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Anal Chem
Anal Chem
ac
ancham
Analytical Chemistry
0003-2700
1520-6882
American Chemical Society

39165174
10.1021/acs.analchem.4c03474
Article
Deciphering the Evolution of Inertial Migration in Serpentine Channels
Liu Yong †‡
https://orcid.org/0000-0003-1113-6264
Zhang Jun §∥
Peng Xiaobo *‡
https://orcid.org/0000-0002-3463-0786
Yan Sheng *†‡
† Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
‡ College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen 518060, China
§ Queensland Micro- and Nanotechnology Centre, Nathan, Queensland 4111, Australia
∥ School of Engineering and Built Environment, Griffith University, Nathan, Queensland 4111, Australia
* Email: pengxb@szu.edu.cn.
* Email: shengyan@szu.edu.cn.
21 08 2024
03 09 2024
96 35 1430614314
05 07 2024
14 08 2024
12 08 2024
© 2024 The Authors. Published by American Chemical Society
2024
The Authors
https://creativecommons.org/licenses/by-nc-nd/4.0/ Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Serpentine channels coupling inertial and secondary flows enable effective particle focusing and separation, showing great potential in clinical diagnostics and drug screening. However, the nonsteady secondary flows in the serpentine channel make the evolution of inertial migration unclear, hindering the development and application of the serpentine channel. Herein, to refine the inertial migration mechanism, we established a model with varying curvature ratios to study the effect of secondary flow on particle migration in the serpentine channel. This method used direct numerical simulation (DNS) to calculate inertial lift, mapped the inertial lift to cross sections of the serpentine channel, and deciphered the inertial migration by using the Lagrangian particle tracking (LPT) method. The inertial migration of microparticles is experimentally investigated to validate the established numerical model. The results indicate that particle migration in serpentine channels follows a two-stage migration. An increase in secondary flow accelerates the second stage of the migration process while slowing the first stage process. Subsequently, we investigated the effects of different parameters, including Reynolds number, aspect ratio, and blockage ratio, on the equilibrium positions of particles, providing guidelines for the high-resolution separation of particles. Taking flow resistance into account, the dimensionless study makes the separation of arbitrary-sized particles possible. This work reveals the migration mechanism in serpentine channels, paving the way for the inertial separation of the particles.

National Natural Science Foundation of China 10.13039/501100001809 52305609 Basic and Applied Basic Research Foundation of Guangdong Province 10.13039/501100021171 2021A1515110277 Pearl River S and T Nova Program of Guangzhou Municipality 10.13039/501100009334 2021QN02Y387 Shenzhen University 10.13039/501100009019 NA document-id-old-9ac4c03474
document-id-new-14ac4c03474
ccc-price
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pmcIntroduction

Inertial microfluidic technologies were extensive applications in cell analysis, drug development, and the chemical industry due to their cost-effectiveness, high-throughput capabilities, and ease of operation.1−3 The manipulation and separation of various types of particles and cells (e.g., droplets, microalgae, platelets) in the mixtures is an important application of inertial microfluidics.4−6 Inertial microfluidics can manipulate particles in Newtonian fluids and non-Newtonian fluids.7 By introducing non-Newtonian fluids to enhance elastic lift forces, high-resolution manipulation of nanoparticles (e.g., DNA, exosomes) can be achieved.8−11 However, preparing non-Newtonian fluids involves adding extra polymer elastic enhancers to Newtonian fluids, which may interfere with the subsequent extraction and analysis of particles.7,12 On the other hand, the inertial microfluidic technique using Newtonian fluids without polymer additives sacrifices separating resolution due to the absence of elastic lift forces but offers superior throughput.3,7

