# A Magnetic Levitation System for Range/Sensitivity-Tunable Measurement of Density

^{1}

^{2}

^{*}

## Abstract

**:**

^{−7}g/cm

^{3}or even higher compared to the existing systems. Meanwhile, the resolution and range of this tunable system can be adjusted to meet different requirements of measurement. More importantly, this system can be operated simply and conveniently. This bundle of characteristics demonstrates that the novel tunable MagLev system could be handily applied in various density-based analyses on demand, which would greatly expand the ability of MagLev technology.

## 1. Introduction

_{2}) is used as a density indicator after loading the test samples. Two indistinguishable NdFeB magnets are separately placed on the top and bottom of the container (the like poles facing each other) [27]. With this vertically aligned configuration, the sample’s density can be measured in a relatively wide range (0.8–3 g/cm

^{3}) with a density resolution in the order of 10

^{−2}to 10

^{−4}g/cm

^{3}per 1 mm for a given concentration of the paramagnetic medium. Subsequently, it is found that by tilting the vertically aligned MagLev structure, the resolution gradually increases with the increase of the tilt angle [29]. At the same time, with the support of the container wall, the maximum measurable sample density of this tilted MagLev could reach 23 g/cm

^{3}. For the limit case, the MagLev system can be tilted to a horizontally aligned configuration, which has the highest resolution down to the order of 10

^{−6}g/cm

^{3}[29]. Nevertheless, the measurable range is only in the order of 10

^{−4}to 10

^{−5}g/cm

^{3}since the supporting force from the container wall can no longer affect the measurement process. This tilted configuration greatly improves the measurement ability of MagLev but also brings great difficulty in operation. Besides this, the ring magnets were also used to construct the axial aligned MagLev structure, which is more convenient for adding samples and paramagnetic medium to the container [31,32]. However, the density resolution of this axial MagLev structure is only in the order of 10

^{−2}to 10

^{−4}g/cm

^{3}, and the range is about 0–4 g/cm

^{3}. Although different kinds of MagLev systems can be applied in different situations, a high-resolution MagLev that can be operated conveniently according to the diverse requirements is more expected in practical applications.

^{−5}to 10

^{−7}g/cm

^{3}, and the range can be changed from 10

^{−3}to 10

^{−5}g/cm

^{3}. At the same time, this system also takes advantage of axial MagLev, which can be operated simply and conveniently. These properties endow the multi-magnet MagLev (M-MagLev) system with great potential to be applied in various density-based analyses on demand.

## 2. Materials and Methods

#### 2.1. Design of M-MagLev System

#### 2.2. Theoretical Method

_{mag}, the downward gravity force F

_{G}, and the upward buoyancy force F

_{f}in the vertical direction. The magnetic force, ${\overrightarrow{F}}_{mag}$, acting on the sample is given as:

^{3})

^{−1}. The larger S, the higher the resolution. So far, two key parameters to evaluate the performance of M-MagLev have been introduced here: measurement range $\Delta \rho $ and sensitivity S. To obtain the best performance, the numerical simulation for the tunable M-MagLev system is first conducted. Here, we use the COMSOL Multiphysics software to carry out numerical simulations and analyses based on theoretical equations.

#### 2.3. Materials

_{2}) aqueous solution prepared by MnCl

_{2}·4H

_{2}O and deionized water for paramagnetic medium. MnCl

_{2}·4H

_{2}O was purchased from Adamas-Beta Co., Shanghai, China. In some cases, NaCl

_{2}could also be used to fine-tune the density of the solution. Therefore, the paramagnetic solution density ${\rho}_{m}$ could be adjusted according to the practical requirement, which determines the density benchmark of measuring objects.

## 3. Results

#### 3.1. Simulation Results

_{z}) between the two housings changed almost linearly with the height of the container. Furthermore, B

_{z}was entirely antisymmetric about the middle point of the centerline where B

_{z}= 0 mT. Therefore, the area between the two housings was set as the density measurement area, and the distance between the two housings was defined as ∆Z. The maximum magnetic flux density (B

_{max}) locates at the top position of the measurement area. In contrast, the minimum magnetic flux density (B

_{min}) locates at the bottom position. They have the same size but opposite directions, i.e., ${B}_{max}=-{B}_{min}$. As we can see from Figure 3a, increasing the number of magnets from two to twelve, the gradient of magnetic flux density along the Z-axis (${B}_{z}^{\prime}$) and two extrema increased gradually. The changes of ${B}_{z}^{\prime}$ and B

