Single-Ring Magnetic Levitation Configuration for Object Manipulation

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A Single Ring Magnetic Levitation Configuration for Object Manipulation and Density-based Measurement Chengqian Zhang, Peng Zhao, Fu Gu, Jun Xie, Neng Xia, Yong He, and Jianzhong Fu Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.8b01724 • Publication Date (Web): 27 Jun 2018 Downloaded from http://pubs.acs.org on July 2, 2018

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Analytical Chemistry

A Single Ring Magnetic Levitation Configuration for Object Manipulation and Density-based Measurement Chengqian Zhang,†, ‡ Peng Zhao,*, †, ‡ Fu Gu,†, § Jun Xie,†, ‡ Neng Xia,†, ‡ Yong He,†, ‡ and Jianzhong Fu†, ‡ † The State Key Laboratory of Fluid Power and Mechatronic Systems, College of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China ‡ The Key Laboratory of 3D Printing Process and Equipment of Zhejiang Province, College of Mechanical Engineering, Zhejiang University, Hangzhou 310027, China § Department of Industrial Engineering, Zhejiang University, Hangzhou 310027, China

Corresponding author: Peng Zhao The State Key Lab of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China. Email: [email protected]

ABSTRACT: Magnetic levitation is a recent research hotspot; however, most of the extant configurations use two magnets with like-poles facing each other. This paper proposes a novel magnetic levitation configuration that is based on a single ring magnet, and this configuration opens a wide operational space that enables object manipulation and density-based measurement. We develop a mathematic model to calculate the magnetic field around the magnet and to numerically correlate the levitation height and density of the object. Experimental results prove that this novel configuration can achieve a high accuracy (±0.0005 ~ ±0.0078 g/cm3) in density measurement for small-sized (~ 5 µL) samples. It can manipulate particles, powders, and oil droplets effectively without any direct contact, and it has high sensitivity in the separation of multiple diamagnetic objects with slight differences in densities as well. The accuracy and sensitivity of the proposed configuration are both higher than those of the extant configurations. All of these results are expected to promote deeper study and applications of the magnetic levitation configuration in the field of density-based characterizations and manipulations. Keywords: Magnetic levitation; ring magnet; density; measurement; manipulation; separation 1

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Magnetic levitation is a recent research topic of interest in many scientific fields, from the studies of solids1-5 and liquids6,7 to the manipulation of particles8-12 or even living cells.13-17 Remarkable achievements of magnetic levitation include the lifting of water,6 gold,18 and the most famous—the stable levitation of a living frog.19 However, magnetic levitation usually needs an extraordinary strong (typically > 10 T) and steep magnetic field gradient,6,19 since the magnetic susceptibility of diamagnetic materials is quite small (-10-5).20 Therefore, only a handful of institutes in the world can realize this configuration,21 and it seems to be impossible to apply magnetic levitation in industrial practices. Recently, a magnetic levitation configuration, denoted as “MagLev”, was proposed to investigate the physical properties of diamagnetic materials.22 This configuration can levitate diamagnetic materials in paramagnetic solutions under an applied magnetic field below 0.5 T, which is created by two rectangular permanent magnets (50 mm × 50 mm × 25 mm) at a distance of 45 mm with like-poles facing each other, as shown in Figure S1 in Supporting Information (SI).22 MagLev is of high density resolution,22 and it has been extensively applied in analyzing forensics,23 biomolecules,24 polymers,4,25 metals,26 and foods.27 Compare to standard Maglev configurations, rotated3 and tilted26 configurations improve measurement sensitivity and range, respectively. Our research group enlarged the two magnets’ separation distance to 60 mm, and employed a curving fitting equation for density measurement.28,29 These two-magnet configurations are essentially based on a similar configuration: two magnets of a predetermined distance (45 mm or 60 mm) and with like-poles facing each another. The constraint of this configuration is obvious, as the space for any operation is extremely restricted. Moreover, the exact analytical 2


