# In-Silico Conceptualisation of Continuous Millifluidic Separators for Magnetic Nanoparticles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}) commonly uses small (i.e., with dimensions similar to the flow channel or separator geometry) permanent magnets. Although such permanent magnets can provide high-gradient magnetic fields in their proximity, their fields propagate over relatively small distances. Hence, permanent magnets are almost exclusively used for HGMS separation in microfluidics [25,26]. Alternatively, HGMS uses fine magnetic structures, such as magnetic fibres or meshes (usually out of magnetically soft materials, i.e., with a small coercivity) being magnetised by an external magnetic field generated by permanent magnets or electromagnets [27,28]. The fine magnetic matrix de-homogenises the magnetic field such that locally high magnetic field gradients occur.

^{−1}) devices are very simple, frequently involving no more than a hand-held magnet to produce the inhomogeneous magnetic field. This simplicity is an advantage over HGMS, which can present challenges in terms of complex set-up [22], operation cost and energy requirements [29], as well as particle recovery or removal [30]. Incomplete particle removal can reduce the available surface area for particle adsorption, which decreases the performance in subsequent separation cycles. A bottleneck of HGMSs for large-scale separation, either continuous or batch, is the loading capacity (maximum volume of material accumulated). MPs or MNPs accumulate first at volumes with high field gradients (e.g., close to the surfaces of magnetically soft fibres). This reduces the separation efficiency over time and can lead to plugging if the fine structures used in many HGMS devices become overloaded with magnetic material. Therefore, LGMS can be preferred for large scale magnetophoretic separation [29]. The design rules for LGMS, however, are less clear and more theoretical and experimental work must be dedicated to realise its true potential [18].

## 2. Concept and Methodology

#### 2.1. Electromagnetic Separator Designs

#### 2.2. Modeling Magnetic Particle Transport

#### 2.2.1. Magnetophoretic Forces

_{3}O

_{4}), i.e., the most magnetic and commonly used iron oxide phase, and the size of the iron oxide nanoparticles (IONPs) considered [41]. It should be noted, however, that using a constant for susceptibility is a simplification, especially for superparamagnetic MNPs (particles smaller ~25 nm for magnetite). In addition, Equation (1) already assumes a non-magnetic medium surrounding the particle. Due to the radial symmetry of the wire’s magnetic field (axial components are zero), the magnetic field gradient in Equation (1) is zero except in the radial direction. Hence, combining Equations (1) and (2) (using cylindrical coordinates) shows that the magnetophoresis acts only radially with a force ${F}_{m.radial}$ of

#### 2.2.2. Drag Forces

#### 2.2.3. Particle Tracking Algorithm

**I:**The axial velocity was given by the annular Poiseuille velocity profile ${v}_{annular}\left(R\right)$, which was determined for the volumetric flow rate $\dot{V}$ and separator dimensions ${r}_{wire}$ and ${r}_{tube}$ by Equations (6) and (7) [42].

**II:**The radial position of each IONP at the separator inlet $R$(L = 0) was initialised randomly. The radially dependent initialisation likelihood corresponded to the axial velocity given by the annular velocity profile. This was to account for velocity-dependent particle flux into the separator when assuming that IONPs are distributed homogeneously in the solution when entering the separator.

**III:**The particle velocity in the axial direction L was governed by the particle’s radial position and the annular velocity. The axial displacement $\Delta L$ per time step $\Delta t$ was updated as

**IV:**The particle velocity in the radial direction $R$ was governed by the particle’s magnetophoretic velocity ${v}_{radial}\left(R\right)$ given by Equation (5). The radial displacement $\Delta R$ per time step $\Delta t$ was updated as

**V:**Following the work of Schaller et al. [43], a stochastic Brownian motion length was added to the updated radial and axial positions to account for diffusive particle transport.

**VI:**The radial and axial particle positions were updated after each period as described in step I–V. As the particles approached the wire, the magnetophoretic forces became dominant and particles remained at the wire surface. To avoid unnecessary computational effort, particle tracking was then terminated, i.e., the particle position was not updated anymore. The tracking of particles was also terminated based on their residence time and after particles exited the separator. The three termination conditions were:

- (1)
- The particle tracking time was a tenfold of the average residence time (referring to the liquid phase) in the separator, which was determined by the flow rate and the separator channel cross section).
- (2)
- The updated axial position exceeded the separator length $\left({L}_{t+\Delta t}>{L}_{separator}\right)$, i.e., the particle exited the separator.
- (3)
- The particle collided with the wire more than 1000 times, which was determined by the collision frequency counter.

