Magnetofluidic micromixer based on a complex rotating magnetic field

Rapid and efficient mixing of particles and fluids in a microfluidic system is of great interest for chemical and biochemical analysis. The present paper investigates magnetofluidic mixing induced by a rotating magnetic field from a number of permanent magnets. Numerical simulation shows the complex magnetic field in the mixing chamber. Simulated particle tracing predicts the trajectories of diamagnetic particles in a paramagnetic medium for the different stationary positions of the magnets. The experimentally obtained trajectories show negative magnetophoresis similar to that predicted by the simulation. However, the static configuration of the magnets cannot achieve mixing of the diamagnetic particles. We demonstrated that a rotating magnetic field could yield up to 86% mixing efficiency at a flow rate of 60 mL min 1 using a diluted ferrofluid of only 1% volume concentration.


Introduction
Microuidics is a powerful tool for handling a small amount of uid. 1,2 One of the ultimate aims of this technology is its implementation in clinical point-of-care applications, where a large number of small samples are evaluated over a short period of time. Mixing is one of the basic sample preparation steps. As the ow is laminar on the microscale and relies on diffusion to mix, a long mixing channel is oen required. A number of active and passive methods exist for improving mixing on the microscale. 3,4 Active mixing disturbs the main ow with secondary ow generated by an external energy source. 5 Among the various external energy sources for active mixing, magnetism has apparent advantages such as contactless manipulation, biocompatibility, simple and robust design. 6,7 Using a magnet and magnetic particles, fast mixing can be achieved in a microuidic device. Yuen et al. utilized two magnetic stirring bars inside a mixing chamber. 8 The sample was delivered from one reaction chamber to another, while the bar was rotated by the external eld of a magnetic stirrer. Lu and Ryu et al. integrated micro stirrers inside the mixing chambers. 9,10 The stirrers were actuated by a rotating magnetic eld. Another magnetouidic approach is injecting magnetic particles into the chip and manipulate the ows from an external magnetic source. For instance, Biswal et al. injected paramagnetic microparticles into a microuidic device to form linear chains and used two pairs of rotating electromagnet for active mixing of two uids. 11 Similarly, Lee et al. demonstrated an active mixing method by aligning ferromagnetic particles (f ¼ 4 mm) to form rod-like structures under the inuence of a moving magnetic eld generated by an electric stirrer. 12 Furthermore, Bau et al. generated stretching and folding of the uids by incorporating an electric and magnetic eld into a micromixer. 13 Yi et al. fabricated a device with an electrode at the sidewall and copper wires at the surface of the chamber. Chaotic mixing was induced by developing a potential difference between the wire electrode to the side wall electrode under a uniform magnetic eld. 14 All above methods have their advantages, but integrating a number of micromagnets or electrode pairs into each of the small device consumes a large amount of time and labors. Furthermore, biocompatibility is a major hurdle for implementation of these techniques in clinical applications.
Magnetophoresis is a promising phenomenon for active ow manipulation. Considering the limitations of the existing methods, positive and negative magnetophoresis as a potential alternative have been explored for the separation, concentration, and transportation of particles. 15 Positive magnetophoresis utilizes magnetic beads as a tag to initiate concentration and separation of biological samples. Under an external magnetic eld, paramagnetic particles in a diamagnetic medium migrate towards the magnetic eld maxima. A number of steps are involved in tagging the beads with the biological sample. The subsequent cleaning steps also considerably increase the process time. [16][17][18][19] As negative magnetophoresis is a label-free approach, where the carrier uid has paramagnetic properties, utilizing this effect drastically reduces the processing time. Diamagnetic particles such as cells or uorescent beads act as magnetic holes in the paramagnetic uid allowing for precise manipulation. 20,21 If an external magnetic eld is applied, the paramagnetic uid particles move towards the magnetic eld maxima. Larger diamagnetic particles migrate towards the magnetic eld minima due to the magnetic susceptibility mismatch with the surrounding medium. 22 Due to these advantages, negative magnetophoresis has been increasingly utilized for the purpose of cell separation, transport, and concentration. The key factors affecting negative magnetophoresis are the physical properties such as ow velocity, volume fraction of the magnetic material in ferrouid, diameter of the magnetic nanoparticle, the magnetic eld strength, as well as the geometrical parameters such as the size of the magnet and its distance from the uidic channel. 23,24 By tuning these properties, magnetic buoyancy force and hydrodynamic drag force can be optimized. Most of the previous reports considered these key factors.
