1 Introduction

Separation of particles based on size is an essential step in numerous industrial and biomedical processes. Filtration, sedimentation, and centrifugation techniques are frequently used to separate and classify particles and cells. But performance of these approaches is generally poor when the particle size and the size difference between target and non-target particles are small (Bhagat et al. 2010b). Furthermore, clogging and fouling of device components is inevitable, which fundamentally limits the efficiency and reliability of the filtration process (Warkiani et al. 2012). The development of microfluidics in the past 15 years has brought about new methods for the separation and selection of microparticles through precise control of fluid flow. Among all existing microfluidic systems, inertial microfluidics (IMF) has experienced substantial growth in many applications such as cell detection and isolation (Bhagat et al. 2010b; Hur et al. 2011), cytometry (Bhagat et al. 2010c), multiplexed bio-assays (Chen et al. 2008; Situma et al. 2006; Wang and Li 2011), and fluid mixing (Sudarsan and Ugaz 2006). As a passive technique, it manipulates cells and particles (by taking advantages of hydrodynamic forces) in microchannels with variety of cross sections (i.e., square, rectangular, trapezoidal, triangular, and circular), in the absence of an externally applied field, and combines the benefits of a passive approach with extremely high-throughput and yield (Amini et al. 2014; Zhang et al. 2016). There are also a plethora of active techniques such as magnetophoresis (Murthy 1999; Pamme and Manz 2004), dielectrophoresis (Kralj et al. 2006; Voldman 2006; Zhu et al. 2010), and acoustophoresis (Destgeer et al. 2013; Ma et al. 2016; Nilsson et al. 2004; Petersson et al. 2007) for particle separation. There are comprehensive and insightful reviews about these different techniques later in this literature (Di Carlo 2009; Sajeesh and Sen 2014; Warkiani et al. 2015).

There are basically two types of IMF including structured and non-structured channels (Amini et al. 2014). Any pattern, in non-structured channels, such as spiral (Bhagat et al. 2008a; Guan et al. 2013; Martel and Toner 2013), asymmetric (Di Carlo et al. 2007), serpentine (Zhang et al. 2014a, b), and contraction/expansion (Lee et al. 2011; Park et al. 2009; Sim et al. 2011; Wu et al. 2016) induces a secondary flow, which is perpendicular to the main flow direction. The channel length in such device can be increased without significantly increasing the size of the device and compromising its portability due to the radial geometry (Yang et al. 2011). Although structured channels have been widely used in recent years, straight channel has some benefits such as small footprint, ease of parallelization, simplicity in fabrication and design principles (Zhang et al. 2014b).

Inertial migration of neutrally buoyant particles across the streamline flowing through a channel was first observed by Segré and Silberberg (1961)—it was called tubular pinch effect (Segré and Silberberg 1961, 1962), formation of focused ring-like particle region that is ~ 60% away from center of channel. Initially, this phenomenon was conceived as the interaction between two major forces: shear-gradient and wall-induced lift forces which were analytically proved by applying perturbation method (Asmolov 1999; Ho and Leal 1974; Schonberg and Hinch 1989; Vasseur and Cox 1976). However, this model is not able to explain the focusing position in non-circular cross-section microchannels. Cross-lateral migration of particles inside square and rectangular microchannels experimentally proved the wall-effect lift force when the inertial effect (i.e., particle Reynolds number RP = Rea2/D 2H , where Re is channel Reynolds number, a is particle diameter, and DH is the hydraulic diameter of channel) goes beyond a certain threshold, RP ~ 1 (Bhagat et al. 2008c; Di Carlo et al. 2007). Accordingly, Papautsky’s group recently proposed a two-step particle focusing model for the rectangular cross-section channels which is in good agreement with the experimental results (Zhou and Papautsky 2013). However, this model fails to explain the inertial focusing for non-rectangular microchannels such as triangular (Kim et al. 2016). The number of equilibrium positions inside a rectangular channel relies on Re number (\(R_{\text{e}} = {{\rho_{\text{f}} U_{ \hbox{max} } D_{\text{H}} } \mathord{\left/ {\vphantom {{\rho_{\text{f}} U_{ \hbox{max} } D_{\text{H}} } \mu }} \right. \kern-0pt} \mu }\), where ρf is the density of fluid, Umax ≈ 2Uavg is the maximum flow velocity, and µ is the dynamic viscosity of fluid), aspect ratio (AR = W/H, where H and W are height and width of the channel, respectively), and particle clogging ratio, K = a/Hmin (Liu et al. 2015). The number of focusing points varies between 2 and 4 at the centers of long and short walls (Bhagat et al. 2008b, c; Chung et al. 2012; Di Carlo et al. 2009; Edd et al. 2008; Hur et al. 2010; Mach and Di Carlo 2010). Mainly two equilibrium positions inside a rectangular microchannel, adjacent to the longer walls, have been utilized for particle filtration (Bhagat et al. 2008b; Zhou et al. 2014), fractionation (Zhou et al. 2013), and flow cytometry (Bhagat et al. 2010a; Oakey et al. 2010) when the AR varies considerably from unity. Recently, experimental results demonstrated that the higher the clogging ratio K (0.3 ≤ K < 0.4), the higher the AR (2 ≤ AR < 5) will be to obtain two strong focusing streaks, i.e., narrower particle band width in the middle of long faces of channel (Liu et al. 2015; Reece and Oakey 2016). Generally, as the ratio of AR/K goes up, the particle band width increases (Liu et al. 2015) within the perspective of inertial microfluidics 10 ≤ Re ≤ 100 (Amini et al. 2014; Bhagat et al. 2008c; Di Carlo 2009). In this study, AR of 2 and 4 is, therefore, chosen at which the best particle focusing is achieved.

