A magnetophoretic microdevice for multi-magnetic particles separation based on size: a numerical simulation study

Due to the outstanding properties, magnetic particles have been widely used as magnetic carriers and adsorbents in a diverse range of fields, such as biomedical diagnostics, food safety, and environmental monitoring, etc. Sorting different magnetic particles from each other based on size simultaneously can be very useful in multiple targets separation. To the best of our knowledge, this research firstly proposes the arc comb structure microfluidic (ACSM) chip for multi-magnetic particles separation based on size by magnetophoresis. And the parameters affecting particle trajectory and separation efficiencies, such as flow velocity, velocity ratio of the two inlets, the remanent flux density of the magnets, and dynamic viscosity of the buffer solution, were investigated by numerical simulations in this study. Particle magnetophoresis spectrum width and general separation ratio are put forward to characterize the particles separation efficiency. Under the recommended simulation condition, six kinds of magnetic beads were separated with a general separation ratio of 0.83 using the newly designed chip. And the separation ratio of the 0.5, 1.8, 2.5, 3, 3.5 and 4.5 µm magnetic particle is 100.00%, 98.18%, 66.36%, 97.25%, 97.27% and 96.36%, respectively. This numerical simulation study provides a theoretical basis for multiple magnetic particles’ separation before experimental trials and also facilitates the utilization and development of microfluidic chips in the future.