For example, Zeng et al. used two attracting magnets placed on the top and bottom of a microchannel to concentrate polystyrene particles and live yeast cells in a ferrouid ow. 25 Furthermore, the authors utilized two offset magnets to separate cells and diamagnetic particles in a sheath-free ferrouid ow. 26 Liang et al. demonstrated the efficacy of ferrouid to trigger negative magnetophoresis for high-throughput particle separation over the diamagnetic medium. 27 In another study, Hejazian et al. deected uorescent polystyrene particles mixed with ferrouid solution by externally arranged permanent magnets to transport nonmagnetic samples. 28 The authors observed that the variation of the deection was inuenced by the ow rate and the distance of the magnet arrangements. Similarly, Zhu et al. observed negative magnetophoresis in a circular chamber to investigate the migration of diamagnetic particle in a weak uniform magnetic eld. 29 Zhu et al. exploited the susceptibility variation to separate the non-magnetic particles of different size in ferrouids under a stationary magnetic eld. 30 Furthermore, the team successfully separated the mixture of uorescent polystyrene microparticles and live cells in ferrouids under a non-uniform magnetic eld. 31 Zhao et al. used label-free, continuous ow magnetophoresis to separate Hela cells from mouse red blood cells. 32 Under an external magnetic eld, the large Hela cells were deected more due to the higher magnetic buoyancy force leading to efficient separation. Pamme et al. demonstrated the continuous separation of magnetic particles of different sizes, and non-magnetic particles based on free-ow magnetophoresis. 33 They further extended the concept to biological cells labeled with magnetic nanoparticles. 34 Simultaneous separation was possible due to the susceptibility mismatch caused by magnetic loading and size differences. Recently, Zhao et al. used negative magnetophoresis to separate low-concentration cancer cells from undiluted white blood cells with customized ferrouids. 35 The approach reduced the exposure time of the cells to ferrouid, thus improves the biocompatibility and cell integrity. Furthermore, the team successfully separated circulating tumor cells (CTCs) from the RBC-lysed blood sample mixed with ferrouid at a high throughput, and a high recovery rate by employing negative magnetophoresis. 36 Most of the studies reported in the literature either require special magnetic particles or arrangement of a number of stationary magnets to achieve uid manipulation. 11,12,37 These magnets make the device bulky and increase the manufacturing cost, especially if further on-chip mixing, separation, trapping, and detection of the biochemical samples are required. [38][39][40][41] In the present study, we address these issues by avoiding direct attachment of the magnets to the chip. The magnets are separately assembled and mounted on a platform. The chip is placed in such a way that the mixing chamber axially aligns with the magnet assembly. The design also offers an optical access for an inverted microscope. In addition, the setup enables the micro-uidic chip to be easily inserted and exchanged. The platform can be extended for on-chip separation to perform the complex biochemical analysis.
Furthermore, we demonstrate here a novel approach for an externally mounted moving magnet for mixing. We carried out a detailed numerical simulation with various magnetic elds in a circular chamber. Associated particle trajectories were observed to predict the mixing performance. Experimental trajectories of uorescent particles subsequently validate the simulation results. Finally, the dynamic behavior of the two uid streams has been analyzed in terms of the mixing efficiency.

Basic design and working principle
Our microuidic device contains two inlets, a circular mixing chamber, and outlet ports. The device was designed with Cor-elDraw (Corel Co., Canada). A commercial CO 2 laser system (Trotec/Rayjet 300) was utilized to fabricate the channel mold from a laminated plastic sheet with the thickness of approximately 250 AE 30 mm. The mixing chamber has a diameter of 2 mm. The microchannel has a width of W ¼ 500 mm, depth of D ¼ 250 AE 30 mm and the total length of L ¼ 20 mm. The length is relatively long to adjust the chip with the mounted platform. The laser machined parts were ultrasonicated for 10 minutes to remove any residual particles. A double-sided adhesive tape (Scotch, 3 M) was used to bond a clean glass slide (sizes of 50 Â 76 mm) with the channel pattern to form a mould. The glass slide ensured that the replicated pattern is at enough to achieve proper plasma bonding. Degassed polydimethylsiloxane (PDMS) prepolymer mixed with a crosslinker was then poured into the mould. The PDMS was further degassed for 30 min to remove any air bubbles and kept in an oven at 80 C for an hour for curing. The cured PDMS replica was peeled off from the mold. The thickness of the PDMS layer was at approximately 3 mm. The inlet and outlet ports were introduced by punching 1.5 mm diameter hole. The replicated PDMS was again cleaned for 10 minutes by ultra-sonication. Finally, the PDMS device was treated with oxygen plasma for 45 seconds and bonded to a clean glass slide. The device was tested for different ow rates with DI water to ensure that the device is free of leakage. Our technique provides an opportunity to redesign and to optimize the device without much delay. Fig. 1 illustrates the schematic diagram of the system under investigation.
