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Lab on a Chip PAPER Fundamentals of inertial focusing in microchannels Cite this: Lab Chip, 2013, 13, 1121

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,

Received 13th November 2012, Accepted 3rd January 2013

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

DOI: 10.1039/c2lc41248a

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

www.rsc.org/loc

cell separations.

Introduction Inertial microfluidics has been attracting considerable interest in recent years due to the promising applications ranging from filtration1–4 to separation5–10 to cytometry of cells.11–13 As a passive technique, it manipulates cells and particles in microchannels without an externally applied field, and combines the benefits of a passive approach with extremely high throughput. In this phenomenon, cells and particles migrate across streamlines and order deterministically at equilibrium positions near channel walls. This behavior is due to the inertial forces, which are typically neglected in the microfluidic low Reynolds number flows. In straight channels, inertial migration is believed to be caused by balance of lift forces arising from the curvature of the velocity profile (the shear-induced lift) and the interaction between particle and the channel wall (the wall-induced lift) as illustrated in Fig. 1a. In square microchannels, the inertial migration of cells or particles leads to focusing in four equilibrium positions centers at the faces of the channels (Fig. 1b).2,3,14–16 Despite the recent interest, the fundamentals of particle ordering in microfluidic systems remain to be elusive.17 Indeed, the majority of work to date has focused on devices and applications. The limited number of studies discussing 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

This journal is ß The Royal Society of Chemistry 2013

the fundamental mechanisms have in fact introduced a number of inconsistencies. For example, in square channels, different particle behaviors have been described2,3,16,18 and not only four but also eight equilibrium positions have been reported by several groups.2,3,14,15 Even further migration into two equilibrium positions has been recently observed in rectangular microchannel.18 These observations cannot be explained from the current understanding of inertial focusing (i.e. balance of two lift forces), which dictates that once particle equilibrates its position should be stable. This incomplete understanding of the mechanism for particle migration in microchannels stems in part from the difficulties in investigating forces that act on particles. Even on the macroscle, the majority of investigations reported to date are numerical in nature. This absence of underlying principles can lead to difficulties in design of devices and impede their development. In this work, we experimentally investigate forces acting on neutrally-buoyant particles flowing through a microchannel. We re-examine the forces that are responsible for particle equilibration and for the first time experimentally measure the lift forces that cause particle equilibration in rectangular microchannels. Previous work by Chun and Ladd14 proposed and showed numerically that particle migration in square channels occurs in two stages. Herein, we experimentally confirm that particle migration occurs in two stages, each dominated by a different lift force balance, and show that the

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Lab on a Chip dominant forces: the viscous drag (FD), which entrains particles along streamlines, and the inertial lift force (FL) that leads to migration across the streamlines. Matas et al. discussed two components of inertial lift that together act to yield an equilibrium position between the channel wall and the centerline.16,21,22 A wall-induced lift force Fw that acts up the velocity gradient away from the wall toward the channel centerline, and a shear-induced lift force Fs that acts down the velocity gradient toward the channel walls (Fig. 1a). It is the net lift force, a balance of the shear-induced and wall-induced lift forces, that is responsible for particle migration into an annulus y0.2D (diameter of the pipe) away from the wall. Early theoretical investigations into inertial migration of particles have identified that for particles of diameter a in a channel 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

Fig. 1 Inertial focusing in rectangular microchannels. (a) Two lift forces orthogonal to the flow direction act to equilibrate microparticles near wall. The shear-induced lift force Fs is directed down the shear gradient and drives particles toward channel walls. The wall-induced lift force Fw directs particles away from the walls and drives particles toward the channel centerline. The balance of these two lift forces causes particles to equilibrate. (b) In square channels, at moderate Re randomly distributed particles focus into four equilibrium positions at the wall centers. (c) Fluorescent images illustrating migration 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 ratio microchannel at moderate Re, microparticles first migrate from the channel bulk toward equilibrium positions near walls under the influence of the shearinduced lift force Fs and the wall-induced lift force Fw. Then, particles migrate parallel to channel walls into wall-centered equilibrium positions under the influence of the rotation-induced lift force FV.

same mechanism is applicable to rectangular microchannels. This work will improve the understanding of the underlying mechanisms of particle inertial migration and will offer a useful guide in the development of inertial microfluidic systems.

