannel sidewall. These particlesexperience doubled wall-added viscous effect due to twoadjacent walls, which significantly increases drag force.25,27

Hence, the net of the drag force and the rotation-induced liftare much smaller in the corner. On the other hand, thedistance from particle to wall significantly affects the dragforce. According to the numerical models by Zeng et al.,27

particle at an intermediate distance to sidewall experiencessmaller drag as compared to one closes to wall, which impliesa larger net force. As a result, the migration velocity of particlesin the corners is much smaller than those in intermediatepositions, leading to formation of two intermediate side-streams. We found that it is easier to observe these side-streams for small particles than for large ones, since smallparticles are generally closer to walls and remain there longer.Increasing Re leads to similar formation of three streams evenfor large particles (e.g., 20 mm diameter) since particlesgenerally move closer to walls with increasing Re.17,22,35,40–42

Indeed, we have observed two side-streams of 20 mm diameterparticles at Re = 180. Therefore, the interaction of drag forceand rotational lift could result in different migration behavior,

Fig. 5 Focusing of microparticles in a low aspect ratio microchannel (25 mm 6 50 mm). (a) Fluorescent images at four downstream positions illustrating ordering of7.32 mm diameter microparticles at Re = 70. Dotted lines represent an approximate position of channel walls. (b) Fluorescent intensity line scans at four downstreampositions (7.32 mm diameter particles) illustrating progressive focusing in to single a band. (c) FWTM as a function of downstream length at various Re (7.32 mmdiameter particles). (d) FWTM as a function of downstream position for four particle sizes at Re = 50. In all experiments, 0.025%. volume fraction was used.


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depending on particle size and Re. We term this as ‘‘cornereffect’’ in this work.

To determine the focusing length under the influence ofthe positive lift coefficient, we once again measured fluor-escent intensities across channel width at consecutive down-stream positions. The data in Fig. 5b illustrate progressivemigration of particles as they flow downstream. The smallpeaks representing side streams disappear at the focusinglength, as indicated by the single peak at L = 20 mm. Due topresence of side streams, we used full width at tenth ofmaximum (FWTM) to quantify the fluorescent line scans. Aswith FWHM in our previous experiments, the reduction inFWTM indicates increasingly tighter focusing of particles.Once peak width stops decreasing and becomes constant, weconclude that particle migration away from the microchannelsidewalls is complete. The downstream position at which thisoccurs is the focusing length.

The progressive evolution of FWTM (Fig. 5c) is similar tothat of the negative lift presented earlier. At first glance, onceagain the figure appears to be counterintuitive since larger theRe the higher the lift force acting on the particle, and hencethe shorter channel downstream length for inertial migrationis expected. However, the lift coefficient exhibits a complex,inverse relationship with Re, as we discussed earlier. Thecurves exhibit a downward slope, until convergence at focusinglength (L+). Again, small shortening of the width followed by asharp drop is apparent. The rapid decrease indicates thatparticles leave the control of corner effect. Opposite to thenegative lift, shear rate decreases when particles approach thecenter of the channel wall where the local velocity ismaximum. This reduced shear rate causes particles to migrateslowly toward the stable equilibrium position, with peak widtheventually reaching a constant value. Note that beforestabilizing, particles appear to oscillate, as indicated by slightfluctuation of the peak width. This is consistent withobservations by others43 and is likely due to particle–particleinteraction as a result of enhanced local concentration.

Smaller particles were found to require longer downstreamlength for complete focusing. Fig. 5d shows results of

experiments with particle suspensions of four different sizes.This focusing behavior is similar to the results for the negativelift we discussed earlier, and of course is expected sinceinertial focusing exhibits a strong dependence on particle size.One notable difference, however, is the absence of a milddecrease (or flat region region) for the 20 and 15.5 mmdiameter particles, which was observed for the other twoparticle sizes. Moreover, the size of the flat region appears tobe inversely dependent on particle diameter. This suggeststhat smaller particles remain near corners longer, which isconsistent with our understanding of the corner effect.