_{max}with the number of magnets are shown in the inset of Figure 3a. Both presented a linear change with the number of magnets. Accordingly, the measurement range $\Delta \rho $ and sensitivity S for the M-MagLev system can be simplified as:

^{3}for two magnets to $2.66\times {10}^{-4}$ g/cm

^{3}for twelve magnets. In other words, the measurement range could be adjusted according to the practical needs just by changing the number of magnets in the M-MagLev system. On the other hand, the sensitivity decreases from $1.0\times {10}^{7}$ mm/(g/cm

^{3}) to $3.0\times {10}^{5}$ mm/(g/cm

^{3}) with the increasing number of magnets. By comparing with the typical horizontal high-resolution MagLev, this tunable system provides not only a higher resolution down to $3.3\times {10}^{-6}~1\times {10}^{-7}$ (g/cm

^{3})/mm, but a wider measurement range.

#### 3.2. Experimental Results

_{2}aqueous solution (0.300 mol/L MnCl

_{2}·4H

_{2}O). The density of this paramagnetic medium ${\rho}_{m}$ can be calculated to be 1.0289 g/cm

^{3}, which is confirmed by the measurement of the U-tube oscillating densitometer. We selected several polystyrene beads for the test, whose densities were about 1.02 to 1.03 g/cm

^{3}, with slight differences in each other. If the bead density was beyond the range of the system, it would float on the top or settle down at the bottom of the container. For the levitated beads, the levitation height indicates the density of the bead: the higher position, the smaller density. As we can see in the left part of Figure 4, only one bead can levitate in the system. For M-MagLev with twelve magnets, this bead levitates near the center of the container. With the decrease in the number of surrounding magnets, the equilibrium position of this bead gradually rises. It indicates that the density of this bead is a little lower than 1.0289 g/cm

^{3}. The right part of Figure 4 presents the relationship between the levitation height and bead density with the number of magnets changing from 4 to 10. The lines are the simulation results, while the points are the experimental results. According to the levitation height, we can infer that the density of this bead is about 1.028891 g/cm

^{3}. Thus, the theoretical results are well-matched with the experimental results and provide reliable conclusions for the experimental results. Consequently, the tunable M-MagLev system is straightforward to operate and has a reasonably strong resolution to screen various samples, such as cells and drugs.

## 4. Discussion

#### 4.1. The Linear Distribution of Magnetic Flux Density

_{z}), which directly affects the accuracy and convenience of operation. Since the number of magnets has little effect on the distribution of B

_{z}(Figure 3a), we use the M-MagLev system with two magnets as an example to illustrate the effects. Firstly, we set the size of the magnet as 10 mm × 10 mm × 60 mm, where the length of the magnet is 60 mm. By numerical simulation, the change of distribution of B

_{z}with the surrounding radius R is plotted in Figure 5a. Naturally, a gradual decline in maximum magnetic flux density B

_{zmax}is presented in this figure as the magnets get farther and farther apart. However, the change of B

_{z}with height Z shows a certain degree of nonlinear distribution for some surrounding radii. Here, we use linear equations to fit these curves. The degree of linearity is evaluated by the coefficient of determination—the closer the value is to 1, the better the linearity of B

_{z}. As the insert in Figure 5a shows, with the increase of surrounding radius R from 21 mm to 30 mm, the coefficient of determination rises rapidly to the maximum value 0.9984 at R = 25 mm and then gradually decreases. That is to say, for the magnet 60 mm in length, a surrounding radius of 25 mm would give the distribution of B

_{z}the best linearity. Subsequently, we summarized the optimal radius for different magnet lengths in Figure 5b. A proportional relationship is presented between the optimized surrounding radius and magnet length. By fitting, it can be found that the linearity of B

_{z}was best when the ratio of surrounding radius to magnet length was about 0.4231. Accordingly, by choosing the appropriate surrounding diameter and length of magnets, better accuracy and convenience could be achieved for the M-MagLev system.

#### 4.2. Measurement Stability

_{2}, C

_{3}, and C

_{6}. Here, C

_{n}is the rotational symmetry of order n, which means rotation by an angle of 360°/n does not change the object. By numerical simulations, we found that the symmetry of the magnet’s distribution does not affect the measurement range and resolution of the M-MagLev system but affects the levitation stability of samples. As shown in Figure 6, the distributions of the magnetic flux density at the middle cross-section (XY plane at the midpoint of centerline) for C

_{3}and C

_{6}symmetries present multiple stable points near the centerline (indicated by the red dot circles), which can easily lead to non-univocal results. For comparison, the distribution of magnetic flux density for C

_{2}symmetry shows only one stable point. It lies on the container’s centerline, which is what operators expect for the practical measurement.

