Magnetic levitation

Abstract

The magnetic levitation system described includes a first cylindrical magnet, a second cylindrical magnet coaxially aligned with the first cylindrical magnet, and a first cavity coaxially aligned with the first cylindrical magnet. The like-polarity surfaces of the first and second cylindrical magnets face each other parallel to one another, generating a linear magnetic field between the first and second cylindrical magnets. Also described is a method of using the magnetic levitation system to analyze diamagnetic or paramagnetic samples.

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims priority to U.S. Provisional Application No. 62 / 738,758, filed September 28, 2018, and U.S. Provisional Application No. 62 / 800,796, filed February 4, 2019, the entire contents of which are incorporated herein by reference.

[0002] Incorporation by Reference All patents, patent applications, and publications cited herein are incorporated by reference in their entirety in order to more fully describe the state of the art as known to those skilled in the art as of the invention date set forth herein. [Background technology]

[0003] This application relates generally to magnetic levitation (MagLev) devices.

[0004] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with Government support under ER45852 awarded by the Department of Energy. The United States Government has rights in the invention. Summary of the Invention

[0005] In one embodiment, a first cylindrical magnet; A second cylindrical magnet arranged coaxially with the first cylindrical magnet; a first cavity formed within the first cylindrical magnet and coaxially aligned with the first cylindrical magnet; A magnetic levitation system is described, comprising: The surfaces of the first cylindrical magnet and the second cylindrical magnet that have the same pole are parallel to each other and face each other, and a linear magnetic field is generated between the first cylindrical magnet and the second cylindrical magnet.

[0006] In one or more embodiments disclosed herein, the first cavity is cylindrical.

[0007] In one or more embodiments disclosed herein, the first cavity spans the entire height of the first cylindrical magnet.

[0008] In one or more embodiments disclosed herein, the linear magnetic field extends into the first cavity.

[0009] In one or more embodiments disclosed herein, the first cylindrical magnet and the second cylindrical magnet have different radii or heights.

[0010] In one or more embodiments disclosed herein, the first cylindrical magnet and the second cylindrical magnet have the same radius, height, or both.

[0011] In one or more embodiments described herein, the system further includes a second cavity formed within the second cylindrical magnet and coaxially aligned with the second cylindrical magnet.

[0012] In one or more embodiments disclosed herein, the second cavity is cylindrical.

[0013] In one or more embodiments disclosed herein, the distance between surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is d, the heights of the first cylindrical magnet and the second cylindrical magnet are h1 and h2, respectively, and the ratio of at least one of h1 and h2 to d is about 0.2:1 to about 10:1.

[0014] In one or more embodiments disclosed herein, the ratio of at least one of h1 and h2 to d is about 1.67:1.

[0015] In one or more embodiments disclosed herein, the first cylindrical magnet has an inner diameter id1, and the ratio of id1 to d is about 0.2:1 to about 10:1.

[0016] In one or more embodiments disclosed herein, the ratio of id1 to d is about 1.67:1.

[0017] In one or more embodiments disclosed herein, the first cylindrical magnet has an outer diameter of od1, and the ratio of od1 to d is about 0.3:1 to about 100:1.

[0018] In one or more embodiments disclosed herein, the ratio of od1 to d is about 5:1.

[0019] In one or more embodiments disclosed herein, the system further includes a second cylindrical cavity formed within and coaxially aligned with the second cylindrical magnet, the second cylindrical magnet having an inner diameter id2, and a ratio of id2 to d between about 0.2:1 and about 10:1.

[0020] In one or more embodiments disclosed herein, the ratio of id2 to d is about 1.67:1.

[0021] In one or more embodiments disclosed herein, the system further includes a second cavity, the second cavity being formed within the second cylindrical magnet and coaxially aligned with the second cylindrical magnet, the outer diameter of the second cylindrical magnet being od2, and the ratio of od2 to d being about 0.3:1 to about 100:1.

[0022] In one or more embodiments disclosed herein, the ratio of od2 to d is about 5:1.

[0023] In one or more embodiments disclosed herein, the first cavity is cylindrical, and the ratio of the inner diameter (id1):outer diameter (od1):height (h1):distance (d) between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet is approximately 1.67:5:1.67:1.

[0024] In one or more embodiments disclosed herein, the system further includes a second cylindrical cavity formed within the second cylindrical magnet and coaxially aligned with the second cylindrical magnet, the ratio of the inner diameter (id2):outer diameter (od2):height (h2):distance (d) between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet being approximately 1.67:5:1.67:1.

[0025] In one or more embodiments disclosed herein, the system further comprises a container configured to hold a paramagnetic medium and disposed at least partially between the first cylindrical magnet and the second cylindrical magnet.

[0026] In one or more embodiments disclosed herein, the vessel further comprises an inlet configured to allow the addition or removal of the paramagnetic medium or paramagnetic sample or diamagnetic sample.

[0027] In one or more embodiments disclosed herein, the container includes two ends, at least one of which extends into or through the first cavity.

[0028] In one or more embodiments disclosed herein, the system further includes a second cavity formed within and coaxially aligned with the second cylindrical magnet, and the other of the two ends of the container extends into or through the second cylindrical cavity.

[0029] In one or more embodiments disclosed herein, the vessel is a cuvette.

[0030] In one or more embodiments disclosed herein, the paramagnetic medium comprises an aqueous solution of a paramagnetic compound.

[0031] In one or more embodiments disclosed herein, the paramagnetic compound is selected from the group consisting of MnCl2, MnBr2, CuSO4, GdCl3, DyCl3, HoCl3, Gd chelates, and combinations thereof.

[0032] In one or more embodiments disclosed herein, the Gd chelate compound is gadolinium(III) diethylenetriaminepentaacetic acid.

[0033] In one or more embodiments disclosed herein, the paramagnetic medium comprises an aqueous solution of MnCl2.

[0034] In one or more embodiments disclosed herein, the paramagnetic medium comprises a hydrophobic paramagnetic medium or a paramagnetic ionic liquid.