Physics of inertial migration Inertial migration of particles was originally discovered in ´ and Silberberg observed cylindrical pipes in 1960s, when Segre that randomly dispersed y1 mm diameter particles formed an annulus in a 1 cm diameter pipe.19,20 With advent of microfluidics, observations of the same phenomenon were confirmed in microchannels in the recent years.17,21,22 Particles flowing in a microchannel are subjected to two

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(1)

where r is fluid density and G is the shear rate. The lift coefficient is a function of the Reynolds number (Re) and has been shown to decrease with increasing Re.24,25 Considering that the shear rate is a function of average flow velocity and Uf and channel dimension (G = 2Uf/Dh),23 the net lift force can be expressed as FL ~

4rCL Uf 2 a4 3pmDh 2

(2)

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

4rCL Uf 2 a3 3pmDh 2

(3)

Thus, particles migrate a lateral distance toward equilibrium position that is proportional of the flow velocity and the downstream position. Recent work by Di Carlo et al.16 has shown that lift force scaling is dependent on the particle position in the channel, suggesting that disparate fluid dynamic effects act to create the inertial lift equilibrium positions. Motion of particles near the microchannel centerline is dominated by the shearinduced lift Fs due to the vorticity v around particle surface.24,26,27 The direction of this force is toward channel walls, determined by the cross-product of the vorticity and the relative velocity Ur (Fs = v 6 Ur). We note here that the relative particle velocity Ur is the difference between the particle velocity and flow velocity, as particles lag behind the flow;21,28,29 the direction of Ur is opposite to the direction of Uf (if we assume as simplified 2-D model; the more complex 3-D case is discussed by Loth and Dorgan26). Early work with numerical models has shown that Fs is strongly dependent on


Lab on a Chip particle diameter and scales as Fs 3 a2.26,30 As particles approach microchannel walls, a wall-induced lift force Fw arises due to proximity to walls.25–27,31–33 Direction of this force can also be described by as a cross-product of the vorticity and the relative particle velocity vectors Fw = v 6 Ur, as both lift forces act orthogonal to channel walls. Numerical models by Zeng et al.24,27 show that vorticity near walls is in the 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 force Fw exhibits even stronger dependence on particle size (compared to the shear-induced lift) and the distance to wall d, scaling as Fw 3 a3/d.34 While the balance of the two lift forces can successfully explain particle focusing in a round duct, square and rectangular channels present a more complex situation due to radial asymmetry. The expressions above could be still applied to estimate the FL and UL by defining hydraulic diameter Dh = 2WH/(W + H) for a channel W wide and H high. Since the shear-induced lift causes particles to migrate away from the channel center, down the shear gradient toward the channel wall, one would expect particles to equilibrate along the perimeter of the channel (including corners) in order to achieve force balance. However, work in microfluidic channels has identified four distinct focusing positions centered at each face in square microchannels, as illustrated in Fig. 1b.2,3,16,35 The absence of particles in corners suggests that additional lateral migration effects take place near channel walls that cause particle migration toward wall centers. Indeed, the early work by Saffman36 proposed a rotationinduced lift force FV. This Saffman lift force plays an important role near channel walls, but was considered negligible compared with the shear-induced force while particles are far away from the channel wall.27,30,31,33,36 Recent numerical simulations by Dorgan et al.37 have confirmed that in most flows particles are more likely to experience a shearing behavior. Consequently, most descriptions of inertial flows do not consider this lift force. While negligible away from the wall, we believe that the rotation-induced lift is significant at the channel wall and can help explain the asymmetric equilibrium positions in rectangular microchannels. Cherukat and Mclaughlin33 have shown that the effect of rotation is very small but becomes important when the shear is large and particle is close to wall. Due to the parabolic nature of the Poiseuille flow, the shear rate increases dramatically near the wall, which satisfies Cherukat and McLaughlin’s conclusion and lead us to believe that the rotation-induced lift is critical. Rubinow and Keller,38 and more recently Loth and Dorgan,26 proposed that the rotationinduced lift scales with particle diameter as FV 3 a3. Analogous to the other lift forces, the direction of this force is determined by the cross-product of the rotation and the relative particle velocity vectors FV = V 6 Ur as illustrated in Fig. 1d.26 Since the rotation is due to shear28,29 and the neutrally buoyant particles always lag behind flow velocity in Poiseuille flow,21,28,29 this rotation-induced lift acts against the