#### 2.2.4. Time-Step

#### 2.2.5. Separation Efficiency Definition

#### 2.2.6. Computation

## 3. Results

#### 3.1. Effect of Design and Operating Parameters on Separator Efficiency

^{−3}(see Equation (3)) compared to the mean residence time which scales with the separator cross section with ~R

^{−2}(see Equations (6) and (7)). Hence, a closer proximity to the wire can be more beneficial to draw MNPs to the wire than a longer residence time.

#### 3.2. Optimum Separation Conditions for 250 nm MNPs

#### 3.3. Optimum Separation Conditions for 500 nm MNPs

## 4. Conclusions and Perspective

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ayansiji, A.O.; Dighe, A.V.; Linninger, A.A.; Singh, M.R. Constitutive relationship and governing physical properties for magnetophoresis. Proc. Natl. Acad. Sci. USA
**2020**, 117, 30208–30214. [Google Scholar] [CrossRef] - Gijs, M.A.M.; Lacharme, F.; Lehmann, U. Microfluidic Applications of Magnetic Particles for Biological Analysis and Catalysis. Chem. Rev.
**2010**, 110, 1518–1563. [Google Scholar] [CrossRef] - Lim, B.; Vavassori, P.; Sooryakumar, R.; Kim, C. Nano/micro-scale magnetophoretic devices for biomedical applications. J. Phys. D Appl. Phys.
**2016**, 50, 033002. [Google Scholar] [CrossRef] - Zaidi, N.S.; Sohaili, J.; Muda, K.; Sillanpää, M. Magnetic Field Application and its Potential in Water and Wastewater Treatment Systems. Sep. Purif. Rev.
**2014**, 43, 206–240. [Google Scholar] [CrossRef] - Castelo-Grande, T.; Augusto, P.A.; Rico, J.; Marcos, J.; Iglesias, R.; Hernández, L.; Barbosa, D. Magnetic water treatment in a wastewater treatment plant: Part I sorption and magnetic particles. J. Environ. Manag.
**2021**, 281, 111872. [Google Scholar] [CrossRef] - Solsona, M.; Nieuwelink, A.-E.; Meirer, F.; Abelmann, L.; Odijk, M.; Olthuis, W.; Weckhuysen, B.M.; Berg, A.V.D. Magnetophoretic Sorting of Single Catalyst Particles. Angew. Chem. Int. Ed.
**2018**, 57, 10589–10594. [Google Scholar] [CrossRef] - Rossi, L.M.; Costa, N.J.S.; Silva, F.P.; Wojcieszak, R. Magnetic nanomaterials in catalysis: Advanced catalysts for magnetic separation and beyond. Green Chem.
**2014**, 16, 2906–2933. [Google Scholar] [CrossRef] - Munaz, A.; Shiddiky, M.; Nguyen, N.-T. Recent advances and current challenges in magnetophoresis based micro magnetofluidics. Biomicrofluidics
**2018**, 12, 031501. [Google Scholar] [CrossRef] - Song, K.; Li, G.; Zu, X.; Du, Z.; Liu, L.; Hu, Z. The Fabrication and Application Mechanism of Microfluidic Systems for High Throughput Biomedical Screening: A Review. Micromachines
**2020**, 11, 297. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xi, H.-D.; Zheng, H.; Guo, W.; Gañán-Calvo, A.M.; Ai, Y.; Tsao, C.-W.; Zhou, J.; Li, W.; Huang, Y.; Nguyen, N.-T.; et al. Active droplet sorting in microfluidics: A review. Lab Chip
**2017**, 17, 751–771. [Google Scholar] [CrossRef] - Banerjee, U.; Mandal, C.; Jain, S.K.; Sen, A.K. Cross-stream migration and coalescence of droplets in a microchannel co-flow using magnetophoresis. Phys. Fluids
**2019**, 31, 112003. [Google Scholar] [CrossRef] - Zhou, R.; Bai, F.; Wang, C. Magnetic separation of microparticles by shape. Lab Chip
**2017**, 17, 401–406. [Google Scholar] [CrossRef] [PubMed] - Lin, G.; Makarov, D.; Schmidt, O.G. Magnetic sensing platform technologies for biomedical applications. Lab Chip
**2017**, 17, 1884–1912. [Google Scholar] [CrossRef] - Chircov, C.; Grumezescu, A.M.; Holban, A.M. Magnetic Particles for Advanced Molecular Diagnosis. Materials
**2019**, 12, 2158. [Google Scholar] [CrossRef][Green Version] - Hejazian, M.; Nguyen, N.