The experiments consisted of two parts. First, particle trajectories caused by negative magnetophoresis were investigated. A water-based ferrouid (EMG 707, Ferrotec, USA) was diluted to 1% volume concentrations (f ¼ 1% vol.) with DI water.
Green uorescent polyethylene microspheres (1.00 g cm À3 , 30 mm, Cospheric, USA) was mixed with the fer-rouid solution. The mixture was then injected by a precision syringe pump (SPM-100, SIMTech Microuidics Foundry) through a Teon tube to one of the inlet ports. In this system, the ferrouid served as the paramagnetic medium. The diamagnetic uorescent beads created magnetic holes. The experiments were recorded over a period of 1 minute to visualize the trajectories caused by negative magnetophoresis. The video was recorded with a USB camera (Edmund Optics, Germany) at a rate of 30 frames per second through an inverted microscope (Nikon Eclipse TE-100). Associated soware (uEye Cockpit) was utilized to save the videos. Initially, a xed ow rate of 15 mL min À1 was initiated for the observations. Magnetophoresis effect was characterised for four different magnet congurations with one to four magnets.
In the second part, mixing efficiency was determined for the different arrangements. Two uid samples were injected through the two syringe pumps (SPM-100, SIMTech Micro-uidics Foundry). Diluted ferrouid (f ¼ 1% vol.) with green uorescent polyethylene microspheres (1.00 g cm À3 , 30 mm, Cospheric, USA) and diluted ferrouid (f ¼ 1% vol.) without uorescent microparticles were delivered to each of the inlets to evaluate the mixing efficiency. Flow rates ranging from 15-60 mL min À1 were considered with rotational speeds ranging from 25 to 200 rpm. Fig. 2 demonstrates the step by step fabrication of the LOC device with the experimental setup. The associated images depict the channel before and aer mixing.
We utilized the laser machining to fabricate the mounting platform of the motor on top of the microuidic device. A cylindrical PMMA base was mounted on the motor sha. A number of holes were implemented for holding the magnets. Cylindrical NdFeB magnet (grading N38, 5 Â 6 mm, AFM Magnetics, Australia) were inserted to induce a magnetic eld. The magnet arrangement was a key factor to improve the magnetic eld distribution. Numerical simulation with COMSOL Multiphysics was considered for optimising the ux distribution over the mixing chamber. The optimised magnet arrangement facilitated increased ux distribution without making the system bulky.
The magnetic eld strengths were calibrated by a Gauss meter (Hirst Magnetic Instruments Ltd., UK). The tip of the Gauss metre was positioned at the center of the cylindrical base that is 1 mm apart from the edge of the magnets. This position represents the centre of the mixing chamber. Fig. 3 shows the measured magnetic ux density versus the distance from the surface of the magnets. The gure demonstrates a gradual decrease of ux density over the distance. Furthermore, the twomagnet conguration indicates an increased ux density as compared to the one-magnet conguration. The three-magnet   and four-magnet congurations show higher ux density than the two-magnet conguration. Interestingly, the three-magnet conguration indicates a slightly higher ux density than the four-magnet conguration. The data depicted in Fig. 3 provided important insight on the magnetic ux distribution for the different congurations. For a detailed magnetic ux distribution analysis, the magnetic base was adjusted manually around the edge of the mixing chamber. The gap between the magnetic surfaces to the circular chamber was approximately 3 mm. We selected 4 positions that are 90 degrees apart from each other. For the analysis, the video was converted into the still images by ImageJ soware (NIH, imagej.net). For better visualisation, background noise and the stationary particles were subtracted to obtain the streamlines of each image using a customised MATLAB code.
