Microparticle sorting in microfluidic Taylor–Couette flows

In recent years, the field of microfluidics has experienced significant advances, especially in the sorting and classification of microparticles.¹ These technologies play a vital role in a variety of applications, including biomedical diagnostics, environmental monitoring, and pharmaceutical manufacturing.² 

The separation of particles and cells is essential in lab-on-a-chip applications across science, engineering, and industry sectors. A broad range of physical mechanisms and principles are employed, from active separation techniques such as acoustic, electric, magnetic, and optical methods to passive techniques that leverage inertial, viscoelastic, and centrifugal effects.³ Additionally, sophisticated geometrical constraints are utilized in techniques such as deterministic lateral displacement (DLD)^4,5 and pinched flow fractionation (PFF).⁶ 

While active methods offer direct control over particle separation, they require specific properties of the particles to be effective.^7,8 Conversely, passive methods depend on hydrodynamic or geometrical effects and do not require specific particle properties other than geometry and density. However, passive methods necessitate sophisticated microchannel designed with specific geometries or additional micrometer-sized obstacles within the channels.⁹ 

A well-known challenge in these systems is that the microchannels must have small cross sections to generate strong hydrodynamic forces.¹⁰ This results in a complex manufacturing process and also increases the likelihood of particle clogging. In general, it is known that the longer the microfluidic channels, and thus the longer the travel time of the particles, the higher the degree of separation. However, increased channel length also increases manufacturing costs, leads to significant pressure losses, and strong hydrodynamic forces increase the risk of damaging biological particles such as cells.^11–13 

To extend the travel time of particles within microchannels without increasing their length, rotating systems such as microfluidic Taylor–Couette devices offer a promising alternative, as will be detailed in this article. These devices allow continuous rotation of either the inner or outer wall, giving hydrodynamic forces ample time to act on the particles, facilitating migration and size segregation under well-set boundary conditions. In addition, the absence of flow constrictions prevents flow circulation regions and reduces particle accumulation or clogging effects. Furthermore, these systems can achieve high particle throughput because their length or diameter can be easily adjusted while maintaining a narrow gap width. In addition, they can be manufactured using conventional techniques available in most workshops.

Generally, rotating systems introduce specific dynamics concerning particle behavior, such as size segregation in granular drum flows and fingering wave patterns in partially filled liquid drum flows.^14,15 When solid particles and liquids are subjected to rotation and gravity, they often form unique structures. One of the most notable is “particle banding,” where initially homogeneously suspended particles are arranged into periodic, axially well-separated bands.

This phenomenon was first observed in partially filled horizontal Taylor–Couette (TC) flows, drum flows with neutrally buoyant particles, and in settling particles in partially filled drum flows.^16–20 Subsequent studies have also observed particle banding in completely filled rotating drum flows with low-viscosity liquids, where particles collected in bands with specific wavelengths.^21,22 Further research has confirmed that inertial waves are the driving mechanism of band formation in the latter scenarios.^23,24 Previous studies primarily focused on monodisperse suspensions. However, the work of Kumar and Singh,²⁵ which considered bidisperse suspensions in rotating drums, observed that particles segregated into bands of alternating particle size. Recently, Brockmann et al.²⁶ extended the parameter space explored in previous studies for the case of low-viscosity fluids.^21–24,27 They discovered a wide range of particle banding patterns and compared banding in drums and co-rotating Taylor–Couette flows. They found that the inner cylinder in the Taylor–Couette flow can stabilize banding patterns, resulting in more defined bands.

However, most studies dealing with Taylor–Couette flows have focused on particle dynamics in the presence of spiral, wavy, and steady Taylor vortices,^28–32 or on the effect of particles on the transition behavior of the flow.^33,34 Few have studied the behavior of neutrally buoyant particles in laminar TC flows.^32,35 To the best of the authors' knowledge, no studies have investigated the behavior of suspensions with settling particles or suspensions with polydisperse particle distribution in (microfluidic) Taylor–Couette devices.

We observed band pattern size segregation in a laminar Taylor–Couette device with an extremely small gap (see Fig. 1). This discovery revealed that laminar Taylor–Couette devices hold potential for particle separation. Motivated by the need for reliable methods for separating microparticles, this paper investigates the banding related size separation of particles in microfluidic Taylor–Couette devices with gap widths in the micrometer and millimeter range (130–1200 µm).