Straight channels,13 spiral channels,14−16 and serpentine channels17,18 are common geometries used for particle manipulation in Newtonian fluids. Among these channels, the serpentine channel has attracted significant attention for particle focusing and separation due to its superior performance, including sheath-less flow, low flow resistance, and a compact device footprint.19 Di Carlo et al.20 first proposed the serpentine microchannel for particle focusing and separation, discovering that the introduction of secondary flows reduces the equilibrium positions of particles. The physical mechanism behind particle focusing positions in serpentine channels is described by the ratio of inertial lift force to Dean drag force.20−23 Dean drag forces dominate the particle mixing, while inertial lift forces dominate the particle focusing.20 The numerical simulations presented by Jiang et al.23 elucidated the impact of flow conditions on the equilibrium positions of particles. According to their results, at lower Reynolds numbers, under the dominance of inertial lift force, particles migrate to two bands near the long sidewalls of the channel.24 Secondary flows then sweep particles toward the short sidewalls, positioning smaller particles closer to the short sidewalls compared to larger particles.22 At reasonably high Reynolds numbers, under the dominance of Dean drag force, both small and large particles concentrate toward the centers of the secondary flows near the outer sidewalls of the channel, leading to undistinguished focusing positions.21,23

Subsequently, novel designs of serpentine geometry were used to enhance particle inertial focusing and separation efficiency.22,25 Cha et al.25 designed periodic concave obstacles on serpentine channels, demonstrating that concave obstacles can more effectively regulate the inertial focusing and separation of particles. Furthermore, the curved profile22,26 and cross-sectional shape27 of the serpentine channel were also considered as geometric optimization targets. Ying et al.26 demonstrated that the design of a zigzag profile enhances the focusing performance of particles. Zhang et al.27 reported that a relatively lower aspect ratio accelerates the process of inertial focusing. These designs of serpentine channels improve the performance of inertial microfluidics by altering the secondary flows. Current mechanistic studies on inertial migration provide some empirical guidance for the design of serpentine channels. However, due to the nonsteady secondary flows in the serpentine channel, the evolution of inertial migration is still not clear. The lack of theoretical investigation will increase trial-and-error costs for the design and development of serpentine channels, hindering their widespread application in particle focusing and separation.

To address this issue, we numerically studied the effect of secondary flow on particle migration in serpentine channels. The inertial lift on particles was calculated by using direct numerical simulation (DNS), and the results were used as boundary conditions in Lagrangian particle tracking (LPT) to obtain the evolution of particle trajectories. We conducted experiments to verify the accuracy of the DNS-LPT model in simulating particle motion. The inertial lift and Dean drag forces were calculated to analyze particle equilibrium positions. Building on the two-stage migration theory in straight channels,28,29 we expanded the understanding of particle migration mechanisms in serpentine channels. Additionally, the influence of dimensionless parameters of serpentine channel geometry and fluid flow, including Reynolds numbers (Re), aspect ratio (AR), and blockage ratio (β), on particle migration was examined to provide guidelines for arbitrary-sized particle separation.

Methods

Theory

When neutral particles migrate in a serpentine channel, their equilibrium positions are determined by the competition between inertial lift force and Dean drag force.20,24,30,31 Ho and Leal32 were the first to theoretically propose an inertial lift force explicit formula for the planar Poiseuille flow1

where a is the particle diameter, L is the characteristic length, Umax is the maximum channel velocity, ρ is the fluid density, and CL is the lift coefficient.

In a serpentine channel, the radial pressure gradient causes fluid near the channel center to flow outward along the circumference, while fluid near the wall flows inward, resulting in two counter-rotating vortices known as secondary flow or Dean flow.33 The magnitude of the secondary flow can be quantified using the dimensionless Dean number (De)2

where R is the radius of curvature. The hydraulic diameter Dh = 2WH/(W + H) for a rectangular channel, with W and H as the width and height of the cross section. The Reynolds number Re = ρUmaxDh/μ (μ is dynamic viscosity).

Ookawara et al.34,35 formulated the expression for the average Dean velocity for a given De as3

The drag force exerted on a particle can then be obtained by assuming Stokes drag4

Numerical Modeling

Three-dimensional DNS is the sole method that can accurately compute inertial lift forces, and it has been employed by our research group and others to obtain detailed lift force fields.25,36−38 Mapping the lift force field calculated for a straight channel to a serpentine channel with the same cross section and using the LPT method can predict particle trajectories. This method (DNS-LPT) has been used by us and other research groups and validated by the experimental results.25,36−38

DNS is the numerical coupling of the incompressible Navier–Stokes (N–S) equations and Newton’s second law to calculate the lift force on finite-sized particles within the cross section of a straight channel. The N–S equations are5

6

where u is the fluid velocity tensor, t is the time, and p is the pressure. The equations for particle motions are7

8

where ωp is the angular velocity vector, 1 is the unit tensor, τ is the shear rate tensor, n is the unit normal vector of the particle surface, I is the moment of inertia tensor of the particle, and xc is the position of the particle centroid.