Despite comprehensive experiments and numerical analysis of the lift forces in a rectangular (Di Carlo et al. 2009; Liu et al. 2015) and non-rectangular microchannels (Kim et al. 2016), which all possess planes of symmetry, less effort is made on the “flow field structure” of trapezoidal straight microchannels and its effects on inertial focusing. On the other hand, a rectilinear channel with trapezoidal cross sections breaks down all planes of symmetry. Though trapezoidal cross section has been exclusively applied in curved microchannels to alter the secondary flow structure—thereby improving the separation resolution (Guan et al. 2013; Warkiani et al. 2014a, b; Wu et al. 2012) and filtration (Rafeie et al. 2016)—the channel AR (AR = W/Hmin) ranged from ~ 7.1 to ~ 12.5 in previous works, which was relatively high, and the slope of slanted wall was limited to low degree (Tan(α) < 0.1). Given the simplicity (operation and manufacturing) of straight microchannels compared to the spiral channels and ease of parallelization, we believe that for certain type of applications, straight microchannels would be better candidates for particle sorting. In this work, inertial focusing was experimentally investigated in a straight microchannel with a trapezoidal cross section—without the presence of curvature-induced secondary flow. The impacts of slanted wall and particle clogging ratio as the two determining factors were mainly studied then. A particle size-dependent lateral focusing shifting toward the longer side wall was found (Re < 100). Its major lateral focusing, nonetheless, moved backward to the shorter side wall for high-Re flows. Remarkably, an almost complete transition of particle focusing with K ~ 0.9 (45 µm) from the longer side wall to the shorter side wall was demonstrated at extremely high particle Re number of RP ~ 85. This phenomenon might shed further light into the mechanism of cross-lateral inertial migration of particles close to the wall (wall-effect lift force) through establishing a comprehensive numerical model in near future. Eventually, as a proof of principle, a trapezoidal straight microchannel was utilized for continuous filtration of a broader range of particle clogging ratio (0.3 < K < 1) in comparison with the rectangular straight microchannels (W > H, 0.3 ≤ K < 0.5 (Zhou et al. 2014)).