All experiments performed in this study are based on different microfluidic Taylor–Couette devices consisting of a transparent acrylic outer cylinder [Fig. 2(a), “1”] and an opaque aluminum inner cylinder (“2”). To create the outer cylinder, a hole is milled in an acrylic block and all surfaces are polished. A bearing system allows the outer and inner cylinders to rotate independently. Two different outer cylinders of 38 and 40 mm diameter and nine different inner cylinders of $L=90$ mm length and diameters between 36 and 39 mm were fabricated and used in the experiments to create different geometries. In addition to shafts with a constant radius along the axis of rotation, shafts with axial varying radii are created, resulting in shafts with conical segments as shown in Fig. 2(b). In addition, an outer cylinder is created with its radius slightly increasing along the axis of rotation. The gap height is determined by measuring the outer cylinder diameter with a 3-point inside micrometer (TESA, 5 µm resolution), while the inner cylinder radius is measured with a high-precision caliper (Mahr, 10 µm resolution). The gap heights measured at different positions A, B, and C (see Fig. 4) are hereafter referred to as $HA$⁠, $HB$⁠, and $HC$⁠. The change in gap width is described as $ΔH=|(Hmin−Hmax)|/(Hmin+Hmax)×2×100$%.

The inner and outer shafts are driven independently by different 12 V DC motors (“3”) with different gear ratios, depending on the desired speed. The motor is connected to the shaft by a timing belt, allowing the motors to be easily changed by simply rearranging the belt. The rotation rate is controlled by adjusting the voltage of the power supply (“4”) and measured by a photoelectric barrier (“5”) and a perforated disk mounted on the shaft, with the information processed by an Arduino MEGA.

The particle distributions are recorded from above using a Nikon AF Nikkor 20 mm 1:28D lens (“6”) connected to an IDS U3-3080CP-C-HQ Rev 2.2 CMOS camera (“7”). A Logitech C920 HD PRO webcam and an Iphone 13 mini (mounted on a stable platform) are used to capture images of the particles from the side. Illumination is provided by a Veritas Constellation 120E high-power LED (“8”).

Distilled water mixed with Pluronic F127 (Merck, Germany) or SDS (Carl Roth) surfactant is introduced into the system through four holes drilled in the top and bottom of the system using a submersible pump. It was found that F127 is more efficient to prevent clustering of small particles (6 µm) compared to SDS. By rotating the shaft, opening and closing the holes, and pumping intermittently, air bubbles are removed from the system. The particles are then filled into a syringe mixed with distilled water and are carefully injected into the system.

To prepare the suspension, we use polymethyl methacrylate (PMMA) particles of 6, 15, 30, 40, and 60 µm in diameter (“Microbeads”). The (bulk) solid volume fraction or particle volume fraction represents the average solid volume fraction in the liquid. It is defined as $Φ=Vparticles/Vtotal$⁠, where $Vparticles$ is the volume of the particles and $Vtotal$ is the combined volume of the liquid and particles (⁠ $Vtotal=Vparticles+Vliquid$⁠). In the case of polydisperse suspensions, the volume fraction was equally distributed among the sizes. To distinguish the particles in the images, they are dyed with different colors, including Rhodamine B and various food dyes.

Before each experiment, the inner cylinder is rotated at high speed so that the flow becomes transient and turbulent such that all the particles are mixed and homogeneously distributed in the system. Before each experiment, great care is taken to ensure that the particles are homogeneously distributed along the cylinder axis and that the system is leveled. The experiment begins with the cylinder at rest and all particles settled at the bottom. The rotation rate is then continuously increased to the target value, within 1 min. The experiments were conducted in the laminar regime, with careful checks to detect any vortices, which were easily identified by visual tracking of the particles. Prior to these tests, the critical Reynolds number was calculated. At Reynolds numbers above this threshold, the flow becomes unstable, potentially leading to vortex formation. The critical Reynolds number was estimated using the code of Brockmann et al.³⁶ and the formula provided by Esser and Grossmann³⁷ and then converted to a rotation rate. It is known from Taylor–Couette devices with pronounced conical shapes in both the outer and inner cylinders that the velocity profile differs from that of a cylindrical Taylor–Couette device. However, in our setup, the outer cylinder is mostly cylindrical and the inner cylinder is less conical compared to, for example, the work of Wimmer.³⁸ Therefore, the conicity of the cylinders was not taken into account when estimating the critical Reynolds number. Instead, the minimum and maximum gap widths were considered using the analytical velocity profile for a cylindrical Taylor–Couette system. Images were captured every 10 min until the system reached equilibrium, at which point no further changes in the particle patterns were observed.