Paper flow velocity gradient and is directed toward the center face of the channel wall. In this case, particles near the channel wall would further migrate to the center of the wall, which is in agreement with the experimental observations in square and rectangular channels. Indeed, our results in this work also suggest that particle spinning is dominant in case of a wall region where shear and wall forces cancel each other. From the introduction of rotation-induced lift force, a more complex model of inertial migration at finite Re emerges, illustrated in Fig. 1d. In rectangular channels, particles first migrate from the channel bulk toward equilibrium positions near walls. Then, particles migrate parallel to channel walls into wall-centered equilibrium positions. Particles in the corner can migrate along either of the two nearby walls depending on the local shear rate,33 which suggests that both channel aspect ratio (AR = height/width) and Re can modify the focusing positions in rectangular channel since both of them can alter the magnitude of local shear rate close to walls. More specifically, if the AR y 1 or at high Re, the four equilibrium positions emerge even in rectangular channels.6,18 Nevertheless, in low aspect ratio channels (AR % 1), particles will focus into two equilibrium positions centered at the top and bottom walls, as demonstrated in Fig. 1c. Similar behavior can be observed for high aspect ratio channels (AR & 1), with two equilibrium positions developing at the center of sidewalls. In summary, the inertial migration of particles in microchannels follows the balance of the lift forces and occurs in two stages. Both shear and wall induced lift forces are due to the vorticity around particle surface, dominating particle migration toward channel walls (stage I). Once the initial equilibrium is reached, near channel walls particle motion is dominated by the rotation-induced lift force. As a result, particles migrate to the center points of walls (stage II). This new model of inertial focusing is generally applicable to rectangular microchannels of any aspect ratio at finite Re, and can be used to aid design of inertial microfluidic systems. In the subsequent sections, we discuss our experimental validation of this two-stage model and for the first time measure the associated lift coefficients.

Experimental approach The lift forces are characterized by their associated lift coefficients, with the sign of coefficients representing direction of the lift force. For particles flowing in a microchannel, the sign of the lift coefficient due to the balance of the shearinduced and wall-induced lift forces is negative, as particles migrate away the channel centerline and orthogonal to the channel wall.3,23 Conversely, the lift coefficient due to rotational is positive, which implies lift up the velocity profile and parallel to channel wall.30 Herein, the two coefficients are denoted as CL2 and CL+ and the corresponding lift forces are FL2 for Fs (since Fs and Fw act against each other) and FL+ for FV respectively.

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The channel length L required to fully focus particles can be written as2,3 L~

Um Lm 3pmDh 2 Lm ~ UL 2rUf CL a3

(4)

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

3pmDh 2 Lm | 2rUf a3 L

(5)

Therefore, as long as we are able to determine the particle migration distance Lm and the focusing length L, the lift coefficient can be obtained. In this work, we use microchannels with rectangular crosssection for experimental determination of the lift coefficients because of the two equilibrium positions that emerge at complete focusing. In a low aspect ratio channel, randomlydistributed particles will rapidly migrate and equilibrate near the top and bottom walls under the influence of negative lift FL2, and thus the migration distance is half of the channel height length (Lm2 = H/2). This initial stage I focusing can be observed experimentally using a high aspect ratio channel (essentially rotating channel by 90u), an approach we successfully used in the past.2,3 In stage II, particles migrate toward the center of the top and bottom walls under the influence of the positive lift FL+ and the transportation length is approximately half of the width (Lm+ = W/2). The stage II focusing in low aspect ratio channels can be observed directly with an inverted microscope. The same approach can be extended to high aspect ratio channels, by appropriately switching the H and W. Experimental determination of the downstream focusing length can then be used in calculation of the appropriate lift coefficients using eqn (5) (i.e. CL2 from L2, and CL+ from L+). Since lift coefficients are a function of particle position within the flow, the measurements in this work are averaged along the migration direction. In the subsequent sections we discuss experimental measurements of both positive and negative lift coefficients and their dependence on the flow parameters and microchannel geometry.