[0035] In one or more embodiments disclosed herein, the maximum magnetic field along a common axis of the first cylindrical magnet and the second cylindrical magnet is about 0.20-0.50T.

[0036] In one or more embodiments disclosed herein, the first cylindrical magnet and the second cylindrical magnet are NdFeB magnets.

[0037] In one or more embodiments disclosed herein, the system may include a second cylindrical cavity that is coaxially aligned with the second cylindrical magnet across the entire height of the second cylindrical magnet, and an inner diameter of the first cylindrical magnet or the second cylindrical magnet is between about 15 mm and about 40 mm.

[0038] In one or more embodiments disclosed herein, the inside diameter of the first cylindrical magnet or the second cylindrical magnet is about 25 mm.

[0039] In one or more embodiments disclosed herein, the system may include a second cylindrical cavity that is coaxially aligned with the second cylindrical magnet across the entire height of the second cylindrical magnet, and an outer diameter of the first cylindrical magnet or the second cylindrical magnet is about 50 mm to about 100 mm.

[0040] In one or more embodiments disclosed herein, the outer diameter of the first cylindrical magnet or the second cylindrical magnet is about 76 mm.

[0041] In one or more embodiments disclosed herein, the height of the first cylindrical magnet or the second cylindrical magnet is about 15 mm to about 50 mm.

[0042] In one or more embodiments disclosed herein, the height of the first cylindrical magnet or the second cylindrical magnet is about 25 mm.

[0043] In one or more embodiments disclosed herein, the distance between surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is about 5 mm to about 50 mm.

[0044] In one or more embodiments disclosed herein, the distance between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet is about 15 mm.

[0045] In one aspect, a method for analyzing a diamagnetic or paramagnetic sample is described, the method comprising: Providing any one of the above systems; disposing a container configured to hold a paramagnetic medium between the first cylindrical magnet and the second cylindrical magnet; adding a paramagnetic medium and a diamagnetic or paramagnetic sample, separately or together, to the vessel; levitating the diamagnetic or paramagnetic sample under the linear magnetic field between the first cylindrical magnet and the second cylindrical magnet; Includes.

[0046] In one or more embodiments disclosed herein, the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is added via a first cylindrical cavity.

[0047] In one or more embodiments disclosed herein, the method further comprises removing the paramagnetic medium or the paramagnetic sample or the diamagnetic sample.

[0048] In one or more embodiments disclosed herein, the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is removed through the first cylindrical cavity.

[0049] In one or more embodiments disclosed herein, the paramagnetic medium comprises an aqueous solution of a paramagnetic compound, a hydrophobic paramagnetic medium, or a paramagnetic ionic liquid.

[0050] In one or more embodiments disclosed herein, the paramagnetic compound is selected from the group consisting of MnCl2, MnBr2, CuSO4, GdCl3, DyCl3, HoCl3, Gd chelates, and combinations thereof.

[0051] In one or more embodiments disclosed herein, the Gd chelate compound is gadolinium(III) diethylenetriaminepentaacetic acid.

[0052] In one or more embodiments disclosed herein, the paramagnetic medium comprises an aqueous solution of MnCl2.

[0053] In one or more embodiments disclosed herein, the paramagnetic medium further comprises cetyltrimethylammonium bromide.

[0054] In one or more embodiments disclosed herein, the paramagnetic sample or the diamagnetic sample comprises a cross-linked polymer.

[0055] In one or more embodiments disclosed herein, the method further comprises determining a density of the cross-linked polymer.

[0056] In one or more embodiments disclosed herein, the method further includes generating a standard curve correlating the floating height of a sample in the system with its density, and determining the density of the cross-linked polymer using the standard curve and the floating height in the system.

[0057] In one or more embodiments disclosed herein, the cross-linked polymer comprises polydimethylsiloxane.

[0058] Any aspect or embodiment disclosed herein can be combined with any other aspect or embodiment disclosed herein. Combinations of one or more embodiments described herein with one or more other embodiments described herein are expressly contemplated.

[0059] The present application will be described with reference to the following figures, which are presented for purposes of illustration and are not intended to be limiting: [Brief description of the drawings]

[0060] [Figure 1A] 1 illustrates a MagLev device including multiple coaxially arranged block magnets according to one or more embodiments described herein. [Figure 1B] 1 illustrates an axial MagLev device including multiple coaxially arranged ring magnets according to one or more embodiments described herein. [Figure 2A] 1 illustrates a pair of indistinguishable magnets arranged coaxially with opposing poles and a diamagnetic sample levitated in a paramagnetic medium, according to one or more embodiments described herein. [Figure 2B]1 illustrates spatial characteristics of a magnetic field in a vertical cross section through a central axis according to one or more embodiments described herein. [Figure 2C] 2C illustrates the magnetic field strength Bz along the vertical dotted line in FIG. 2B, according to one or more embodiments described herein. [Figure 3A] 1 illustrates the calibration of a MagLev device according to one or more embodiments described herein using an array of droplets of a non-aqueous organic liquid levitated in an aqueous solution of a paramagnetic medium. [Figure 3B] 3B illustrates a calibration curve resulting from the calibration shown in FIG. 3A, according to one or more embodiments described herein. [Figure 4] 13 illustrates the extended range of densities that can be analyzed using Axial MagLev, according to one or more embodiments described herein. [Figure 5A] 1 illustrates a droplet of polydimethylsiloxane prepolymer levitating in a paramagnetic solution, according to one or more embodiments described herein. [Figure 5B] 1 shows a plug of Vaseline® gel floating in a paramagnetic solution, according to one or more embodiments described herein. [Figure 5C] 1 illustrates the removal of a sub-population of density standard beads floating in a paramagnetic solution, according to one or more embodiments described herein. [Figure 6A] FIG. 2 is a schematic diagram illustrating four independent physical parameters used to define the configuration of a MagLev device, according to one or more embodiments described herein. [Figure 6B] 1 shows an example of a linear gradient of maximum magnetic field strength of .about.0.4 T along the central axis between the magnets (dotted line in inset), according to one or more embodiments described herein. [Figure 6C] 1 shows an example of a non-linear gradient of maximum magnetic field strength of .about.0.5 T along the central axis between the magnets (dotted line in inset), according to one or more embodiments described herein. [Figure 6D] 1 illustrates fine adjustment of the separation between two magnets used in a MagLev device, according to one or more embodiments described herein. [Figure 6E] 1 illustrates the magnetic field strength Bz along the central axis between two magnets at varying separation distances according to one or more embodiments described herein. [Figure 6F] 6E shows a plot of the curve at d=0.6 in FIG. 6E according to one or more embodiments described herein. [Figure 7A] 1 illustrates an assembly of a MagLev device according to one or more embodiments described herein. [Figure 7B] FIG. 7B illustrates a collapsed view of the MagLev device of FIG. 7A according to one or more embodiments described herein. [Figure 8A] 1 illustrates magnetic levitation using a single ring magnet, according to one or more embodiments described herein. [Figure 8B] 8B illustrates the magnetic field of the single ring magnet of FIG. 8A according to one or more embodiments described herein. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0061] Magnetic levitation (MagLev) is a simple and useful technique that exploits density, a universal physical property of all matter, and has applications in various fields such as chemistry, biochemistry, and materials science. The MagLev configuration shown in Figure 1A shows a spatial arrangement of magnets in which a sample vessel (e.g., a 45 mm tall rectangular cuvette) is sandwiched between two block NdFeB permanent magnets. In this configuration, the user cannot (i) easily add or remove paramagnetic media or levitated samples (especially viscous or sticky samples), (ii) observe the levitated sample from above or below, (iii) move the paramagnetic media in the vessel along the central axis between the magnets, and (iv) accommodate sample vessels that are taller than the separation between the magnets, e.g., tall vials and test tubes.