To perform microscopic particle image velocimetry (µPIV), a custom-built microscope was designed with an optical tube (InfiniTube Special, Infinity Photo-Optical) illuminated by a 7 W high-power green LED [ $λ≈0.532$ nm, ILA iLA.LPS v3, “10” in Fig. 2(a)]. A dichroic mirror and two bandpass filters mounted on a filter cube (Thorlabs DFM1/M, “11”) direct the light through the objective and filter the fluorescence signal. The system uses an Infinity Photo-Optical IF-3 objective (“12”) with 1× magnification and a Nikon Cfi60 objective with 20× magnification (“12”). Images are captured using a 12-bit, 2048 × 1952 pixel high-speed CMOS camera (Phantom T1340, Vision Research, “13”) at 3273 frames per second. The optical system is traversed in 50 µm steps to scan the gap [arrow in Fig. 2(a)]. The depth of correlation (DOC) is estimated to be 46 µm³⁹ for 20× magnification and 6 µm particles. Similar configurations have been shown to be reliable for microscopic flows.^40–42 

The µPIV analysis is performed using DaVis 10 (LaVision). First, a geometric mask filter is applied, followed by subtraction of a 5 px $×$ 5 px sliding average, and finally, a 5 px $×$ 5 px median filter is used. The PIV cross correlation is performed with a 128 px $×$ 128 px correlation window, starting with two initial passes with 50% overlap, followed by another pass with 75% overlap. The window size is gradually reduced to 64 px $×$ 64 px. Adaptive correlation windows are used, and the vector field is generated by summing the correlations of 307 frames. To assess the uncertainty in the velocity data, the scanning procedure was repeated three times. The velocity data from each measurement plane are averaged to produce the velocity profiles in the z-direction.

PIV analysis was also performed on data obtained from the macroscopic view [camera “7” in Fig. 2(a)] to gain insight into the band evolution process. The post-processing included a time series subtract-average filter followed by a 5 px $×$ 5 px sliding average subtract filter. The initial correlation window is successively reduced from 96 px $×$ 96 px to 16 px $×$ 16 px, with two initial passes with 50% overlap and two additional passes with 75% overlap. The vector field was generated by summing the correlations of 1000 frames.

Equation (1) accounts for inertia, pressure-, lift-, drag-, and effective buoyancy force. Here, $ρ$⁠, $ρp$⁠, $U$⁠, $u$⁠, V, A, and g denote the density of the fluid, the density of the particle, the velocity of the fluid, the velocity of the particle, the particle volume, and the projected surface area in the direction of the flow of the particle and the gravitation, respectively. $CA$ = 0.5 is the added mass coefficient that account for the mass of fluid that is accelerated with the particle.

The Reynolds number is defined as $Rep=dp|U−u|ρ/µ$⁠. Equation (1) is integrated numerically using Matlab (ode15s). Interaction with the inner and outer cylinder walls is realized by introducing a perfectly elastic spring force with a spring constant of $cSpr=10g{(ρp−ρ)/(ρCA+ρp)}(1/0.125dp)$⁠.

To validate the code, a reference case²⁴ was considered. Figure 3(b) shows the numerical results of the present work (blue, green and black lines) compared with the numerical (red line) and experimental data of the reference case²⁴ (circles). Unlike the reference case, the governing equations in this work also include the virtual mass force and the lift force induced by system and particle rotation. When the virtual mass and lift forces are neglected in our simulations (blue line), there is excellent agreement with the numerical reference data²⁴ (red line). However, when the virtual mass force (green line) and lift force (black line) are included, the resulting orbit shows slight deviations. Nevertheless, all scenarios—whether including or neglecting the virtual mass and lift forces—effectively capture the particle dynamics, as evidenced by the close agreement with the experimental results (circles) in Fig. 3(b). In the following sections, all forces are considered in the simulations.

