Di Jiang†
a,
Wenlai Tang†a,
Nan Xiang*ab and
Zhonghua Ni*a
aSchool of Mechanical Engineering, Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, Southeast University, Nanjing, 211189, China. E-mail: nan.xiang@seu.edu.cn; nzh2003@seu.edu.cn
bState Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China
First published on 6th June 2016
This paper elucidates the particle focusing mechanisms in a symmetrical serpentine microchannel. The particle trajectories and the fluid fields in serpentine microchannels are explored via the numerical simulation based on a lattice Boltzmann method (LBM)-immersed boundary method (IBM) model. Experiments are also performed to verify the obtained simulation results. The investigation results help us to clearly understand the important role of the Dean flow for particle focusing in symmetrical serpentine microchannels. The balance of the inertial lift force and the Dean flow drag force at lower flow intensities leads the particles to flow near the sidewalls. As the flow intensity becomes stronger, the alternation of the Dean flow direction has special hydrodynamic effects to focus or separate particles of different sizes. Because of the volume effects, large particles are more prone to rotate with the Dean vortex and can be separated from the small particles at specific flow intensities. We envision that the simulation results of the focusing behaviors of particles would be of great significance for efficiently designing and optimizing inertial microfluidics.
Di Carlo9 firstly proposed the serpentine microchannel for particle focusing, and found that in an asymmetrical serpentine channel, the four equilibrium positions observed in a common straight channel with a square cross-section could be reduced to one certain equilibrium position. Gossett et al.10 further investigated the variation of particle focusing positions from the top view under various flow intensities. However, the separation effect of particles by size in the asymmetrical serpentine microchannel is not obvious.11 Instead, the symmetrical serpentine channel may be a good candidate for the separation of differently sized particles. It has been confirmed in Zhang's experiments12 that the square-wave shaped serpentine channel could be employed to separate the particles of different sizes according to their different lateral focusing positions. Although a great success has been achieved in the application of serpentine channels, the particle focusing or separation mechanisms in these channel geometries remain unclear. Specifically, the above-mentioned experimental studies are incapable of showing the particle focusing behaviors in the cross-section and are difficult to study the interaction between the particle and the flows. These limitations can be easily overcome by using the numerical simulation.
The simulation research in this paper is implemented by using the lattice Boltzmann method coupled with immersed boundary method (LBM-IBM). The mesoscopic kinetic LBM has been paid increasing attentions nowadays for its complicated boundary adaptation, parallelizability13 and low memory consuming.14 Ladd15,16 firstly employed the LBM for the simulation of particle migration behaviors in straight channel flows and calculated the interaction forces through the momentum transfer between flow and solid nodes. Since the uniform mesh is employed to both flow field and solid particles in this method, the particle motion stability or simulation effectiveness is sacrificed. To overcome these shortcomings, Feng and Michaelides17 coupled the IBM18 with the LBM. In this method, the particle boundary is represented by the Lagrange nodes and the deformation between instantaneous and reference boundary define the force imposed on the particle, which guarantees the stability of the particle motion simulation.
This paper intends to study the particle focusing behaviors in a symmetrical serpentine microchannel via the numerical simulation and the experiments. In our numerical simulation, the particle structure is expressed by the finite element method (FEM) and the flow field is calculated by the lattice Boltzmann method (LBM). Then, the fluid part and the particle part are coupled using the immersed boundary method (IBM). Through applying the developed IB-LBM codes, the inertial focusing behaviors of 10 μm and 5 μm particles in a serpentine microchannel are analyzed under various flow intensities, and the interactions between particle and surrounding fluid field are also explored. Then, the experiments are performed to verify the findings obtained from the simulation.
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Fi(x, t) is the external force term and can be defined as
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The kinematic viscosity can be calculated via the relaxation time τ as follows:
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In order to learn the effects of forces on particles, a 3D finite element membrane model with the energy method21 is applied. Specifically, the particle surface is divided into triangular faces and the vertices of each triangular faces are the boundary nodes. External work W is calculated as W = ES + EB + EA + EV22,23 where the ES is the strain energy, EB is the bending energy, EA is the area energy and EV is the volume energy. The strain energy EST of a single deformed triangular face can be calculated as24
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The bending energy EB describes the curvature contribution and can be written as23
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Similarly, EA and EV describe the surface and volume energy stored in the deformed particles and can respectively be described as23,25
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f(xn) = −∂W/∂xn. | (12) |
In IBM method,18 the force of a fluid node xf transferred from neighbor particle boundary nodes xn is represented as
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Dirac delta function D(xn − xf) = δ(xn − xf)δ(yn − yf)δ(zn − zf), | (14) |
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Using the no slip boundary condition, the velocity of the particle membrane node can be obtained from its neighbor fluid nodes. Fluid velocity uf(xf) is calculated from LBM and the solid node position xn can then be updated by ∂xn/∂t = up(xn). The particle nodes velocity is then determined as .