[0062] In some embodiments described herein, a MagLev system using a cylindrical magnet (e.g., a ring magnet) is described. These "axial" MagLev devices allow access to the sample along the central axis of the magnetic field, which is not possible with block magnet MagLev devices (e.g., the device of FIG. 1A). In some embodiments, axial MagLev allows density-based magneto-Archimedean analysis, separation, and other manipulations in a simple procedure. In some embodiments, density can be used to perform, for example, (i) separation or analysis of various types of non-biological and biological materials (e.g., glass, metals, crystalline polymorphs, polymer particles, mammalian cells, yeast, and bacteria), (ii) monitoring chemical processes (e.g., chemical reactions and polymerizations on solid supports) as density changes, monitoring binding events (e.g., ligand-enzyme binding, antibody-antigen binding, antibody-cell binding), (iii) non-contact object orientation, self-assembly in three dimensions, and (iv) quality control of injection molded plastic parts.

[0063] In one embodiment, a first cylindrical magnet; A second cylindrical magnet arranged coaxially with the first cylindrical magnet; A first cavity aligned coaxially with the first cylindrical magnet; A magnetic levitation system is described, comprising: The surfaces of the first cylindrical magnet and the second cylindrical magnet that have the same pole are parallel to each other and face each other, and a linear magnetic field is generated between the first cylindrical magnet and the second cylindrical magnet.

[0064] In some embodiments, the first cavity is cylindrical. In other embodiments, the cross section of the first cavity is a symmetric shape, such as a triangle, a square, a hexagon, etc. In some embodiments, applicants have surprisingly found that a pair of cylindrical magnets (e.g., ring magnets) with like poles facing each other (as in an anti-Helmholtz configuration using electromagnets) can provide a linear, axially symmetric magnetic field to levitate diamagnetic and weakly paramagnetic materials (i.e., materials with relatively weak paramagnetic properties and relatively low magnetic susceptibility, e.g., aluminum) in a paramagnetic medium (e.g., an aqueous solution of MnCl2 or GdCl3) for density-based analysis, separation, and manipulation. In some embodiments, the linear magnetic field between the ring magnets can be used to efficiently perform procedures for calibrating and performing density measurements. In some embodiments, the nonlinear portion of the magnetic field, particularly the magnetic field within the cavity of the ring magnet, can be used to perform density-based separation and manipulation.

[0065] In some embodiments, an axial MagLev system is described with reference to FIG. 1B. As shown in FIG. 1B, according to one or more embodiments, the axial MagLev system 100 includes a first cylindrical magnet 101, a second cylindrical magnet 103 coaxially aligned with the first cylindrical magnet 101, and a first cylindrical cavity 105 coaxially aligned with the first cylindrical magnet 101. In some embodiments, the first cylindrical cavity 105 spans the entire height of the first cylindrical magnet 101. In other embodiments, the first cylindrical cavity 105 spans a portion of the height of the first cylindrical magnet 101. In some embodiments, the magnetic levitation system 100 includes a second cylindrical cavity 111 coaxially aligned with the second cylindrical magnet 103. In some embodiments, the second cylindrical cavity 111 spans the entire height of the second cylindrical magnet 103. In other embodiments, the second cylindrical cavity 111 spans a portion of the height of the second cylindrical magnet 103. In some embodiments, the first cylindrical magnet 101 and the second cylindrical magnet 103 are NdFeB magnets. In some embodiments, the first cavity 105 and the second cavity 111 are each cylindrical. In other embodiments, the cross section of the first cavity and / or the second cavity is a symmetrical shape, such as a triangle, a square, a hexagon, etc.

[0066] Applicants have surprisingly found that one or more embodiments of the MagLev system disclosed herein (comprising two cylindrical magnets with opposing like poles) provide a linear magnetic field between the first and second magnets. In some embodiments, the magnet size, magnet aspect ratio, and / or distance between the two magnets are further optimized to provide a linear magnetic field. In some embodiments, the optimized linear magnetic field generated between two ring magnets (coaxially arranged with opposing like poles) allows levitation of diamagnetic and weakly paramagnetic materials in a paramagnetic suspension medium, making density measurements easier.