We utilised an Arduino board (Duinotech Uno-XC 4410, China) to precisely control the rotational speed of the motor. The microprocessor was coded to provide a predened speed ranging from 20 to 220 rpm. A xed time frame has been considered to ensure that at least two circulations of the magnet around the chamber were observed. Each time frame consisted of at least 16 still images, for the different rotational speeds and ow rates. A customized MATLAB (MathWorks) code was used to analyse the mixing efficiency from the converted grayscale images. A total of 2400 still images were evaluated to determine the average mixing efficiency.

Theoretical background
The ow pattern of particles in an external magnetic eld is affected by the magnetic force, hydrodynamic force, and drag forces. 42 Under a complex magnetic eld, the trajectory of particles depends on the balance of these forces. In the presence of a paramagnetic uid, the manipulation of diamagnetic particles such as polystyrene is initiated by negative magnetophoresis, where the particles are pushed to the magnetic eld minima. 28 The magnetic eld can be described as: here, m 0 ¼ 4 Â 10 À7 is the permeability of the vacuum, M represents the magnetization of the material under a magnetic eld of H, and V m is the scalar magnetic potential. 43 Laminar ow follows the continuity equation and the Navier-Stokes equation for the particle trajectories. 28 Under steady state condition, the transport of uids from the inlet to the outlet is governed by the continuity equation: where r f and u f are respectively the density and velocity of the uids, and the Navier-Stokes equation: where p f is the pressure of the uids, I is the identity matrix, m f is the dynamic viscosity of the uids, and D is the thickness of the channel. 44 The motion of the particle is governed by the force balance: where m p is the mass of the particle, n is the velocity of the particle, F m and F d are the magnetophoretic and uids drag force. 45 The drag force can be determined by Stokes law as: where s p is the particle velocity response time (s). The particle velocity response time of a spherical particle within a laminar ow can be written as: where m is the uid viscosity (Pa s), r p is the particle density, and d p is the particle diameter. The non-magnetic particle experiences a magnetophoretic force that pushes it away from the magnetic eld: where m r is the relative permeability of the uid, and m r.p is the particle relative permeability. Mixing efficiency was calculated by evaluating the intensity of the recorded uorescent images using customised MATLAB code. 46 The mixing index is determined by measuring the pixel intensity over a cross-section area: where I i is the selected pixel intensity and I is the average intensity of the selected cross-sectional area, N represents the total number of pixels. The values vary between zero and one. Zero represents the no mixing condition, and one represents the complete mixing conditions.

Numerical simulations
COMSOL Multiphysics 5.2 (COMSOL Inc, USA) was utilized to observe the complex magnetic eld of the different magnet congurations. The magnetic eld with no current (MFNC) physics was utilized to model the magnetic ux density. The magnets with a diameter of f ¼ 5 mm and a thickness of h ¼ 6 mm were placed in 4 different locations and 90 apart around the mixing chamber. The magnetization of the magnet was 1.6 Â 10 6 A m À1 . A circular magnetic insulator with a diameter of 2 cm was imposed around the model. The relative permeability of the uid was 1.05. For investigating the particle trajectories, low Reynolds number (creeping ow, SPF2) and particle tracing for the uid ow (FPT) through the circulating chamber was considered. The model consists of an inlet, a circular mixing chamber, and an outlet. The edges from the magnet to the edges of the circular chamber had a minimum distance of 1 mm. An incompressible uid (diluted ferrouid in DI water) was introduced into the inlet with a ow rate of 15 mL min À1 . The chamber was considered as a shallow channel with a thickness of 250 mm. For simplicity, no-slip boundary condition was selected. Six particles with a diameter of 30 mm and a particle density of 1200 kg m À3 ware considered in the simulation. The trajectory was initiated for a time period of 100 seconds with a time steps of 0.01 seconds.