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expression describes the magnetic field in MagLev is fairly complicated.30,31 Whitesides and his co-workers correlated the levitation height to the magnetic field at the surface of magnets in a two-magnet system with a separation distance of 45 mm, and found that the magnetic field along the centerline is approximated of a linearity.22 For the MagLev configurations of different distances, standard density samples are always required for numerical fitting to obtain the correlation equation; this benchmarking process is considered to be tedious and laborious.28 Here, we demonstrate a novel magnetic levitation configuration that is based on a single permanent ring magnet and provide a mathematic model for this particular configuration to correlate the levitation height and the density of the suspended object. The configuration developed in this work requires only a ring magnet with axial magnetization and a glass tube of paramagnetic solution. Compared to the existing superconducting magnets with similar ring shape,6,21,32-34 the proposed configuration is simple, portable, inexpensive and needs no power. This single ring magnet configuration can be exploited in many potential applications, such as density measurement and object manipulation, and its effectiveness has been proved via a series of designed experiments. Without being limited in the operational space as in the two-magnet configurations do, this configuration improves visualization and manipulation on the materials immersed in the solution and would therefore definitely extend the potential applications in fields such as solid/liquid studies and particle/cell manipulation. In sum, the proposed single ring magnet configuration is simple, portable, low-cost, highly accurate, needs no external energy supply, and is convenient to operate.

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THEORY AND IMPLEMENTATION Configuration of Magnetic Levitation. As shown in Figure 1, the proposed configuration is only composed of a single ring magnet and a glass tube that contains paramagnetic solution, which are placed coaxially to each other. The ring magnet with axial magnetization is used to create the magnetic field. Two different ring magnets were employed in this study, i.e., N35-H20 (inner radius r1 = 20 mm, outer radius r2 = 30 mm, height h = 20 mm, and surface magnetic pole density σ = 1.23 T) and N35-H10 (r1 = 12.5 mm, r2 = 25 mm, h = 10 mm, and σ = 0.88 T). Compared to the superconducting magnet configuration, the utilization of permanent ring magnet makes the experiment fairly simple and portable, and it grants more spaces for operation instead of being restricted in iron core and other ancillary equipment.34 Besides, a camera and prisms are always required for recording experimental procedure because the vision of the operational zone in superconducting magnet is blocked by the iron core. Moreover, the superconducting magnet and its ancillary equipment consume a great amount of electricity to generate sufficient magnetic field (~ 10T)6 and to cool down the magnet. In magnetic levitation configurations, strong magnetic field is no longer required due to the use of paramagnetic solutions, typically, aqueous solutions of paramagnetic salts (MnCl2 or GdCl3)26 or paramagnetic ionic liquids (e.g., Gd·DTPA, a biocompatible solution).35 MnCl2 aqueous solution and MnCl2 methanol solution were adopted in this study. Under the influence of a gradient magnetic field in the axial direction, the diamagnetic objects (e.g., cells, carbon materials, and polymers) immersed in the paramagnetic solution will be relocated at an equilibrium levitation position along the centerline of the magnet. The transparent paramagnetic solutions allow a good visualization of the 4


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submerged objects. Each levitation experiment is photographed using a digital camera, and then the levitation height z is calculated via counting the number of pixels in the corresponding photograph.28,29 The computing equation is

,

where Ns is the number of pixels between the object and ring magnet, and Nh is the number of pixels of the magnet height, h. Further information of magnets calibration, magnetic susceptibility calculation, and safety considerations is shown in SI.

Figure 1. The proposed configuration of magnetic levitation using a single ring magnet. (a) 3D diagram of the magnetic levitation configuration. This configuration consists of a tube of paramagnetic solution placed coaxially with the ring magnet. (b) Schematic diagram of the magnetic levitation configuration and the forces involved in levitating material. The diamagnetic object is levitated in the solution due to the balance of Fmag (magnetic force) and Fg (resultant force of gravity and buoyancy), i.e., Fmag + Fg = 0.