Introduction
In recent years, the application of magnetic particles as magnetic carriers and adsorbents on medical diagnosis, bioanalysis, or therapy (drug delivery, cell separation, bacterial and viral detection, DNA isolation, protein purification, and rare cancer cell detection, etc.) has become very popular owing to its good biocompatibility, chemical stability, high loading capacity, mild operating conditions and low cost (Shasha & Krishnan, 2021). Sorting different magnetic particles from each other based on size can help target biomolecules or cells separate from the complex solution and enhance bioanalytical efficiency distinctly.
Magnetic force is one of the most commonly used and effective way to control and separate magnetic particles. Conventional magnetic separation is achieved by supplying an external magnetic field to retain the magnetic particles in the reaction vessel and remove the remainder of the reaction mixture. However, multiple magnetic targets separation requires repetitive sequential separation and washing steps, which is very time-consuming CONTACT Jian Jiang 5767241@qq.com; Huilan Su suhuilan1986@163.com Supplemental data for this article can be accessed here. https://doi.org/10. 1080/19942060.2022.2109064 (Plouffe et al., 2015). High gradient magnetic separation (HGMS) was also developed to separate magnetic materials from large-scale flowing fluids and has been applied in industrial applications for more than 40 years (Gómez-Pastora, . With the progress of nanotechnology, the HGMS system was reported to separate biomagnetic particles, such as nucleic acid (Pearlman et al., 2020), monoclonal antibody (Ebeler et al., 2018), and bacteria (Xue et al., 2019). Magnetic field flow fractionation (MFFF) has been shown as an effective way to separate magnetic particles (Orita et al., 2013;Williams, 2005). Zborowski et al. (1999) first developed a magnetic quadrupole field flow fractionation (MQFFF) method for continuous separation of human peripherical lymphocytes labeled with magnetic colloids. Moore et al. (2014) tested the feasibility of magnetic red blood cell separation using an approach based on the mechanism of MFFF. In general, these methods mentioned above always need large equipment and are not suitable for small volumes of medical and biological samples.
Miniaturization has progressed significantly in the last decade which in turn enabled the development of microfluidic devices. Microfluidic devices can handle small volumes of liquids (10 −9 -10 −18 L) within an area with dimensions of tens to hundreds of micrometers. The separation process can be carried out in a short time with low sample and reagent consumption. Thanks to these features, the microfluidic device has been a satisfactory method for the separation of magnetic particles and has gained significant attention from the researchers (Cao et al., 2020;Zhao et al., 2016). Many groups performed intensive works in both experimental and numerical areas and presented various microfluidic devices for magnetic particles or biomagnetic particles separation during the last few years (Khashan et al., 2013;Kwak et al., 2017;Poudineh et al., 2016;Su et al., 2021;Sun et al., 2020). For example, Kwak et al. (2017) designed a Mag-Gradient Chip, which used a serpentine channel with five straight channel segments parallel to the magnet surface. The magnetically labeled circulating tumor cells (CTCs) were captured in a series of small chambers on the side of a microfluidic channel by application of a magnetic field gradient across the channel breadth. The cells could be separated according to their epithelial cell adhesion molecule (EpCAM) expressing level. Other microfluidic device, like the magnetic ranking cytometry (MagRC) device, was also developed for separating CTCs carrying immunomagnetic beads (Poudineh et al., 2016). The MagRC device contains 100 distinct zones with varied magnetic capture zones. An array of X-shaped structures in the channel generates regions of locally low velocity and circular nickel micromagnets patterned within the channel enhance the externally applied magnetic field. Increasing the size of the micro magnets along the channel increases their region of influence, where high magnetic field gradients lead to efficient CTCs capture. Apart from these microfluidic devices, more and more studies employ the technology of free-flow magnetophoresis to separate magnetic particles in recent years.
Magnetophoresis describes the behavior of magnetic particles moving through a viscous medium under the influence of an external magnetic field. In continuous flow, magnetic particles were deflected from the direction of laminar flow by a perpendicular magnetic field depending on particle size, magnetic susceptibility, the flow rate, and so on (Pamme & Manz, 2004). Many microfluidic devices with different geometrical structures were reported to separate magnetic particles. Some of the microfluidic devices were employed to separate one kind of magnetic target from non-magnetic material. For example, Cardoso et al. (2018) fabricated an X-shaped PDMS microfluidic system to clean and separate the superparamagnetic iron oxide nanoparticles from the synthesis solution. Under the optimized geometrical configuration of the microfluidic channels and experimental conditions, the cleaning efficiency could reach 99.7%. Wang et al. (2017) designed a novel three-dimensional printed magnetophoretic system for the continuous flow separation of the H5N1 virus. Lin et al. (2019) presented a continuous-flow two-stage separation strategy of lateral separation and vertical separation in a flyover style microfluidic chip to realize high-purity WBCs separation. And the final separation purity of WBCs reached up to 93 ± 1.67%. Besides the one target separation, a variety of studies on two different kinds of magnetic particles separation were also reported. For instance, Ngamsom et al. (2016) employed a rectangular microfluidic chip in combination with immunomagnetic labeling to sort Salmonella typhimurium and E. coli from mixed cultures or food pre-enrichment broth by free-flow magnetophoresis. Zhang et al. (2020) proposed a strategy based on lateral magnetophoresis to separate magnetic nanobeads with either a strong or weak magnetic responsiveness on a chip system. The system was applied to sort DNA from hepatitis B virus and complementary DNA from hepatitis C virus simultaneously. Alnaimat and Mathew (2020) developed a new microfluidic device with two inlets as well as two outlets and employed a soft magnet element subjected to a uniform magnetic field to create the desired spatially varying magnetic field inside the microchannel. Then the simulation-based parametric analysis of a continuous flow magnetophoretic by the microfluidic device to separate two different magnetic particles (1 and 10 μm) was carried out by their team. Next, the microfluidic devices for three or more different kinds of magnetic targets separation by magnetophoresis will be presented. Pamme et al. developed a rectangular microfluidic chip to separate magnetic beads with diameters of 2.0 and 4.5 μm and polystyrene beads (6 μm) by free-flow magnetophoresis in the early time (Pamme & Manz, 2004). In Lee's paper, they reported a microfluidic device that integrated physical separation (a slanted weir structure) with magnetophoretic separation to separate the CTCs, RBC, and WBC in whole blood. The experimental results demonstrated that the microfluidic device achieved good separation efficiency (more than 93.3%) (Lee et al., 2020). Samanta et al. (2016) presented a numerical study, characterizing the performance of a magnetophoretic hybrid microfluidic device having two inlets and three outlets for immunomagnetic isolation of three different particles from a continuous flow. By optimizing the geometrical parameters of the microfluidic device, the maximum capture efficiency and separation index were obtained. Kumar and Rezai (2017) devised a microfluidic device based on inertial focusing and positive magnetophoresis for the fractionation of four particles (5, 11, and 35 μm magnetic and 15 μm non-magnetic particles). Shen and Park (2018) developed a microfluidic magnetophoresis device consisting of a trapezoidal channel containing five side outlet branches and a narrow rectangular channel with threeoutlet branches to separate the cells precisely based on iron content. Their magnetophoresis microfluidic device was able to sort cells accurately. Consequently, the technology of on-chip continuous flow magnetophoresis was proved to be a unique and useful method for separating magnetic particles from each other as well as from nonmagnetic materials.
Although lots of work on the separation of magnetic particles by continuous flow magnetophoresis has been carried out, a small number of researches on the efficient separation of multi-magnetic particles (more than three kinds) were reported. In the present study, a newly designed microfluidic chip with an arc comb structure was proposed to separate six sorts of magnetic particles based on size by continuous flow magnetophoresis. As far as we know, the ACSM chip for multi-magnetic particles separation by continuous flow magnetophoresis has not been reported. For a newly designed microfluidic chip, numerical simulation can be used for a preliminary study to prove the concept of this study's work (Balakrishnan et al., 2021). The numerical simulation software can provide an interactive environment for modeling, and the model enables a better understanding of the physical phenomena involved in continuous magnetic particle separation (Xu et al., 2022). Numerical simulation is convenient for one to analyze and optimize experimental parameters and is flexible to track the behavior of each particle at any time (Issakhov et al., 2019;Thune et al., 2021;Yan et al., 2021). And it has been widely used to optimize the chip geometrical structure or investigate the effect of different factors on particle separation (Alnaimat & Mathew, 2020;Samanta et al., 2016;Shen & Park, 2018;Xu et al., 2020). For example, Alnaimat and Mathew (2020) developed a mathematical model of a microfluidic device to obtain good performance metrics by optimizing the geometric structure and operating parameters. Samanta et al. optimized the channel geometry (2016) as well as the separation parameters (2020) by numerical simulation using an Eulerian-Lagrangian model. Wu et al. (2017) used a finite element software package (COMSOL 4.4) to establish the numerical model and optimized the trajectories of polydisperse Fe 3 O 4 nanospheres in a microfluidic channel equipped with a single permanent magnet. In the research of Outokesh et al. (2022), the Newtonian model was carried out to effectively predict the particle's trajectory in negative magnetophoresis, and the simulation results well matched the experimental results. Cao et al. (2018) applied the divergence of the Maxwell stress tensor and hydrodynamic stress to calculate the magnetic and hydrodynamic forces acting on particles, respectively. They used the finite element model to simulate the particle motion behavior and separated two types of magnetic particles (4 and 8 μm) from each other in a microfluidic flow under an externally applied gradient magnetic field. Consequently, numerical simulation is proved to be a visualized, convenient, helpful, and reliable technique for multiplex magnetic particles separation.
In this research, the simulation-based parametric analysis of the new ACSM chip for multi-magnetic particles separation is carried out. According to investigating and optimizing the separation conditions, including flow velocity, velocity ratio of the two inlets, the remanent flux density of the magnets, and dynamic viscosity of the buffer solution, six kinds of magnetic particles were separated based on size by the ACSM chip. A continuous flow magnetophoresis method conducted on an ACSM chip used to separate multiplex magnetic particles simultaneously is very practical and can be widely used in bioseparation, drug discovery, imaging, environmental remediation, etc.