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

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focusing length, we measured fluorescent intensity across channel width at successive downstream positions. Progressive entrainment of particles as they flow downstream is shown in Fig. 2b. The intensity of the middle region decreases, suggesting depletion of particles and their migration away from the center. The intensity near sidewalls grows, forming two peaks, due to migration of particles toward channel sidewalls. Quality of particle focusing can be judged by measuring full width at half maximum (FWHM) of these intensity peaks. Progressive reduction in the FWHM indicates progressively tighter focusing. The downstream position where migration toward the channel wall stops and FWHM value stabilizes is the focusing length. We should note here that particle migration along the vertical sidewall does not affect the peak width in this case. Plotting focusing quality data at multiple Re as a function of downstream length (Fig. 2c) reveals that complete focusing can be achieved as indicated by values approaching particle diameter. Increasing Re leads to longer channel length needed for particle focusing, suggesting lift coefficient is not constant. Indeed, similar observations have been reported by others on the macroscale.23,27 At first glance, Fig. 2c appears to be counterintuitive since larger the Re (Uf), the higher the lift force acting on the particle, and hence the shorter channel downstream length for inertial migration. However, the lift coefficient CL in eqn (2) is also a function of Uf, and thus the relationship between the focusing length and flow velocity (Re) is more complex. Previous numerical results24,25,27 have shown CL to decrease with increasing Re, leading to an optimal flow Re that offers the minimum focusing length. Close examination of Fig. 2c shows that particle migration is not uniform, as FWHM first decreases slowly, followed by a sharp drop indicating development of two peaks near sidewalls. This can be partially attributed to the parabolic nature of the velocity profile which results in a shear rate lowest in the channel center. Since shear force is dependent on the second power of shear rate, as expressed in eqn (1), the lateral migration velocity accelerates in a quadratic manner which leads to fast migration. As particles approach channel walls, the migration velocity reduces exponentially due to the logarithmic increase of the drag force, and the wall induced lift force counteracting the shear induced lift force.25,39 Further experiments were performed to explore effects of particle 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 flow conditions and their focusing progress is shown in Fig. 2d. As expected, large particles migrate much faster due to the larger lift force according to eqn (1). Thus, they require shorter channel length for focusing. Decreasing particle diameter, however, does not lead to drastic increase in focusing length but rather acts in a more linear way with respect to reciprocal of particle size. Presenting focusing length as a function of Re illustrates the non-linear behavior and validates our hypothesis of a nonconstant lift coefficient (Fig. 3a). The separation between


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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 of 7.32 mm diameter microparticles at Re = 120. Dotted lines represent an approximate position of channel walls. (b) Fluorescent intensity line scans at four downstream positions (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 of downstream position for four particle sizes at Re = 50. In all experiments, 0.025%. volume fraction was used.

adjacent curves representing particles of different size increases linearly as particle size reduces. Small particles (e.g., 7.32 mm diameter) are more easily affected by the flow Re as the curve is less flat than that of the larger (e.g., 20 mm diameter) 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 trend and magnitude have been reported by others based numerical models.25 Presenting our experimental results as a function of Re20.5 (Fig. 3b inset) shows a strong linear relationship (R2 . 0.99). This is in agreement with the recent proposition by Loth and Dorgan26 that Saffman lift is dependent on Re20.5. The same relationship may also be inferred from the earlier analysis 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 diameter when related to Re20.5.


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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 (7.32 mm 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 lift coefficient as a function of Re for microchannels with width fixed at W = 50 mm (7.32 mm diameter particles).

is the first experimental determination of CL2 and it confirms the speculation and numerical models recently proposed by others. Smaller particles exhibit a larger negative lift, suggesting an inverse relationship. The negative lift coefficient for particles at multiple Re in Fig. 4a presents the experimental data as a function of a22. Again, low Re exhibits higher CL2, which is consistent with our previous observations. From these results, we conclude that CL2 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 This agreement with numerical results also indicates that Asmolov’s equation for parabolic flow in macroscale channels is applicable in on the microscale. We next investigated the effects of channel dimensions on the negative lift coefficient. First we fixed the channel aspect ratio (AR y 2), but scaled the cross-section and measured focusing length for various particle suspensions. From these measurements we calculated CL2 for each channel (Fig. 4b). As expected, the coefficient increased with increasing channel size. Comparison of the slopes shows that CL2 scales approximately with cross-sectional area. Additional experiments which fixed channel width showed that the lift coefficient varies with channel width (CL2 3 W2) which is the smaller dimension, as data in Fig. 4c show. Plotting data for varying heights (Fig. 4d) shows that the two channels, which have similar width, are approximately the same. From

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these results, we conclude that the negative lift coefficient is dominated by the smaller dimension in rectangular channel. This actually has been implied previously by us2,3 and by others12 when channel width was used instead of hydraulic diameter as the characteristic dimension of a high AR channel. The predominant effect of width here is related to the coincidence of migration direction and the higher shear rate along the small dimension. Overall, our experimental results show that in high aspect ratio microchannels negative lift coefficient is strongly dependent on the particle diameter and channel width, but is only weakly dependent on the flow Re. These experiments therefore lead us to an expression indicating that the negative lift coefficient scales as CL { !

W2 pffiffiffiffiffiffi ,HwW Re

a2

(6)

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


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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 of 7.32 mm diameter microparticles at Re = 70. Dotted lines represent an approximate position of channel walls. (b) Fluorescent intensity line scans at four downstream positions (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 mm diameter 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.