All the results presented below pertain to scenarios where the inner cylinder is rotated while the outer cylinder remains stationary. Our experiments have demonstrated that in these systems, particles initially distributed homogeneously tend to migrate axially within conical annular gaps. Depending on the setup, this migration can occur either toward the narrower or wider sections of the gap when the inner cylinder is rotated in a laminar regime without the presence of vortices. The temporal evolution of particle migration is exemplified in Figs. 4(a) and 4(c) for a monodisperse suspension. Shortly after the experiment begins, the particle distribution shifts rapidly, within seconds, to the right end of the system. After 10 min, all particles are collected in a band close to the system's end. In most cases involving a rotating inner cylinder and a conical inner shaft, we observed that particles typically migrate toward the narrower section of the gap, suggesting a preference for this position in such configurations. Conversely, when pairing a conical outer cylinder with a perfectly cylindrical inner shaft, we sometimes noted particle migration toward the wider section of the gap. This trend was also occasionally observed in systems with shafts of varying slopes. It is also important to note that in the majority of these experiments, the particles did not migrate completely to one end of the system; rather, they stopped at some distance from the system's end point. Moreover, we also found that the rotation rate typically affects the position of the particle band. An example for particles of size 60 µm and a conical gap with changing outer cylinder diameter (“SCOO”) is given in Figs. 4(a) and 4(c). As can be seen, the equilibrium particle concentration is shifted to the right corner, where the gap width is wider than in the left corner. Similarly, in systems with different slopes, such as a quadruple cone (QCO) and a double cone (DCO), particles are observed to migrate in the axial direction and accumulate in bands [Fig. 4(b)]. In Fig. 4(d), where the slope varies along the axial direction, two bands can be found resulting from particles from the left and right sides of the cylinder. Here, the very narrow gap width present in the middle section (⁠ $HA≈224$ µm) seems to act as a particle barrier that prevents the two separate bands from merging.

When several particle sizes are present in the system, they can be arranged in staggered bands according to their size, as shown in Fig. 5 for a single cone (SCO) and for a quadruple cone (QCO) in Fig. 6. From the time series shown in Figs. 5(a) and 5(d), we can see how particles of different sizes migrate into their respective bands over time. For polydisperse suspensions, reaching equilibrium can take significantly longer than for monodisperse ones, requiring times of up to 1–2 h, especially when smaller particle sizes are involved [Figs. 5(a) and 5(d)].

Within our experiments, we could not find a clear trend in which direction particles migrate and in which size order they finally arrange. They may be arranged in symmetrical patterns or in asymmetrical patterns, as can be seen from Figs. 7(a) and 7(c). The experiments indicate that symmetric patterns are more likely to occur at lower rotation rates, while asymmetric patterns occur more likely at higher rotation rates. This can be seen exemplarily in Fig. 8, where we show the effect of increasing rotation rate on observed patterns in a bidisperse suspension. Asymmetric patterns have also been observed in systems with pronounced conical shape of the inner cylinder. However, more systematic experiments are required here for some definite conclusion.

We also could not find a general rule for the particle size order in asymmetrical systems. While most of the experiments showed that the largest particles are closer toward the ends of the system, we also could find cases where smaller species where found to occupy the most outward bands.

We observed that particle size separation can occur in bidisperse, tridisperse, and quadrodisperse suspensions, separating particle sizes from 60 to 6 µm. In general, the larger the particles and the smaller the number of particle sizes involved, the sharper the separation. Also, the time to reach equilibrium state is significantly shorter for larger particles than for smaller particles.

Separation of bidisperse suspensions (60  and 40 µm, 60 and 30 µm, and 15 and 6 µm) appeared as a very robust phenonema and could be achieved easily in most cases. However, separation of quadrodisperse suspensions appeared to be very sensitive to disturbances such as clustering of smaller particles, dirt, and airbubbles and selection of the correct rotation rate.

In general, the separation process is very susceptible to imperfections of the experiment. Typical sources of disturbance are airbubbles which have been trapped in the bearings and then get into the flow as well as abrasive material which originates from the bearings and sealings. Also sometimes we observed that after several hours particles tend to form clusters, resulting in a different effective particle diameter.

