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

Magnetic levitation (MagLev) is a promising density-based analytical technique with numerous applications. Several MagLev structures with different levels of sensitivity and range have been studied. However, these MagLev structures can seldom satisfy the different performance requirements simultaneously, such as high sensitivity, wide measurement range, and easy operation, which have prevented them from being widely used. In this work, a tunable MagLev system was developed. It is confirmed by numerical simulation and experiments that this system possesses a high resolution down to 10−7 g/cm3 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.


Introduction
Density is a fundamental property of materials that is critical in various physical and chemical processes. Its significance in applications such as separation [1][2][3], chemical reactions [4][5][6], quality control [7,8], and blood tests [9] has made it an essential topic of interest in materials science and engineering [10][11][12][13][14][15]. Despite the several techniques proposed to determine material density, they suffer from significant drawbacks. Certain well-known density measurement methods, such as hydrometers, density-gradient columns, pycnometers, oscillating-tube densitometers, suspended microchannel resonators, and hydrostatic weighing balances, require large sample volumes and have limited precision, making them incompatible with certain sample types such as gels, pastes, and gums [16][17][18][19][20][21][22]. Additionally, the required equipment for some of these techniques is bulky, expensive, and challenging to use in certain environments. Therefore, the need for a portable, convenient, low-cost, and accurate density measurement method that can be used in resource-limited environments or field settings is still apparent.

Design of M-MagLev System
The M-MagLev system is designed based on the horizontally aligned MagLev structure to take advantage of high-resolution and straightforward operation. Here, an upright cylindrical container and several indistinguishable magnets make up the tunable system. As shown in Figure 1a,b, the paramagnetic medium and test samples can be conveniently added to the nonmagnetic container from the top opening. The detachable magnets surround the container symmetrically and are fixed by the magnet housing against the interaction between the magnets since the like-poles of the magnets face each other.

Theoretical Method
The samples levitated in a paramagnetic medium are subjected to the magnetic force Fmag, the downward gravity force FG, and the upward buoyancy force Ff in the vertical direction. The magnetic force, ⃗ , acting on the sample is given as: where is the volume of the sample; , the magnetic permeability of vacuum; ⃗ is the

Theoretical Method
The samples levitated in a paramagnetic medium are subjected to the magnetic force F mag , the downward gravity force F G , and the upward buoyancy force F f in the vertical direction. The magnetic force, → F mag , acting on the sample is given as: where V is the volume of the sample; µ 0 , the magnetic permeability of vacuum; → B is the magnetic flux density; ∇ represents the gradient operator, i.e., ∇B(x, y, z) = ∂B ∂x + ∂B ∂y + ∂B ∂z ; and ∆χ is the magnetic susceptibility difference between the sample and paramagnetic medium. Here, we defined χ s and χ m as the magnetic susceptibility of the levitating sample and medium, respectively, so we have ∆χ = χ s − χ m . We set the centerline of the container as the Z-axis in the Cartesian coordinate system. As indicated by our previous work, the direction of the magnetic flux density at the centerline of the container is along the Z-axis, which is the best levitation position for the test samples [29]. According to the equilibrium conditions of force on the sample, we have: where F f and F G , respectively, are buoyancy force and gravity force on the sample; ρ s and ρ m the density of the sample and medium, respectively; g gravitational acceleration; B z the gradient of magnetic flux density B along the centerline. According to the equilibrium equation (Equation (2)), the density of the sample can be expressed as: Under the effect of surrounding magnets, the magnetic flux density B along the centerline of the container will achieve two extremums: B min and B max . Therefore, the maximum density ρ smax and the minimum density ρ smin that can be measured by the M-MagLev system could be determined as: Here, the difference between the maximum density ρ smax and the minimum density ρ smin is defined as ∆ρ, which is the range of density measurements for the M-MagLev system with a certain number of magnets. After substituting ρ smax and ρ smin , we have the expression of ∆ρ as: For a given concentration paramagnetic solution, the value of ∆χ(µ 0 g) −1 remains constant. We introduce another important characteristic parameter, sensitivity (S), for the M-MagLev system to quantitatively indicate the resolution. The sensitivity is defined as the levitation height variation per unit density difference of sample with a unit of mm (g/cm 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 ∆ρ 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.