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

Determination of the positive lift coefficient Particle rotation-induced lift force (positive lift) has not been described in microchannels, yet it is associated with migration behavior of particles.14–17 Unlike the negative lift that directs particles along the velocity gradient, the positive lift guides particle migration against the gradient. It is therefore responsible for particle migration along microchannel perimeter toward sidewall centers. The result is the four equilibrium positions near sidewall centers in a square channel or two stable positions along the middle of the longer sidewalls in a rectangular channel. To investigate the effect of positive lift, we performed experiments in low aspect ratio (AR , 1) microchannels. As in the experiments described above, we measured fluorescent intensity across microchannel width at successive downstream positions. The results for 7.32 mm diameter particles (Fig. 5a) show three streams – one clearly-visible, broad stream in the center and two weaker streams near sidewalls, which disappear after sufficient downstream length. This development of three streams is a new phenomenon, previously not


observed in low aspect ratio channel. In fact, based on our previous work, we expected a single broad band spanning nearly the entire width of the microchannel as we have reported recently.2,3 This new phenomenon may be explained by considering particles flowing near microchannel sidewall. These particles experience doubled wall-added viscous effect due to two adjacent walls, which significantly increases drag force.25,27 Hence, the net of the drag force and the rotation-induced lift are much smaller in the corner. On the other hand, the distance from particle to wall significantly affects the drag force. According to the numerical models by Zeng et al.,27 particle at an intermediate distance to sidewall experiences smaller drag as compared to one closes to wall, which implies a larger net force. As a result, the migration velocity of particles in the corners is much smaller than those in intermediate positions, leading to formation of two intermediate sidestreams. We found that it is easier to observe these sidestreams for small particles than for large ones, since small particles are generally closer to walls and remain there longer. Increasing Re leads to similar formation of three streams even for large particles (e.g., 20 mm diameter) since particles generally move closer to walls with increasing Re.17,22,35,40–42 Indeed, we have observed two side-streams of 20 mm diameter particles at Re = 180. Therefore, the interaction of drag force and rotational lift could result in different migration behavior,

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Paper depending on particle size and Re. We term this as ‘‘corner effect’’ in this work. To determine the focusing length under the influence of the positive lift coefficient, we once again measured fluorescent intensities across channel width at consecutive downstream positions. The data in Fig. 5b illustrate progressive migration of particles as they flow downstream. The small peaks representing side streams disappear at the focusing length, as indicated by the single peak at L = 20 mm. Due to presence of side streams, we used full width at tenth of maximum (FWTM) to quantify the fluorescent line scans. As with FWHM in our previous experiments, the reduction in FWTM indicates increasingly tighter focusing of particles. Once peak width stops decreasing and becomes constant, we conclude that particle migration away from the microchannel sidewalls is complete. The downstream position at which this occurs is the focusing length. The progressive evolution of FWTM (Fig. 5c) is similar to that of the negative lift presented earlier. At first glance, once again the figure appears to be counterintuitive since larger the Re the higher the lift force acting on the particle, and hence the shorter channel downstream length for inertial migration is expected. However, the lift coefficient exhibits a complex, inverse relationship with Re, as we discussed earlier. The curves exhibit a downward slope, until convergence at focusing length (L+). Again, small shortening of the width followed by a sharp drop is apparent. The rapid decrease indicates that particles leave the control of corner effect. Opposite to the negative lift, shear rate decreases when particles approach the center of the channel wall where the local velocity is maximum. This reduced shear rate causes particles to migrate slowly toward the stable equilibrium position, with peak width eventually reaching a constant value. Note that before stabilizing, particles appear to oscillate, as indicated by slight fluctuation of the peak width. This is consistent with observations by others43 and is likely due to particle–particle interaction as a result of enhanced local concentration. Smaller particles were found to require longer downstream length for complete focusing. Fig. 5d shows results of