2 Theoretical background

The lateral migration of neutrally buoyant particles dispersed in a fluid flowing through a channel originates from inertial lift force which is decomposed to shear-gradient (FS) and wall-induced (FW) lift forces. The inertial effects begin to be dominant if the particle size is comparable to the characteristic channel length, a/DH > 0.07 (Di Carlo et al. 2007). Shear-gradient lift force arises from curvature of velocity profile inside the channel although formation of parabolic velocity profile cannot occur without the presence of the wall (Amini et al. 2014; Zhang et al. 2016). The difference in velocity around the finite-sized particle (Martel and Toner 2014; Zhang et al. 2016) induces a dominating shear-gradient lift force across the channel height (W > H). Particles, therefore, migrate toward the top and the bottom wall in the rectangular channels (Fig. 1a). In contrast, due to the dissymmetry in the corresponding trapezoidal channels, the equilibrium plane near the slanted wall is spoiled and thus particles are mainly driven toward the center and the bottom wall (Fig. 1b). In this zone, adjacent to the wall, a cross-lateral wall-effect lift (FCL) is dominant, driving particles toward the center of long wall in the rectangular channels (Zhou and Papautsky 2013; Zhou et al. 2013). Nonetheless, altering the rectangular cross section to the trapezoidal causes an asymmetry in the velocity profile while the core of maximum velocity declines toward the longer side wall, resulting in off-center turning point in the shear field, whereby lateral focusing shifts toward the longer side wall (Fig. 1b).

Fig. 1
figure 1

Schematic representations of shear-gradient lift force (FS), wall-induced lift force (FW) and cross-lateral wall-effect lift force (FCL) in a a rectangular and b a trapezoidal microchannel. c Rectangular and trapezoidal straight microchannels designed; slope of the slanted wall is Tan(α)

Full size image

According to analytical models developed earlier to justify the particle migration across the streamline (Feng et al. 1994; Ho and Leal 1974), shear-gradient forces are independent of particle rotation (Martel and Toner 2014) and it should be noted that the rotation-induced lift force is 3 orders of magnitude smaller than the shear-gradient lift force (Ho and Leal 1974; Rubinow and Keller 1961). The wall-induced lift force is triggered when the particle approaches the wall. A dissymmetry in pressure distribution around the particle adjacent to the wall pushes back the particle toward the center (Martel and Toner 2014). This dissymmetry in pressure originates from a decelerated fluid flow between the particle and side wall which leads to an increase in pressure (Bernoulli’s principle).

The net inertial lift force scales with a4 which was derived using the perturbation method (Asmolov 1999) based on point-particle approximation, i.e., neglecting finite-sized particle (Di Carlo et al. 2009) as follows:

$$F_{\text{L}} = f_{\text{L}} \frac{{\rho U_{ \hbox{max} }^{2} a^{4} }}{{D_{\text{H}}^{2} }}$$
(1)

where fL is a dimensionless lift coefficient which is a function of particle position for a definite channel cross section and Re number (Di Carlo 2009) which can be decreased by increasing Re number (Lee and Balachandar 2010; Zeng et al. 2009). Recently, it was experimentally shown that fL in a channel with rectangular cross section can be expressed as follows (Zhou and Papautsky 2013), where W > H:

$$f_{\text{L}} \propto \frac{{H^{2} }}{{a^{2} \sqrt {\text{Re}} }}$$
(2)

Therefore, the net inertial lift force FL scales with a2 (FL ∝ a2).

3 Materials and methods

3.1 Design and fabrication

To study the new configuration of the inertial lift forces and their impacts on particle focusing within a microchannel with a trapezoidal cross section, we designed a series of trapezoidal channels with AR = 2 to 4 while the slopes of upper wall varied from ~ 0.2 to ~ 0.65 (Fig. 1c, tan(α) = 0.2–0.65). Also, a microchannel with rectangular cross section (tan(α) = 0) was fabricated not only to compare the results but also as a benchmark to verify the well-recognized particle focusing phenomenon for the designed rectangular channel. All the microchannels’ length is 40 mm, with one inlet and one outlet. The length of microchannels having bifurcation is increased to 60 mm. The height of shorter side wall in the trapezoidal channels which is equal to the rectangular height was chosen to be 50 µm (Fig. 1c). Rectangular microchannels were fabricated using standard soft lithography techniques described elsewhere (Warkiani et al. 2011), while trapezoidal channels were built using micro-milling technique described in our previous works (Warkiani et al. 2014a). The cross sections of all fabricated channels can be seen in Fig. S1. Having been cured in an oven, the polydimethylsiloxane polymer (PDMS, Sylgard 184 Silicone Elastomer Kit, Dow Corning) replica was bonded to another thick layer of PDMS. First, the degassed mixed PDMS and curing agent (ratio 10:1) were cured for 1.5 h in an oven at 80 °C. Next, after replica and blank layer of PDMS were peeled off, the inlet and outlet were pierced using a Uni-Core™ puncher. Finally, the replica and blank layer were bonded irreversibly by oxygen plasma cleaner (PDC-002, Harrick Plasma) and cured for 3 h in an oven at 70 °C. A 1/50″ tubing (Tygon ND-100-80, USA) was used for the inlet and outlet ports.

