# Experimental Study of the Deposition of Magnetic Particles on the Walls of Microchannels

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{5}particles/µL and 1.45 × 10

^{4}particles/µL, for the smaller and larger particles, respectively. For both magnetic particle sizes, the corresponding volume fraction in the solutions used for the experiments carried out (with only magnetic particles) is 6.33 × 10

^{−4}.

^{5}particles/µL) and four different concentrations of non-magnetic particles with values 1.74 × 10

^{3}particles/µL, 3.47 × 10

^{3}particles/µL, 6.95 × 10

^{3}particles/µL, and 1.04 × 10

^{4}particles/µL. In these solutions, the volume fraction of magnetic particles was kept constant and the corresponding volume fractions of non-magnetic particles were 9.10 × 10

^{−4}, 1.82 × 10

^{−3}, 3.64 × 10

^{−3}, and 5.46 × 10

^{−3}.

_{r}) of 1.32–1.37 T and 1.29–1.32 T, respectively [36]. The magnet was installed, as shown in Figure 1, near transparent square capillaries manufactured in borosilicate glass (Vitrocom Inc., Mountain Lakes, NJ, USA [37]). Two different sizes of microchannels were used with hydraulic diameters of 300 µm and 600 µm. The length of the microchannels was 100 mm.

## 3. Results

- PA or PE indicate the parallel or the perpendicular direction of the magnetization vector, respectively (see Figure 1).
- M3 or M5 indicate the size of the magnet (3 or 5 mm).
- P1 or P4, the size of the magnetic particles.
- C3 or C6 the hydraulic diameter of the microchannel (300 or 600 μm).
- QL, QM, and QH, the low, medium or high flow rate.
- PNM indicates the parts per ten thousands volume fraction of non-magnetic particles in the solution (09, 18, 36, or 55).

#### 3.1. Effect the Orientation of the Magnetization Vector and of the Flow Rate

^{4}, 1.2 × 10

^{4}, and 1.7 × 10

^{4}, for the low, medium, and high flow rates, respectively. Note that the dimensionless time used in the plots is defined using the average fluid velocity. When the magnetization vector is parallel to the microchannel wall, the particles start to accumulate near the center of the magnet and then around the leading and trailing edges for all flow rates, Figure 5a–c shows this. For this orientation of the magnetization vector and at the lowest flow rate, the thicknesses of the deposits near the leading edge (red curve in Figure 5a) and near the center (blue curve in Figure 5a) of the magnet decrease for $t{}^{*}$ > 1.5 × 10

^{4}indicating that the deposited particles are dragged by the action of the fluid velocity toward the trailing edge (green curve in Figure 5a) of the magnet because the relatively small magnetization force. This phenomenon is not observed for the perpendicular orientation of the magnetization vector that generates a much stronger magnetic force on the particles.

#### 3.2. Effect of the Size of the Magnet

#### 3.3. Effect of the Diameter of the Particles

#### 3.3.1. Magnet Size of 5 mm

#### 3.3.2. Magnet Size of 3 mm

#### 3.4. Effect of the Hydraulic Diameter of the Microchannel

#### 3.5. Effect of the Concentration of Non-Magnetic Particles on the Deposition of the Magnetic Beads

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

g | [m/s^{2}] | Gravity acceleration |

L | [m] | Length |

S | [m] | Width of the accumulation |

W | [m] | Width of the channel |

A | [m^{2}] | Area of the accumulation |

M_{s} | [A/m] | Magnetization of the magnet |

$\overline{V}$ | [m/s] | Mean Velocity |

x | [m] | Cartesian axis direction |

y | [m] | Cartesian axis direction |

z | [m] | Cartesian axis direction |

$\Delta t$ | [s] | Time between images |

nframes | [-] | Number of images |

Non Dimensional Numbers | ||

${t}^{*}$ | [-] | ${t}^{*}=\left(\Delta t\cdot nframes\cdot \overline{V}\right)/W$ |

${A}^{*}$ | [-] | ${A}^{*}=A/\left({L}_{m}W\right)$ |

${S}^{*}$ | [-] | ${S}^{*}=S/W$ |

Subscripts | ||

m | Magnet, magnetic |

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**Figure 1.**Physical model showing the position of the magnet and the orientation of the magnetization vector with respect to the flow direction. (