It may be mentioned that when both cylinders had an almost perfect cylindrical form, e.g., with constant gap width in axial direction, no particle migration or separation of particles could be observed. Instead, particles distribute along the gap and tend to form bands resembling the “axial band patterns type 1” observed by Brockmann et al.²⁶ [see Fig. 9(a)]. However, the number of these bands reduces over time and particles tend to agglomerate at the ends of the cylinder axis. As can be seen from Figs. 9(a) and 9(b) in an almost perfect cylindrical gap, shortly after the start of the experiment the particles collect into several bands. A separation of particles within these bands can also be seen [Fig. 9(a)]. Over the time, these bands travel outwards and collect into particle accumulations at the ends of the cylinder. Interestingly, in these accumulations a size segregation occurs, which could be attributed to the “brazil nut effect.”⁴⁹ This behavior was observed for both higher [Figs. 9(a) and 9(b), 1200 µm] as well as for lower gap widths [Fig. 9(c), 400 µm].

We also performed experiments with the outer cylinder at rotation. Compared to the cases with rotating inner cylinder, we found that the particle bands were much more compact, as can be seen exemplarily in Fig. 10(a). As can also be seen from Fig. 10(a), the bands show some degree of particle separation. Occasionally, as exemplified in Fig. 10(a), two bands were observed; over longer periods (60+ min), these sometimes merged into a single band.

However, when smaller particle sizes were involved, we could not find a rotation rate providing stable bands or some clear size segregation. Instead, it was often observed that particles migrate toward the end of the system, as can be seen exemplarily in Fig. 10(b). In contrast, when the same system was subjected to inner cylinder rotation, a clear particle separation could be observed over a range of 10–40 rpm, as shown exemplarily for 40 rpm in Fig. 10(b).

To explore the process of particle separation as well as band formation, we conducted particle image velocimetry (PIV) measurements on a specific case. A monodisperse suspension containing 60 µm particles was prepared, ensuring homogenous distribution before initiating the experiment, which resulted in the formation of a distinct band. We did not wait for all particles to aggregate within the band to capture particle velocity data across the system. Notably, the position of the band remained relatively stable compared to its final equilibrium state, where all particles are collected within the band. Hence, this measurement also captures the dynamics of the band formation process. The velocity field analysis indicated that the particles velocity varies significantly along the axial direction, specifically in the band. As depicted in Fig. 11, the image shows the particle band in the upper half of the system, contrasting with particle velocities in the lower half. Remarkably, the velocities peaked to the left and right of the band, where particles in these regions move faster than those in the band center. This velocity disparity leads to collisions among particles moving along different trajectories. We infer that these collisions promote shear-induced migration, generating a driving force in suspensions of monodisperse particles.⁵⁰ This phenomenon will be explored in more detail later in the text. The measurements presented in Fig. 11 demonstrate that particle velocity varies throughout the system during the band formation process. To provide a more comprehensive analysis, we will now discuss additional experiments that not only present the velocity profile across the gap obtained via µPIV but also reveal the shear rate distribution and the influence of particle concentration on the direction of migration.