3.2 Sample preparation

Different neutrally buoyant particle sizes (polystyrene ρ ~ 1.05 g/mL), with mean diameters of 10, 20, 45 µm (Fluoresbrite® YG Microspheres, Polysciences Inc, USA), and 15 µm (Dragon green, Bangs Lab Inc, USA), were used to investigate the effect of particle clogging ratio (K) under different flow conditions. Fluorescent particles were diluted with DI water to reach ~ 0.01–0.02% volume fraction to decrease the particle–particle interaction. Tween 20 (Sigma-Aldrich, Singapore) was added, at a ratio of ~ 0.1% volume fraction to avoid particle clogging or particle aggregation.

3.3 Numerical simulations

To investigate the flow field structure within different channel cross-section profiles, a 3D microchannel model was numerically developed using commercial software ANSYS Fluent (Version 15). The Navier–Stokes governing equations were solved for laminar flow using SIMPLE scheme. No-slip boundary condition was applied to wall. The velocity inlet and atmospheric pressure were set to the inlet and outlet, respectively. The physical properties of liquid water, i.e., density of 998.2 kg/m3 and dynamic viscosity of 0.001002 kg/m/s, were used for flow simulation.

3.4 Experimental setup and measurement

Each suspension was pumped through the channel using a syringe pump (Chemyx, F200, USA) at different flow rates from 0.05 to 5 mL/min. The channel was mounted on an inverted epifluorescence microscope (Olympus IX71, Olympus Inc., USA) equipped with 16-bit CMOS camera (optiMOS, QImaging). The images were taken at downstream close to the outlet (~ 40 mm). To capture the trace of fluorescent particles, exposure time was set to 1 s. To statistically find particle streak distribution, on average 100 fluorescent image sequences were stacked using IMAGEJ software to create a composite image along with an intensity line scan across the channel width. Counting particles using hemocytometer could not be carried out accurately for the dilute suspension in an order of 104 particles/mL (10, 15 and 20 µm) and 103 particles/mL (45 µm). Thus, a smaller sample volume of 1 µL (10–20 µm particle size) and 50 µL (45 µm particle) was taken for enumerating particles. Having been collected from each outlet, samples were agitated by using vortex mixer and pipetting and then the smaller sample volume was taken and dispersed into a Petri dish. Subsequently, the fluorescent particles were counted manually on the microscope stage using fluorescent illumination.

To eliminate formation of air bubbles inside the channels, all channels required to be first primed by 70% ethanol for ~ 5 min before pumping particle suspension.

4 Results and discussion

4.1 Inertial focusing in the trapezoidal straight channel with AR of 2

To investigate the particle equilibrium position inside a trapezoidal cross section, the effect of trapezoidal cross section on axial velocity and vorticity profile was first studied as the main factors in the first step of particle migration undergoing shear-gradient lift force (Zhou and Papautsky 2013). Figure 2 shows axial velocity contour and two main vorticities whose directions in the plane are perpendicular to the main flow. Shear-gradient inertial lift force originates from curl velocity (Zhou and Papautsky 2013) (FS = ω × US, where US is the particle slip velocity and ω is the curl velocity). ω x and ω y are the representative of two major shear rates (\(\omega_{x} \approx \frac{{\delta V_{\text{Axial}} }}{\delta y}\) and \(\omega_{y} \approx - \frac{{\delta V_{\text{Axial}} }}{\delta x}\)) which drive particles toward the top/bottom wall and the side walls at the primary stage of lateral migration. The axial velocity contour displays deviation of the maximum velocity inside the trapezoidal cross section toward the longer side wall. This deviation is one of the main factors which results in a new configuration of net inertial force and new equilibrium positions. ω x contour in Fig. 2a shows no significant change in the symmetrical condition when the rectangular cross section is altered to the trapezoid, the ω y contour, nonetheless, incurs a noticeable asymmetry along the channel width, particularly when AR increases from 2 to 4 (Fig. 2a, \(\omega_{y}\)), leading to the shear-gradient lift forces, pushing particles toward the longer side wall for the trapezoidal cross section (Sect. 4.2), dash lines show zero-shear-rate boundary where direction of shear-gradient lift force alters. Similarly, the corresponding normalized axial velocity profiles show that elevated AR, from 2 to 4, intensifies the asymmetry of the velocity contour for the trapezoidal channels (Fig. 2b). But, increasing the slope of the slanted wall, Tan(α), from 0.2 to 0.65, cannot change the asymmetry of axial velocity profile considerably; for the AR = 4 (Fig. 2c).