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Fig. 1 (a) Sketch of the symmetrical serpentine microchannel and (b) curved microchannel unit for simulation. |
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Fig. 3 Particle trajectories from the top view under (a) Re = 25, (b) Re = 50, (c) Re = 75 and (d) Re = 100. |
The cross-section views of the stacked particle trajectories were also drawn to find the equilibrium positions as illustrated in Fig. 4. Specifically, at Re = 25, small particles migrate close to the sidewalls with circle trajectories in the cross-section while large particles focus along the sidewalls with their equilibrium positions closer to the channel center, as compared with the small ones. There are four clear equilibrium rings in the cross-section for small and large particles due to the vertical and horizontal symmetry of the fluid field in the serpentine microchannel at low flow intensities. When Re = 50, particles focus close to the horizontal center of the channel. At Re = 75, the observed four equilibrium positions further reduce to two and the particle trajectories in the cross-section appear as “V” shapes. The symmetrical “V” leads the differently sized particles to be aligned from the top view. However, when the Re increases to 100, trajectories of large particles are stretched in the cross-section and small particles are defocused into four equilibrium positions once more. Compared with the large particles, small particles are more easily dragged by the strong effects of Dean flow with two counter-rotating vortices, and the direction alternating Dean flow also focus the small particles at two symmetrical trajectories from the top view. The simulation results of Reynolds number-focusing position agree well with the bifurcation focusing pattern in symmetrical serpentine channel from Gossett and Di Carlo's experiments.10
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Fig. 4 Particle trajectories in the cross-section under (a) Re = 25, (b) Re = 50, (c) Re = 75 and (d) Re = 100. The directions of small particle trajectories in (a) are shown by orange arrows. |
In order to explain the above particle focusing phenomena, we quote the magnitude ratio of inertial lift force FL to Dean drag force FD,9,27
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The competition of FL and FD decides the focusing pattern of various sized particles. We drew the trajectories of a pair of particles in symmetric positions and the fluid field around the lower particle (see Fig. 5) to study the basic rule of particle focusing. The serpentine channel is divided into four sections to analyze particle migration. As cross-referenced in Fig. 4, at Re = 25, small particles flowing through Section 1 are mainly driven by the Dean flow and move close to the sidewall. When the small particles enter into Section 2, the Dean flow direction turns opposite, but the high flow intensity here strengthen the FL, which slows down the rate of particle going towards the vertical center plane and keeps the particle flowing at the corner of the channel. In Sections 3 and 4, FD plays the dominant role again and the particles flow towards the vertical center plane first and flow in the opposite direction. When Re increases to 50 and 75, small particles migrate to the center of the channel under FD, which swings the particles horizontally in the cross-section. FL here has relatively slight influence on the particle migration in vertical direction. At Re = 100, the strong Dean flow drags the small particles to the sidewall again, and the FL in the high flow intensity area has obvious effects on the small particles.
Compared with the small particles, large particles in Fig. 5(c) have similar behaviors in the serpentine channel. However, along with the increase of Re, the side to center migration processes of large particles are much faster. When the Re is up to 100, large particles are more difficult to be separated into multi-lines than small particles. At low flow rates, particles flow along two sidewalls under the mixing effects of FL and FD28 and focus at channel midline by the strong direction alternating FD29 at high flow rates. However, the size difference also plays an important role for focusing pattern, and it is found that the large particles are more likely to focus at the channel horizontal center. From eqn (16), we found that the small particles are much easier to be influenced by FD, which explains why the small particles separate easily under a high Re. Considering the volume effects of large particles in a low AR channel, rotation of the particle in the Dean vortex is a possible explanation for the faster side to center migration. Large particles are more likely to be rotated by the counter-direction flow of the Dean vortex and sucked in it. When the flow intensity is weak, the rotation strengthens the Dean flow effects on the large particle and focus them closer to the channel center. The particle rotation here can be defined by the vector p. This vector p starts from the particle center and points to a fixed node on the particle surface. φ is defined as the angle between the projection of p on the channel cross-section plane and the z-axis. The rotations of the small and large particles are shown in Fig. 5(b) and (d). Blue lines are the trajectories of particle centers and the red lines are the trajectories of the fixed nodes. The rotation direction of the small particles has no obvious rules, but the large particles, meanwhile, rotate with the direction of Dean flow vortex as expected. The rotation of the large particles is also strengthened with the increase of Re. When the Re is higher, the rotation directions of large particles agree better with the Dean flow vortex.