**a**) Parallel (PA), (

**b**) perpendicular (PE). Gravity acts along the negative z direction.

**Figure 2.**Experimental setup. (

**a**) General view. The red rectangle indicates the zone shown in (

**b**) with the cubical magnet and the microchannel.

**Figure 3.**Example of a recorded and a processed image (

**a**) before mask subtracting with the particle deposit in black and (

**b**) after mask subtracting with the particle deposit in white.

**Figure 4.**Snapshots of the deposition for experiment PA-M5-P1-C3-QL (

**a**–

**d**) and for PE-M5-P1-C3-QL (

**e**–

**h**). The non-dimensional time are for (

**a**,

**e**) $t{}^{*}$ = 4.47 × 10

^{1}, for (

**b**,

**f**) $t{}^{*}$ = 4.47 × 10

^{3}, for (

**c**,

**g**) $t{}^{*}$ = 8.95 × 10

^{3}and for (

**d**,

**h**) ${t}^{*}$ = 1.79 × 10

^{4}. The lines in (

**a**,

**e**) indicate the positions where the accumulation is monitored, ${{z}^{\prime}}_{\frac{1}{4}}$ (red line), ${{z}^{\prime}}_{0}$ (blue line) and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green line).

**Figure 5.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green). Low flow rate, (

**a**) PA-M5-P1-C3-QL and (

**d**) PE-M5-P1-C3-QL. Medium flow rate (

**b**) PA-M5-P1-C3-QM and (

**e**) PE-M5-P1-C3-QM. High flow rate, (

**c**) PA-M5-P1-C3-QH and (

**f**) PE-M5-P1-C3-QH. Vertical lines indicate times ${t}^{*}$ = 4.47 × 10

^{1}, ${t}^{*}$ = 4.47 × 10

^{3}, ${t}^{*}$ = 8.95 × 10

^{3}, and ${t}^{*}$ = 1.79 × 10

^{4}which correspond to the snapshots of Figure 4.

**Figure 6.**Snapshots of the deposition for experiments PE-M5-P1-C3-QM (

**a**–

**d**) and PE-M3-P1-C3-QM (

**e**–

**h**). The non-dimensional time are for (

**a**,

**e**) $t{}^{*}$ = 4.47 × 10

^{1}, for (

**b**,

**f**) $t{}^{*}$ = 4.47 × 10

^{3}, for (

**c**,

**g**) $t{}^{*}$ = 8.95 × 10

^{3}, and for (

**d**,

**h**) $t{}^{*}$ = 1.79 × 10

^{4}.

**Figure 7.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green). Medium flow rate (

**a**) PE-M5-P1-C3-QM and (

**b**) PE-M3-P1-C3-QM. Vertical lines corresponds to times $t{}^{*}$ = 4.47 × 10

^{1}, ${t}^{*}$ = 4.47 × 10

^{3}, $t{}^{*}$ = 8.95 × 10

^{3}, and $t{}^{*}$ = 1.79 × 10

^{4}which correspond to the snapshots of Figure 6.

**Figure 8.**Snapshots of the deposition for the experiment PE-M5-P4-C3-QM at times, (

**a**) $t{}^{*}$ = 4.47 × 10

^{1}, (

**b**) $t{}^{*}$ = 4.47 × 10

^{3}, (

**c**) $t{}^{*}$ = 8.95 × 10

^{3}, (

**d**) $t{}^{*}$ = 1.79 × 10

^{4}.

**Figure 9.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue) and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green) for PE-M5-P4-C3, (

**a**) low flow rate, (

**b**) medium flow rate, and (

**c**) high flow rate.

**Figure 10.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green) for (

**a**) PE-M5-P1-C3-QM and (

**b**) PE-M5-P4-C3-QM.

**Figure 11.**Snapshots of the deposition Particle accumulations for the experiments PE-M3-P1-C3-QM (

**a**–

**d**) and PE-M3-P4-C3-QM (

**e**–

**h**). The non-dimensional time are for (

**a**,

**e**) $t{}^{*}$ = 4.47 × 10

^{1}, for (

**b**,

**f**) $t{}^{*}$ = 4.47 × 10

^{3}, for (

**c**,

**g**) $t{}^{*}$ = 8.95 × 10

^{3}and for (

**d**,

**h**) ${t}^{*}$ = 1.79 × 10

^{4}.