_{2}symmetrical distribution of magnets makes all the levitated beads lie on the container’s centerline. In contrast, some beads are levitated off the centerline in the case of the C

_{6}symmetrical distribution of magnets. This result is consistent with our numerical simulation and confirms that the C

_{2}symmetrical distribution of magnets is beneficial to improve the measurement stability and accuracy of the M-MagLev system.

## 5. Conclusions

^{−7}g/cm

^{3}. By changing the number of magnets, the resolution can also be changed from 10

^{−7}to 10

^{−5}g/cm

^{3}, and the corresponding measurement range varies from 10

^{−5}to 10

^{−3}g/cm

^{3}, which provides excellent convenience for connecting with other density measuring instruments with lower accuracy. With the systematic discussions on the basic parameters of the M-MagLev system, it is found that when the magnet length and surrounding radius reach a proportion of 0.4231, the magnetic flux density on the container’s centerline shows the best linearity with height. Meanwhile, the C

_{2}symmetrical distribution of magnets can significantly improve the levitation stability of samples. These distinguished and adjustable advantages make this tunable high-resolution M-MagLev system suitable for a wide variety of density-based applications on demand.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Overview of M-MagLev system. Photographs of (

**a**) an M-MagLev system with 6 magnets and (

**b**) the container in the M-MagLev system. (

**c**) The corresponding simulation model of the M-MagLev system.

**Figure 2.**Simulation of the M-MagLev system with six magnets. (

**a**) Contour plots of magnetic flux density distribution in horizontal slices of system. (

**b**) Contour plots of magnetic flux density distribution in axis section of system. The red arrows indicate the direction of magnetic flux density.

**Figure 3.**(

**a**) Magnetic flux density profile along the centerline for the different numbers of magnets. The two gray areas at the top and bottom represent the parts blocked by the magnet housing. The inset presents the changes of B

_{max}and ${B}_{z}^{\prime}$ with the number of magnets. (

**b**) Changes of range and sensitivity with the different numbers of magnets of the M-MagLev system.

**Figure 4.**Density measurements using the M-MagLev system. Left part is captured photographs of the polystyrene bead levitated in M-MagLev with 4, 6, 8, and 10 magnets, respectively. Right part is the mapping of levitation height and the density of the bead for the corresponding M-MagLev according to the theoretical model. Symbols are the experimental results, while lines are the simulation results.

**Figure 5.**(

**a**) Magnetic flux density profile along the centerline for different surrounding radii R. Points are the simulation results, and the corresponding lines were obtained by linear fitting. The coefficients of determination for the linear fitting are listed in the insert. (

**b**) Relationship between the optimized surrounding radius and magnet length.

**Figure 6.**Magnetic flux density distribution in middle cross-section (x-y plane) of 6-magnet M-MagLev system with C

_{2}-symmetry (

**a**), C

_{3}-symmetry (

**b**), and C

_{6}-symmetry (

**c**). The red dotted circles in (

**b**,

**c**) indicate the multiple stable positions.

**Figure 7.**Captured photographs of polystyrene beads levitated in 6-magnet M-MagLev system with (

**a**) C

_{6}-symmetry distribution of magnets and (

**b**) C

_{2}-symmetry distribution of magnets. The insets are the distribution diagram of six magnets. The red dashed lines represent the different positions of the same ball under different magnetic distributions.

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**MDPI and ACS Style**

Yu, J.; Li, D.; Zhu, C.; Ouyang, Q.; Miao, C.; Yu, H.
A Magnetic Levitation System for Range/Sensitivity-Tunable Measurement of Density. *Sensors* **2023**, *23*, 3955.
https://doi.org/10.3390/s23083955

**AMA Style**

Yu J, Li D, Zhu C, Ouyang Q, Miao C, Yu H.
A Magnetic Levitation System for Range/Sensitivity-Tunable Measurement of Density. *Sensors*. 2023; 23(8):3955.
https://doi.org/10.3390/s23083955

**Chicago/Turabian Style**

Yu, Junhui, Donghai Li, Chengxian Zhu, Qiran Ouyang, Chunyang Miao, and Haidong Yu.
2023. "A Magnetic Levitation System for Range/Sensitivity-Tunable Measurement of Density" *Sensors* 23, no. 8: 3955.
https://doi.org/10.3390/s23083955