Different magnetic pole arrangements for the magnets around the mixing chamber were investigated to optimise the magnetic ux gradient, Fig. 4. We expected that one of the different magnet arrangements will provide the optimal ux eld gradient. 1,18 Some magnet arrangements provide a stronger ux gradient than others. 47,48 For example, two-magnet congurations with both the positive pole alignments provide more ux gradient into the mixing chamber then the positive and negative pole alignments due to their repulsive behaviour, Fig. 4B and C. However, in the three-magnet arrangements, the only positive or negative pole alignment repulses the ux lines in such a way that there is no effective eld gradient into the mixing chamber. Besides, moving the magnet into a close setup improves the eld gradient by many folds, Fig. 4F. For the fourmagnet setup, an alternate magnetic pole alignments give the highest magnetic eld gradient. For the simulation of particle trajectory, the congurations of Fig. 4A, B, F and G-H were considered. The next step was observing the effect of magnetic ux gradient for different positions of the magnets around the mixing chamber, Fig. 5. The gure represents the magnetic ux maxima and minima in the mixing chamber. With a clear understanding of magnetic ux distribution, it is possible to predict the negative magnetophoresis of the uorescent particles. Fig. 6 shows the deection of the diamagnetic particles in the mixing chamber. In Fig. 5A, the magnet is positioned close to the inlet region. The ux lines are aligned with the ow streamlines, and the ux minima are visible close to the outlet region. Hence, the particles tend to have straight trajectories from the inlet to the outlet, Fig. 6A. If the magnet moves 90 degrees clockwise, the ux lines become perpendicular to the streamlines. Moreover, the eld maxima shied to the top of the mixing chambers. As the particle follows the eld minima, the trajectory bends towards this region, Fig. 6B. In Fig. 5C, the magnet was further moved by 90 and positioned close to the outlet. The eld maxima again shi oppose to the hydrodynamic ows, and the eld minima are visible close to the inlet. Therefore, the particles tend to accumulate at the inlet and are deected at the outlet, Fig. 6C. The magnet again moved by 90 and positioned at the bottom of the chamber and subsequently the eld minima region transferred to the upper part of the chamber. Therefore, the trajectories of the particle shied towards the eld minima as depicted in Fig. 6D.
For the two-magnet congurations, the setup was arranged in such a way that the like poles were facing towards the surface of the mixing chamber. The ux eld gradient increases compared to the one-magnet setup. 47 Fig. 5E shows the two magnets positioned 180 apart. The magnetic eld minima can be observed at the top and the bottom of the mixing chamber. Furthermore, resultant ux elds are parallel to the streamlines of the ow. Therefore, the particles in Fig. 6E are projected through these eld minima regions. Furthermore Fig. 5F and H show identical eld pattern that is opposite. However, the eld minima in Fig. 5F is at the bottom whereas the eld minima in Fig. 5H is at the top of the inlet. Therefore, most particles tend to move toward these regions, Fig. 6F and H. Nonetheless, some of the particles could overcome the magnetic force and successfully pass through the mixing chamber by leaving a bending trail around the eld minima region. Fig. 5G shows that the eld minima were predominantly close to the inlet and the outlet. Some particle remains at the inlet region and is not be able to travel towards the outlet as they achieve zero velocity, Fig. 6G. However, other particles pass through the mixing chamber by maintaining possible eld minima towards the outlet ports.
For the three-magnets conguration depicted in Fig. 5I-L, the ux gradient becomes stronger and only one eld minimum could be observed. The particle follows these eld as shown in Fig. 6I-L. Nonetheless, as the eld minima are entirely covered by the inlet region, particle accumulated there, Fig. 6J. Fig. 5L indicates that the ux density gradually reduces from the inlet to the outlet port. Minimum deection is observed for this gradual reduction of magnetic eld strength. For the fourmagnet conguration depicted in Fig. 5M-P, the magnetic ux gradient is stronger. In addition, the eld minima disappear. Particle tracing shows only two patterns, Fig. 6M-P. Furthermore, it was interesting to observe that the trajectories have a tendency to move toward the bottom parts of the mixing chambers.
The detailed investigation suggested that the deection pattern of the particles predominantly depends on the magnet position around the mixing chamber. Moreover, the dynamics movements by a DC motor could initiate similar patterns which have been utilized for active mixing in the subsequent experiments.

Negative magnetophoresis in the stationary magnet elds
Fluorescent particles of 30 mm diameter were mixed with a diluted ferrouid of 1% vol. concentration. As discussed in the simulation, the different magnet congurations were considered. Due to the diamagnetic nature of the uorescent particles, negative magnetophoresis was observed in the experiments. 28 The particles tend to accumulate in the magnetic ux minima region. Furthermore, due to the paramagnetic nature of the ferrouid, the uid experience a bulk force that creates a secondary ow in addition to the main hydrodynamic ow.