[0067] In some embodiments, the linear magnetic field is present in the space 107 between the first magnet 101 and the second magnet 103. In some embodiments, the linear magnetic field extends into the first cylindrical cavity 105 and / or the second cylindrical cavity 111. In some particular embodiments, the linear magnetic field extends into at least about half of the height of the first cylindrical cavity 105 and / or the second cylindrical cavity 111. According to one or more embodiments, the term "linear magnetic field" as used herein generally refers to a magnetic field whose vector varies linearly along the central axis of the configuration, such that the magnetic field has a linear gradient.

[0068] In some embodiments, the cylindrical magnet is also referred to as a "ring magnet." In some embodiments, the first cylindrical magnet and the second cylindrical magnet have different radii or heights. In some embodiments, the first cylindrical magnet and the second cylindrical magnet have the same radius, the same height, or both.

[0069] In some embodiments, the axial MagLev system further includes a vessel (e.g., 109 in FIG. 1B) configured to hold a paramagnetic medium and disposed at least partially between the first and second cylindrical magnets. In some embodiments, the vessel 109 further includes an inlet 113 configured to allow addition or removal of a paramagnetic medium or a paramagnetic or diamagnetic sample. As shown in FIG. 1B, in certain embodiments, the vessel 109 includes two ends (top and bottom), at least one of which (e.g., the top end of the vessel 109) extends into or through the first cylindrical cavity 105. In some embodiments, the first cylindrical cavity 105 allows for easy placement of the vessel 109 between the two opposing magnets 101 and 103, and easy removal of the vessel 109 from the axial MagLev system 100, both of which can be accomplished via the first cylindrical cavity 105. In some embodiments, the other end of the vessel (e.g., the bottom end of the vessel 109) extends into or through the second cylindrical cavity 111 (not shown in FIG. 1B). Similarly, in some embodiments, the second cylindrical cavity 111 allows for easy placement of the vessel 109 between the two opposing magnets 101 and 103, and easy removal of the vessel 109 from the axial MagLev system 100, both of which can be accomplished via the second cylindrical cavity 111. Non-limiting examples of vessels include cuvettes and test tubes.

[0070] In some embodiments, the size and / or aspect ratio of the first and / or second cylindrical magnets provide a linear magnetic field between the two cylindrical magnets. In some embodiments, the distance between the like-polarized surfaces of the first and second cylindrical magnets is d, the heights of the first and second cylindrical magnets are h1 and h2, respectively, and the ratio of at least one of h1 and h2 to d is about 0.2:1 to about 10:1. In some embodiments, the ratio of at least one of h1 and h2 to d is about 0.5:1, 0.8:1, 1:1, 1.5:1, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, 8:1, 9:1, or 10:1, or within a range bounded by any of the values ​​of the two ratios disclosed herein.

[0071] Thus, in some embodiments, the ratio of h1 to d or the ratio of h2 to d is about 0.5:1 to about 10:1, about 1:1 to about 10:1, about 2:1 to about 10:1, about 3:1 to about 10:1, about 4:1 to about 10:1, about 5:1 to about 10:1, about 6:1 to about 10:1, about 7:1 to about 10:1, about 8:1 to about 10:1, or about 9:1 to about 10:1. In some embodiments, the ratio of h1 to d or the ratio of h2 to d is about 0.5:1 to about 9:1, about 1:1 to about 9:1, about 2:1 to about 9:1, about 3:1 to about 9:1, about 4:1 to about 9:1, about 5:1 to about 9:1, about 6:1 to about 9:1, about 7:1 to about 9:1, or about 8:1 to about 9:1. In some embodiments, the ratio of h1 to d or the ratio of h2 to d is about 0.5:1 to about 8:1, about 1:1 to about 8:1, about 2:1 to about 8:1, about 3:1 to about 8:1, about 4:1 to about 8:1, about 5:1 to about 8:1, about 6:1 to about 8:1, or about 7:1 to about 8:1.

[0072] In some embodiments, the ratio of h1 to d or the ratio of h2 to d is about 0.5:1 to about 4:1, about 0.75:1 to about 3:1, about 1:1 to about 2:1, about 1.5:1 to about 2:1, or about 1.6:1 to about 1.8.

[0073] In some embodiments, the ratio of at least one of h1 and h2 to d is about 1.67:1.

[0074] In some embodiments, the inner diameter of the first cylindrical magnet is id1, and the ratio of id1 to d is about 0.2:1 to about 10:1. Thus, in some embodiments, the ratio of id1 to d is about 0.5:1 to about 10:1, about 1:1 to about 10:1, about 2:1 to about 10:1, about 3:1 to about 10:1, about 4:1 to about 10:1, about 5:1 to about 10:1, about 6:1 to about 10:1, about 7:1 to about 10:1, about 8:1 to about 10:1, or about 9:1 to about 10:1. In some embodiments, the ratio of id1 to d is about 0.5:1 to about 9:1, about 1:1 to about 9:1, about 2:1 to about 9:1, about 3:1 to about 9:1, about 4:1 to about 9:1, about 5:1 to about 9:1, about 6:1 to about 9:1, about 7:1 to about 9:1, or about 8:1 to about 9:1. In some embodiments, the ratio of id1 to d is about 0.5:1 to about 8:1, about 1:1 to about 8:1, about 2:1 to about 8:1, about 3:1 to about 8:1, about 4:1 to about 8:1, about 5:1 to about 8:1, about 6:1 to about 8:1, or about 7:1 to about 8:1.

[0075] In some embodiments, the ratio of id1 to d is about 0.5:1 to about 4:1, about 0.75:1 to about 3:1, about 1:1 to about 2:1, about 1.5:1 to about 2:1, or about 1.6:1 to about 1.8.

[0076] In some embodiments, the ratio of id1 to d is about 1.67:1.

[0077] In some embodiments, the outer diameter of the first cylindrical magnet is od 1, and the ratio of od 1 to d is about 0.3:1 to about 100:1. Thus, in some embodiments, the ratio of od 1 to d is about 0.5:1 to about 50:1, about 1:1 to about 50:1, about 2:1 to about 50:1, about 3:1 to about 50:1, about 4:1 to about 50:1, about 5:1 to about 50:1, about 6:1 to about 50:1, about 7:1 to about 50:1, about 8:1 to about 50:1, about 9:1 to about 50:1, or about 10:1 to about 50:1. In some embodiments, the ratio of od1 to d is about 1:1 to about 10:1, about 2:1 to about 9:1, about 3:1 to about 8:1, about 4:1 to about 7:1, about 4:1 to about 6:1, or about 4:1 to about 5:1.