To understand the force and rotation of particles deeper, particles and the fluid field in the channel cross-section A–A marked in Fig. 5(c) are extracted and illustrated in Fig. 6 to study the interaction between particles and the flow around. It can be concluded that at Re = 25, the 5 μm particle near the corner is balanced under FL and FD, which are in opposite directions. The FL acted on the small particle points to the channel corner, which holds the particle focusing close to the sidewall. In the case of 10 μm particles, there is more interaction between the particle and fluid, and the particles are too large to balance as the small particles close to the sidewall. The Dean flow easily rotates the large particles horizontally with its counter-direction flow near the Dean flow vortex and therefore the large particles focus closer to the channel center than small ones. From the eqn (16), the larger particle has heavier FL influence and that is why the large particles usually flow closer to the inner wall of a spiral channel.30 What is interesting is that smaller particles here flow closer to the sidewalls in the serpentine channel, which means that there should be more consideration when analyzing particle behaviors in complicated fluid field. Due to the volume effects, different sized particles may flow under different fluid field environment in the same channel.
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Fig. 6 The cross-section view of the fluid field in the Fig. 5 A–A section and tangent vectors around 5 μm and 10 μm diameter particles. |
When Re increases up to 50, 75 and 100, the Dean flow in the serpentine microchannel is stronger and particles are driven left and right by the direction alternating Dean flow. Even if at a higher Re, the FL here is still in a considerable quantity. The FL leads the particles to flow towards the upper and bottom walls, which forms the bottom of the “V” trajectory in the cross-section at Re = 75. Large particles rotate in the Dean flow vortex center, which strengthens the particle focusing performance. Instead, the 5 μm particle is too small to rotate with the Dean flow, which makes the focusing of small particles in the serpentine channel more unstable.
The Dean flow is the main motivation for the particles focusing. To analyze the Dean flow vortex's effects on the particles migration in the whole serpentine channel, the vertical velocity fluid field in z-direction is drawn to show the Dean flow distribution from the top view. Fig. 7 shows typical upper Dean vortex flow fields in a serpentine channel. Positive value means that the flow rises up and negative value shows that the flow goes down. The vertical rotation flow around the small particle is also shown in the figure. The fluid velocity gratitude of the main flow rotates the particle and the rotation flow is stronger than the Dean flow when the flow intensity is weak. However, at Re = 25, small particles focus at the vortex center when passing the inner wall side in a curved channel. As the flow intensity in the serpentine channel is stronger, particles begin to focus near the Dean flow vortex center in the whole serpentine channel, which means that the Dean flow plays the dominant role for particle focusing in this situation.
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Fig. 7 The flow velocity field in vertical direction of upper Dean flow vortex in the serpentine channel. |
From the analysis above, the effect of FD is to regularly swing the particles in the cross-section of the serpentine channel and the FL which is relatively strong at lower flow intensity is more like a trap to keep the particle flow near the sidewalls. Stronger flow intensity frees the particle from the FL trap and intensifies the swing effects of FD. Large particles are more likely to rotate in the cross-section with Dean flow vortex, which makes the focusing of large particles is more stable than small ones. In the focusing process, the equilibrium positions are dynamically changing with the fluid condition, not fixed at a point in the cross-section as the equilibrium positions in a straight microchannel.
Polystyrene fluorescent particles with average diameters of 4.8 μm (G0500, Thermo Fisher Scientific Inc.) and 9.9 μm (G1000, Thermo Fisher Scientific Inc.) were analyzed in this experiment. The 4.8 μm particle sample of 1% solid content was diluted 200 times with phosphate-buffered saline (PBS) solution (Sigma-Aldrich) and the 9.9 μm particle sample of 1% solid content was diluted 40 times. Tween 20 (Sigma-Aldrich) was added to prevent the particle from aggregation. Mixed particle suspension with different particle sizes was the mixture of equal volume of two sized particle suspension. The ethyl alcohol with a low surface energy was loaded into the symmetrical serpentine channel firstly before the particle focusing experiments to clean the channel walls and prevent the bubbles from producing to disturb the particle flowing. In the particle focusing experiment, the flow in serpentine microchannel was driven by a micro-injection pump (KDS270, KD Scientific Inc.) and observed by a fluorescence inverse microscope (IX71, Olympus). 10× objective lens (UPLFLN, 10×/0.3) and fluorescence excitation module were applied to observe the particle behavior in light field and dark field. The equipped 14 bit high-speed CCD camera (Exi Blue, Qimaging) and software IMAGE-PRO EXPRESS were used to capture the particle behaviors as video sequences.