We next compared the focusing length for particles ofdifferent diameter at various Re (Fig. 6a). While the curvesresemble those for negative lift (Fig. 3a), the focusing length L+

is much longer than L2, indicating slow migration velocityalong microchannel walls. Furthermore, the spacing betweenthe adjacent curves here is increasing faster as particlediameter decreases, indicating a stronger dependence onparticle size. As we will see below, the focusing length hereexhibits an inverse square relationship with particle diameter,which is different from the inverse linear relationship wefound for the negative lift. Smaller particles exhibit a largerpositive lift coefficient, suggesting an inverse relationship,analogous to the negative lift. We calculated the positive liftcoefficient for particles as a function of Re (Fig. 6b), and foundit to scale as CL

+ 3 Re20.5. This is not surprising consideringthat both positive and negative lift forces are related to theshear rate.

The magnitude of positive lift coefficient is y106 smallerthan that of the negative lift. Regardless of particle position,these results are in agreement with Saffman’s conclusion ofignorable rotational lift which is an order of magnitudesmaller as compared to shear induced lift in the unboundedcase.30,33,36 Furthermore, the magnitude of the positive liftcoefficient is comparable to the numerical results by Kuroseand Komori (note that our Rep = 0.4–30).30 From ourexperimental results, we conclude that rotational lift (positivelift) dominates particle motion near walls.

Fig. 6 Single stream focusing length (a) and the corresponding positive lift coefficient (b) as a function of Re. The inset reveals the linear dependence on Re20.5 for allfour particles used.


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Presenting the positive lift coefficient as a function ofparticle diameter reveals a direct inverse relationship (Fig. 7a).Thus, the dependence of positive lift coefficient on particlesize is not as strong as that of the negative lift, which is afunction of the second power (CL

2 3 a22). Since CL+ 3 a21,

substituting it into eqn (2), we find that positive lift force ishighly dependent on particle diameter, as FL

+ 3 a3. Thisrelationship is consistent with both Saffman’s higher ordercomponent and Rubinow and Keller’s expression.26,30,36,38 Theagreement with previous works again indicates that particlemigration near walls is dominated by rotational lift.Considering that shear induced lift (negative lift) is balancedby wall induced lift, this is also reasonable.

We next investigated the effects of channel dimensions onthe positive lift. Increasing channel cross-section led toapproximately parabolic increase in slope (Fig. 7b). However,it is not obvious which dimensional parameter (cross-sectionalarea, channel height, or channel width) is the majorcontributor. Our experiments show that positive lift coefficientexhibits dependence on H2 (Fig. 7d). This is analogous to thedependence of the negative lift coefficient on W2 (Fig. 4c). Inboth cases, the lift coefficient is dependent on the smallerchannel dimension. In the case of the positive lift coefficient,however, the underlying mechanism is different. The changein the positive lift is based on that of the negative lift. Whilethe negative lift acts parallel to the smaller dimension, thepositive lift acts orthogonal to the smaller dimension. In the

case of the negative lift, expanding the smaller dimensionrenders to an increase of coefficient due to reduced flowvelocity at a given Re. But negative lift decreases according toeqn (2). Diminished lift force then modifies the focusingposition relative to channel walls. More specifically, it causesthe position shift to channel center, which has beendemonstrated in several previous investigations17,22,35,40–42

When particles are far away from channel wall, the drag forcereduces rapidly.25,27 Therefore, in the case of positive lift, thenet force increases and thus the positive lift coefficientincreases.

We also investigated the long dimension and AR effect onpositive lift coefficient (Fig. 7c). As channel width increasesfrom 50 to 200 mm (height fixed at H y 27 mm), the slopes ofthe linear fitting curves also show slight change accordingly. Itcan be easily explained in terms of shear rate along the longdimension (W) which is parallel to the positive lift. Increasingthe long dimension alters the local shear rate near channelwidth and thus affects the lift force. Expanding dimensionleads to low velocity at given Re, which also impacts the lift.However, the effect of long dimension is minor compared tothat of the short dimension.

Microchannel AR has a non-linear effect on the positive liftcoefficient (Fig. 7c). To reduce the possibility of four focusingpositions, which may introduce potential errors, we investi-gated the effects of microchannel aspect ratio for values AR %1 based on the rotation effect described in the theory section.

Fig. 7 (a) Positive lift coefficient is inversely proportional to the particle diameter. The measurements were made 50 mm 6 27 mm microchannels. (b) Positive liftcoefficient as a function of Re at fixed aspect ratio (20 mm diameter particles). (c) Positive lift coefficient as a function of Re for microchannels with height fixed at H =27 mm (20 mm diameter particles). (d) Positive lift coefficient as a function of Re for microchannels with width fixed at W = 100 mm (20 mm diameter particles).