[0078] In some embodiments, the ratio of od1 to d is about 5:1.

[0079] In some embodiments, the ratio of the inner diameter (id1) of the first cylindrical magnet to the outer diameter (od1) to the height (h1) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-5:1-10:1-5:1. In some embodiments, the ratio of the inner diameter (id1) of the first cylindrical magnet to the outer diameter (od1) to the height (h1) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-3:2-8:1-3:1. In some embodiments, the ratio of the inner diameter (id1) of the first cylindrical magnet to the outer diameter (od1) to the height (h1) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-2:3-6:1-2:1. In some embodiments, the ratio of the inner diameter (id1):outer diameter (od1):height (h1) of the first cylindrical magnet:distance (d) between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet is approximately 1.67:5:1.67:1.

[0080] In some embodiments, the inner diameter of the second cylindrical magnet is id2, and the ratio of id2 to d is about 0.2:1 to about 10:1. Thus, in some embodiments, the ratio of id2 to d is about 0.5:1 to about 10:1, about 1:1 to about 10:1, about 2:1 to about 10:1, about 3:1 to about 10:1, about 4:1 to about 10:1, about 5:1 to about 10:1, about 6:1 to about 10:1, about 7:1 to about 10:1, about 8:1 to about 10:1, or about 9:1 to about 10:1. In some embodiments, the ratio of id2 to d is about 0.5:1 to about 9:1, about 1:1 to about 9:1, about 2:1 to about 9:1, about 3:1 to about 9:1, about 4:1 to about 9:1, about 5:1 to about 9:1, about 6:1 to about 9:1, about 7:1 to about 9:1, or about 8:1 to about 9:1. In some embodiments, the ratio of id2 to d is about 0.5:1 to about 8:1, about 1:1 to about 8:1, about 2:1 to about 8:1, about 3:1 to about 8:1, about 4:1 to about 8:1, about 5:1 to about 8:1, about 6:1 to about 8:1, or about 7:1 to about 8:1.

[0081] In some embodiments, the ratio of id2 to d is from about 0.5:1 to about 4:1, from about 0.75:1 to about 3:1, from about 1:1 to about 2:1, from about 1.5:1 to about 2:1, or from about 1.6:1 to about 1.8.

[0082] In some embodiments, the ratio of id2 to d is about 1.67:1.

[0083] In some embodiments, the outer diameter of the second cylindrical magnet is od2, and the ratio of od2 to d is about 0.3:1 to about 100:1. Thus, in some embodiments, the ratio of od2 to d is about 0.5:1 to about 50:1, about 1:1 to about 50:1, about 2:1 to about 50:1, about 3:1 to about 50:1, about 4:1 to about 50:1, about 5:1 to about 50:1, about 6:1 to about 50:1, about 7:1 to about 50:1, about 8:1 to about 50:1, about 9:1 to about 50:1, or about 10:1 to about 50:1. In some embodiments, the ratio of od2 to d is about 1:1 to about 10:1, about 2:1 to about 9:1, about 3:1 to about 8:1, about 4:1 to about 7:1, about 4:1 to about 6:1, or about 4:1 to about 5:1.

[0084] In some embodiments, the ratio of od2 to d is about 5:1.

[0085] In some embodiments, the ratio of the inner diameter (id2) of the second cylindrical magnet to the outer diameter (od2) to the height (h2) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-5:1-10:1-5:1. In some embodiments, the ratio of the inner diameter (id2) of the second cylindrical magnet to the outer diameter (od2) to the height (h2) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-3:2-8:1-3:1. In some embodiments, the ratio of the inner diameter (id2) of the second cylindrical magnet to the outer diameter (od2) to the height (h2) to the distance (d) between the surfaces of the same poles of the first cylindrical magnet and the second cylindrical magnet is about 1-2:3-6:1-2:1. In some embodiments, the ratio of the inner diameter (id2):outer diameter (od2):height (h2):distance (d) between like-polarized surfaces of the first and second cylindrical magnets is approximately 1.67:5:1.67:1.

[0086] In some embodiments, the second cylindrical cavity 111 spans the entire height of the second cylindrical magnet 103 and is coaxially aligned with the second cylindrical magnet 103, and the inner diameter of the first cylindrical magnet and / or the second cylindrical magnet is about 15 mm to about 40 mm. In some embodiments, the inner diameter of the first cylindrical magnet or the second cylindrical magnet is about 15 mm to about 40 mm, about 20 mm to about 30 mm, or about 22 mm to about 27 mm. In some embodiments, the inner diameter of the first cylindrical magnet and / or the second cylindrical magnet is about 25 mm.

[0087] In some embodiments, the outer diameter of the first cylindrical magnet and / or the second cylindrical magnet is about 50 mm to about 100 mm. In some embodiments, the outer diameter of the first cylindrical magnet and / or the second cylindrical magnet is about 60 mm to about 90 mm. In some embodiments, the outer diameter of the first cylindrical magnet and / or the second cylindrical magnet is about 70 mm to about 80 mm. In some specific embodiments, the outer diameter of the first cylindrical magnet or the second cylindrical magnet is about 75, 76, or 77 mm.

[0088] In some embodiments, the height of the first cylindrical magnet and / or the second cylindrical magnet is about 15 mm to about 50 mm, about 15 mm to about 40 mm, about 20 mm to about 30 mm, or about 20 mm to about 25 mm. In some embodiments, the height of the first cylindrical magnet or the second cylindrical magnet is about 25 mm.

[0089] In some embodiments, the distance between the surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is about 5 mm to about 50 mm. In some embodiments, the distance between the surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is about 15 mm.