Theoretical formulations
Continuous flow magnetophoresis separation is a process involving several forces acting on the particles. The magnetophoretic particle transport in a continuous fluid is governed by a multitude of factors including the magnetic force (F mag ), hydrodynamic forces (F drag ), gravity, buoyancy, inertia, particle-fluid interactions (perturbation to the flow field), and interparticle effects (such as Brownian motion, magnetic dipole interactions, Helmholtz double-layer interactions, and Van der Waals forces) (Gómez-Pastora, . In this study, only F mag and F drag were taken into account, and other forces were neglected. This is because other forces were generally considered as second order in effect, and considering all these forces is complex and unnecessary for the purpose of this work (Gómez-Pastora, . Therefore, F mag and F drag were considered as the dominant forces acting on a magnetic particle traveling through the magnetophoretic chip in the present study. In Figure 1(a), the trajectories of the magnetic particles of different sizes are drawn in different colors. The trajectories are scattered and parabolic in the separation chamber under the F drag in the horizontal direction and the F mag in the vertical direction. The reported microfluidic devices for multiple magnetic particles separation by Force analyses on a magnetic particle suspended in pure water. The background shows the surface of the flow in the absence of the magnetic particle. F mag and F drag denote the magnetic and Stokes drag forces, respectively. u f and u p denote the fluid and particle velocities, respectively. The microchannel and magnet are not drawn to scale. magnetophoresis are mainly rectangular chips. For example, Pamme and Manz (2004) proposed a vertical configuration for the outlet channels (plan A), and Shen and Park (2018) suggested a horizontal configuration (plan B). In this study, the ACSM chip including a semicircle separation chamber and quarter arc outlets is proposed. The separation efficiency of the magnetic particles with different sizes using the newly designed ACSM chip will be investigated by numerical simulation in this study.
The detail ACSM chip size can be seen in supplementary material. A homogeneous aqueous buffer suspension carrying the magnetic particles to be separated is introduced through inlet 1. And inlet 2 allows only the aqueous buffer solution. As shown in Figure 1(b), two cubic neodymium iron boron (NdFeB) permanent magnets which formed a Kittel domain are placed beside the semicircle separation chamber. The offset between the center of the magnets and the center of the chip chamber is defined as d, namely d = x magnet − x chip , when the magnets are placed on the right side of the chip chamber, d > 0, and vice versa.
According to Newton's Laws of Motion, particles are considered discrete elements, and their trajectories are estimated as follows : where m p and du p /dt are the bead mass and acceleration respectively, and F total is the resultant force vector exerted on a particle. F total is dominated by the magnetic force (F mag ) and the hydrodynamic force (F drag ) (Gómez-Pastora, Karampelas, et al., 2017).

Magnetic force
In the case of a nonconducting particle, in a static, irrotational external applied field, H ext , the magnetophoretic force is given by (Bruus, 2008;Kleinehanding et al., 2013): R p (in m) is the radius of the particle's equivalent homogeneous sphere. μ 0 is the vacuum permeability (in N/A 2 ). μ f (μ f is dimensionless) is the relative permeability of the medium. C (C is dimensionless) is the Clausius-Mossotti function. ∇ is the nabla operator, ∇ ≡ [∂/∂x, ∂/∂y, ∂/∂z], which produces a vector when acting on a scalar quantity. And the external magnetic field, H ext , should be distinguished from the local field which includes contributions from the particle itself. In this case, the Clausius-Mossotti function is given by: where μ r,p (μ r,p is dimensionless) is the relative permeability of the particle. Since the permeability of fluid is equivalent to that of vacuum, the relative permeability of fluid μ f = 1. The value of H ext (in A/m) can be computed from a 'Magnetic Fields' or a 'Magnetic Fields, No Currents' interface. Many studies show the relative permeability of magnetic particles μ r,p 1 (Aissa et al., 2015;Mikkelsen et al., 2005;Wu et al., 2021;Xu et al., 2022;Zhu et al., 2020), hence C ≈ 1.