Materials
We use the NdFeB bar-magnets (10 mm × 10 mm × 100 mm, strength N35) to structure the M-MagLev system in our validation experiments. They were purchased from Genchang Magnet Material Co., Ltd., Shanghai, China. We use the Gauss meter (HT108, Hengtong magnetoelectric technology Co., Ltd., Shanghai, China) to measure the magnets' surface magnetic flux density (Bs). The magnet housings of M-MagLev system (the white polymer parts shown in Figure 1a) were fabricated by 3D printing (Shanghai Yinmeng Intelligent Technology Co., Ltd, Shanghai, China) with Nylon.
We used manganese chloride (MnCl 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 ρ m could be adjusted according to the practical requirement, which determines the density benchmark of measuring objects.
Polystyrene beads were used as the test samples, which were purchased from Golden Ball Industry Co., Ltd., Ningbo, China. They were all about the same size (about 6 mm in diameter) and weight (about 0.11 g for each bead). We used a U-tube oscillating densitometer (DMA 35, Anton-Paar, Shanghai, China) to measure the density of the paramagnetic medium. Deionized water (18.4 MΩ/cm) used for all experiments was obtained from Milli-Q system (Millipore, Bedford, MA, USA). A digital single-lens reflex camera was used to record the levitation height.

Simulation Results
The M-MagLev system with six NdFeB bar-magnets (10 mm × 10 mm × 100 mm) was chosen first for simulation. Six bar-magnets were arranged around the container, maintaining the same distance from the center of the container, with a radius (R, the distance between the magnets center and container center) of 42 mm. In order to simplify the analysis, we have imposed the restriction that the system exhibits only the simplest form of rotational symmetry, namely a C2 rotation symmetry, which implies that the system remains invariant under a rotation of 180 degrees around the center axis of the container. This assumption reduces the complexity of the problem and allows us to focus on the essential features of the system's behavior. Furthermore, this symmetry allows the magnetic field distribution in the container to exhibit only one extremum point. Additional discussion on the distribution of the magnets can be found in Section 4. The magnetic flux density distribution of the M-MagLev system with six magnets is presented in Figure 2. To clearly illustrate the distribution of the magnetic flux density at different heights, we extracted three slices of the magnetic flux density distribution along the vertical direction, as shown in Figure 2a. It can be observed that the magnetic flux emanates from all magnets, and as it propagates, the magnetic flux density progressively declines. Inside the container, the magnetic fluxes mutually superimpose. At the central axis position of the container, due to the symmetrical distribution of the surrounding magnets, the in-plane magnetic flux components cancel out, forming a minimum point of magnetic flux density in the container region. Only the axial component along the container remains. Figure 2b shows the axial profile of the magnetic flux density norm in the container, where the red arrows indicate the direction of magnetic flux density. The distribution of magnetic flux density was found to be symmetric about the central axis, providing favorable magnetic field distribution conditions for accurately measuring the density of samples.
magnetic flux components cancel out, forming a minimum point of magnetic flux density in the container region. Only the axial component along the container remains. Figure 2b shows the axial profile of the magnetic flux density norm in the container, where the red arrows indicate the direction of magnetic flux density. The distribution of magnetic flux density was found to be symmetric about the central axis, providing favorable magnetic field distribution conditions for accurately measuring the density of samples. According to the distribution of magnetic flux density, the central axis of the container is the most ideal measurement position. The distribution profiles of magnetic flux density along the centerline for a different number of magnets are summarized in Figure  3a. Except for the area hidden by the magnet housing (about 10 mm in thickness), the magnetic flux density along the Z-axis (Bz) between the two housings changed almost linearly with the height of the container. Furthermore, Bz was entirely antisymmetric about the middle point of the centerline where Bz = 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 (Bmax) locates at the top position of the measurement area. In contrast, the minimum magnetic flux density (Bmin) locates at the bottom position. They have the same size but opposite directions, i.e., = − . 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 ( ) and two extrema increased gradually. The changes of and Bmax 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 ∆ and sensitivity S for the M-MagLev system can be simplified as: According to the distribution of magnetic flux density, the central axis of the container is the most ideal measurement position. The distribution profiles of magnetic flux density along the centerline for a different number of magnets are summarized in Figure 3a. Except for the area hidden by the magnet housing (about 10 mm in thickness), the magnetic flux density along the Z-axis (B 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 ) and two extrema increased gradually. The changes of B z 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 ∆ρ and sensitivity S for the M-MagLev system can be simplified as: For the M-MagLev system with a different number of magnets, we calculated and summarized the range ∆ρ and sensitivity S with Equations (7) and (8) in Figure 3b. The increase in the number of magnets can greatly extend the measurement range of the M-MagLev, from 7.52 × 10 −6 g/cm 3 for two magnets to 2.66 × 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 × 10 7 mm/(g/cm 3 ) to 3.0 × 10 5 mm/(g/cm 3 ) with the increasing number of magnets. By comparing with the typical horizontal highresolution MagLev, this tunable system provides not only a higher resolution down to 3.3 × 10 −6 ∼ 1 × 10 −7 (g/cm 3 )/mm, but a wider measurement range.
MagLev, from 7.52 × 10 g/cm for two magnets to 2.66 × 10 g/cm for twelve mag-nets. 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 × 10 mm/(g/cm 3 ) to 3.0 × 10 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 × 10 ~1 × 10 (g/cm 3 )/mm, but a wider measurement range.