[0090] In some embodiments, the maximum magnetic field along the common axis of the first and second cylindrical magnets is about 0.20-0.50 T. In some particular embodiments, the maximum Bo along the central axis between the first and second cylindrical magnets is 0.33 T, with the linear magnetic field extending into the cavity of the ring magnet (about half the separation distance between the magnets). Without wishing to be bound by any particular theory, it is believed that the geometry of the device described in one or more embodiments herein facilitates performing density-based analysis and separation of diamagnetic and weakly paramagnetic samples, and facilitates exchanging the paramagnetic medium surrounding the levitated object (e.g., by moving the sample vessel relative to the magnet).

[0091] In some embodiments, the paramagnetic medium comprises an aqueous solution of a paramagnetic compound. In some embodiments, the paramagnetic compound is selected from the group consisting of MnCl2, MnBr2, CuSO4, GdCl3, DyCl3, HoCl3, a Gd chelate compound, and combinations thereof. In some particular embodiments, the Gd chelate compound is gadolinium(III) diethylenetriaminepentaacetate.

[0092] In some embodiments, the paramagnetic medium comprises an aqueous solution of MnCl. In some embodiments, the paramagnetic medium comprises a hydrophobic paramagnetic medium or a paramagnetic ionic liquid.

[0093] In another aspect, a method for analyzing a diamagnetic or paramagnetic sample is described, the method comprising: Providing a system according to any one of the embodiments described herein; disposing a container configured to hold a paramagnetic medium between the first cylindrical magnet and the second cylindrical magnet; adding a paramagnetic medium and a diamagnetic or paramagnetic sample, separately or together, to the vessel; levitating the diamagnetic or paramagnetic sample under the linear magnetic field between the first cylindrical magnet and the second cylindrical magnet; Includes.

[0094] In some embodiments, the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is added via a first cylindrical cavity. In some embodiments, the method further comprises removing the paramagnetic medium or the paramagnetic sample or the diamagnetic sample. In some embodiments, the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is removed via the first cylindrical cavity.

[0095] In some embodiments, the paramagnetic medium comprises an aqueous solution of a paramagnetic compound, a hydrophobic paramagnetic medium, or a paramagnetic ionic liquid. In some embodiments, the paramagnetic compound is selected from the group consisting of MnCl2, MnBr2, CuSO4, GdCl3, DyCl3, HoCl3, a Gd chelating compound, and combinations thereof. In some embodiments, the Gd chelating compound is gadolinium(III) diethylenetriaminepentaacetic acid. In some embodiments, the paramagnetic medium comprises an aqueous solution of MnCl2. In some embodiments, the paramagnetic medium further comprises cetyltrimethylammonium bromide. In some particular embodiments, the paramagnetic sample or the diamagnetic sample comprises a crosslinked polymer.

[0096] In some embodiments, the method further comprises determining a density of the cross-linked polymer. In some embodiments, the method further comprises generating a standard curve correlating a flying height of a sample in the system with its density, and determining a density of the cross-linked polymer using the standard curve and the flying height in the system. In some embodiments, the cross-linked polymer comprises PDMS.

[0097] In this specification, terms such as first, second, third, etc. may be used to describe various elements, but these elements are not limited by these terms. These terms are used simply to distinguish one element from another. Thus, a first element described below can be called a second element without departing from the teachings of the exemplary embodiment. Spatially relative terms such as "upper", "lower", "left", "right", "front", "rear", etc. are used herein to facilitate description and describe the relationship of one element to another, as illustrated. It will be understood that the spatially relative terms and the illustrated configurations are intended to encompass different orientations of the device in use or operation in addition to the orientations described herein and depicted in the figures. For example, if the device in the figures is turned over, elements described as "below" or "beneath" other elements or features will be oriented "above" the other elements or features. Thus, the exemplary term "upper" can encompass both an upward and downward orientation. Devices may be otherwise oriented (e.g., rotated 90 degrees or placed in other orientations) and the spatially relative descriptors described herein interpreted accordingly. Additionally, in this disclosure, when an element is referred to as being "coupled," "on," "connected," "coupled," "in contact," or the like to another element, unless specified otherwise, it may be directly coupled, on, connected, coupled, or in contact with the other element, or there may be intervening elements.

[0098] The terms used herein are for the purpose of describing particular embodiments and are not intended to limit the exemplary embodiments. In particular embodiments, singular forms such as "a" and "an" are intended to include the plural unless the context indicates otherwise. Furthermore, the terms "includes," "including," "comprises," and "comprising" specify the presence of stated elements or steps and do not exclude the presence or addition of one or more other elements or steps.

[0099] example Certain embodiments will now be described in the following non-limiting examples.

[0100] FIG. 1A illustrates the configuration of a MagLev device according to some embodiments. The MagLev device comprises two block magnets (e.g., NdFeB permanent magnets, W×L×H: 50.8 mm×50.8 mm×25.4 mm) arranged coaxially with a separation of 45.0 mm, with the same poles facing each other. In some embodiments, the cuvette (e.g., 45 mm high) is a typical container used to levitate diamagnetic samples (represented by 3 mm spheres in FIG. 1A) in a paramagnetic medium (e.g., an aqueous solution of MnCl2).

[0101] In some embodiments, the axial MagLev device includes a cylindrical magnet, e.g., a ring magnet, in a configuration that removes physical barriers to physical sampling in the magnetic field present in the MagLev device of FIG. 1A. Thus, in some embodiments, the axial MagLev simplifies the procedure for performing density-based analyses, separations, and manipulations. In certain embodiments, an optimized linear magnetic field generated between two ring magnets (coaxially arranged with the same poles facing each other) can levitate diamagnetic materials (and weakly paramagnetic materials, e.g., aluminum) in a paramagnetic suspension medium, making density measurements easier to perform. In certain embodiments, this axial configuration allows (i) the addition of sample and / or paramagnetic medium through an opening in the vessel and removal of the sample (e.g., a subpopulation of small particles) from the vessel while suspended in the magnetic field; (ii) the ability to view the sample 360° around the sample vessel and from above and below; and (iii) facilitating density measurements of small sample volumes (as small as a single sub-millimeter particle, as demonstrated). The compact design, portability, affordability, and ease of use of the Axial MagLev device will broaden the application of magnetic methods in analyzing, separating, and / or manipulating various types of samples (e.g., solids, liquids, powders, pastes, gels, and biological materials) in fields such as materials science, chemistry, and biochemistry.