Hydrodynamic force
The hydrodynamic force is mainly influenced by the flow velocity as well as the cross section of the particle and can be calculated from Stokes' law (Batchelor, 1967): where η (in Pa·s) is the dynamic viscosity of the fluid, u f and u p are the velocity of the fluid and the particles, respectively (in m/s). f D is a coefficient representing the impact of microchannel wall on beads movement. f D is equal to 1 when the magnetic particle is far away from the microchannel wall, while it is a little bigger than 1 when the particle is close to the wall (Wu et al., 2011).

Particles deflection in the ACSM chip
The particle trajectories are calculated using a Newtonian formulation with the Stokes drag law (Furlani, 2010), where τ p is the particle velocity response time: (Furlani, 2010) where ρ p is the density of the particle, taking Fe 3 O 4 magnetic beads with a diameter of 2 μm as an example, ρ p = 5000 kg/m 3 (Furlani, 2010), and η = 1 × 10 −3 Pa·s for buffer. τ p can be calculated as 69.44 nanoseconds, which is an extremely short time before the beads flow together with the fluid velocity. Hence, it can be considered that the motion of the magnetic bead in the microchannel is always quasi-equilibrium. In this case, we can ignore particle inertia (m p du p /dt → 0) in Equation (5). And the motion equation of the magnetic particles can be simplified. Although deflection is caused by the magnetic field, the degree of deflection depends on the F mag and F drag acting on the particles. The particle velocity can be calculated from Equation (7) (Jo et al., 2017): Therefore, u p depends only on the sizes of the particles when the magnetic field, magnetic susceptibilities of the particles, and medium viscosity hold constant. The coefficient α is related to the diameter of the magnetic particles and is impervious to magnetic fields.

Numerical simulations
Considering the dominant F mag and F drag , COM-SOL Multiphysics (versions 5.5, Burlington, MA), the finite element method combined with fourth-order Runge-Kutta method was used to establish the numerical model to predict and optimize the trajectories of magnetic particles (Wu et al., 2011). According to Reynolds (Carlo et al., 2007) (where w and h denote the width and depth of the microchannel, L H and U m = 2υ average are hydraulic diameter in the microchannel and the maximum channel velocity, respectively), Re is 1.4 when u f is 10 mm/s. As a result, the flow is in a laminar state. And along the microchannel wall, the non-slip condition (zero velocity) is applied. The outflow boundary condition (usually used to restrict the flow of incompressible fluid) is adopted at the outlet (Wu et al., 2011).
The numerical simulation consists of three application modes, including 'magnetic field, no current', 'particle tracing for fluid flow (fpt)' and 'laminar flow'. The simulations were conducted on an 8-core laptop with 8 GB of RAM. Two dimensional (2D) simulations were carried out and the mesh is composed of about 115 000 units. Table 1 lists the values of the parameters utilized in the numerical simulation. Particles are delivered into the fluid region at an initial velocity equal to that of the buffer solution. The total simulation time is kept at 240 s. And the position of the particles is tracked every 0.04 s.

Magnetophoresis spectrum width Θ and general separation efficiency P G
In order to characterize the landing position and width of the particles at the outlets, the scale parameter particle magnetophoresis spectrum width Θ (the unit, mm) is proposed. Θ is defined as the difference between the maximum and the minimum position value of the particles, that is: the stretched arc length at the outlets, and the arc outlet closing to the magnet direction is defined as 0. δ(r -q j ) is a function of the landing area to determine whether the j th particle falls in the current mesh cell. S M is the average of the particle magnetophoresis spectrum s M for K independent tests. In this program, K is kept as 10 to eliminate the potential randomness in particles release. Θ reflects the focusing degree of the magnetic particles with the same diameter after the deflection in the separation chamber. Figure 2 displayed the magnetophoresis spectrum width Θ of the particles with different diameters under a give simulation conditions. As can be seen, the smaller Θ is, the better focusing tendency will get. The upper and lower boundaries of the particle magnetophoresis spectrum can be obtained accurately at the same time.
In addition, in order to measure the multi-particles separation efficiency of the newly designed chip, the general separation efficiency P G is defined as follows: where p i is the percentage of the i th particle that enters into the desired outlet, and N denotes the kind of particles that need to be separated. P G reflects the separation efficiency of the whole system instead of the separation effect of a single kind of particle. Figure 2. The 3, 4, and 5 μm particle magnetophoresis spectrum width Θ when the magnet remanent flux density is 1.2 T, the υ 1 is 2.9 mm/s, the υ 2 is 8.7 mm/s, and the η of liquid is 1.1 × 10 −3 Pa·s.