To avoid the random factor, multiple image frames under the same operating condition were stacked to obtain the statistical particle distribution during a certain period of time. The video files were loaded to free ImageJ software (version 1.48v, NIH) and the image sequences were stacked by “Sum Slices” method to obtain the composite fluorescence image illustrating the particle trajectories. As for the bright field images, the obtained sequences were stacked by “Min Intensity” method to obtain the statistical particle distribution across the channel width.
Particle focusing patterns were recorded under increasing flow intensities in the experiment and the results are shown in Fig. 8. When the volume flow rate Q is 80 μL min−1 (Re ≈ 22.2), both the 4.8 μm and 9.9 μm particles focus into two symmetrical curved lines along the microchannel sidewalls and the space between the trajectories of large particles is narrower than that of small particles. At Q = 180 μL min−1 (Re ≈ 50), 9.9 μm particles focus to the channel center, but the 4.8 μm particles obviously focus in two lines at the microchannel center and there are some small particles still remain in the close wall area. It can be seen that before all of the small particles focus into a line in the center of the channel, the particles in center already begin to be separated by the Dean flow. The small particles near the center are easily controlled by the Dean flow and separate into two symmetrical trajectories and others close to the walls are still under the influence of FL. Complicated flow condition in the serpentine channel makes different dominant forces act on the small particles at different positions. With the stronger Dean flow at increased Re, the small particles begin to focus in clear two curved lines (Q = 220 μL min−1, Re ≈ 61.1), and the two symmetric focused trajectories of large particles appear when the Re is higher (Q = 300 μL min−1, Re ≈ 83.3). The Dean flow has more obvious influence on the small particles. However, the focusing performance of large particles is much better, which perhaps can be ascribed to the rotation in the Dean vortex as predicted in the simulation. At a high Re, the rotation of large particles can absorb parts of the energy that drags the particle towards the sidewalls. Because of the gravitation to the Dean vortex center, the large particles have a more stable focusing performance and flow closer to the channel center than small particles. Some stubborn small particles here are still unfocused and form the background of the focused trajectories.
To compare the particle distribution of simulation and experiments, we draw the particle focusing positions in Fig. 8(b). The normalized fluorescence intensity data are extracted from Fig. 8(a) cross-section B–B. Since the relationship between driving force and Re was calculated form the Poiseuille's law, which is defined for a straight channel, the Re in the simulation is higher than the actual value. In the particle distribution comparison of simulation and experiments, the Re of the simulation has been divided by 1.5. For large particles, the simulation focusing positions match the experimental results well as a bifurcation pattern. The small particles in the experiment have more complicated patterns. Before small particles appear at the channel center, they focus well along the sidewalls. When majority of small particles focus at the channel center at higher Re, there is still a small part of them focus close to the sidewalls. These small particles may still be trapped by the FL when the flow intensity is relatively low (Q = 180 μL min−1). When the FD is the dominant force in the channel (Q = 220 μL min−1, Q = 300 μL min−1), these small particles may flow at the channel vertical center plane and are pushed by the Dean flow further to the sidewalls.
Dark and bright field photographs of the mixture of differently sized particles focused at serpentine microchannel outlet are shown in Fig. 8(c). As the prediction in the simulation, the large particles focus closer to the channel center when the flow intensity is slow (Q = 80 μL min−1). At Q = 140 μL min−1 (Re ≈ 38.9), the large particles focus in the channel center first and the small ones still in two curved lines aside. This time difference of different sized particles focusing is useful for the particle separation. Here FD is the relatively strong force for the large particles which are trapped in the Dean flow vortex and the small particles are still controlled by the balance of FL and FD. When Q = 180 μL min−1, mixed particles focus in a trajectory and some small particles still flow near the sidewalls. However, the separation of differently sized particles is not obvious under the stronger Dean flow (Q = 220 μL min−1) because their trajectories are close and the disturbance of the defocused small particles is difficult to avoid.
Footnote |
† These authors contributed equally to this work. |
This journal is © The Royal Society of Chemistry 2016 |