The diamagnetic particles follow and pass through eld minima region. As the magnet is positioned close to the inlet, the particle passes through the chamber without any deections, Fig. 7A. However, if the magnet is positioned close to the outlet port (Fig. 7C), the magnetic eld opposes the ow. The particle velocity is reduced while passing through the mixing chamber. Moreover, a number of particles are trapped in the mixing chamber due to the dominating magnetic forces. The particle stream demonstrates the phenomena where the longer stream observed close to the inlet region denotes a higher velocity, while it becomes shorter at the outlet indicating a relatively slower velocity. Fig. 7B and D show the particle deection mostly towards the magnetic eld minima due to negative magnetophoresis.
For the two-magnet congurations, particles tend to move more to the top and bottom where the eld minima are, Fig. 7E. Fig. 7F and H show mirrored and slightly twisted trajectories. In both cases, the particles move through the minimum eld close to the inlet and outlet. In Fig. 7G, the eld minima are close to the inlet and outlet, because the magnets are positioned at the top and bottom of the mixing chamber. Thus, the particles mostly deect near the inlet while owing towards the outlet ports.
In a three-magnet conguration, identical and opposite deections of the particles were observed, Fig. 7I and K. However, in Fig. 7J, the particle deects towards the top and bottom region of the mixing chamber as the magnetic eld minima are close to the inlet. Furthermore, no particles were observed at the center of the chamber. The simulated trajectory also provides an identical pattern, where the particles follow the side walls. A completely opposite phenomenon can be observed in Fig. 7L, where the particles mostly aggregated at the center region due to the higher magnetic eld gradient all over the chamber.
In the four-magnet congurations depicted in Fig. 7M-P, only two patterns of particle trajectories were observed. For example, Fig. 7M and O indicates that the trajectories shied towards the bottom region. When the magnet was placed 45 degrees further, the particles migrate towards the centre of the chamber, Fig. 7N and P.
The experimental data agree with the simulated trajectories. However, only six particles were considered in the simulation.
In the experiments, we observed more complex secondary ows on top of the chamber. This could be due to a more complex and three-dimensional nature of the magnetic ux gradient acting on the uid ows. 28 The secondary ow is relatively slower and sometimes circulates opposing the main hydrodynamic ow. The deection of the uorescent particle relies on the active magnetic force perpendicular to the hydrodynamic force.

Mixing enhancement with multi-magnet setup
Mixing phenomena occur due to the susceptibility mismatch between the paramagnetic uids and diamagnetic particles. The susceptibility mismatch generates a magnetoconvective ow of the paramagnetic uids towards the magnetic eld maximum. The competition between the hydrodynamic force and the magnetophoretic force determines the mixing performance. 6,49 Fig. 8 shows the mixing efficiency of the multimagnet congurations with rotational speed ranging from 25 to 200 rpm. Fig. 8A shows the mixing efficiency for a ow rate of 15 mL min À1 for different rotational speeds of the one to four magnet congurations. Mixing of 1% vol. ferrouid solution with 30 mm uorescent particle, and 1% vol. ferrouid solution (without uorescent particle) was evaluated. The mixing efficiency was 28% in the chamber without any magnetic arrangement is due to the molecular diffusion. The mixing efficiency increases to 32% when the one-magnet (0 rpm) was mounted on top of the chamber. The efficiency increases due to the static magnetic eld of the magnets acting on the bulk uids. As a result, the magnetic nanoparticles of the ferrouid solution slightly migrate towards the magnetic eld maxima region.
For the one-magnet conguration, the mixing efficiency dramatically increases from 32% to approximately 68% due to the actuation of the magnets (25 rpm). The actuation changes the orientation of the magnetic ux lines. As the magnet revolves, the uorescent particles are dragged along following the eld minima. The mixing efficiency gradually increases to 79% with increasing rotational speed. The mixing efficiency reaches its maximum at 100-125 rpm due to the right force balance condition. 28,42 Aerwards, the efficiency reduces as the uid ow cannot follow the change of the magnetic eld. 11 For the two-magnet setup, the magnetic force becomes more dominant. The efficiency improves to 73-83% when the magnet rotates between 25 and 200 rpm, respectively. For the three, and four magnet setup, the magnetic ux becomes stronger but eld minima are missing in the chamber, Fig. 5. Thus, mixing efficiency reduces to 64-76%, and 62-75% for the three, and four magnet setup respectively, as the diamagnetic particle follow the magnetic eld minima.