Fig. 2
figure 2

a Normalized axial velocity and vorticity profile at Re = 100 for channels with rectangular and trapezoidal cross sections with tan(α) = 0.2. All contours are plotted under the same condition to be comparable qualitatively. Dash lines show zero-shear rate boundary where direction of shear-gradient lift force alternates. b The corresponding normalized axial velocity profile for the rectangular and the trapezoidal channel at the middle of channel height coincided with dash lines at ω x contour. It shows that the deviation of maximum velocity toward the longer side wall is enhanced by increasing channel AR from 2 to 4. c Normalized axial velocity profiles of trapezoidal straight microchannels for increased slopes of the slanted wall at the middle of channel height

Full size image

The equilibrium position of 10-µm and 15-µm fluorescent particles (K ~ 0.2-0.3) is shown for the trapezoidal channel (HSW50 µm- HLW70 µm-W100 µm) with AR = 2 in Fig. 3. As can be seen in Fig. 3a, particles start gradually focusing and migrating toward the longer side wall when Re number increases (20 ≤ Re < 240). Further increasing Re causes another lateral focusing next to the short side wall (Re ≥ 240, K ~ 0.3), whereas the smaller particles with K ~ 0.2 are mostly dispersed due to the lower shear force. These minor lateral particle streams next to the side walls can be seen in Fig. 3b at high Re = 320 using intensity line scans. The main lateral focusing peak of K ~ 0.3 particles is slightly deviated toward the longer side wall (80 ≤ Re ≤ 160); when exceeding a critical threshold, it causes particles to move back to the middle of the channel. This movement of major lateral focusing point in the trapezoidal channel is illustrated in Fig. 3b by using fluorescence intensity line scan across the channel width. This low off-center movement of the major lateral focusing (~ 10% of the channel width at Re ~ 80) in the trapezoidal channel with AR = 2 compared to the trapezoidal channel with AR = 4, to be investigated next, has qualitatively similar trends to the numerical simulation, showing less dissymmetry in the velocity profile (or shear field).

Fig. 3
figure 3

a Fluorescent images of 10-µm (K ~ 0.2) and 15-µm (K ~ 0.3) microparticles display lateral distribution of particles in the trapezoidal channel with AR = 2 and Tan (α) = 0.2, b corresponding intensity line scan of 15-µm particles across the channel width (Letter L and S show the longer and the shorter side wall of the trapezoidal channel, respectively)

Full size image

4.2 Inertial focusing in the trapezoidal straight channel with AR of 4

Particle focusing inside the rectangular channel (H50 µm × W200 µm) with AR of 4 was first investigated as a benchmark (Tan(α) = 0). Figure 4a shows the lateral particle stream with K ~ 0.4 (20 µm) equilibrated at the centers of the long face walls for Re < 212. As shown, further increasing Re number causes unstable equilibrium and consequently turns the lateral focusing at the middle into two lateral focusing streams symmetrically (Re ≥ 212), which can also be seen in the corresponding intensity line scan profile (Fig. 4b). Smaller particle clogging ratio of K ~ 0.3 (15 µm) forms a comparatively broader particle band width [See supplementary information Fig. S2(a)] due to less wall-effect lift forces, FCL ~ a3 (Zhou and Papautsky 2013). These behaviors are in agreement with previous works (Liu et al. 2015; Reece and Oakey 2016).