In this article, we first use COMSOL Multiphysics 6.0 software to establish a straight channel model and perform DNS-based calculations for inertial lift forces. The inertial lift force experienced by the particles is normalized using the following inertial lift coefficient9

Next, we established a serpentine channel model (Figure 1 A). The mesh of the serpentine channel was uniformly divided (Figure 1B). Volumetric flow rate Q was specified by setting inlet or outlet boundary conditions to fully develop the laminar inflow or outflow. The pressure constraint of p + ∂p/∂n = 0 was applied at the outlet. The steady-state flow field was solved by using the N–S equations. The DNS-based lift field was mapped onto the cross section of the serpentine channel. The Newtonian formulation was used for Lagrangian particle tracing3910

where up is the particle velocity, SD is the Stokes drag coefficient, Rer is the relative Reynolds number, and ρp is the particle density. The first term on the right-hand side of eq10 is the drag force, which can drive particles to move along with fluid flow. The second term is the virtual mass force. The third term is the inertial lift from DNS-based calculations.

Figure 1 (A) Illustration of periodic serpentine channel model. (B) Geometry (i) and mesh generation (ii) of the channel. (C) The vector diagram of the inertial lift at a quarter of the cross section of the straight channel in this study (ii) is consistent with that from Hu et al.37 (i) AR = 4, β = 0.2, and Re = 100. Reproduced from ref (37) with permission from the Royal Society of Chemistry. The equilibrium positions are marked with circles. (D) Top view of the positions of 10 and 15 μm particles at the outlet of the serpentine microchannel (W × H × R = 200 μm × 50 μm × 250 μm) at flow rates ranging from 300 to 1800 μL/min. Left: simulation results; right: experimental results.

The channel dimensions, particle diameter, and volumetric flow rate are nondimensionalized into the channel-aspect ratio (AR = W/H), blockage ratio (β = a/H), Re, and channel-curvature ratio (γ = Dh/2R). The vertical and horizontal positions of the particles in the channel cross section are quantified by the channel heights as 2y/H and 2z/H, respectively. To compare the inertial lift force and Dean drag force on particles, the Dean drag force was nondimensionalized on the same scale (eq 9)11

CLy/CDy and CLz/CDz are the two components of CL/CD in the y and z direction, respectively.

Experimental Section

We fabricate microfluidic chips using standard photolithography and polydimethylsiloxane (PDMS) soft lithography techniques.40 10, 15, and 20 μm spherical polystyrene microbeads were dispersed in deionized (DI) water, and the particle weight ratios in the suspensions were 0.05–0.1%. Tween 20 (Sigma-Aldrich, product no. P9416) was added as a surfactant at a weight ratio of 0.1% to avoid particle aggregation. A high-speed CCD camera (Photron, FASTCAM SA3) was mounted on the microscope, and the videos of the single-particle movement were recorded using an ultrashort exposure time. The frame rate of a high-speed camera for recording particle migration is 3000 fps. The open-source software ImageJ (National Institutes of Health) was used to postprocess and analyze the captured videos.