Numerical model verification
The numerical model in this research was validated with experimental results of particle capture efficiency in freeflow magnetophoresis by Pamme and Manz (2004). The parameters are consistent with those used in the literature. Ten independent experiments were carried out by the Monte-Carlo method to eliminate the randomness of the experiments.

Grid independence verification
As the finite element method is used to solve both the fluid flow and magnetic field in the channel, the grid independence verification experiments should be conducted to confirm the results' independence from mesh size. In this study, the grid independence was investigated by observing the variation of F drag acting on the particles and the particle magnetophoresis spectrum width Θ with the change of maximum mesh cell size m max . The maximum mesh cell size m max was set as 1/4, 1/5, 1/10, 1/15, 1/20, 1/25, and 1/40 mm (that is, 4, 5, 10, 15, 20, 25, and 40 grids in each 1 mm), respectively. And other parameters used in the grid independence simulation experiments were fixed.

Statistical analysis
Every simulation experiment on particle separation was repeated at least 5 times in this research. The mean value was calculated and used to evaluate the separation efficiency (see Section 3.1 for detail).

Numerical model verification result
The distributions of the magnetic particles with a diameter of 4.5 μm at the outlets in both simulation and practical experiments were shown in Figure 3. It can be seen that the profile of the distribution by numerical simulation is roughly consistent with the results of Pamme and Manz (2004). The recovery difference value DV of the 4.5 μm particle between the two studies was calculated by DV = |p lt − p nm |. The p lt and the p nm is the recovery of the 4.5 μm particles in the literature and the numerical model, respectively. The maximum difference value is 9.75% at outlet 4, and the minimum difference value is 0.55% at outlet 5. The average value is 5.21%.

Grid independence verification result
The simulation results of the grid independence verification are plotted in Figure 4. From Figure 4, part A shows the F drag acting on the particles in a transversal line that is 18 mm away from the x-axis (the transversal line can be seen in Figure 6(b)). To observe the changes of the F drag under different grid refinement more clearly, the top of the curves in part A is enlarged and plotted in part B. When the m max was set as 1/4 mm, the F drag acting on the particles is not smooth with obvious fluctuations. In the process of densifying the meshing, it is found that the F drag acting on the particles is gradually smooth and tends to be stable. What's more, the variation of the particle magnetophoresis spectrum width Θ with different grid refinements is also investigated, and the results are shown in part C (Figure 4). It can be observed that when the mesh number increases from 4 Figure 4. Grid-independence verification simulation results. Part A shows the F drag acting on particles in a transversal line that is 18 mm away from the x-axis with different grid refinement, and the top of the curves is enlarged (part B). Part C shows the variation of particle magnetophoresis spectrum width Θ with different grid refinements. The parameters are setting as follows: B r = 1.3 T, υ 1 = 0.9 mm/s, κ = 10, η = 1 × 10 −3 Pa·s, D p = 4.5 μm, c = 2 × 10 6 /mL, and d = 1 mm.
cells per millimeter to 40 cells per millimeter, the particle magnetophoresis spectrum width Θ oscillates from 1.04 mm down to 0.85 mm gradually. And when the mesh number is 15, the Θ tends to be stable. To balance the computational efficiency and the accuracy of the simulation, we defined the mesh number as 15 cells per millimeter.

Magnetic field and velocity field calculation
Before analyzing the deflection of particles in the chip chamber, the magnetic field and velocity field are calculated by COMSOL Multiphysics 5.5, and the results are shown in Figure 5. The square of the gradient of the flux density mode ∇|H| 2 , that is, the direction of magnetic force has been calculated and shown in red arrows in Figure 5. It is indicated that when the particles enter into the separation chamber, the magnetic force is along the red arrow direction. Meanwhile, particles will also be subject to the drag force of the velocity field, which is perpendicular to the velocity field isoline direction, almost towards the x-direction. Magnetic particles with different sizes will move along different directions according to the F mag and F drag .

F mag and F drag acting on the particles
Four transversal lines (as can be seen in Figure 6(b)) were set to investigate the F mag and F drag acting on the particles. The magnetic field intensity, specifically, the B x , B y , and |B| on the transversal lines are shown in Figure 6(a). And the forces acting on the particles are given in Figure 6(b), on a given group of two-dimensional transversals, respectively.
The magnetic field strength has the same fluctuating tendency between different transversal lines, as shown in Figure 6(a). Take transversal 1 as an example to illustrate the forces acting on particles (Figure 6(b)). The magnetic force acting on particles increases slowly with closing to x = 1 in the x-direction. The drag force increases at first (near the intersection of the narrow channel and the separation chamber), then decreases (closing to the outlets). For transversal 1 or 2, drag force is stronger than magnetic force across the whole measuring range. Transversal 4 shows differently. When x > −5 mm, the Figure 5. Distribution of magnetic field and velocity field in the separation chamber. The parameters are setting as follows: B r = 1.2 T, υ 1 = 2.9 mm/s and υ 2 = 8.7 mm/s. magnetic force becomes stronger than the drag force, which is beneficial for the deflection of particles. Meanwhile, it can be seen from Figure 6(b). For transversal line 4, the F drag is discontinuous when x is closing to 1 mm. This is mainly due to the partition of the chip material at the outlets. With the downward shift of the transversals (from line 1 to line 4), the magnetic force increases while the drag force decreases in general. The magnetic force is gradually overtaking drag force in its new dominant role.