**Figure 12.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green) for (

**a**) PE-M3-P1-C3-QM and (

**b**) PE-M3-P4-C3-QM.

**Figure 13.**Snapshots of the depositions for PE-M5-P4-C3-QL and PE-M5-P4-C6-QL at different times. For real time $t=$ 200 s ((

**a**)$t{}^{*}$ = 4.47 × 10

^{3}and (

**e**) $t{}^{*}$ = 4.88 × 10

^{2}), $t=$400 s ((

**b**) $t{}^{*}$ = 8.95 × 10

^{3}, and (

**f**) $t{}^{*}$ = 9.76 × 10

^{2}), $t=$800 s ((

**c**) ${t}^{*}$ = 1.79 × 10

^{4}and (

**g**) $t{}^{*}$ = 1.95 × 10

^{3}) and $t=$1000 s ((

**d**) $t{}^{*}$ = 2.24 × 10

^{4}and (

**h**) $t{}^{*}$ = 2.44 × 10

^{3}).

**Figure 14.**Time-evolutions of the non-dimensional thickness of the magnetic particle deposits at positions ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green) for PE-M5-P4-C6-QL.

**Figure 15.**Time-evolutions of the non-dimensional area of the deposited magnetic particles according to Set 6 of Table 1.

**Figure 16.**Length accumulation at ${{z}^{\prime}}_{\frac{1}{4}}$ (red),${{z}^{\prime}}_{0}$ (blue), and ${{z}^{\prime}}_{-\frac{1}{4}}$ (green) for experiments (

**a**) PE-M5-P1-C3-QM-PM; (

**b**) PE-M5-P1-C3-QM-PNM09; (

**c**) PE-M5-P1-C3-QM-PNM18; (

**d**) PE-M5-P1-C3-QM-PNM36; (

**e**) PE-M5-P1-C3-QM-PNM55.

**Figure 17.**Accumulation of magnetic particles comparison at non-dimensional times $t{}^{*}$ = 1, 3000, 6000, 9000, 12,000 for experiments (

**a**) PE-M5-P1-C3-QM-PM; (

**b**) PE-M5-P1-C3-QM-PNM09; (

**c**) PE-M5-P1-C3-QM-PNM18; (

**d**) PE-M5-P1-C3-QM-PNM36; (

**e**) PE-M5-P1-C3-QM-PNM55.

**Table 1.**Experimental conditions. Magnetization vector, MS: magnet width, CD: capillary diameter, PD: magnetic particle diameter, and FR: flow rate.

Set | Experiment Label | Magnetization Vector Direction | MS (mm) | CD (μm) | PD (μm) | FR (mL/min) |
---|---|---|---|---|---|---|

1 | PA-M5-P1-C3-QL | $\parallel $ | 5 | 300 | 1.14 | 0.03 |

PA-M5-P1-C3-QM | 0.06 | |||||

PA-M5-P1-C3-QH | 0.12 | |||||

2 | PE-M5-P1-C3-QL | $\perp $ | 5 | 300 | 1.14 | 0.03 |

PE-M5-P1-C3-QM | 0.06 | |||||

PE-M5-P1-C3-QH | 0.12 | |||||

3 | PE-M3-P1-C3-QM | $\perp $ | 3 | 300 | 1.14 | 0.06 |

PE-M3-P4-C3-QM | 4.37 | |||||

4 | PE-M5-P4-C3-QL | $\perp $ | 5 | 300 | 4.37 | 0.03 |

PE-M5-P4-C3-QM | 0.06 | |||||

PE-M5-P4-C3-QH | 0.12 | |||||

5 | PE-M5-P4-C6-QL | $\perp $ | 5 | 600 | 4.37 | 0.03 |

6 | PE-M5-P1-C3-QM-PM | $\perp $ | 5 | 300 | 1.14 | 0.06 |

PE-M5-P1-C3-QM-PNM09 | ||||||

PE-M5-P1-C3-QM-PNM18 | ||||||

PE-M5-P1-C3-QM-PNM36 | ||||||

PE-M5-P1-C3-QM-PNM55 |

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**MDPI and ACS Style**

Varela, S.; Rivas, A.; Vernet, A.; Pallarès, J.
Experimental Study of the Deposition of Magnetic Particles on the Walls of Microchannels. *Micromachines* **2021**, *12*, 712.
https://doi.org/10.3390/mi12060712

**AMA Style**

Varela S, Rivas A, Vernet A, Pallarès J.
Experimental Study of the Deposition of Magnetic Particles on the Walls of Microchannels. *Micromachines*. 2021; 12(6):712.
https://doi.org/10.3390/mi12060712

**Chicago/Turabian Style**

Varela, Sylvana, Antonio Rivas, Anton Vernet, and Jordi Pallarès.
2021. "Experimental Study of the Deposition of Magnetic Particles on the Walls of Microchannels" *Micromachines* 12, no. 6: 712.
https://doi.org/10.3390/mi12060712