[0102] In some embodiments, a pair of ring magnets are arranged with like poles facing each other (as in an anti-Helmholtz configuration using electromagnets) to create an axially symmetric linear magnetic field that is used to levitate diamagnetic and weakly paramagnetic materials (e.g., aluminum) in a paramagnetic medium (e.g., aqueous solutions of MnCl2 or GdCl3) for density-based analysis, separation, and manipulation (FIG. 1B). In some embodiments, the linear magnetic field between the ring magnets can be used to simplify procedures for calibrating and performing density measurements. In other embodiments, the nonlinear portion of the magnetic field, particularly the magnetic field within the cavity of the ring magnets, can be used to perform density-based separations and manipulations.

[0103] 1B illustrates an axial MagLev device including two ring magnets (e.g., NdFeB permanent magnets, OD×ID×H: 76.2 mm×25.4 mm×25.4 mm) with the same poles facing each other and arranged coaxially with a separation of 15.0 mm, according to one or more embodiments. In these embodiments, a cuvette containing a sample (represented by a 3 mm sphere in FIG. 1B) is included to illustrate the size of the device.

[0104] In some embodiments, the size and aspect ratio of the magnets are optimized to provide a linear magnetic field. In certain embodiments, the maximum Bo along the central axis between the magnets is about 0.3T, 0.4T, 0.5T, 0.6T, 0.7T, 0.8T, 0.9T, 1.0T, or within a range bounded by any two values ​​disclosed herein. In certain embodiments, the maximum Bo along the central axis between the magnets is 0.33T, and the linear magnetic field extends into the cavity of the ring magnet (about half the separation distance between the magnets). In these embodiments, this shape facilitates performing density-based analysis and separation of diamagnetic and weakly paramagnetic samples, and facilitates replacing the paramagnetic medium surrounding the levitated object (e.g., by moving the sample container relative to the magnet).

[0105] In some embodiments, the axial MagLev configuration using two ring magnets makes many new procedures available or easier than traditional MagLev procedures. In some embodiments, this axial configuration allows easy access to both the top and bottom of the sample vessel, allowing easy sample recovery and replacement of the paramagnetic medium surrounding the levitation without removing the sample vessel from the magnetic field.

[0106] Device Design In some embodiments, two ring magnets (e.g., NdFeB permanent magnets) were coaxially arranged with like poles facing each other such that the gradient of the magnetic field between the magnets was linear, with the central axis aligned with the gravity vector. As shown in FIG. 2A, magnetic field simulations were performed using Comsol® to optimize the shape of the ring magnets, including the inner diameter (id), outer diameter (od), magnet height (h), and distance (d) between the magnets, to maximize the linear magnetic field strength between the magnets. In some embodiments, the "homothetic" property of the magnetic field generated by one or more permanent magnets was exploited, in that the spatial profile of the magnetic field (i.e., the spatial distribution of the magnetic field) does not change as the physical size of the magnets increases or decreases. In some embodiments, the "aspect ratio" (i.e., id:od:h:d) of the multiple magnets was optimized to maximize the magnetic field strength (see FIGS. 6A-6F and Table 2 for details). Results from these simulations suggest that in some embodiments, the magnetic field strength between the magnets may be linear with Bo up to about 0.4T or 0.5T. Although we have focused on the linear gradient in the gap between the magnets, the full extent of the linear gradient, in some embodiments, extends slightly into the cavity of the ring magnet, which for the configuration shown in FIG. 1B is approximately the same size as the inner diameter of the ring magnet.

[0107] 2A-2C show a design of a MagLev device using ring magnets, according to one or more embodiments. FIG. 2A shows a pair of indistinguishable magnets arranged coaxially with the same poles facing each other (ID:OD:H:Separation=1:3:1:0.6) and a diamagnetic object (sample) levitating in a paramagnetic medium, according to one or more embodiments. In these embodiments, the z-axis of the setup was directed against the gravity vector. FIG. 2B shows the spatial characteristics of the magnetic field in a vertical cross section through the central axis (the central axis overlaps with the vertical dotted line), according to one or more embodiments. The magnetic field strength Bz is shown along the vertical dotted line (in FIG. 2B), according to one or more embodiments. Here, a setup using two N45 grade NdFeB permanent magnets achieves Bo=~0.33T. In this plot, the z-axis is unitless.

[0108] In some embodiments, two NdFeB ring magnets of the same shape (76 mm outer diameter, 25 mm inner diameter, 25 mm height) were placed 15 mm apart. Without wishing to be bound by any particular theory, in some embodiments, it is believed that this configuration (i) produces a strong linear magnetic field (Bo=~0.33 T) between the magnets, (ii) has a large working distance (15 mm) between the magnets, (iii) is compact and therefore easy to observe the sample (unlike configurations using magnets with a large od to id ratio), (iv) uses commercially available magnets, and (v) uses relatively inexpensive magnets (~$50 per magnet). In some embodiments, 3D printed plastic housings, metal rods, and screws were used to mechanically fix the magnets in space (Figures 7A-7B). In some embodiments, rings of different sizes can be combined with one or more flux concentrators (e.g., soft iron plates) to create a magnetic field to levitate diamagnetic objects.

[0109] As shown in FIG. 8A, according to one or more embodiments, a single ring magnet with two magnetic minima along the central axis was used to levitate diamagnetic material in a paramagnetic medium. In these embodiments, it was found that the spatial characteristics of this magnetic field result in a nonlinear magnetic field along the left-right central axis, rather than a linear field as produced in the axial MagLev systems disclosed herein (e.g., the axial MagLev system of FIG. 1B). In some embodiments, the optimized linear magnetic field simplifies the process of calibrating and performing density measurements. Furthermore, in some embodiments, the axial MagLev configuration allows (i) easy addition of sample and / or paramagnetic medium through the opening of the vessel and easy removal of the sample (e.g., a subpopulation of small particles) suspended in the magnetic field from the vessel, (ii) viewing of the sample 360° around the sample vessel and from above and below, and (iii) facilitating density measurements of small samples (as small as a single sub-millimeter particle, as demonstrated).