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At given Re, decrease of AR (increase width W) first acts toincrease the coefficient. This is primarily due to the effect of along dimension, which increases hydraulic diameter butreduces the flow velocity, as discussed above. Furtherincreases in W offer only a smaller contribution to the Dh,since Dh = 2HW/(H + W) = 2H/(H/W + 1). For an infinitely longW, the maximum of Dh is 2H. Thus, further changes in ARshow only minor changes to flow velocity, leading tostagnation in the lift coefficient values. Conversely, reductionof flow velocity couple with increased W tremendouslymodifies the velocity profile along the large dimension. Morespecifically, the local shear rate (G = 2Uf/W) reduces substan-tially, which weakens the rotation-induced lift as large sheardrives this force. In short, the nonlinear manifestation of theAR effect on the positive lift coefficient depends on theinteraction between velocity and the rotation-induced lift,implying that an optimal AR exists for particle migration alongthe long dimension. In this work, AR = 0.2 (equivalent AR = 5in a high AR channel) is optimal.

Overall, our results show that lateral migration alongmicrochannel wall is primarily dominated by the rotationinduced lift (positive lift force FL

+ 3 a3). The positive liftcoefficient has shown the dependence on Re20.5, analogous tothe negative lift coefficient. It also appears to strongly dependon H and scale inversely with particle diameter a. Theseexperiments therefore lead us to an expression for the positivelift coefficient as

CLz!

H2

affiffiffiffiffiffi

Rep ,WwH (7)

Since we have already provided the positive coefficients forfour particles, it is easy to estimate the coefficients for otherparticles in different channels using this expression. Particlebehavior in the microchannel is then completely predicable bythe combination of negative and positive coefficients. Next, wefurther examine the relationship between the two liftcoefficients.

Discussion

Our experimental results conclusively show that particlefocusing in rectangular microchannels occurs in two stages(Fig. 1d). First, randomly distributed particles flowing throughthe channel migrate from the bulk toward the longer sidewallsforming two broad bands near these walls, dominated byshear-induced lift force. Second, particles in these two bandsthen further migrate to the middle of the long wall and getstabilized forming tight streams, undergoing rotation-inducedlift force. From our experimental results, the first step occursrapidly, while the second step is slow. Our previous work2,3 hasalready pointed out that negative lift (i.e., shear induced lift) isresponsible for the lateral migration toward the microchannelwall. As particles migrate closer to walls, positive lift (i.e., therotation induced lift) gains strength and drives particles from

corners against the velocity gradient toward the middle pointof the wall where velocity profile is symmetric and equilibriumposition is stable.

Our experimental results show that both the negative andthe positive lift coefficients exhibit a weak inverse dependenceon flow (CL 3 Re20.5) and a strong dependence on the smallestmicrochannel dimension (CL 3 W2 or CL 3 H2). However,particle diameter exhibits different influences—the negativecoefficient scales with a22 while the positive coefficient scaleswith a21. This difference leads to a distinct particle-sizedependent feature of the negative (FL

2 3 a2) and the positive(FL

+ 3 a3) lift forces.We compared the negative and the positive lift coefficients

for migration of 10, 15, and 20 mm diameter particles to betterunderstand their relationship. The negative lift coefficient wasbased on the experimental results obtained in a 50 mm 6 100mm channel. The positive lift coefficient was based on theexperimental results obtained in a 100 mm 6 50 mm channel.The data presented in Fig. 8 show that the negative liftcoefficient is much larger in magnitude (y106) than thepositive lift coefficient for all particle diameters tested and thecorresponding shear and rotation induced lift forces arecalculated to be y10 nN vs. y1 nN. This is in agreement withthe accepted view that the rotation induced lift is usuallysmall, but becomes significant near channel walls. This resultis also in agreement with Saffman’s results which show thatshear induced lift is generally much larger than the rotationallift.30,36 We note, however, that the coefficients obtained inthis work are averaged along the particle migration path, sincea lift coefficient is as a function of position in the flow.23,25–27

While at first glance one may consider neglecting thepositive lift due to its significantly smaller size, it becomescritical in narrow microchannels since it dominates anddictates the length needed for complete focusing. Consideringthat low aspect ratio microchannels are much easier tofabricate and are frequently used in microfluidic systems,the positive lift coefficient indeed plays an important role.Based on our two-stage model of focusing behavior inrectangular microchannels (Fig. 1d), the complete focusing

Fig. 8 Comparison of the negative and positive lift coefficients for migration ofparticles (10, 15, and 20 mm in diameter) in a low aspect ratio (50 mm 6 100mm) microchannel.


