1 Introduction

Particles in confined, finite Reynolds number (~ 1 < Re < ~ 1000) flows exhibit lateral focusing and longitudinal ordering patterns. This behavior is utilized for label-free cell manipulation and organization in numerous biomedical systems, and is widely known as inertial microfluidics. In inertial microfluidics, cells experience forces lateral to their flow direction, which lead to stable and precise equilibrium positions in the cross section of the channel (Segré and Silberberg 1961; Di Carlo et al. 2007; Martel and Toner 2014). The location of the equilibrium positions is affected by the particle Reynolds number (ReP = Ua2/νDh, where U is the characteristic velocity of the fluid, a is the particle diameter, ν is the kinematic viscosity, and Dh is the hydraulic diameter of the channel), the three-dimensional geometry of the channel, and the particle diameter (Di Carlo et al. 2009; Yuan et al. 2018; Mukherjee et al. 2019). This precise particle and cell ordering is exploited in various applications such as cell separation and enrichment (Fachin et al. 2017; Syed et al. 2017; Kim et al. 2018), rare cell isolation (Ozkumur et al. 2013; Shen et al. 2014), and cell-bead co-encapsulation (Lagus and Edd 2013a; Moon et al. 2018).

In addition to focusing, particles self-assemble into longitudinal pairs and trains via flow-mediated interactions (Lee et al. 2010). The perturbed Poiseuille flow around a particle is reflected at the channel walls which results in a secondary flow that is largely responsible for the particles’ interactions and their consequential longitudinal organization (Matas et al. 2004; Humphry et al. 2010; Lee et al. 2010; Pan et al. 2018). At sufficiently high concentrations, particles form coherent trains with equivalent inter-particle distances (Matas et al. 2004; Humphry et al. 2010; Lee et al. 2010; Kahkeshani et al. 2015; Gao et al. 2017; Del Giudice et al. 2018). However, if the concentration is further increased, the focusing of particles is impaired and no stable train can be formed (Kahkeshani et al. 2015; Reece and Oakeya 2016; Gupta et al. 2018). The presence of unevenly spaced or unstable trains is detrimental to various biomedical applications which rely on precise cell ordering (Edd et al. 2008; Di Carlo 2009; Hur et al. 2010; Lagus and Edd 2012, 2013b). Therefore, precise determination of particle ordering is of significant theoretical and practical interest.

Oscillatory inertial microfluidics is a recent technique which enables inertial focusing at extremely low particle Reynolds number flows, and can be used to manipulate submicron-scale particles and bacteria (Mutlu et al. 2018). The method relies on switching the direction of the flow field at a sufficiently high frequency, such that the suspended particles travel distances which are significantly (i.e, orders of magnitude) longer than the finite length of the microchannel. As inertial lift forces are independent of the flow direction, the particles focus to the same position regardless of the flow direction. Furthermore, this method also enables continuous observation of particles during their inertial focusing and longitudinal ordering process, which is extremely useful for making extended observations without sacrificing spatial resolution.

In this study, we investigated the dynamic formation, the precision and the stability of particle pairs and trains in an oscillatory inertial microfluidics system. We observed that the magnitude of the particle–particle interactions and the stability of the resulting pairs are significantly affected by the cross-sectional geometric arrangement of the particle pair. Focused particles attracted, repelled, or minimally interacted with each other based on their geometric arrangement and the inter-particle distance. We identified different regimes of particle interactions with respect to particle concentration, based on theoretical limitations and experimental findings. Consequently, we developed and experimentally verified a correlation between the inter-particle distance and the concentration of the particles. We identified the ratio between the short dimension of the channel and the particle size as the determining parameter in particle spacing, and proposed a correlation to estimate the inter-particle distance which we also experimentally verified.

2 Experimental procedure

The oscillating flow was generated by two three-way valves from the Lee Company (LHDA 0533315H) each connected by Tygon tubing (ID: 0.02″, OD: 0.06″, Cole Palmer) to a compressor, the microfluidic device, and the atmosphere (sink), as previously reported by Mutlu et al. (2018) (Fig. 1a). A high-speed camera (Phantom v4.2, Vision Research) was used to record images of the suspended monodispersed particles (ρ = 1.05 g/cm3, a = 3–20 µm) via a microscope (Eclipse 90 i, Nikon) equipped with a 10 × and 20 × magnification. The particles from Thermo Scientific (Fluoro-Max) were suspended at volume concentrations (cv) of 0.0001 < cv < 0.02 in phosphate buffered saline and density matched with OptiPrep (Sigma-Aldrich). The different devices were fabricated using standard soft lithography (Duffy et al. 1998) with the following dimensions: Device A had a length L = 3 cm, a width W = 40 µm, and a height H = 24 µm (in short: 3 × 40 × 24). Devices B–E had the dimensions 3 × 80 × 24, 6 × 40 × 29, 3 × 130 × 40, and 3 × 130 × 80, respectively. The ratio between the width (W) and the height (H) of all channels was greater than 1.4 to guarantee focusing of the particles to two focusing positions (instead of fours in a squared channel).