Results and Discussion

Model Establishment and Validation

We performed DNS-based calculations to determine the distribution of the inertial lift coefficient CL across the cross section for AR = 4, β = 0.2/0.3, and γ = 0.16. Figure 1C shows a comparison between the DNS calculations implemented in this study using COMSOL and those implemented by Hu et al.37 using the Overture object-oriented framework (β = 0.2). The results at a quarter of the cross section of the straight channel show that the distributions of the two inertial lift vectors and the equilibrium positions of the particles match well. LPT-based calculations were used to determine the particle trajectories with β = 0.2 and 0.3 under different volumetric flow rates. Simultaneously, suspensions of 10 and 15 μm particles were injected separately at the inlet of the serpentine channel (W × H × R = 200 μm × 50 μm × 250 μm). The equilibrium positions at the channel outlet from the numerical simulation were consistent with the experimental results (Figure 1D). At different flow conditions, the coupling of inertial lift and Dean drag force on the two types of particles altered their inertial focusing and equilibrium positions.20,31 As the flow rate increased, the two-position focusing pattern observed in the top view of the channel merged into a single-position focusing pattern. The transition occurred at a lower flow rate for the 15 μm particles compared to the 10 μm particles; 15 μm particles transitioned at 900 μL/min, while 10 μm particles transitioned at 1200 μL/min. This is consistent with the phenomenon first observed by Di Carlo et al. in the symmetric serpentine microchannel.20 The additional inertial effect (i.e., Dean drag force) in the width direction alters the two stable equilibrium positions. This bias becomes more pronounced for smaller particle Reynolds numbers (RP = β2Re). Subsequently, this mechanism has been widely applied for biological particle focusing and separation.18,22,24,27,41 Low aspect ratio serpentine channels can suppress mixing effects,41 making them favorable for particle separation,18,42 while high aspect ratio serpentine channels are more suitable for particle focusing.12 In this paper, we investigate particle migration in serpentine channels with low aspect ratios of AR = 2, 3, and 4 to reveal the mechanisms of particle separation.

Effect of Secondary Flow on Particle Migration

We used the channel-curvature ratio (γ) as a variable in our simulations to study the effect of the secondary flow on particle migration. In our previous study, we found that particles with β = 0.1 can stably exhibit the two-position focusing pattern in the serpentine channel over a wide range of Re.31 In contrast, larger particles (β = 0.2/0.3) are more likely to migrate to the center of the channel and exhibit a single-position focusing pattern. To better distinguish the single-position focusing pattern in a straight channel while ensuring the inertial manipulation criterion of β > 0.07,43 we used relatively small particles with β = 0.1.

Particle motion in serpentine channels was simulated at AR = 3, β = 0.1, and Re = 50 with γ = 0, 0.02, 0.06, and 0.16. Figure 2A shows that the increase in γ leads to greater horizontal particle bias from the center of the channel. In a straight channel (γ = 0) without the influence of the Dean drag force, particles stably equilibrate at two equilibrium positions along the horizontal center. This is because, for low AR, the horizontal shear gradient effect is eliminated.44 Under strong wall effects, the particles are unstable in their horizontal distribution within the channel.

Figure 2 (A) Equilibrium positions of particles in the serpentine channel for different channel-curvature ratios (γ = 0, 0.02, 0.06, and 0.16), with γ = 0 representing a straight channel. (i) Model. (ii) 3D view. (iii) Cross-sectional view. (B) The migration of two stages in the channel of γ = 0.16 and 0.02. The initiation (i) and completion (ii) of the first stage of migration; the initiation (ii) and completion (iii) of the second stage of migration. (C) Time history of the vertical position of the particles (γ = 0.02, 0.06, and 0.16). Illustration: completion time of the first migration stage (t1), total time to migrate to equilibrium (t2), and the ratio of these two times (t1/t2) for different γ values. (D) Spatial evolution of particles in the serpentine flow channel. Horizontal position of particles at cross sections aa′ and ee′ in the 1st (#1), 10th (#10), and 19th (#19) periods of the serpentine channel. The error bars indicate the width of the horizontal distribution.