At a ow rate of 30 mL min À1 , the hydrodynamic force became more dominant, Fig. 8B. The mixing efficiency for the one, and two magnet conguration gradually decreases to 64-78%, and 68-82%, respectively. The maximum efficiency was observed between 100 and 125 rpm for both the cases. For the three-magnet setup, the mixing efficiency improves to 73-85% between 25 and 125 rpm, respectively. The peak efficiency was 85% at 100 rpm due to the right force balance condition.
However, for the four-magnet setup, the mixing efficiency reduced to 67-83%. The maximum recorded efficiency was 83% between a range of 100-125 rpm. The reduced efficiency is again caused by the lack of eld minima, Fig. 5.
At a ow rate of 45 mL min À1 , the mixing efficiency dramatically drops to 61-72%, and 64-75% for the one, and two magnet conguration, Fig. 8C. As the hydrodynamic force dominates over the magnetic force, the diamagnetic particle associated with the ferrouid cannot follow the moving magnetic eld, Fig. 5. The three, and four magnet conguration offers a stronger eld gradient with reduced eld minima in the mixing chamber. The efficiency improves to 68-84%, and 66-81% for the three, and four magnet conguration, subsequently. The peak efficiency was at 100 rpm due to the right force balance condition. As the ow rate further increases to 60 mL min À1 , the mixing efficiency for the one, and two magnet conguration further reduces to 54-67%, and 60-72%, respectively, Fig. 8D. Improved mixing performance was observed for three, and four magnet conguration at higher ow rates. The improved efficiency was 73-86%, and 70-83% for the three, and four magnet conguration. The four-magnet conguration shows reduced efficiency over three-magnets as the magnet works against each other, Fig. 3. Another observation was that at low speed, the efficiency uctuates more compared to a higher speed for the one, and two magnet conguration, Fig. 8. The ow stream does not deect, if the magnet is closer to the inlet or the outlet ports, Fig. 7A, C and E. At a higher speed, the deviation decreases and a more stable mixing performance can be observed. Moreover, the uctuation of the efficiency for the three, and four magnet conguration reduces at a lower speed, due to the lack of magnetic eld minima, indicating a more stable mixing pattern.

Conclusion
The concept reported here is suitable for mixing of two or more samples. Our main motivation was to observe the phenomena of negative magnetophoresis and improved mixing with rotating magnets. Numerical simulation was conducted to observe the magnetic eld distribution over the mixing chamber. Simulated particle tracing predicts trajectories of 30 mm uorescent particles in ferrouid stream for the four stationary positions of the magnet assemblies. The experimental trajectories show negative magnetophoresis similar to the predicted simulations. Both numerical and experimental data conrm that mixing of diamagnetic particles can be achieved by negative magnetophoresis.
Efficient mixing was demonstrated with a low concentration of only 1% vol. ferrouid. One, and two-magnet congurations show mixing efficiencies of approximately 79% and 83% at a ow rate of 15 mL min À1 . The mixing efficiency gradually reduces with ow rates ranging from 30-60 mL min À1 . Interestingly, improved mixing performance was observed at higher ow rates (45-60 mL min À1 ) for the three, and four magnet conguration. The trends were caused by the higher hydrodynamic force that allows the diamagnetic particle to balance with the stronger moving magnetic eld. The highest mixing efficiency was 86% at a ow rate of 60 mL min À1 using the three-magnet conguration. The mixing efficiency reduced to 83% for the four-magnet conguration at a ow rate of 60 mL min À1 .
The efficiency of the mixing concept presented here can be further improved using higher-grade magnets. A larger magnet will also improve the efficiency but may not suit the small microuidic device. Moreover, the magnetic platform was mounted with a relatively large gap of almost 3 mm. The gap might be reduced to further improve the mixing efficiency. The multi-magnet conguration shows similar mixing performance at different ow rates for the microuidic device. Multiple chambers with multiple motors in a series would allow for cascaded mixing. The same mixing concept can apply to paramagnetic particles functionalized with the antibody for diagnostic applications. 50 Future works will address issues such as immobilizing biomarker on the beads, their on-chip mixing and separation. This journal is © The Royal Society of Chemistry 2017