Experimental Results
To verify the simulation results, we conducted proof-of-concept experiments of the M-MagLev system with a different number of magnets by levitating polystyrene beads in a container filled with MnCl2 aqueous solution (0.300 mol/L MnCl2·4H2O). The density of this paramagnetic medium 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

Experimental Results
To verify the simulation results, we conducted proof-of-concept experiments of the M-MagLev system with a different number of magnets by levitating polystyrene beads in a container filled with MnCl 2 aqueous solution (0.300 mol/L MnCl 2 ·4H 2 O). The density of this paramagnetic medium ρ 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. 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 theoretica results are well-matched with the experimental results and provide reliable conclusion for the experimental results. Consequently, the tunable M-MagLev system is straightfor ward to operate and has a reasonably strong resolution to screen various samples, such a cells and drugs.

The Linear Distribution of Magnetic Flux Density
After systematic simulations, we found that the magnet's length and the surrounding radius of magnets greatly influence the linear distribution of magnetic flux density along the centerline of the container (Bz), which directly affects the accuracy and convenience o operation. Since the number of magnets has little effect on the distribution of Bz ( 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 Bz with the sur rounding radius R is plotted in Figure 5a. Naturally, a gradual decline in maximum mag netic flux density Bzmax is presented in this figure as the magnets get farther and farthe apart. However, the change of Bz with height Z shows a certain degree of nonlinear distri bution 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 i to 1, the better the linearity of Bz. As the insert in Figure 5a shows, with the increase o 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 distribu tion of Bz the best linearity. Subsequently, we summarized the optimal radius for differen magnet lengths in Figure 5b. A proportional relationship is presented between the opti mized surrounding radius and magnet length. By fitting, it can be found that the linearity of Bz was best when the ratio of surrounding radius to magnet length was about 0.4231

The Linear Distribution of Magnetic Flux Density
After systematic simulations, we found that the magnet's length and the surrounding radius of magnets greatly influence the linear distribution of magnetic flux density along the centerline of the container (B 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. Accordingly, by choosing the appropriate surrounding diameter and length of magnets, better accuracy and convenience could be achieved for the M-MagLev system.

Measurement Stability
To ensure that the direction of magnetic flux density on the container's centerline is along the Z-axis, the magnets must be arranged around the container symmetrically. This part will discuss the influence of different symmetries on the M-MagLev system measurement results. Here, we still take the M-MagLev system with six magnets as an example. There are three kinds of axisymmetric distribution for the six magnets: C2, C3, and C6. Here, Cn 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 C3 and C6 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 C2 symmetry shows only one stable point. It lies on the container's centerline, which is what operators expect for the practical measurement.

Measurement Stability
To ensure that the direction of magnetic flux density on the container's centerline is along the Z-axis, the magnets must be arranged around the container symmetrically. This part will discuss the influence of different symmetries on the M-MagLev system measurement results. Here, we still take the M-MagLev system with six magnets as an example. There are three kinds of axisymmetric distribution for the six magnets: C 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. Accordingly, by choosing the appropriate surrounding diameter and length of magnets, better accuracy and convenience could be achieved for the M-MagLev system.

Measurement Stability
To ensure that the direction of magnetic flux density on the container's centerline is along the Z-axis, the magnets must be arranged around the container symmetrically. This part will discuss the influence of different symmetries on the M-MagLev system measurement results. Here, we still take the M-MagLev system with six magnets as an example. There are three kinds of axisymmetric distribution for the six magnets: C2, C3, and C6. Here, Cn 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 C3 and C6 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 C2 symmetry shows only one stable point. It lies on the container's centerline, which is what operators expect for the practical measurement.  The corresponding experiments were also performed to verify the results. As indicated in Figure 7, the C 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 Sensors 2023, 23, 3955 9 of 11 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. The corresponding experiments were also performed to verify the results. As indicated in Figure 7, the C2 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 C6 symmetrical distribution of magnets. This result is consistent with our numerical simulation and confirms that the C2 symmetrical distribution of magnets is beneficial to improve the measurement stability and accuracy of the M-MagLev system.

Conclusions
In summary, a novel tunable high-resolution MagLev system with multiple magnets was developed. By combining theoretical simulation and experiments, the mechanism and performance of this system were well demonstrated and studied. Compared with other MagLev structures, this M-MagLev possesses the highest resolution-down to 1 × 10 −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 C2 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.

Conclusions
In summary, a novel tunable high-resolution MagLev system with multiple magnets was developed. By combining theoretical simulation and experiments, the mechanism and performance of this system were well demonstrated and studied. Compared with other MagLev structures, this M-MagLev possesses the highest resolution-down to 1 × 10 −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.