Fig. 1
figure 1

a Two three-way solenoid valves were connected via a Tygon tubing to the pressurized sample, the microfluidic device, and the sink. In the microfluidic device, the suspension of particles was oscillated. b Typical particle velocity profile (\( u \)) was measured and normalized by the mean flow velocity [\( u \)M = p/(Rh A), where Rh is the hydraulic resistance, A is the channels’ cross-sectional area, and p is the applied pressure]. The dashed line depicts the mean flow profile \( u \)P = \( u_{\text M} \) sgn[sin(2πtf)] normalized by \( u \)M over two oscillations. The theoretical particle velocity (\( u \)T ≈ 1.34 \( u \)M, solid line) is greater than the velocity of the mean flow due to their focusing position. c Focusing positions in the cross section of a microchannel are shown. d Depending on the particles focal position two distinct pair arrangements exist. Particles occupying the same focal side (SS) or opposing focal sides (OS)

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The frequency of the oscillation (f) was selected, such that the travel distance of particles during an oscillation is smaller than half of the channel length (f  > 2 \( u \)/L, where \( u \) is the average velocity of the particles). This led to frequencies between f = 2 Hz and f = 10 Hz, and ensured that the particles observed during a measurement stayed inside the microfluidic channel. In this frequency range, the particles experienced a strain per oscillation (S = L/H) in the range of 375 < S < 2000 and the system had a Womersley number [α = Dh(2πf/ν)1/2] in the range 0.1 < α < 0.8. These parameters indicate that a full velocity profile is established during every oscillation and that the flow can be treated as unidirectional (Loudon and Tordesillas 1998; Morris 2001). Therefore, the oscillation frequency range of this study is expected to have no effect on the dynamic particle pair formation and inter-particle distance, which was also verified experimentally. The particle velocity profile in each oscillation (Fig. 1b) was consistent with the theoretical velocity of inertially focused particles (Di Carlo et al. 2009).

At the initialization of the experiments, the particle suspension was oscillated in the microfluidic channel until all particles inertially migrated to one of the two focus positions (Fig. 1c). Due to the rectangular channel geometry, adjacent particles in the longitudinal direction could have only two geometric arrangements based on occupied focus positions: (1) same side (SS) and (2) opposite sides (OS) (Fig. 1d). The inter-particle distance (δ) was measured from center-to-center of adjacent particles and was normalized by the particle diameter (δ*=δ/a) (Matas et al. 2004; Humphry et al. 2010; Kahkeshani et al. 2015; Gao et al. 2017). The normalized inter-particle distances of particle pairs in the SS and OS arrangements are denoted as SSδ* and OSδ*, respectively.

3 Results and discussion

We observed that the dynamics of pair formation differed significantly based on the pair’s geometric arrangement (SS or OS). Pairs in OS arrangement, pair formation occurred rapidly when the non-dimensional inter-particle distance (δ*) reached a critical value (Fig. 2a). After formation, spacing was robust as the OSδ* remained constant with diminutive fluctuations. In contrast, pair formation occurred more gradually in the SS arrangement, and SSδ* was subject to substantial fluctuations after pair formation. To better illustrate this phenomenon, we calculated the relative velocity of particles in a pair (\( u \)δ = dδ/dt). A negative \( u \)δ signifies that particles are moving closer and a positive \( u \)δ indicates that they are moving apart. The non-dimensional relative velocity (\( u \) *δ ) was calculated by dividing \( u \)δ by the average velocity (\( \bar{u} \)= (\( u \)1+ \( u \)2)/2). Independent of the pair arrangement, particles below a certain inter-particle distance (OSδ*< 1.8 and SSδ*< 3.7) moved away from each other (Fig. 2b, \( u \) * δ  > 0). At intermediate δ* (1.8 < OSδ*< 5.5 and 3.7 < SSδ*< 7), particles in a pair moved towards each other (\( u \) * δ  < 0). At higher δ*, particles in both pair arrangements minimally interacted (\( u \) * δ  ≈ 0). Note that the magnitude of \( u \) * δ of pairs in OS arrangement was an order of magnitude greater than in the SS arrangement.