For this, an experiment was performed with the quadruple cone geometry in the steady state when the band formation process was completed. This geometry was chosen because it restricts the particles to smaller sections, accentuating particle migration effects. The system was filled with 6 and 60 µm particles at overall volume fractions of 0.1% and 1.44%, respectively. The experiment was also repeated with 6 µm particles only, to approximate the flow of the pure liquid and use the 6 µm particles as tracers to extract velocity information. While great care was taken to homogeneously distribute the small particles, intentionally more 60 µm particles were introduced on the right side of the system to highlight differences in band formation caused by the influence of volume fraction, such that approximately 80%–90% of 60 µm particles were in the right half of the system. The inner cylinder was rotated at 46.5 rpm. Shortly after the initiation of rotation (1–2 min), band formation of 60 µm particles was observed in both the left and right sections. Figure 12(a) shows the steady-state result of the band formation after 2 h. It can be seen that on the left side of the system, the particle band migrated in the direction of decreasing gap width; while on the right side, the particles migrated in the direction of increasing gap width. This indicates that the direction of particle migration also depends on the volume fraction and underlines that particles do not always travel in the direction of decreasing gap width. The azimuthal velocity profile measured with µPIV right beside the band in ROI1, as indicated in Fig. 12(a), shows no significant deviation from the analytical solution for a non-conical Couette flow, as given in the literature.⁴⁶ In general, the agreement with the analytical solution is excellent [Fig. 12(b)]. However, in the case of 60 µm particles, it appears that the azimuthal fluid velocity is slightly slowed down due to the presence of the particles. This reveals that indeed the presence of 60 µm particles affects the velocity profile. The kink observed in Fig. 12(b) for $z′>600$ µm is related to measurement errors due to out-of-plane correlation effects in the µPIV evaluation. This deviation is also visible for red data points in Fig. 12(c). In Fig. 12(c), where we show the axial velocity profile, it can be seen that the presence of 60 µm particles induces a significant axial Poiseuille-like flow in the direction of the band. This axial flow can explain how the particles migrate to the location of the band. A slight axial flow can also be seen with only 6 µm particles. From Taylor–Couette flows with a significant conical shape it is known that centrifugal effects induce a meridional velocity in the direction of the axis of rotation.³⁸ However, this velocity exhibits maxima at the walls, while in our case the component exhibits maxima in the channel center. We therefore conclude that in our case, where the outer cylinder is not conical, the conical shape does not induce most of the axial flow. Instead, the presence of particles (a particle band) induces an axial Poiseuille-like flow component. However, the slight backflow observed in Fig. 12(c) (green data point) could be related to the meridional component induced by the conical shape. Due to mass conservation, the axial net flow must be zero in the system, which means that the integrated axial flow has to cancel out, implying that there must be an axial flow in the negative direction somewhere along the circumference. Hence, depending on their azimuthal position, particles could be transported either toward or away from the band, suggesting that the exact migration dynamics are yet to be fully explained. To understand the distribution of particle velocities within the band, we performed µPIV on the 60 µm particles in ROI2 (Fig. 12). The resulting velocity and shear rate distributions are shown in Figs. 12(e) and 12(f). As can be seen, the azimuthal velocity $uφ$ decreases significantly (by 20%) toward the center of the band. This results in significant gradients of $duφ/dx$ on either side of the band center. When the particle band consists of polydisperse particles, both large and small particles exhibit varying shear-induced migration rates. This is because the driving force behind shear induced migration scales approximately with $dp2$⁠.⁵¹ This difference in rate of migration then can lead to particle separation.^52–54 

To gain a deeper insight into the dynamics of small and large particles, we further analyze the computed single particle trajectories shown in Fig. 13. All particle sizes exhibit clockwise orbital motion, with their radial positions within the gap fluctuating with time. Once the particles reach a steady state, the shape of their orbits becomes consistent. This steady state is reached when the particles make contact with the outer cylinder, where wall contact up to $φ=180°$ forces them into a stable trajectory. This occurs in less than 80 s for 60 µm particles and less than 240 s for 15 µm particles, regardless of their initial positions. In contrast, 6 µm particles remain dispersed in the gap for a significantly longer time (>600 s) before reaching a steady state orbit, highlighting their suitability as tracers for µPIV measurements. The following discussion will focus exclusively on these steady-state orbits. At $φ=180°$ all particles touch the outer wall. Larger particles, whose centers are closer to the inner cylinder, move faster than medium and small particles at this point. For $φ>180°$⁠, all particles detach from the outer wall and begin to settle into the gap [see Figs. 13(a)–13(c)]. At $φ=90°$⁠, the largest particles are near the center of the channel, while the smaller particles remain closer to the outer wall. All particles reach their closest point to the inner cylinder at $φ=0°$⁠. Note that the largest particles are much closer to the inner cylinder wall compared to the medium and small particles. Throughout the orbit, the largest particles remain much closer to the inner cylinder. As a result, they are exposed to higher fluid velocities. Consequently, as shown in the polar plot in Fig. 13, they reach the highest velocity. The calculated velocities for 60 µm particles agree well with the measured velocities in the band shown in Fig. 12(e). These trajectories show that the particles occupy different radial positions and reach different velocities within the gap depending on their size.