The dynamic behavior of the particles in the ACSM chip
In this part, the particle initial trajectories of entering the chip from inlet 1 and the trajectories in the semicircular separating chamber were simulated and depicted in Figures 7 and 8, respectively. When the particles are released randomly in inlet 1, observations of the particle trajectories at different times are shown in Figure 7. Moreover, the trajectories of the particles in the separation chamber were also captured at t = 15, t = 20, t = 30, and t = 40 s, respectively. And the results are shown in Figure 8.
The particle separation efficiency mainly depends on the magnetic field intensity, the fluidic field (inlet velocity, the velocity ratio of the two inlets and the fluid viscosity), and the characters of the magnetic particles (diameter, density and susceptibility), etc. It's necessary to investigate the effect of the main parameters on particle separation. Therefore, the main separation conditions, including the fluid velocity, magnetic field strength and buffer dynamic viscosity, were analyzed by COMSOL Multiphysics software platform.

Effect of flow velocity on particle separation
The flow velocity ratio κ = υ 2 /υ 1 should be kept at a high level, as the buffer solution introduced into the channel at inlet 2 is used for generating a sheath flow to  The concentration of the particles is 1 × 10 5 /mL, κ = 10, η = 1 × 10 −3 Pa·s, and d = 3 mm. In the figure, different colors indicate different particle sizes, corresponding to 0.5, 1.8, 2.5, 3, 3.5, and 4.5 μm, respectively. (The size of the particles and the chip are not drawn to actual scale. And the particles are magnified.) focus the particles stream (Alnaimat & Mathew, 2020). In this research, the spectrum width Θ and the general separation efficiency P G were investigated under the different flow velocity ratio κ. The κ was changed by keeping υ 1 at 1 mm/s and altering υ 2 . The results are shown in Figure 9(a). With the increase of υ 2 , the particle (1 μm) magnetophoresis spectrum width Θ becomes narrower, while the particle deflection decreases. The particle recoveries p i are rising and they are all above 90%. Narrow magnetophoresis spectrum width Θ and larger deflection facilitate multiple particle separation. When the κ is 10 (i.e. υ 2 = 10 mm/s), the particle magnetophoresis spectrum width Θ tends to be stable and the recovery of magnetic beads is 95.2%.
Apart from the flow velocity ratio κ, the flow velocity υ 1 was also explored. This simulation was conducted at κ = 10, and υ 1 was 500, 1000, and 1500 μm/s respectively. Five different particles (the diameter is 1, 3, 5, 10, and 15 μm, respectively) were selected to be tested. The results are shown in Figure 9(b). When κ is fixed at 10, the particle magnetophoresis spectrums for all particle sizes are uplifted about 1.5-2 mm with the increase of the flow velocity υ 1 from 500 to 1500 μm/s. According to Equation (4), the drag force (F drag ) acting on particles will enhance with the increase of υ 1 , and the deflection of the particles will decrease, which is consistent with the simulation results. A high υ 1 is not conducive to separate multi-particles. However, the low flow velocity will lead to a long separation time. The calculation time is about 35, 21, and 12 min, when υ 1 is 500, 1000, and 1500 μm/s, respectively. Considering the separation efficiency as well as the separation time, υ 1 is recommended as 1000 μm/s.

Effect of magnetic field intensity on particle separation
The magnetic field intensity affects the deflection of the particles. Firstly, the magnetic field intensity was changed by adjusting the relative position between the magnets and the microfluidic chip in the x-direction, namely Figure 9. (a) The effect of flow velocity ratio κ (υ 1 = 1 mm/s) on the particle magnetophoresis spectrum width Θ. (b) The effect of υ 1 (κ = 10) on the particle magnetophoresis spectrum width Θ. The parameters are setting as follows: B r = 0.8 T, η = 1 × 10 −3 Pa·s and c = 2 × 10 6 /mL. changing the value of d (d can be seen in Figure 1(b)). As shown in Figure 10(a), when the magnets are far away from the center of the separation chamber (7 < d < 10 mm), the L s of the particles is kept at 10 around. The magnetic field is barely working on the particles, and the particles have no deflection. When the magnets are moved to the center of the separation chamber (d from 7 to 1), the L s of the particles (5 and 10 μm) drops drastically, which means the particle deflection increases. What's more, the particle magnetophoresis spectrum of the small particle with a diameter of 1 μm is different from the big ones (5 μm or 10 μm). When the d is changed (d from 10 to 1), the L s of the small particles (1 μm) barely changes and remains at 10 mm. The magnetic field has a weak effect on the small particles under a certain separation condition. When 2 ≤ d ≤ 4 mm, the particle separation efficacy can be guaranteed.
Next, the effect of remanent flux density B r on particle magnetophoresis spectrum width was investigated. Keep other conditions constant (κ = 10, υ 1 = 1 mm/s, η = 1 × 10 −3 Pa·s, c = 2 × 10 6 /mL, d = 3 mm) and change the remanent flux density B r . The results are shown in Figure 10(b), the L s of the particles (4 μm) decreases with increase of remanent flux density. When the remanent flux density is 1.2 T, the distribution of the particle magnetophoresis spectrum is the narrowest.