-T. Negative magnetophoresis in diluted ferrofluid flow. Lab Chip
**2015**, 15, 2998–3005. [Google Scholar] [CrossRef] [PubMed][Green Version] - Munaz, A.; Kamble, H.; Shiddiky, M.J.A.; Nguyen, N.-T. Magnetofluidic micromixer based on a complex rotating magnetic field. RSC Adv.
**2017**, 7, 52465–52474. [Google Scholar] [CrossRef][Green Version] - Pamme, N.; Wilhelm, C. Continuous sorting of magnetic cells via on-chip free-flow magnetophoresis. Lab Chip
**2006**, 6, 974–980. [Google Scholar] [CrossRef] [PubMed] - Hejazian, M.; Li, W.; Nguyen, N.-T. Lab on a chip for continuous-flow magnetic cell separation. Lab Chip
**2015**, 15, 959–970. [Google Scholar] [CrossRef][Green Version] - Zhang, Q.; Yin, T.; Xu, R.; Gao, W.; Zhao, H.; Shapter, J.G.; Wang, K.; Shen, Y.; Huang, P.; Gao, G.; et al. Large-scale immuno-magnetic cell sorting of T cells based on a self-designed high-throughput system for potential clinical application. Nanoscale
**2017**, 9, 13592–13599. [Google Scholar] [CrossRef][Green Version] - Ngamsom, B.; Esfahani, M.M.N.; Phurimsak, C.; Lopez-Martinez, M.J.; Raymond, J.-C.; Broyer, P.; Patel, P.; Pamme, N. Multiplex sorting of foodborne pathogens by on-chip free-flow magnetophoresis. Anal. Chim. Acta
**2016**, 918, 69–76. [Google Scholar] [CrossRef] [PubMed] - Myklatun, A.; Cappetta, M.; Winklhofer, M.; Ntziachristos, V.; Westmeyer, G.G. Microfluidic sorting of intrinsically magnetic cells under visual control. Sci. Rep.
**2017**, 7, 6942. [Google Scholar] [CrossRef][Green Version] - Ge, W.; Encinas, A.; Araujo, E.; Song, S. Magnetic matrices used in high gradient magnetic separation (HGMS): A review. Results Phys.
**2017**, 7, 4278–4286. [Google Scholar] [CrossRef] - Lim, J.; Yeap, S.P.; Low, S.C. Challenges associated to magnetic separation of nanomaterials at low field gradient. Sep. Purif. Technol.
**2014**, 123, 171–174. [Google Scholar] [CrossRef] - Leong, S.S.; Yeap, S.P.; Lim, J. Working principle and application of magnetic separation for biomedical diagnostic at high- and low-field gradients. Interface Focus
**2016**, 6, 20160048. [Google Scholar] [CrossRef][Green Version] - Zeng, L.; Chen, X.; Du, J.; Yu, Z.; Zhang, R.; Zhang, Y.; Yang, H. Label-free separation of nanoscale particles by an ultrahigh gradient magnetic field in a microfluidic device. Nanoscale
**2021**, 13, 4029–4037. [Google Scholar] [CrossRef] - Chen, Q.; Li, D.; Lin, J.; Wang, M.; Xuan, X. Simultaneous Separation and Washing of Nonmagnetic Particles in an Inertial Ferrofluid/Water Coflow. Anal. Chem.
**2017**, 89, 6915–6920. [Google Scholar] [CrossRef] [PubMed] - Eskandarpour, A.; Iwai, K.; Asai, S. Superconducting Magnetic Filter: Performance, Recovery, and Design. IEEE Trans. Appl. Supercond.
**2009**, 19, 84–95. [Google Scholar] [CrossRef] - Moeser, G.D.; Roach, K.A.; Green, W.H.; Hatton, T.A.; Laibinis, P.E. High-gradient magnetic separation of coated magnetic nanoparticles. AIChE J.
**2004**, 50, 2835–2848. [Google Scholar] [CrossRef] - Toh, P.Y.; Yeap, S.P.; Kong, L.P.; Ng, B.W.; Chan, D.J.C.; Ahmad, A.L.; Lim, J. Magnetophoretic removal of microalgae from fishpond water: Feasibility of high gradient and low gradient magnetic separation. Chem. Eng. J.
**2012**, 211–212, 22–30. [Google Scholar] [CrossRef] - Mayo, J.T.; Yavuz, C.T.; Yean, S.; Cong, L.; Shipley, H.; Yu, W.; Falkner, J.; Kan, A.; Tomson, M.; Colvin, V.L. The effect of nanocrystalline magnetite size on arsenic removal. Sci. Technol. Adv. Mater.