Lab on a Chip experiments with particle suspensions of four different sizes. This focusing behavior is similar to the results for the negative lift we discussed earlier, and of course is expected since inertial focusing exhibits a strong dependence on particle size. One notable difference, however, is the absence of a mild decrease (or flat region region) for the 20 and 15.5 mm diameter particles, which was observed for the other two particle sizes. Moreover, the size of the flat region appears to be inversely dependent on particle diameter. This suggests that smaller particles remain near corners longer, which is consistent with our understanding of the corner effect. We next compared the focusing length for particles of different diameter at various Re (Fig. 6a). While the curves resemble those for negative lift (Fig. 3a), the focusing length L+ is much longer than L2, indicating slow migration velocity along microchannel walls. Furthermore, the spacing between the adjacent curves here is increasing faster as particle diameter decreases, indicating a stronger dependence on particle size. As we will see below, the focusing length here exhibits an inverse square relationship with particle diameter, which is different from the inverse linear relationship we found for the negative lift. Smaller particles exhibit a larger positive lift coefficient, suggesting an inverse relationship, analogous to the negative lift. We calculated the positive lift coefficient for particles as a function of Re (Fig. 6b), and found it to scale as CL+ 3 Re20.5. This is not surprising considering that both positive and negative lift forces are related to the shear rate. The magnitude of positive lift coefficient is y106 smaller than that of the negative lift. Regardless of particle position, these results are in agreement with Saffman’s conclusion of ignorable rotational lift which is an order of magnitude smaller as compared to shear induced lift in the unbounded case.30,33,36 Furthermore, the magnitude of the positive lift coefficient is comparable to the numerical results by Kurose and Komori (note that our Rep = 0.4–30).30 From our experimental results, we conclude that rotational lift (positive lift) 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 all four particles used.

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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 lift coefficient 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).

Presenting the positive lift coefficient as a function of particle diameter reveals a direct inverse relationship (Fig. 7a). Thus, the dependence of positive lift coefficient on particle size is not as strong as that of the negative lift, which is a function of the second power (CL2 3 a22). Since CL+ 3 a21, substituting it into eqn (2), we find that positive lift force is highly dependent on particle diameter, as FL+ 3 a3. This relationship is consistent with both Saffman’s higher order component and Rubinow and Keller’s expression.26,30,36,38 The agreement with previous works again indicates that particle migration near walls is dominated by rotational lift. Considering that shear induced lift (negative lift) is balanced by wall induced lift, this is also reasonable. We next investigated the effects of channel dimensions on the positive lift. Increasing channel cross-section led to approximately parabolic increase in slope (Fig. 7b). However, it is not obvious which dimensional parameter (cross-sectional area, channel height, or channel width) is the major contributor. Our experiments show that positive lift coefficient exhibits dependence on H2 (Fig. 7d). This is analogous to the dependence of the negative lift coefficient on W2 (Fig. 4c). In both cases, the lift coefficient is dependent on the smaller channel dimension. In the case of the positive lift coefficient, however, the underlying mechanism is different. The change in the positive lift is based on that of the negative lift. While the negative lift acts parallel to the smaller dimension, the positive lift acts orthogonal to the smaller dimension. In the


case of the negative lift, expanding the smaller dimension renders to an increase of coefficient due to reduced flow velocity at a given Re. But negative lift decreases according to eqn (2). Diminished lift force then modifies the focusing position relative to channel walls. More specifically, it causes the position shift to channel center, which has been demonstrated in several previous investigations17,22,35,40–42 When particles are far away from channel wall, the drag force reduces rapidly.25,27 Therefore, in the case of positive lift, the net force increases and thus the positive lift coefficient increases. We also investigated the long dimension and AR effect on positive lift coefficient (Fig. 7c). As channel width increases from 50 to 200 mm (height fixed at H y 27 mm), the slopes of the linear fitting curves also show slight change accordingly. It can be easily explained in terms of shear rate along the long dimension (W) which is parallel to the positive lift. Increasing the long dimension alters the local shear rate near channel width and thus affects the lift force. Expanding dimension leads to low velocity at given Re, which also impacts the lift. However, the effect of long dimension is minor compared to that of the short dimension. Microchannel AR has a non-linear effect on the positive lift coefficient (Fig. 7c). To reduce the possibility of four focusing positions, which may introduce potential errors, we investigated the effects of microchannel aspect ratio for values AR % 1 based on the rotation effect described in the theory section.