Fig. 2
figure 2

a High-speed images of two particle pairs are shown. Their corresponding normalized inter-particle distance (δ* = δ/a) is plotted against the time. The shade of the marker indicates the focusing side of the particle. b Initial attraction and repulsion of particle pair formation was investigated with the non-dimensional relative velocity (\( u \) * δ  = \( u \)δ/\( \bar{u} \)) as a function of δ*. The solid lines on OS represent a moving average of the data values. The error bars indicate the standard deviation of multiply observed particle pairs. The experiments were conducted with a =10 µm particles in device A with a cross section of 40 × 24 µm2

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Fig. 3
figure 3

δP is depicted as a function of Rep for OS pairs (top) and SS pairs (bottom). The dashed line is an exponential least square regression fit to the data values (OSδP = − 6.4 Re −0.16p + 25.3 µm and SSδP = − 3.4 Re −0.48p + 40.4 µm). The error bars indicate the standard deviation of different experiments (n ≥ 3). The experiments were conducted in device A with a cross section of 40 × 24 µm2

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Fig. 4
figure 4

a δP of differently sized particles in different channel geometries are depicted. Devices A–E have the dimensions 3 × 40 × 24, 3 × 80 × 24, 6 × 40 × 29, 3 × 130 × 40, and 3 × 130 × 80, respectively. b δP is depicted as a function of the channel height (H). The equation shown is linear least square regression fits to the data values (dotted lines). Inset: δP as a function of the channel width (W). c δ *P as a function of the non-dimensional channel height (H*=H/a) is depicted for the Rep range of 0.1 < Rep < 1. The equation shown in b is least square regression fits to the data values (dotted lines). The results are compared to the works of Humphry et al. 2010, Edd et al. 2008 and Lagus and Edd 2012 (stars). The error bars indicate the standard deviation of different experiments (n ≥ 3)

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The sign and magnitude of the non-dimensional relative velocity (\( u \) * δ ) indicate the nature and the scale of the particle interactions. A \( u \) * δ  > 0 signifies that the particles in a pair move apart and thus exhibit repulsive interactions. In contrast, \( u \) * δ  < 0 means that particles approach each other and exhibit attractive interactions. Both pair arrangements exhibited repulsive interactions at small δ* (Fig. 2b), and the magnitude of these repulsive interactions increased when particles got closer (i.e., δ* got smaller). At intermediate δ*, particle pairs showed an attractive interaction. The interplay between attractive and repulsive interactions led to an equilibrium distance (δ *E ) between pairs at which \( u \) * δ was zero. For particles in OS arrangement, the strong interactions led to a robust equilibrium at OSδ *E  = 1.8, whereas the weak interactions in SS arrangement led to a weak equilibrium at SSδ *E ≈  3.7. At higher δ*, both pair arrangements exhibited weak interactions that were neither unambiguously repulsive nor attractive.

Our results showed that the particle–particle interactions (as measured by \( u \) * δ ) correlate linearly with the flow velocity (U), and may impair the inertial focusing of particles at sufficiently low flow velocities. We observed that \( u \) * δ stayed constant when U was increased to adjust Rep from 0.1 to 0.7 (Fig. 2b). This is only possible if \( u \) δ is linearly correlated with U, due to the mathematical definition of \( u \) * δ (= \( u \)δ/\( \bar{u} \)). Consequentially, for constant fluid properties and a fixed geometry, particle–particle interactions correlate linearly with Rep, whereas the lateral drift velocity induced by inertial lift forces correlate with U2 and thus Re 2p (Di Carlo et al. 2007). In Stokes flow, the secondary flow around a single particle is linearly correlated with the rotational velocity of the particle, which is linearly correlated with the flow velocity (U) (Mikulencak and Morris 2004). The inertia of the flow only marginally affects the secondary flow around a single particle in a parabolic channel flow (Mikulencak and Morris 2004; Lee et al. 2010). Therefore, we postulate that particle–particle interactions correlate with the secondary flow and are predominantly unaffected by the inertia of the fluid. This indicates that when the flow velocity (U) decreases, particle–particle interactions diminish less compared to inertial lift forces, which may eventually impair the focusing and ordering behavior of particles. In our experimental setup, we observed defocusing of particles at flow velocities below U < 0.1 cm/s (a = 10 µm in device A); however, further studies are needed to investigate these critical flow velocities.