Fig. 4
figure 4

Fluorescent images and the corresponding intensity line scans of the rectangular channel a–b and the trapezoid with Tan(α) = 0.2 c–d for 20-µm particles (K ~ 0.4). e Fluorescent images of various particle sizes in the trapezoidal channel with Tan(α) = 0.45 and the corresponding line scans for f 20-µm particles (K ~ 0.4) and g 45-µm particles (K ~ 0.9), displaying lateral distribution of particles in the trapezoidal channel. The channel aspect ratio of all channels is AR = 4 (Letter L and S show the longer and the shorter side wall of the trapezoidal cross sections, respectively)

Full size image

Figure 4c shows that changing the rectangular channel (Tan(α) = 0) to the trapezoidal channel with Tan(α) = 0.2 (HSW50 µm- HLW90 µm-W200 µm) slightly enhances the deviation of lateral focusing (K ~ 0.4, 20 µm) toward the longer side wall for low Re ~ 49, compared with the trapezoidal channel having AR = 2 (Fig. 3), but the main lateral focusing gradually gets back to the center of long face wall when Re increases to ~ 196. Further increasing the channel Re number leads to the migration of main lateral focusing close to the shorter side wall while some other weaker lateral particle streams simultaneously start to emerge around the middle and the longer side wall (Re ~ 733). The movement of major lateral focusing toward the longer side wall and its backward shifting can be seen in Fig. 4d. Also, the minor equilibrium is developed next to the longer side wall as Re exceeds ~ 196 illustrated by small peak adjacent to the wall at Fig. 4d. Particles with lower clogging ratios, less than or equal to K ~ 0.2 (10 µm), have no clear lateral focusing and are almost dispersed along the channel width [Fig. S2(b)].

Interestingly, elevating the slope of slanted wall up to Tan(α) ~ 0.45 (HSW50 µm- HLW140 µm-W200 µm) further enhances the migration of particles with K greater than or equal to ~ 0.3 toward the longer side wall (Fig. 4e) and boosts their focusing quality (Fig. 4f) for low-Re flows (Re < 100). However, minor peaks in Fig. 4f at Re > 100 demonstrate that going beyond this threshold of channel Re number intensifies particle dispersion around the channel center and the longer side wall compared to the trapezoidal channel with lower slope (Tan(α) ~ 0.2). Due to the comparatively large slant, it could handle a broader range of particle size without clogging. Figure 4e shows that the lateral focusing of particles with K ~ 0.9 (45 µm) sustains higher Re ~ 264 while neighboring the longer side wall. Remarkably, further increasing Re to ~ 654 demonstrates an almost complete shift of the lateral focusing toward the shorter side wall which can also be observed using the intensity line scan (Fig. 4g). It should be noted that the intensity line scan does not approach zero near the longer wall, representing a few random particles passing by near the longer side wall.

To further clarify the location of particle focusing inside the trapezoidal channels, a side view channel was fabricated. Its fabrication process can be found elsewhere (Kim et al. 2016). Figure 5a shows a side view of two particle streams for large-sized particles of K ~ 0.9 (Re < 264), fluctuating near the center and the bottom wall (Smaller particle clogging ratio K ~ 0.4 shown at Fig. S3 displays clearly the side view of the lateral streams. For low-slant trapezoid Tan(α) ~ 0.2 particles are mainly equilibrated near the bottom wall and increasing slope of the slanted wall to Tan(α) ~ 0.45 drives particle streams to both the center and the bottom wall.) These two streams gradually merge close to the bottom wall when Re increases to ~ 654. Figure 5b schematically displays the corresponding shifting of K ~ 0.9 particles from the longer side wall to the shorter side wall when Re goes up.

Fig. 5
figure 5

a Top and side views of the particle focusing (K ~ 0.9) in the trapezoidal channel with Tan(α) = 0.45 and b the corresponding schematic particle movements by increasing Re. c Transient lateral particle focusing along the channel length and the corresponding line scans for Re ~ 545 and K ~ 0.9. It demonstrates a two-step lateral movement while it first moves toward the longer side wall and then starts shifting back toward the smaller side wall (Letter L and S show the longer and the shorter side wall of the trapezoidal cross sections, respectively)