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At given Re, decrease of AR (increase width W) first acts to increase the coefficient. This is primarily due to the effect of a long dimension, which increases hydraulic diameter but reduces the flow velocity, as discussed above. Further increases in W offer only a smaller contribution to the Dh, since Dh = 2HW/(H + W) = 2H/(H/W + 1). For an infinitely long W, the maximum of Dh is 2H. Thus, further changes in AR show only minor changes to flow velocity, leading to stagnation in the lift coefficient values. Conversely, reduction of flow velocity couple with increased W tremendously modifies the velocity profile along the large dimension. More specifically, the local shear rate (G = 2Uf/W) reduces substantially, which weakens the rotation-induced lift as large shear drives this force. In short, the nonlinear manifestation of the AR effect on the positive lift coefficient depends on the interaction between velocity and the rotation-induced lift, implying that an optimal AR exists for particle migration along the long dimension. In this work, AR = 0.2 (equivalent AR = 5 in a high AR channel) is optimal. Overall, our results show that lateral migration along microchannel wall is primarily dominated by the rotation induced lift (positive lift force FL+ 3 a3). The positive lift coefficient has shown the dependence on Re20.5, analogous to the negative lift coefficient. It also appears to strongly depend on H and scale inversely with particle diameter a. These experiments therefore lead us to an expression for the positive lift coefficient as H2 CL z ! pffiffiffiffiffiffi ,W wH a Re

(7)

Since we have already provided the positive coefficients for four particles, it is easy to estimate the coefficients for other particles in different channels using this expression. Particle behavior in the microchannel is then completely predicable by the combination of negative and positive coefficients. Next, we further examine the relationship between the two lift coefficients.

corners against the velocity gradient toward the middle point of the wall where velocity profile is symmetric and equilibrium position is stable. Our experimental results show that both the negative and the positive lift coefficients exhibit a weak inverse dependence on flow (CL 3 Re20.5) and a strong dependence on the smallest microchannel dimension (CL 3 W2 or CL 3 H2). However, particle diameter exhibits different influences—the negative coefficient scales with a22 while the positive coefficient scales with a21. This difference leads to a distinct particle-size dependent feature of the negative (FL2 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 better understand their relationship. The negative lift coefficient was based on the experimental results obtained in a 50 mm 6 100 mm channel. The positive lift coefficient was based on the experimental results obtained in a 100 mm 6 50 mm channel. The data presented in Fig. 8 show that the negative lift coefficient is much larger in magnitude (y106) than the positive lift coefficient for all particle diameters tested and the corresponding shear and rotation induced lift forces are calculated to be y10 nN vs. y1 nN. This is in agreement with the accepted view that the rotation induced lift is usually small, but becomes significant near channel walls. This result is also in agreement with Saffman’s results which show that shear induced lift is generally much larger than the rotational lift.30,36 We note, however, that the coefficients obtained in this work are averaged along the particle migration path, since a lift coefficient is as a function of position in the flow.23,25–27 While at first glance one may consider neglecting the positive lift due to its significantly smaller size, it becomes critical in narrow microchannels since it dominates and dictates the length needed for complete focusing. Considering that low aspect ratio microchannels are much easier to fabricate 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 in rectangular microchannels (Fig. 1d), the complete focusing

Discussion Our experimental results conclusively show that particle focusing in rectangular microchannels occurs in two stages (Fig. 1d). First, randomly distributed particles flowing through the channel migrate from the bulk toward the longer sidewalls forming two broad bands near these walls, dominated by shear-induced lift force. Second, particles in these two bands then further migrate to the middle of the long wall and get stabilized forming tight streams, undergoing rotation-induced lift force. From our experimental results, the first step occurs rapidly, while the second step is slow. Our previous work2,3 has already pointed out that negative lift (i.e., shear induced lift) is responsible for the lateral migration toward the microchannel wall. As particles migrate closer to walls, positive lift (i.e., the rotation induced lift) gains strength and drives particles from

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


Lab on a Chip length for particle migration to their two stable equilibrium positions at finite Re can now be derived from eqn (4) as 3pmDh 2 H W ,W wH (8) z L~ 4rUf a3 CL { CL z where W is the longer channel dimension, and H is the shorter channel dimension. Thus, eqn (8) is directly applicable to low aspect ratio channels, which are the most common in microfluidic systems. For high aspect ratio channels, H and W must be swapped to represent channel height. Since CL+ , CL2 as in Fig. 8 and W . H in a rectangular microchannel, W H w . Thus, positive lift is dominating the channel length CL z CL { required for complete focusing. While eqn (8) as derived for a rectangular microchannel, it can be easily applied to a square channel by setting W = H or a round microchannel by setting W = 0 and H to diameter. Note that in round channel, only the first stage of particle equilibration occurs, dominated by the shear´ annulus near induced negative lift force; the result is the Segre capillary walls. Thus, eqn (8) is generally applicable. Ultimately, our measurements allow a more precise calculation of channel length necessary for complete focusing. It has been a common practice to estimate focusing length using approximation of eqn (4), and then over-engineer devices by fabricating a much longer channel since experimental results previously did not completely agree with the focusing length calculations. For example, using the common approximation for CL = 0.5, at the 7.32 mm diameter particles are expected to focus in y1 mm in our microchannels at Re = 120; our experimental results, however, show a much longer y9 mm length for complete focusing. With the improved understanding of inertial migration and the accurate lift coefficients, eqn (8) can be used to aid design of microchannels with enhanced performance in a wide range of applications. When designing channels to separate particles or cells of different size, being able to accurately calculate focusing length for each particle or cells size will permit proper placement of outlets for maximizing separation efficiency and sample purity. For other applications, such as flow cytometry based on inertial microfluidics as recently proposed by Hur et al.,12 our two-step inertial focusing model will allow a more precise downstream positioning of the readout optics, without the need for large imaging windows or excessively long microchannels.