We observed that the stability of pairs and the precision of the ordered particle spacing (δ*) also differed significantly between the two particle arrangements (i.e., OS and SS). The concentration of inertially focused particles in a rectangular microchannel was described by the line concentration cL=aN/LF (Di Carlo 2009), where a is the particle diameter, LF is the length of the field of view and N is the number of particles within the field of view. δ* was analyzed by the probability distribution of δ* [p(δ*)] at different cL. Particles in OS arrangement formed precisely spaced pairs and were minimally affected by cL. This can be seen at high and narrow p(δ*) peaks over a wide cL range (Fig. 3a). In contrast, pair spacing in the SS arrangement was significantly affected by cL, which can be seen by the shift of p(δ*) with an increase in cL. An increase in cL led to a decrease in the spread of p(δ*) and an increase in the maximum p(δ*). We defined δ *P as δ* with the highest probability [peak value of p(δ*)] at a specific cL. The most probable normalized inter-particle distance in the OS arrangement (OSδ *P ) was in good agreement with the OS equilibrium distance (OSδ *E , see Fig. 2b) at OSδ *P  = OSδ *E  = 1.8 (Fig. 3b, closed diamonds). In contrast for SS pairs, SSδ *P was initially higher than SSδ *E , but progressively decreased with an increase in cL until SSδ *P  = SSδ *E  = 3.7 at cL = 0.22 (Fig. 3b, open diamonds). At higher cL values, both pair arrangements showed the same downwards trend in δ *P with an increase in cL. For both arrangements, their δ *P values were below the respective equilibrium positions.

Fig. 5
figure 5

a Probability distribution of the normalized inter-particle distance [p(δ*)] is formed by measuring δ* for all particle pairs in every frame, followed by creating a histogram with the obtained data (bin width of 0.5 µm). p(δ*) is depicted at different line concentrations (cL). δ* with the highest probability [peak value of p(δ*)] is denoted δ *P . b δ *P of particles on the SS (open) as well as of particles on OS (filled) is depicted. The saturation limit is defined by δ*=1/cL. The error bars indicate the standard deviation of different experiments (n = 4)

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At high line concentrations (cL), we found that the normalized inter-particle distance (δ*) is directly correlated with cL. An equally spaced train with N particles has a length of

$$ L_{t} \, = \,(N - 1)a\delta^{*} . $$
(1)

The line concentration of this train is given by

$$ c_{\text{L}} \, = \,(N - 1)a/L_{t} . $$
(2)

The equations resolved for δ* result in the geometrical dependency δ*=1/cL which describe a saturation limit (Fig. 3b). For a train of equally spaced particles, δ *P above the saturation limit is geometrically impossible. At approximately 80% of the saturation limit (80% saturation), SSδ *P is equal to SSδ *E . Above the 80% saturation, δ *P of both arrangements is equal or smaller than δ *E and is eventually influenced by the concentration of particles. Therefore, we define the area between the 80% saturation and the saturation limit as the ‘saturated regime’ (Fig. 3b, dark shaded area). In this regime, the particle–particle interactions are dominated by repulsive interactions and most particles are part of one coherent particle train. In turn, the strong increase in repulsive interactions for a decrease of δ* below δ *E leads to a very precise δ *P in this regime. OS pairs reach an inter-particle accuracy of up to δ*< 0.1 (1 < µm) at cL= 0.65, while SS pairs show a significantly reduced scatter of p(δ*) at cL= 0.35 (Fig. 3a). Below the 80% saturation, the low concentration of particles and the interplay of attractive and repulsive interactions prevent the formation of a coherent particle train. This area is defined as ‘dilute regime’ (Fig. 3b, white area). In this regime, particles on OS form stable particle pairs at OSδ * P = OSδ *E due to their strong interactions. The weaker interactions between particles on the SS form less stable pairs. They are insufficient to uphold a precise SSδ *P at SSδ *E . This can also be observed at the large spread of p(δ*) of SS pairs (Fig. 3a). To negate the influence of the concentration, δ *P at 80% saturation is used for the remainder of the study.

We observed an increase in the inter-particle distance (δP) with increasing Rep, over our entire experimental parameter range (0.01 < Rep < 1.0) (Fig. 4). To cover this broad Rep range, we used differently sized particles (a = 3, 5, and 10 µm) in the same channel. In the subrange 0.1 < Rep < 1.0, OSδP increased only minimally (≈ 2 µm) and SSδP stayed approximately constant. This result agrees with our findings in Fig. 2, specifically that the effect of Rep on δ *E is minimal in the Rep range of 0.1 < Rep < 0.7.