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length for particle migration to their two stable equilibriumpositions at finite Re can now be derived from eqn (4) as

L~3pmDh

2

4rUf a3

H

CL{ z

W

CLz

� �

,WwH (8)

where W is the longer channel dimension, and H is the shorterchannel dimension. Thus, eqn (8) is directly applicable to lowaspect ratio channels, which are the most common inmicrofluidic systems. For high aspect ratio channels, H andW must be swapped to represent channel height. Since CL

+ ,

CL2 as in Fig. 8 and W . H in a rectangular microchannel,

W

CLz w

H

CL{. Thus, positive lift is dominating the channel length

required for complete focusing. While eqn (8) as derived for arectangular microchannel, it can be easily applied to a squarechannel by setting W = H or a round microchannel by setting W =0 and H to diameter. Note that in round channel, only the firststage of particle equilibration occurs, dominated by the shear-induced negative lift force; the result is the Segre annulus nearcapillary walls. Thus, eqn (8) is generally applicable.

Ultimately, our measurements allow a more precisecalculation of channel length necessary for complete focusing.It has been a common practice to estimate focusing lengthusing approximation of eqn (4), and then over-engineerdevices by fabricating a much longer channel since experi-mental results previously did not completely agree with thefocusing length calculations. For example, using the commonapproximation for CL = 0.5, at the 7.32 mm diameter particlesare expected to focus in y1 mm in our microchannels at Re =120; our experimental results, however, show a much longery9 mm length for complete focusing. With the improvedunderstanding of inertial migration and the accurate liftcoefficients, eqn (8) can be used to aid design of micro-channels with enhanced performance in a wide range ofapplications. When designing channels to separate particles orcells of different size, being able to accurately calculatefocusing length for each particle or cells size will permitproper placement of outlets for maximizing separationefficiency and sample purity. For other applications, such asflow cytometry based on inertial microfluidics as recentlyproposed by Hur et al.,12 our two-step inertial focusing modelwill allow a more precise downstream positioning of thereadout optics, without the need for large imaging windows orexcessively long microchannels.

Conclusions

In summary, in this work we describe and experimentallyvalidate the two-stage model inertial focusing in microchan-nels. We also, for the first time, experimentally demonstratethat rotation-induced lift plays an important role in theinertial migration of particles. Compared with previousnumerical results, our experimental results suggest thatshear-induced lift (negative lift) is the leading force that drivesparticles toward the channel walls. As particles approaching

channel wall, a counteracting wall-induced lift arises primarilydue to the vortices generated according to wake. Once thesetwo forces balance each other, the rotation-induced lift(positive lift) takes control and acts on the particle resultingin a net force along the wall toward its center. Particles thenmigrate to the stable positions centered at the faces wherelittle spinning presents due to minimum shear rate.

Moreover, for the first time, we experimentally measure liftcoefficients, which previously could only be obtained fromnumerical simulations. Measurements of these key parametersnot only validate the previous theoretical analyses of lift forceson the macroscale and confirm the numerical outcomes (interms of dependence on flow Re and particle size), but alsopermits precise control of particle behavior within therectangular microchannels, offering new insights into designand optimization for inertial microfluidic devices. Moreimportantly, our two-stage model of inertial focusing isbroadly applicable to cross-sectional geometries beyondrectangular microchannels. While channel cross-section canmodulate their magnitude, the three lift forces remain presentas focusing forces. By correlating these magnitudes with flowvelocity field, one can determine equilibrium positions inarbitrary microchannel cross-sections. For example, one wouldexpect three equilibrium positions in a microchannel withequilateral-triangle cross-sectioned. Or, in a semicircularchannel, one would expect particles to form semi-Segreannulus at first but finally occupy two positions at the centersof both walls where velocity profile is symmetric and rotation-induced lift disappears.