Additionally, in low aspect ratio straight channels at low Re, the two-stage migration theory, where particles first quickly migrate to the two long walls (the first migration stage) and then slowly to the center (the second migration stage), has been reported by many research groups.28,29,31,44 We found that the presence of secondary flow in the serpentine channel follows a two-stage migration pattern distinct from that in straight channels. For the channel with γ = 0.16 (De = 20), during the first migration stage, the shear gradient lift force moves the particles to the wall, while simultaneously, under the influence of the Dean drag force, most particles migrate away from the center of the channel (Figure 2B). Under the influence of these two forces, the particles migrated to the long side of the channel by the ninth period and completed the first stage of migration. In the second migration stage, under the predominant influence of the Dean drag force, the particles equilibrated at 2z/H ≈ ± 2. In contrast, in the channel with γ = 0.02 (De = 7.07), due to less interference from Dean drag, the particles quickly complete the first stage of migration (the second period) and eventually focus at the center of the channel after completing the two-stage migration. However, through our analysis of the stacked cross sections aa′ and ee′ of each serpentine channel period and the video of particle spatial evolution, we found that the focus in the serpentine channel is not stationary (see Videos S1 and S2). Instead, particles oscillate continuously within a certain region and converge to varying degrees, according to the secondary flow.

The time history of the vertical position of the particles shows that the oscillating trajectory is determined by the initial release position and the secondary flow (Figure 2C). After migrating to a certain vertical position in the first stage, the particles maintain continuous horizontal and vertical reciprocating motion in the second stage (Figure 2D). The particles experience a series of serpentine periods, converging to 2y/H = ± 0.81 at De = 7.07 (γ = 0.02) and oscillating within 2y/H = ± (0.55–0.85) at De = 20 (γ = 0.16). The duration of the first migration stage (t1) is positively correlated with γ, while the second stage duration (t2) is negatively correlated with γ. The increase in the secondary flow leads to a greater drag in the central region of the serpentine channel, and the decrease in γ causes the secondary flow to alternate more quickly in the periodic direction due to volume effects. When γ = 0.16, the secondary flow direction alternates every 5.12 × 10–4 s on average; while when γ = 0.02, it alternates every 4.45 × 10–2 s on average, obtained using a transient solver. At high γ, the Dean drag (eq 4) changes more rapidly in the plus and minus z directions, causing particles to stay longer in the central region of the flow channel (t1 is longer). However, most of the particles that stagnate in the first stage have already migrated to the horizontal sides of the channel, significantly shortening the migration time in the second stage (t2 is shorter).

Analysis of Particle Forces

The constantly changing direction of Dean drag forces caused unstable particle focusing in the serpentine channel. Analyzing particle behavior using the superposition method might not have provided a convincing explanation. However, the competition between inertial lift and the Dean drag force during the second migration stage determines the focusing position of the particles. Even though particles oscillated in the vertical direction during the second migration stage, we selected the vertical oscillation center as the position for force analysis to elucidate the particle migration mechanism (Figure 3). The oscillation centers of particles in the serpentine channels are located at 2y/H = 0.70 for γ = 0.16 (De = 20) and at 2y/H = 0.81 for γ = 0.02 (De = 7.07).

Figure 3 Distribution of inertial lift (A) and Dean drag (B) at the serpentine channel with γ = 0.16. The inertial lift, Dean drag, and net force on particles at the oscillation center during the second stage of migration were at γ = 0.16 (C) and γ = 0.02 (D).

The inertial lift and Dean drag forces on the particles at various positions in the ee′ cross section in the 20th period are shown in Figure 3A,B. The inertial lift forces (CLz), Dean drag forces (CDz), and net force (CNz) on particles at the vertical oscillation center in the channels with γ = 0.16 (De = 20) and γ = 0.02 (De = 7.07) are shown in Figure 3C,D, respectively. The Dean drag force at γ = 0.16 (De = 20) is much greater than that at γ = 0.02 (De = 7.07), while the inertial lift force remains the same. This causes the zero point of CNz at γ = 0.16 (De = 20) to deviate further from the center. A positive slope of the CNz indicates a relatively stable equilibrium, with greater slopes indicating higher stability, whereas a negative slope indicates an unstable position.44,45 The relatively stable position for γ = 0.16 (De = 20) is at 2z/H = 2, while for γ = 0.02 (De = 7.07), the relatively stable position is at 2z/H = 0.2. This is consistent with the particle focusing positions obtained from the particle tracking (Figure 2D). Specifically, during the second migration stage, particles at γ = 0.16 (De = 20) focus on the sides of the channel under the dominance of the Dean drag force, while particles at γ = 0.02 (De = 7.07) focus on the center of the channel under the dominance of the inertial lift force.