We found that the inter-particle distance (δP) is linearly correlated with the channel height (H, i.e. the shorter channel dimension), which enabled us to formulate a non-dimensional correlation for δ *P . The δP of differently sized particles (various a) in various channel geometries were measured (Fig. 5a). The OSδP increased from OSδP = 16.4 µm in channel A (H = 24 µm and a = 10 µm) to OSδP = 77.9 µm in channel E (H = 80 µm and a = 10 µm). The same increase was observed on the SS, where SSδP increased from SSδP = 36.4 µm in channel A (a = 10 µm) to SSδP = 121.9 µm in channel E (a = 10 µm). Specifically, the δP was linearly correlated to H and minimally influenced by the channel width (W) (Fig. 5b). Varying the particle size (a) in a given channel showed only a minor effect on δP compared to the effect of H on δP (Fig. 5a). Additionally, the effect of Rep in the range of 0.1 < Rep < 1.0 were also negligible. Based on our results, we fit (least-squares regression) a non-dimensional linear curve to predict OSδ *P and SSδ *P as a function of the non-dimensional channel height (H*= H/a) as follows: OSδ *P =1H* − 0.8 and SSδ *P =1.5H*+0.2 (Fig. 5c). In confined channel flows the flow disturbance induced by a particle is reflected by the channel walls and causes the characteristic secondary flow (Zurita-Gotor et al. 2007; Lee et al. 2010). We hypothesize that the linear nature of this reflection is the cause for the linear correlation between the channel height and the inter-particle distance.

Comparison with the previous reports reveal that our results obtained in the oscillatory inertial microfluidic system can also be extended to unidirectional, steady-flow inertial microfluidics. The focusing positions and expected particle velocity observed during our measurements are in good agreement with the previously reported inertial, steady-flow focusing positions in rectangular microchannels (Di Carlo et al. 2007, 2009; Lee et al. 2010; Kahkeshani et al. 2015). Furthermore, our inter-particle distance (δP) results agree with the results (Fig. 5c, stars) from the previous reports using steady-flow inertial microfluidics (Humphry et al. 2010; Edd et al. 2008; Lagus and Edd 2012). Using an oscillatory flow system, Morris 2001 and Butler et al. 1999 previously demonstrated that particles show the same migration behavior as a unidirectional flow when the strains per oscillations (S) is greater than 1.5. This is consistent with our results, despite the significantly higher range of S (375 < S < 2000) investigated in our experiments and the different underlying particle migration physics based on inertial lift forces.

The experimental setup and the method presented in this study could also be used for the investigation of flow focusing dynamics in non-Newtonian fluids. In non-Newtonian fluids such as aqueous viscoelastic solutions, particle-focusing positions switch from the sides of the microchannel to the centerline and to the corners (Yang et al. 2011; D’Avino et al. 2012; Kang et al. 2013). Furthermore, depending on the Reynold’s number, inertial forces also contribute to particle migration, resulting in inertia-elastic flow focusing (Lim et al. 2014). We hypothesize that the dynamics of particle ordering will be significantly different than oscillatory inertial microfluidics explored in this study, due to the combined effect of different migration physics and geometrical arrangements of focused particles in non-Newtonian fluids.

4 Conclusion

In summary, we investigated the dynamic formation, the precision, and the stability of particle ordering in an oscillatory inertial microfluidics system, and the extremely low Rep range enabled by the technique. We quantified the pair interactions based on their geometric arrangement, and determined that pairs in the OS arrangement interact significantly stronger than pairs in the SS arrangement. Consequently, the stronger interactions in the OS arrangement led to more stable, robust, and precisely spaced particle pairs. We determined that the particle–particle interactions correlate linearly with the flow velocity, and postulated that this correlation will lead to impaired inertial focusing and ordering at sufficiently low flow velocities. We identified a saturation limit, and found a geometrical correlation to estimate the inter-particle distance. In addition, we developed a linear correlation between the two non-dimensional parameters: inter-particle distance and channel height (i.e., shorter channel dimension), which can be used to develop applications that rely on precise inter-particle distance of particle trains. Expanding the available parameter space of inertial particle manipulation in terms of particle size and Rep range has the potential to lead to novel biomedical applications for smaller bioparticles such as bacteria and exosomes, as well as characterization and phenotyping applications for larger cells. Therefore, we expect that the results of this study will be of significant theoretical and practical interest.