**2007**, 8, 71–75. [Google Scholar] [CrossRef][Green Version] - Surenjav, E.; Priest, C.; Herminghaus, S.; Seemann, R. Manipulation of gel emulsions by variable microchannel geometry. Lab Chip
**2009**, 9, 325–330. [Google Scholar] [CrossRef] [PubMed] - Li, P.; Kilinc, D.; Ran, Y.-F.; Lee, G.U. Flow enhanced non-linear magnetophoretic separation of beads based on magnetic susceptibility. Lab Chip
**2013**, 13, 4400. [Google Scholar] [CrossRef][Green Version] - Zhang, X.; Zhu, Z.; Xiang, N.; Long, F.; Ni, Z. Automated Microfluidic Instrument for Label-Free and High-Throughput Cell Separation. Anal. Chem.
**2018**, 90, 4212–4220. [Google Scholar] [CrossRef] [PubMed] - Banis, G.; Tyrovolas, K.; Angelopoulos, S.; Ferraro, A.; Hristoforou, E. Pushing of Magnetic Microdroplet Using Electromagnetic Actuation System. Nanomaterials
**2020**, 10, 371. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, T.; Li, J.; Morozov, K.I.; Wu, Z.; Xu, T.; Rozen, I.; Leshansky, A.M.; Li, L.; Wang, J. Highly Efficient Freestyle Magnetic Nanoswimmer. Nano Lett.
**2017**, 17, 5092–5098. [Google Scholar] [CrossRef] [PubMed] - Yu, H.; Tang, W.; Mu, G.; Wang, H.; Chang, X.; Dong, H.; Qi, L.; Zhang, G.; Li, T. Micro-/Nanorobots Propelled by Oscillating Magnetic Fields. Micromachines
**2018**, 9, 540. [Google Scholar] [CrossRef][Green Version] - TSSF005.00 Wire Size & Current Rating Guide. Available online: www.jst.fr/doc/jst/pdf/current_rating.pdf (accessed on 31 October 2021).
- Jackson, J.D. Classical Electrodynamics, 3rd ed.; Wiley: Hoboken, NJ, USA, 1998; Available online: https://www.wiley.com/en-gb/Classical+Electrodynamics%2C+3rd+Edition-p-9780471309321 (accessed on 22 September 2021).
- Natukunda, F.; Twongyirwe, T.M.; Schiff, S.J.; Obungoloch, J. Approaches in cooling of resistive coil-based low-field Magnetic Resonance Imaging (MRI) systems for application in low resource settings. BMC Biomed. Eng.
**2021**, 3, 3. [Google Scholar] [CrossRef] [PubMed] - Ansorge, R. Magnetic Field Generation. Phys. Math. MRI 2016. [CrossRef]
- Heider, F.; Zitzelsberger, A.; Fabian, K. Magnetic susceptibility and remanent coercive force in grown magnetite crystals from 0.1 μm to 6 mm. Phys. Earth Planet. Inter.
**1996**, 93, 239–256. [Google Scholar] [CrossRef] - Sparrow, E.; Chen, T.; Jónsson, V. Laminar flow and pressure drop in internally finned annular ducts. Int. J. Heat Mass Transf.
**1964**, 7, 583–585. [Google Scholar] [CrossRef] - Schaller, V.; Kräling, U.; Rusu, C.; Petersson, K.; Wipenmyr, J.; Krozer, A.; Wahnström, G.; Sanz-Velasco, A.; Enoksson, P.; Johansson, C. Motion of nanometer sized magnetic particles in a magnetic field gradient. J. Appl. Phys.
**2008**, 104, 093918. [Google Scholar] [CrossRef] - Sinha, A.; Ganguly, R.; Puri, I.K. Magnetic separation from superparamagnetic particle suspensions. J. Magn. Magn. Mater.
**2009**, 321, 2251–2256. [Google Scholar] [CrossRef] - Orenstein, W.A.; Bernier, R.H.; Dondero, T.J.; Hinman, A.R.; Marks, J.S.; Bart, K.J.; Sirotkin, B. Field evaluation of vaccine effi-cacy. Bull. World Health Organ.
**1985**, 63, 1055–1068. [Google Scholar] - Leong, S.S.; Ahmad, Z.; Low, S.C.; Camacho, J.; Faraudo, J.; Lim, J. Unified View of Magnetic Nanoparticle Separation under Magnetophoresis. Langmuir
**2020**, 36, 8033–8055. [Google Scholar] [CrossRef] [PubMed] - Leong, S.S.; Ahmad, Z.; Lim, J. Magnetophoresis of superparamagnetic nanoparticles at low field gradient: Hydrodynamic effect. Soft Matter
**2015**, 11, 6968–6980. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Concept of (continuous) magnetic separator using a tube with a current-carrying wire in its centre and (