Parametric Study on Size-based Particle Separation

The separation of particles depends on their equilibrium position after the second stage of the migration. To guide the separation of particles, we numerically studied the effects of AR and Re on the equilibrium positions of particles (β = 0.1/0.2) in serpentine channels (γ = 0.16). Figure 4 shows the distribution of particles at cross sections aa′ in the 20th period of the serpentine channel. In the vertical direction, β = 0.1 particles are closer to the wall than β = 0.2 particles. This is consistent with previous numerical simulations and experimental observations.36,38,46 As AR increases, the shear rate across the channel cross section is enhanced, causing particles to migrate toward the channel walls (Figure 4A). The increase in shear rate shortens the time for the first stage of migration, leading to a reduction in the length required for particle focusing in high AR channels, which is advantageous for the portability of microfluidic devices.47 Additionally, particles can achieve better separation in channels with AR = 4 compared to those with a lower AR (Figure 4B). In high AR channels, during the second stage of particle migration, the range of the Dean force acting near the walls broadens, causing β = 0.1 particles to bias more significantly. The inertial lift and Dean force experienced by β = 0.2 particles are both significantly greater than those experienced by β = 0.1 particles. However, as the particle size increases from β = 0.1 to β = 0.2, the inertial lift increases more significantly than the Dean force (eqs 1 and 4), causing β = 0.2 particles to remain in equilibrium at the center of the channel.

Figure 4 Effect of the aspect ratio and Reynolds number on the equilibrium position of particles with blockage ratios of 0.1 and 0.2. The equilibrium position of particles is obtained from the cross sections aa′ in the 20th period of the serpentine channel. Distribution of particles’ vertical (A) and horizontal (B) positions at Re = 50 with AR = 2, 3, and 4. Distribution of particles’ vertical (C) and horizontal (D) positions at AR = 4 with Re = 50, 100, 150, and 200.

A serpentine channel with AR = 4 was used to analyze the equilibrium of particles at different Re. In the vertical direction, as the Reynolds number increases, the lift force caused by the shear gradient becomes significant, surpassing the lift force caused by wall effects, pushing the particles toward the wall (Figure 4C).48 In the horizontal direction, which is of interest for particle separation, the increase in the level of Re leads to smaller differences in particle positions (Figure 4D). Specifically, as Re increases, particles with β = 0.2 shift toward the outer wall of the channel, while the biases of smaller particles on both sides converge near the center of the channel and then shift toward the outer wall. This phenomenon has also been observed in previous experiments.20,23 Excessive Dean force in the horizontal direction causes particles to mix together during the first stage of migration and then oscillate significantly under the larger secondary flows near the center of the channel. This is unfavorable for particle separation in serpentine channels. In conclusion, particles with β = 0.1 and β = 0.2 can be separated at the working condition of AR = 4 and Re = 50. Next, we utilize these parameters for separating arbitrary-sized particles.

Inertial Separation in Serpentine Channels

The inertial separation concept is based on the theory of two-stage inertial migration and parametric study, which permits precise prediction of particle positions within the serpentine channel. We injected binary mixtures of 10 μm (β = 0.1) and 20 μm (β = 0.2) spherical polystyrene microbeads into a serpentine channel with dimensions of W × H × R = 400 μm × 100 μm × 500 μm (γ = 0.16, AR = 4). At Re = 50, the trajectories of the particles captured at the outlet matched well with the simulation results (Figure 5A,B).

Figure 5 Inertial separation in serpentine channels. λ is the scaling factor. Top view of 10 and 20 μm particle trajectories at the outlet of a W × H × R = 400 μm × 100 μm × 500 μm channel (γ = 0.16, AR = 4, Re = 50): Experimental results (A) and numerical simulation (B). 20 μm particles are near the center of the channel, while 10 μm particles are near the sidewalls. The scaled-down 20 μm particles are denoted as a1, and the scaled-down 10 μm particles are denoted as a2. After shrinking the 10 and 20 μm particles by factors of 1, 50, and 100, their trajectories (C) and equilibrium positions (D) in the serpentine channels. (E) Brownian motion forces on particles at equilibrium positions under different shrinking factors. (F) Maximum pressure in the channel during scaling and the pressure that three types of chips can withstand.