Effect of magnetic field together with flow velocity on particle separation
Flow velocity and remanent flux density B r are two significant factors affecting particles separation effects. In this study, the effect of the flow velocity together with remanent flux density on particle separation was investigated. Set the parameters as follows: η = 1.1 × 10 −3 Pa·s, κ = 10, d = 3 mm. Increase B r from 0.2 to 1.4 T with an interval of 0.2 T. And change υ 1 from 0.8 to 1.2 mm/s. The L s of the 2 μm particle under different B r and υ 1 was shown in Figure 11. The simulation results were similar to the results of the single parameter analysis. The deflection of the particles increases with the increase of B r and the decrease of υ 1 . What's more, when the influence of B r and υ 1 on particle separation was investigated, the value of B r and υ 1 was also recorded once the general separation ratio P G is above 0.9. And the simulation tests were repeated 10 times. Then the quadratic curve fitting for these discrete data (B r and υ 1 ) was conducted. And the results are drawn in Figure 11 with the blue dotted line. The fitting quadratic curve of B r and υ 1 can provide useful data reference in an actual experiment. If υ 1 is confirmed, the general range of B r can be inferred based on the fitting quadratic curve, and vice versa.

Effect of flow dynamic viscosity on particle separation
The fluid dynamic viscosity η also has an effect on particle separation. And η is not a constant, which is determined by the specific buffer solution (including the solvent and the solute) and the environmental conditions (temperature, pressure, etc.). It's necessary to analyze the effect of η on particle separation when using the newly designed ACSM chip.
The result is shown in Figure 12. It is shown that with the increasing of υ 1 and η in a non-uniform way, the deflection of the particles is decreasing. According to Equation (4), the flow velocity at the inlet 1 and buffer dynamic viscosity are both positively related to the drag Figure 11. The effect of remanent flux density together with flow velocities at inlet 1 on the particle magnetophoresis spectrum width. Only one kind of magnetic particle (2 μm) has been examined, which is illustrated against the left vertical axis. The relationship between remanent flux density and velocity is depicted with the right vertical axis. The blue dot line is for the fitting results, while the blue diamonds is for the experimental data. force. The flow field plays a more dominant role with increase of υ 1 and η, which will lead to a smaller deflection for the particles. Meanwhile, the particles magnetophoresis spectrum overlapped, which is not beneficial for separating multi-particles with different sizes. Hence, the υ 1 and η is recommended as 1.1 mm/s and 1.1 × 10 −3 Pa·s, respectively.

Effect of flow dynamic viscosity together with remanent flux density on particle separation
In this research, we investigate the particle separating effects when the η is 0.9 × 10 −3 , 1.0 × 10 −3 , 1.1 × 10 −3 , 1.2 × 10 −3 , and 1.3 × 10 −3 Pa·s and remanent flux density is 0.4, 0.6, 0.8, 1.0, and 1.2 T correspondingly. The Figure 13. The separation effect of two particles when remanent flux density B r and dynamic viscosity η increase at the same time.
simulation is carried out five times, and the result is shown in Figure 13.
From Figure 13, when the viscosity and remanent flux density are small (η = 0.9 × 10 −3 Pa·s, B r = 0.4 T), the two particles' magnetophoresis spectrum overlapped. It's hard to separate them from each other. With the increasing of remanent flux density and η, the deflection of the particles becomes strong, and the differences of the L s between the two particles become more and more obvious. When the remanent flux density and viscosity are in the range of 0.8-1.2 T and 1.1 × 10 −3 -1.3 × 10 −3 Pa·s, respectively, a good separation effect can be ensured.