**b**) front view with nomenclature.

**Figure 2.**Alternative separator designs with external electromagnets (arrows point in the direction of the current) using (

**a**) spaced single coils (radial gradient between single coils) and (

**b**) horizontal coils in the Golay arrangement (vertical gradient). All field lines shown (dashed lines) are for illustration.

**Figure 3.**(

**a**) Annular velocity profile ($\dot{V}=0.5\text{}\mathrm{mL}/\mathrm{min}$, ${r}_{tube}=500\text{}\mu m$, ${r}_{wire}=350\text{}\mu m$ (${r}_{wire}$/${r}_{tube}$ = 0.7) and concept of particle-tracking algorithm. (

**b**) 20 trajectories of 250 nm IONPs in the magnetic separator of the same dimensions at a flow rate of 0.15 mL/min in the absence of a magnetic field.

**Figure 4.**(

**a**) 20 particle trajectories of 250 nm IONPs in the magnetic separator operated at 0.15 mL/min, ${r}_{tube}$ = 500 µm and ${r}_{wire}$/${r}_{tube}$ = 0.7. Separator efficiency for IONP sizes as indicated at the insets for varying (

**b**) flow rates (${r}_{tube}$ = 500 µm, ${r}_{wire}$/${r}_{tube}$ = 0.7), (

**c**) tube radii (flow rate = 0.5 mL/min, ${r}_{wire}$ = 350 µm, ${r}_{tube}$ = 390–3500 µm, ${r}_{wire}$/${r}_{tube}$ = 0.1–0.9) and (

**d**) wire-to-tube radius ratios (flow rate = 0.5 mL/min, ${r}_{tube}$ = 500 µm, ${r}_{wire}$ = 50–450 µm, ${r}_{wire}$/${r}_{tube}$ = 0.1–0.9).

**Figure 5.**(

**a**) 20 particle trajectories (out of 10,000) for the optimum separation condition for 250 nm MNPs yielding > 80% separator efficiency at 0.09 mL/min (${r}_{tube}$ = 555 µm, ${r}_{wire}$ = 500 µm, ${r}_{wire}$/${r}_{tube}$ = 0.9). Separator efficiency of 250 nm IONPs for different separator dimensions operated at (

**b**) 0.01 mL/min, (

**c**) 0.1 mL/min, and (

**d**) 1 mL/min.

**Figure 6.**(

**a**) 20 particle trajectories (out of 10,000) for the optimum separation condition for 500 nm MNPs yielding >80% separator efficiency at 0.37 mL/min (${r}_{tube}$ = 555.6 µm, ${r}_{wire}$ = 500 µm, ${r}_{wire}$/${r}_{tube}$ = 0.9). Separator efficiency of 500 nm IONPs for different separator dimensions operated at (

**b**) 0.01 mL/min, (

**c**) 0.1 mL/min, and (

**d**) 1 mL/min.

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**MDPI and ACS Style**

Wen, Y.; Jiang, D.; Gavriilidis, A.; Besenhard, M.O. In-Silico Conceptualisation of Continuous Millifluidic Separators for Magnetic Nanoparticles. *Materials* **2021**, *14*, 6635.
https://doi.org/10.3390/ma14216635

**AMA Style**

Wen Y, Jiang D, Gavriilidis A, Besenhard MO. In-Silico Conceptualisation of Continuous Millifluidic Separators for Magnetic Nanoparticles. *Materials*. 2021; 14(21):6635.
https://doi.org/10.3390/ma14216635

**Chicago/Turabian Style**

Wen, Yanzhe, Dai Jiang, Asterios Gavriilidis, and Maximilian O. Besenhard. 2021. "In-Silico Conceptualisation of Continuous Millifluidic Separators for Magnetic Nanoparticles" *Materials* 14, no. 21: 6635.
https://doi.org/10.3390/ma14216635