5



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Theory of Magnetic Levitation Configuration. For a diamagnetic object that is levitated in the paramagnetic solution, the magnetic force applied on the object can be calculated by eq 1,22 where χs (dimensionless) and χm (dimensionless) are the magnetic susceptibilities of the diamagnetic object and the paramagnetic solution, respectively. µ0 = 4π × 10-7 (N·A-2) is the magnetic permeability of free space, V (m3) is the volume of the object, and B is the applied magnetic field. Eq 2 gives the numeric correlation between the resultant force of gravity and buoyancy,22 where ρs is the density of the diamagnetic object and ρm is the density of the solution, and g is the vector of gravity. According to eqs 1 and 2, it can be deduced that levitation occurs at any point where the vector sum of the forces equals to zero, i.e., + 0.

( − ) ( · )

( − )

(1)

(2)

According to the work of Babic,36 the magnetic field created by an axially magnetized permanent magnet ring can be calculated via the Coulombian method37 (see the experimental image in Figure S2). The z-axis is the axis of symmetry for the magnet, and r1, r2, and h denote the inner radius, outer radius, and height of the magnet, respectively. The upper face is charged with a surface magnetic pole density +σ; and the lower face is charged with a surface magnetic pole density -σ. P1+ is a point on the ring upper face and P1- is a point on the lower face of the magnet, the magnetic field B created by the ring magnet at any point M(r,z) of the space is given by eqs 3 and 4.36 Integration of eqs 3 and 4 leads to the magnetic field components along the following three directions: Hr(r, z), Hθ(r, z), and Hz(r, z), and the detailed descriptions of which are given by eqs S2 to S5 in SI. Combining eqs 1, 3 and 4, we 6


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can obtain the mathematic model of the proposed magnetic levitation configuration (see SI for more details).

(, )

3045 ./ 0.1 #$% & #$* & ! ! " ) − ) + ,) ,- ( 4 30 ./0.2 │#$% &│ │#$* &│(

(3)

(4)

Density Measurement Equation. As shown in Figure 1, the diamagnetic object would be pushed along the centerline of the magnet, and it is only subjected to the axial magnetic force where + 0 , as shown in eq 5. ( · ) at the centerline can be calculated using the levitation height and the mathematic model of the magnetic field. We thereby obtain the density of the object based on the magnetic field created by the ring magnet. The density distribution in the z direction at the centerline of the proposed magnetic levitation configuration can be calculated using eq 6. The calculated density and levitation height of the object at the centerline are plotted in Figure 2. We divide the area into five regions, denoted as region I to region V. In region I and region V, the magnetic forces are insufficient to levitate any objects. The vision of the objects is blocked by the magnet in region III. In regions II and IV, the objects of densities within the available range (1.2461 ~ 1.3387 g/cm3) can be levitated at the centerline stably. Figure 2 also implies that objects of lower densities than the solution (< 1.2920 g/cm3) can be levitated below the ring magnet (region II), while objects of higher densities (> 1.2920 g/cm3) are levitated above the ring magnet (region IV). Hence, we can easily separate objects below or above the ring magnet through adjusting the density of the paramagnetic solution. As shown in eq 6, the density of the diamagnetic object (ρs) is related to the magnet’s parameters (inner radius r1, outer radius r2, height h and strength σ), the paramagnetic solution’s 7



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parameters (ρm and χm), and the magnetic susceptibility of the diamagnetic object (χs). Notably, the proposed configuration does not require the sample’s volume to be determined during density measurement.

(s − m ) + +

(6 − 7 ) ( ∙ ) 0

4 ( − ) ( ( ((:$, − :4, ) − (:$,; − :4,; ) − 4 ( − )

Single-Ring Magnetic Levitation Configuration for Object Manipulation

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