Separation performance of the newly designed ACSM chip
After the investigation of the main parameters on particle separation, the simulation was carried out to separate six kinds of particles with different sizes under the recommended conditions using the newly designed ACSM chip. The parameters were set as follows: υ 1 = 1.1 mm/s, κ = 10, η = 1.1 × 10 −3 Pa·s, B r = 1.2 T, c = 2 × 10 6 /mL, and d = 3 mm. In Figure 14, the p i of different particles is drawn with the height of different segment lines. And widths of the segment lines occupied in the x-axis are the desired outlet widths of the particles with a specific size. Under the recommended separation condition, P G keeps a high level and reaches 0.83.
In the numerical simulation study, the ACSM chip separated 6 sorts of magnetic particles. Under the recommend separation condition, the p i of the 0.5, 1.8, 2.5, 3, 3.5, and 4.5 μm magnetic particle is 100.00%, 98.18%, 66.36%, 97.25%, 97.27%, and 96.36%, respectively. Compared with both the published simulation research (Alnaimat & Mathew, 2020;Samanta et al., 2016) Figure 14. Multi-magnetic particles separation results. Particles position at the outlets and the recoveries p i of the 6 different kinds of magnetic particles. and the actual experiments (Pamme & Manz, 2004;Shen & Park, 2018), the sorts of the separated magnetic beads of the ACSM chip are improved, and the separation efficiencies are good, either.
What's more, the magnetic particles that are smaller than 500 nm were not involved in this study. Under the recommended separation condition, the F mag and the F drag will decrease exponentially and linearly with the decrease of particle size, respectively. Take the 100 nm magnetic particle as an example, the F mag acting on the 4.5 μm particles is about 5-35 pN, and the F mag of 100 nm particles is about 7 × 10 −5 -38 × 10 −5 pN. Meanwhile, the F drag acting on the 4.5 μm particles is about 2-78 pN, and the F drag of 100 nm particles is about 0.1-1.7 pN. When the diameter of the particles is smaller than 500 nm, the small F mag cannot deflect the particles and all the particles will flow out from outlet 1. For magnetic particles with different size ranges, the separation conditions (mainly the magnetic field and the flow field) are quite different. Therefore, the separation conditions should be selected and optimized according to the particle sizes.

Conclusion
In this study, we proposed a novel ACSM chip for multiple magnetic particles' simultaneous separation based on size by continuous flow magnetophoresis. The numerical simulation was applied to investigate the main parameters affecting particle separations. Numerical simulation is conducive to parameter investigations for a newly designed microfluidic chip. The most important results are as follows.
(1) The numerical model in this research was validated by the free-flow magnetophoresis experiments of Pamme and Manz (2004). And the simulation results showed good consistency with the experimental results. Meanwhile, the grid verification experiments proved that the grid refinement in this research meets the grid independence.
(2) In this research, the main parameters affecting particle separation, including magnetic field intensity, flow velocity, and dynamic viscosity of the buffer solution, were investigated by numerical simulations. Particle magnetophoresis spectrum width Θ and general separation ratio P G are put forward to characterize the particles' separation efficiency. The magnetic field intensity mainly influences the deflection of the particles. The deflection of the particles increases with the increase of the magnetic field intensity. Specifically, the magnetic field has a weak effect on the small particles (1 μm) compared with the bigger ones (5 and 10 μm). The velocity has a great influence on the deflection of the particles and the particle magnetophoresis spectrum width Θ. In this research, the buffer solution introduced into the channel at inlet 2 is used for generating a sheath flow to focus the particle stream. With the increase of velocity ratio κ, the particle magnetophoresis spectrum width Θ becomes narrower, while the particle deflection decreases. What's more, both the fluid velocity υ 1 and the fluid dynamic viscosity η are positively correlated with the F drag acting on the particles. And a strong F drag will result in a decrease in particle deflection, which is not benefit for multiparticle separation.
(3) Under the recommended separation condition, six kinds of magnetic beads were separated with a general separation ratio P G of 0.83. And the separation ratio p i of the 0.5, 1.8, 2.5, 3, 3.5, and 4.5 μm magnetic particle is 100.00%, 98.18%, 66.36%, 97.25%, 97.27%, and 96.36%, respectively. The particles with a diameter difference of 0.5 μm can be separated efficiently under the recommended separation condition using the ACSM chip.
The proposed ACSM chip can offer an improved design for multiple magnetic particles separation and can increase the sorts of the separated magnetic particles. The relevant simulation results have a significant effect on the fundamental understanding of particle dynamic behavior in the chip chamber. Multiplex magnetic particles separation can improve the bioanalysis efficiently, which can be used in a diverse range of fields. In addition, it should be noted that this model is only applicable for low particle concentrations cases. Interparticle interactions can influence the separation significantly when dense suspensions are employed. Therefore, the extended model should be developed for concentrated solutions in future work.

B
magnetic flux density (T or mT) H magnetic field intensity (A/m) c concentration of beads(/mL) D diameter (μm) d Distance (mm) F force (N) g gravity (9.81 m/s 2 ) h height of the microchannel (m or mm) K times of simulation for calculating the S (-) L length (mm) N number of particles to be released (-) P separation efficiency (-) p recovery for single kind of particle (%) R radius (m or mm) Re Reynolds number (-) S average of particle magnetophoresis spectrum (-) s magnetophoresis spectrum for single simulation (-or %) t time (s) u velocity vector (m/s) w width of the microchannel (m or mm) m mesh size in the meshing (mm) C Clausius-Mossotti function (-) Greek alphabets κ ratio of velocity at inlet 2 and inlet 1 (