Scaling of inertial lift and Dean drag forces under finite-particle size effects enables the separation of arbitrary-sized particles.21 Keeping the relative size of particles and serpentine channels constant, we shrink the 10 and 20 μm particles by a factor of 1 to 100. The shrinking factor is denoted by λ. The scaled-down 20 μm particles are denoted as a1, and the scaled-down 10 μm particles are denoted as a2. Considering the diffusion movement of particles scaled to the nanometer size, we incorporated boundary conditions for Brownian motion forces in the LPT. The trajectories of particles under the influence of diffusion and the Brownian motion forces experienced at focusing positions were computed. Numerical simulations show that even with λ = 100 (a1 = 100 nm, a2 = 200 nm), the particle motion is only locally affected and the particle’s overall trajectories (Figure 5C) or equilibrium positions (Figure 5D) are not affected. This is attributed to that the Brownian motion forces (∼10–5 pN, Figure 5E) are negligible compared to inertial lift (10–100 pN) and Dean drag forces (1–10 pN) in separating 100 and 200 nm particles (λ = 100).

However, the small cross section leads to high fluidic resistance, which requires a microfluidic chip that can withstand higher pressures (Figure 5F). PDMS-glass is a commonly used bonding strategy for sealing channels but with a burst pressure of less than 0.5 MPa, limiting its usage for nanoparticle separation.49 Mair et al.50 developed a plastic microfluidic chip, where one bonding surface is exposed to solvent vapor to enable reversible oxidation. The chip is then bonded and exposed to UV light, allowing it to withstand pressures of up to 34.6 MPa. This chip can meet our needs for scaling down the channel by 40 times, enabling the sorting of 2.5 and 5 μm particles. However, the separation of nanoparticles requires more advanced materials and manufacturing techniques. Andersson et al.51 reported a transparent borosilicate glass chip that can withstand the pressure of up to 254 MPa. This high-pressure glass microfluidic device provides the possibility for the inertial separation of 100 and 200 nm particles.

Taken together, our exploration of particle separation for arbitrary sizes extends inertial microfluidic particle manipulation to the nanoscale down to 100 nm. Human blood contains an abundance of nanoparticles,52 such as exosomes, viruses, DNA, and RNA. For example, exosomes (with an average diameter of 100 nm) serve as biomarkers for various diseases, including immune deficiency,53 cancer,54 and cardiovascular diseases.55 The inertial microfluidic technology we proposed offers the potential to separate biomarkers at the nanoscale for clinical diagnosis.

Conclusions

This work revealed the effect of nonsteady secondary flows on the evolution of inertial migration in a serpentine channel, refining and extending the two-stage migration theory from straight channels to serpentine channels. We found that in the first migration stage particles migrated to two long sidewalls due to inertial lift. In the second stage, the particles were biased to varying degrees toward the channel sidewalls depending on the relative magnitudes of the Dean drag force and inertial lift. We quantified that the secondary flow accelerated particle focusing. Specifically, the acceleration of particle migration due to secondary flow was observed only in the second stage, whereas the migration time in the first stage is actually prolonged. Additionally, the effects of AR, Re, and β on inertial migration were studied to guide arbitrary-sized particle separation. Deciphering the evolution of particles in the serpentine channels can help guide the inertial focusing and manipulation of particles, potentially achieving a manipulation down to 100 nm.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.analchem.4c03474.The evolution of particles in serpentine channels under curvature ratios of 0.16 (Video S1) (MP4)

The evolution of particles in serpentine channels under curvature ratios of 0.02 (Video S2) (MP4)

Supplementary Material

ac4c03474_si_001.mp4

ac4c03474_si_002.mp4

The authors declare no competing financial interest.

Acknowledgments

S.Y. acknowledges the financial support from the National Natural Science Foundation of China (52305609), Zhujiang Perl River Talent Program (2021QN02Y387), and Guangdong Basic, Applied Basic Research Foundation (2021A1515110277) and Shenzhen University 2035 Program for Excellent Research.
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