Full size image

Moreover, the transient lateral particle focusing along the channel length and the corresponding line scan of K ~ 0.9 particles are shown in Fig. 5c. Generally, particles are first pushed toward the middle and the longer side wall (see Fig. S4, Re ~ 440 for smaller particle sizes). Afterward, they start to be driven toward the shorter side wall. The developed minor peak next to the longer side wall that is shown in the intensity line scan at L = 20 mm gradually disappears, corresponding to particles migrating across streamlines toward the shorter side wall as travel to the channel downstream (L = 40 mm). The larger the particle size, the more particles will drift toward the shorter side wall. It should be noted that the particle Re number is considerably high (Rp ~ 85), on the grounds of high Re ~ 654 and large particle size ~ 45 µm. We would have interestingly reached the same Rp for the smaller particle size ~ 20 µm if the corresponding channel Re number could go up to ~ 3300—close to the turbulent flow regime—which is impossible to experimentally explore due to a high pressure drop. Although the unperturbed velocity profile and shear force configuration, when flow is laminar, will remain similar, regardless of its magnitude, the alternation of lateral focusing implies the key role of particle–flow interaction. The particle-induced lateral flow may also play an important role at high-Rp flows (Amini et al. 2012). A comprehensive numerical model which can simulate both particle and flow inertia may lead to an improved understanding of the mechanism of cross-lateral particle movement.

Finally, further increasing the slope of the slanted wall to Tan(α) ~ 0.65 (HSW50 µm- HLW180 µm-W200 µm) caused a broader range of lateral focusing overall (Fig. S5) due to the increased hydraulic diameter of channel cross section.

4.3 Particle filtration

Our experimental results proved the size-dependent cross-lateral migration of particles with K > 0.3 toward the longer side wall for Re < 100. Therefore, a trapezoidal channel with bifurcation was fabricated (AR ~ 4 and Tan(α) ~ 0.45) for filtration purposes. Figure 6a depicts the filtration of a broad range of particle sizes through the longer wall (LW) outlet. As can be seen, particles with K ~ 0.3 (15 µm) were approximately equilibrated at the middle, but the majority still went to the LW outlet. The larger particle sizes such as K ~ 0.4–0.9 (20–45 µm) were laterally focused farther away from the middle of channel cross section neighboring the LW and were subsequently collected from the LW outlet. Figure 6b displays the collected samples from the LW and the shorter wall (SW) outlet for Re ~ 88 (0.4 mL/min). To quantitatively measure the filtration performance of the trapezoidal straight channel, the efficiency is defined as follows:

$${\text{Efficincy}} = \frac{\text{Collected particles of interest}}{\text{Total number of particles}}$$
(3)
Fig. 6
figure 6

a Lateral focusing and filtration of particles through the longer wall outlet in the trapezoidal straight channel (tan(α) ~ 0.45 and AR ~ 4), and b the corresponding samples collected from the shorter wall (SW) outlet and the longer wall (LW) outlet at Re ~ 88 (0.4 mL/min). c–e Concentration of various particle sizes (15, 20 and 45 µm) collected from the SW outlet and the LW outlet. f Efficiency of particle filtration versus flow rate

Full size image

Total number of particles is the sum of particles collected from both outlets. It should be noted that the ratio of collected volumes from LW to SW outlet is 2:1 due to the skewed maximum velocity toward the LW. The concentration of samples collected from both outlets can be seen in Fig. 6c–e for 15, 20, and 45 µm, respectively. It demonstrates a large gap between concentration of particles collected from LW and SW outlet for particles with K > 0.3 (a > 15 µm). The corresponding efficiency, Fig. 6f, shows the high efficiency of ~ 99 and ~ 87% for particles with K ~ 0.4–0.9 and K ~ 0.3, respectively, when flow rate is 0.4 mL/min (Re ~ 88). The smaller particle size of 10 µm (K ~ 0.2) cannot be focused, and ~ 33% of it is diminished through the SW outlet. Thus, filtration resolution of the trapezoidal channel can reach down to ~ 5 µm difference in size, i.e., the trapezoidal channel developed here can focus and filter ~ 15-µm particle or greater in contrast to smaller particle size of 10 µm or less dispersed. Also, the 1.5-fold concentration of the trapezoidal straight channel can potentially increase to twofold by simply cascading two channels, while the whole unit has only one inlet and two outlets, relatively easier than that of the rectangular channels with trifurcation used for filtration (Bhagat et al. 2008b; Mach and Di Carlo 2010; Zhou et al. 2014). In a recent work, a rectangular channel with AR = 2 (W > H) with higher concentration of threefold to filter K ~ 0.4 particles/cells from the center outlet was used (Zhou et al. 2014); nonetheless, the particle clogging ratio could not exceed K ~ 0.5 due to the clogging of the channel. Moreover, the rectangular channels with AR < 1 (H > W) holding similar concentration of 1.5-fold can drive a broad range of particle sizes toward the side walls, K > 0.07 (Bhagat et al. 2008c), disregarding being cross-laterally focused. Nonetheless, because of dominating shear-gradient lift force along the channel width, it is likely suitable for extraction of all particle sizes from side outlets while either a particle-free sample or relatively small non-focused particles (such as bacteria in blood (Mach and Di Carlo 2010)) are collected from the central outlet.