Ultimately, control of equilibrium positions would advancethe research of inertial microfluidics. Although there hasalready been intense attention to the inertial microfluidics inthe recent years, the primary focus has been on developmentof devices for manipulation of cells, with less emphasis paid toelucidation of particle focusing mechanics. The incompleteunderstanding, and sometimes confusing experimental resultsthat indicate a different number of focusing positions insquare or rectangular microchannels under similar flowconditions, has led to poor guidelines and difficulties indesign of inertial microfluidic systems. With elucidation of theequilibration mechanism, we envision better guidelines forthe inertial microfluidics community, ultimately leading toimproved performance and broad acceptance of the inertialmicrofluidic devices in a wide range of applications, fromfiltration to cell separations.

Experimental

Our experimental approach was to visualize flows of thefluorescently-labeled microparticles in microchannels at suc-cessive downstream positions using an inverted epifluores-cence microscope (Olympus IX71) equipped with a 12-bit CCDcamera (Retiga EXi, QImaging). Analogues to microparticlestreak velosimetry (m-PSV), flowing particles generated streaksacross each frame, and we analyzed fluorescent intensities and


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locations of these particle streaks. At least 100 frames wereobtained and stacked using ImageJ1 at each downstreamposition. Fluorescence intensity linescans were used inquantitative analyses.

Fluorescently-labeled, neutrally buoyant polystyrene parti-cles 7.32 to 20 mm in diameter were used in this work. Particleswere mixed at 0.025% volume fraction in deionized water tominimize the particle–particle interactions. Particles werepurchased from a number of vendors, depending on size(Bangs Lab Inc., Polyscience Inc., and Life Technologies Inc.). Asmall drop (y1% volume fraction) of Tween-20 was added toavoid clogging channels. A 1/1699 PEEK tubing and fittings(IDEX) were used to connect to device ports. A syringe pump(NE-1000X, New Era Pump Systems, Inc.) was used to providestable flow rates (10 , Re , 120) in all particle flow experiments.

We used standard soft lithography methods to fabricatemicrochannels in this work. Briefly, we used dry resist PerMX3050 (DuPont) to form masters on 399 silicon wafers byconventional photolithography. Polydimethylsiloxane (PDMS,Down Corning) was cast on the master, degassed, and curedfor 2 h on hotplate at 80 uC. Replicas were peeled and bondedto 199 6 399 standard microscope glass slides (Fisher Scientific)using a surface treater (Electro-Technic Products Inc.). Theinlet and outlet ports were cored manually with flat-headstainless needles.

Acknowledgements

We gratefully acknowledge partial support by the DefenseAdvanced Research Projects Agency (DARPA) N/MEMS S&TFundamentals Program under grant no. N66001-1-4003 issuedby the Space and Naval Warfare Systems Center Pacific (SPAWAR)to the Micro/nano Fluidics Fundamentals Focus (MF3) Center.

References1 J. Seo, M. H. Lean and A. Kole, Appl. Phys. Lett., 2007, 91,

033901.2 A. A. Bhagat, S. S. Kuntaegowdanahalli and I. Papautsky,

Phys. Fluids, 2008, 20, 101702.3 A. A. Bhagat, S. S. Kuntaegowdanahalli and I. Papautsky,

Microfluid. Nanofluid., 2008, 7, 217.4 A. J. Mach and D. Di Carlo, Biotechnol. Bioeng., 2010, 107,

302–311.5 D. R. Gossett, W. M. Weaver, A. J. MacH, S. C. Hur, H. T.

K. Tse, W. Lee, H. Amini and D. Di Carlo, Anal. Bioanal.Chem., 2010, 397, 3249–3267.

6 S. C. Hur, A. J. Mach and D. Di Carlo, Biomicrofluidics, 2011,5, 022206.

7 A. A. S. Bhagat, H. W. Hou, L. D. Li, C. T. Lim and J. Han,Lab Chip, 2011, 11, 1870–1878.

8 A. A. Bhagat, H. Bow, H. W. Hou, S. J. Tan, J. Han and C.T. Lim, Med. Biol. Eng. Comput., 2010, 48, 999–1014.

9 J. Park and H. Jung, Anal. Chem., 2009, 81, 8280–8288.10 S. L. Stott, C. H. Hsu, D. I. Tsukrov, M. Yu, D. T. Miyamoto,

B. A. Waltman, S. M. Rothenberg, A. M. Shah, M. E. Smas,G. K. Korir, F. P. Floyd Jr, A. J. Gilman, J. B. Lord,

D. Winokur, S. Springer, D. Irimia, S. Nagrath, L.V. Sequist, R. J. Lee, K. J. Isselbacher, S. Maheswaran, D.A. Haber and M. Toner, Proc. Natl. Acad. Sci. U. S. A., 2010,107, 18392–18397.