Based on both experimental and numerical results, we conclude that mutual interactions drive banding and species separation in polydisperse suspensions. The particles themselves generate an axial flow that drags other particles along the axial direction, ultimately leading to the formation of particle bands. As shown by simulations, larger particles tend to occupy positions closer to the gap center at $φ=90°$ and consequently experience different axial velocities according to the profile shown in Fig. 12(c). Thus, larger particles move at higher axial velocities and accumulate faster, explaining why bands of larger particles appear first during the banding process in polydisperse suspensions. Within the particle bands, the velocities of the particles in the center are significantly lower than those at the edges. This velocity difference leads to particle collisions and subsequent shear-induced migration toward the center of the band, contributing to band stabilization. In addition, since the migration rates of large and small particles differ, this results in particle separation, consistent with previous observations in duct flow experiments.⁴¹ Simulations also show that particles of different sizes experience markedly different velocities, which may further contribute to segregation processes such as “lane formation”—a phenomenon where particles with different velocities segregate into distinct lanes, as seen in gravity-driven colloidal flows.⁵⁵ 

In the present work, we present a novel method for the separation of micrometer-sized particles using a microfluidic Taylor–Couette device with gap widths in the range of about 120–1200 µm at volume fractions ranging from $Φ=2.2$% to $Φ=11$%. The Taylor–Couette apparatus is operated under laminar conditions, below the rotational speed at which Taylor vortices occur. Our experiments demonstrate that this method can effectively separate suspensions containing up to four different particle sizes, ranging from 60 to 6 µm.

Our study revealed that particle migration and separation occur only when there is a variation in the gap height along the axis of the system. In scenarios where the cylinders are nearly perfectly cylindrical, we observed the formation of band patterns that migrate bidirectionally toward both ends of the annular gap. In the case of a conical inner shaft paired with a cylindrical outer shaft, we observed that particles migrate preferentially in the direction of decreasing gap width.

While rotation of the outer cylinder produced sharply focused bands and facilitated some separation of larger particles in bidisperse suspensions, rotation of the inner cylinder proved more effective in separating smaller particles.

Given the wide range of parameters involved (particle size, gap width, cone shape, particle volume fraction, and rotation rates), we could not find a universal critical dimensionless number for particle separation. In general, it can be said that a cylinder diameter of about 40 mm, a gap width in the range of 120–800 µm, and gap width changes $ΔH$ in the range of 30%–100% can produce particle separation of particles in the range of 6–60 µm at rotation rates of about 20–140 rpm, given a laminar flow regime.

In general, the observed particle separation appears to be a very robust phenomenon, as it was observed in Taylor–Couette Systems with cylinder radii of $Ri≈18–19$ mm and $Ri≈61.22$ mm.

Our experiments demonstrated that dividing the shaft into multiple conical sections impacts both the total number of bands and their positioning. We observed that particles tend to be confined within these individual sections. Consequently, future designs could leverage specific shaft configurations to direct particle bands to designated locations, facilitating targeted extraction.

Micro-PIV measurements revealed that the presence of particle bands induces a significant axial flow velocity component with a maximum at the center of the gap. This axial velocity component, in turn, induces axial migration of the particles, resulting in a reciprocal process of axial flow generation, banding, and particle migration. Single particle simulations showed that particles of different sizes occupy different radial positions, resulting in differences in their axial transport velocities. In addition, the particle velocity throughout the orbital motion is highly size dependent, resulting in different collision rates between small and large particles. Furthermore, experimental results have shown a pronounced shear gradient in particle velocity across the particle bands in the axial direction. This gradient leads to frequent particle collisions within the band. We expect that this will promote shear-induced migration, acting as a mechanism for size-based separation. This is consistent with the findings of Brockmann et al.,⁴¹ who demonstrated size-based particle separation in a tridisperse suspension within microchannel flows.

In summary, this paper presents a novel method for particle separation and explores different scenarios. It also provides initial insights into the underlying particle dynamics, providing preliminary explanations for axial migration and particle segregation. Future research should focus on further investigating the driving mechanisms responsible for the diverse particle separation results observed in this study.

The first author was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Project No. 265191195-SFB 1194, sub-project A03).

The authors have no conflicts to disclose.

Philipp Brockmann: Conceptualization (lead); Data curation (lead); Investigation (lead); Methodology (lead); Project administration (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Christoph Symanczyk: Methodology (equal); Validation (supporting); Writing – review & editing (supporting). Xulan Dong: Data curation (supporting); Investigation (supporting); Visualization (supporting); Writing – review & editing (supporting). Yashkumar Kagathara: Data curation (supporting); Investigation (supporting); Visualization (supporting). Lukas Corluka: Data curation (supporting); Investigation (supporting). Jeanette Hussong: Resources (lead); Supervision (lead); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.