Cross-lateral movement of major equilibrium position toward the shorter side wall in the trapezoidal channel can be utilized in filtration applications. A non-symmetrical bifurcation was used to further enhance the concentration of samples collected from the SW outlet, up to ~ threefold, through increasing the volume of carrier fluid extracted from the LW outlet, though being slightly affected. Figure 7a depicts the transition of lateral particle focusing, K ~ 0.9 (45 µm), from the middle of the channel (Re ~ 440) toward the shorter side wall (Re ~ 1100), and correspondingly changing the filtration of particles through the LW outlet to the SW outlet at Re ~ 440 (2 mL/min) and Re ~ 1100 (5 mL/min), respectively (See the Video S1-3†). The transition condition occurred at Re ~ 880 (4 mL/min), while the lateral focusing fluctuated around the middle of the channel. Further increasing Re led to migration of particles toward the shorter side wall till Re reached ~ 1100 (5 mL/min); exceeding Re ~ 1100 brought about a formation of another lateral focusing near the longer side wall (Re ~ 1320). The soaring concentration of collected samples from the SW outlet (~ 2.5-fold) at Re ~ 1100 (5 mL/min) can be seen in Fig. 7b and the corresponding efficiency in Fig. 7c, as high as ~ 82%. It should be noted that the threshold of critical Re number from which particles began to migrate to the shorter side wall was increased for the trapezoidal channel having bifurcation, mainly due to the slight alteration of cross-section dimensions, which arose from the ± 10 µm resolution of micro-milling technique, and the channel deformation due to the high working pressure. The performance of the device, i.e., efficiency and concentration factor, can be further enhanced by fabricating more precise channels and manipulating the bifurcation design. Also, the filtration resolution can be further enhanced by using a symmetrical trapezoidal cross section, thereby boosting inertial focusing off the channel center even at higher Re number expected.

Fig. 7
figure 7

a Transition of lateral focusing from the middle of channel cross section toward the shorter side wall as the Re number increases. b Concentration of samples collected from the SW and the LW outlet and c the corresponding efficiency (Letter LW and SW show the longer and the shorter wall outlet of the trapezoidal cross section (tan(α) ~ 0.45 and AR ~ 4), respectively)

Full size image

5 Conclusion

In this work, we investigated the inertial particle focusing dynamics in trapezoidal straight microchannels for a broad range of channel Reynolds number (20 < Re < 800) and various slopes of slanted walls. Increasing the aspect ratio (AR) from 2 to 4 enhanced the off-center lateral focusing position from the middle of channel cross section, up to ~ 20% of the channel width (Fig. 4f). A size-dependent lateral focusing position first shifted toward the longer side wall and then moved back toward the shorter side wall when Re was progressively increased (Re > 50). It was found that the focusing point was spoiled near the slanted wall due to the dissymmetry; the focusing was mainly developed near the bottom wall or fluctuated between the channel center and the bottom wall, depending on the slanted wall and Re (Re < 100, channel aspect ratio 4:1). Remarkably, an almost complete shift in major focusing from the longer side wall to the shorter side wall was found for the large particle clogging ratio of K ~ 0.9 (K = a/Hmin) when Re increased noticeably to ~ 650. Eventually, the trapezoidal straight channel (AR ~ 4 and Tan(α) ~ 0.45) along with a bifurcation was used for continuous filtration of a broad range of particle size (0.3 < K < 1) exiting through the longer wall outlet with an efficiency of ~ 99% (Re < 100). Finally, filtration of large-sized particles with K ~ 0.9 through the shorter wall outlet was shown at extremely high Re ~ 1100 with ~ 82% efficiency and ~ 2.5-fold concentration.