11 J. Oakey, R. W. Applegate, E. Arellano, D. D. Carlo, S.W. Graves and M. Toner, Anal. Chem., 2010, 82, 3862–3867.

12 S. C. Hur, H. T. K. Tse and D. Di Carlo, Lab Chip, 2010, 10,274–280.

13 A. A. Bhagat, S. S. Kuntaegowdanahalli, N. Kaval, C.J. Seliskar and I. Papautsky, Biomed. Microdevices, 2010,12, 187–195.

14 B. Chun and A. J. C. Ladd, Phys. Fluids, 2006, 18, 031704.15 D. Di Carlo, D. Irimia, R. G. Tompkins and M. Toner, Proc.

Natl. Acad. Sci. U. S. A., 2007, 104, 18892–7.16 D. Di Carlo, J. F. Edd, K. J. Humphry, H. A. Stone and

M. Toner, Phys. Rev. Lett., 2009, 102, 094503.17 Y. Choi and S. Lee, Microfluid. Nanofluid., 2010, 9, 819–829.18 E. J. Lim, T. J. Ober, J. F. Edd, G. H. McKinley and

M. Toner, Lab Chip, 2012, 12, 2199–210.19 G. Segre and A. Silberberg, Nature, 1961, 189, 209–210.20 G. Segre and A. Silberberg, J. Fluid Mech., 1962, 14,

136–157.21 J. P. Matas, J. F. Morris and E. Guazzelli, Oil Gas Sci.

Technol., 2004, 59, 59–70.22 J. Matas, J. F. Morris and E. Guazzelli, J. Fluid Mech., 2004,

515, 171–195.23 E. S. Asmolov, J. Fluid Mech., 1999, 381, 63–87.24 L. Zeng, F. Najjar, S. Balachandar and P. Fischer, Phys.

Fluids, 2009, 21.25 H. Lee and S. Balachandar, J. Fluid Mech., 2010, 657,

89–125.26 E. Loth and A. J. Dorgan, Environ. Fluid Mech., 2009, 9,

187–206.27 L. Zeng, S. Balachandar and P. Fischer, J. Fluid Mech., 2005,

536, 1–25.28 J. Feng, H. H. Hu and D. D. Joseph, J. Fluid Mech., 1994,

261, 95–134.29 J. Feng, H. H. Hu and D. D. Joseph, J. Fluid Mech., 1994,

277, 271–301.30 R. Kurose and S. Komori, J. Fluid Mech., 1999, 384, 183–206.31 P. Cherukat and J. B. McLaughlin, Int. J. Multiphase Flow,

1990, 16, 899–907.32 J. B. McLaughlin, J. Fluid Mech., 1993, 246, 249–265.33 P. Cherukat and J. B. McLaughlin, J. Fluid Mech., 1994, 263,

1–18.34 P. S. Williams, Chem. Eng. Commun., 1994, 130, 143–166.35 Y. Choi, K. Seo and S. Lee, Lab Chip, 2011, 11, 460–465.36 P. G. Saffman, J. Fluid Mech., 1965, 22, 385–400.37 A. J. Dorgan, E. Loth, T. L. Bocksell and P. K. Yeung, AIAA

Journal, 2005, 43, 1537–1548.38 S. I. Rubinow and J. B. Keller, J. Fluid Mech., 1961, 11,

447–459.39 F. Takemura and J. Magnaudet, J. Fluid Mech., 2003, 495,

235–253.40 J. A. Schonberg, J. Fluid Mech., 1989, 203, 517–524.41 X. Shao, Z. Yu and B. Sun, Phys. Fluids, 2008, 20, 103307.42 Y. W. Kim and J. Y. Yoo, J. Micromech. Microeng., 2008,

18(6), 065015.43 W. Lee, H. Amini, H. A. Stone and D. Di Carlo, Proc. Natl.

Acad. Sci. U. S. A., 2010, 107, 22413–22418.


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