Paper channel wall, a counteracting wall-induced lift arises primarily due to the vortices generated according to wake. Once these two forces balance each other, the rotation-induced lift (positive lift) takes control and acts on the particle resulting in a net force along the wall toward its center. Particles then migrate to the stable positions centered at the faces where little spinning presents due to minimum shear rate. Moreover, for the first time, we experimentally measure lift coefficients, which previously could only be obtained from numerical simulations. Measurements of these key parameters not only validate the previous theoretical analyses of lift forces on the macroscale and confirm the numerical outcomes (in terms of dependence on flow Re and particle size), but also permits precise control of particle behavior within the rectangular microchannels, offering new insights into design and optimization for inertial microfluidic devices. More importantly, our two-stage model of inertial focusing is broadly applicable to cross-sectional geometries beyond rectangular microchannels. While channel cross-section can modulate their magnitude, the three lift forces remain present as focusing forces. By correlating these magnitudes with flow velocity field, one can determine equilibrium positions in arbitrary microchannel cross-sections. For example, one would expect three equilibrium positions in a microchannel with equilateral-triangle cross-sectioned. Or, in a semicircular ´ channel, one would expect particles to form semi-Segre annulus at first but finally occupy two positions at the centers of both walls where velocity profile is symmetric and rotationinduced lift disappears. Ultimately, control of equilibrium positions would advance the research of inertial microfluidics. Although there has already been intense attention to the inertial microfluidics in the recent years, the primary focus has been on development of devices for manipulation of cells, with less emphasis paid to elucidation of particle 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, has led to poor guidelines and difficulties in design of inertial microfluidic systems. With elucidation of the equilibration mechanism, we envision better guidelines for the inertial microfluidics community, ultimately leading to improved performance and broad acceptance of the inertial microfluidic devices in a wide range of applications, from filtration to cell separations.

Conclusions In summary, in this work we describe and experimentally validate the two-stage model inertial focusing in microchannels. We also, for the first time, experimentally demonstrate that rotation-induced lift plays an important role in the inertial migration of particles. Compared with previous numerical results, our experimental results suggest that shear-induced lift (negative lift) is the leading force that drives particles toward the channel walls. As particles approaching


Experimental Our experimental approach was to visualize flows of the fluorescently-labeled microparticles in microchannels at successive downstream positions using an inverted epifluorescence microscope (Olympus IX71) equipped with a 12-bit CCD camera (Retiga EXi, QImaging). Analogues to microparticle streak velosimetry (m-PSV), flowing particles generated streaks across each frame, and we analyzed fluorescent intensities and

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Paper locations of these particle streaks. At least 100 frames were obtained and stacked using ImageJ1 at each downstream position. Fluorescence intensity linescans were used in quantitative analyses. Fluorescently-labeled, neutrally buoyant polystyrene particles 7.32 to 20 mm in diameter were used in this work. Particles were mixed at 0.025% volume fraction in deionized water to minimize the particle–particle interactions. Particles were purchased from a number of vendors, depending on size (Bangs Lab Inc., Polyscience Inc., and Life Technologies Inc.). A small drop (y1% volume fraction) of Tween-20 was added to avoid 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 provide stable flow rates (10 , Re , 120) in all particle flow experiments. We used standard soft lithography methods to fabricate microchannels in this work. Briefly, we used dry resist PerMX 3050 (DuPont) to form masters on 399 silicon wafers by conventional photolithography. Polydimethylsiloxane (PDMS, Down Corning) was cast on the master, degassed, and cured for 2 h on hotplate at 80 uC. Replicas were peeled and bonded to 199 6 399 standard microscope glass slides (Fisher Scientific) using a surface treater (Electro-Technic Products Inc.). The inlet and outlet ports were cored manually with flat-head stainless needles.

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

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