[0110] Selection of paramagnetic media In some embodiments, aqueous solutions of paramagnetic species (e.g., MnCl2, MnBr2, CuSO4, GdCl3, DyCl3, HoCl3, Gd chelates (e.g., gadolinium(III) diethylenetriaminepentaacetate)) are used to levitate objects. In some embodiments, these paramagnetic species are inexpensive, transparent in the visible region of the spectrum (even at high concentrations), and commercially available. In some embodiments, MagLev also works with hydrophobic paramagnetic media and paramagnetic ionic liquids.

[0111] Selection of crosslinked polymeric materials and solvents for measuring swelling ratios In some embodiments, axial MagLev was used to measure the swelling ratio of crosslinked polymeric materials, particularly amorphous and small amounts of crosslinked polymeric materials in hydrophobic solvents, by measuring the density of both dried and fully swollen samples using aqueous MnCl2 suspension media.

[0112] In some embodiments, the swelling ratio (volume ratio) of a crosslinked polymeric material characterizes its tendency to swell upon solvent absorption (i.e., absorption of a solvent into the crosslinked network of the material). In some embodiments, this ratio reflects, in part, the crosslink density present in the polymeric material and the way in which the solvent interacts with the polymer chains. Thus, this ratio is a useful parameter for characterizing crosslinked polymers in various fields such as solid phase organic synthesis, development of superabsorbent materials, and use of polymeric materials for drug release.

[0113] In some embodiments, the swelling ratio of crosslinked polymeric materials can be measured using a variety of techniques, including graduated cylinders (e.g., to observe the bulk volume of a collection of particles), gravimetry (to measure the weight of a sample), optical microscopy (to measure the dimensions of a sample), and specialized equipment. However, these techniques, in some embodiments, can be tedious, require large amounts of sample (in the order of grams), and / or have limited compatibility with various sample types (e.g., irregularly shaped samples, powders, and delicate or gel-like materials).

[0114] In some embodiments, Axial MagLev was developed as a simple and versatile tool to measure the swelling ratio of cross-linked polymeric materials in solvents. In some embodiments, cross-linked PDMS was chosen as a model material. The swelling behavior of PDMS in various organic solvents was characterized (in the context of developing PDMS-based microfluidic devices).

[0115] In some embodiments, the axial MagLev technique used an aqueous solution of MnCl2 to levitate the sample. The solvent used to swell the PDMS sample is not dissolved in the aqueous solution. Of the 39 solvents characterized in previous studies, three exemplary hydrophobic solvents are studied here: chloroform, chlorobenzene, and toluene. In some embodiments, the use of a water-miscible solvent may require solubility compatibility between the sample (PDMS immersed in the solvent) and the suspension medium (e.g., using a hydrophobic Gd-chelate dissolved in the hydrophobic solvent).

[0116] calibration In some embodiments, hydrophobic organic liquids were used in the construction of the device because they (i) have known density, (ii) are available as small droplets (e.g., 1-2 mm diameter) facilitating precise positioning of the center of gravity (compared to commonly used irregularly shaped standard glass beads of ~4 mm), and (iii) are commercially available. In some embodiments, the solubility of water in hydrophobic solvents has negligible effect on the density of the solvent. For example, in some embodiments, the solubility of water in chlorobenzene is 0.3 mol% at room temperature, resulting in less than 0.01% density change in dissolved water.

[0117] 3A-3B show calibration of the MagLev device with a non-aqueous organic liquid, according to one or more embodiments. FIG. 3A shows a train of 5 drops (~3 mL) of organic liquid, according to one or more embodiments, being sequentially added via pipettor to a square cuvette and levitating in an aqueous solution of 0.5 M MnCl2. A calibration curve is shown obtained using a MnCl2 solution, according to one or more embodiments. In these embodiments, a ruler with millimeter graduations was placed next to the cuvette (reading ±0.1 mm) and a photograph of the droplet was taken with a digital camera. In some embodiments, the levitation height h is the distance between the center of gravity of the droplet and the top surface of the lower magnet (see FIG. 2A showing h). In some embodiments, the organic liquid used to calibrate the 3.0 M MnCl2 solution was cyclohexane (ρ=0.779 g / cm 3 ), fluorobenzene (ρ = 1.024 g / cm 3 ), dichloromethane (ρ = 1.325 g / cm 3 ), 1,1,2-trichlorotrifluoroethane (ρ = 1.57 g / cm 3 ), FC40(ρ=1.85g / cm 3 In these embodiments, the equation for the linear fit was h=-61.5ρ+71.9(R 2 = 0.9997), and for 3.0 M MnCl2, h = -11.4ρ + 21.9 (R 2 =0.9999). In these embodiments, all densities were taken from values ​​reported by Sigma-Aldrich. Data are plotted as mean ± standard deviation (N=7 for all measurements). Error bars are much smaller than the size of the symbols and therefore not visible on the plots. In some embodiments, the floating height of the droplets did not change over a 30 minute period.

[0118] Measurable density range In some embodiments, a concentrated aqueous solution of 3.0 M DyCl containing 0.01% (v / v) Tween-20 was used to generate a 100% aqueous solution of air bubbles (ρ≈0 g / cm) as shown in FIG. 3 ) to zirconium silicate (ρ ≒ 3.7 g / cm3 ) were levitated. In these embodiments, the samples were all spherical, except for an irregularly shaped aluminum sample (cut from a 250 mm thick sheet of foil). In some embodiments, a square cuvette (path length 1 cm) was used to levitate the samples, and cyclohexane (ρ = 0.779 g / cm) was used to calibrate the device. 3 ), dichloromethane (ρ = 1.325 g / cm 3 ), 1,2-dibromoethane (ρ = 2.18 g / cm 3 ), tribromoethane (ρ = 2.89 g / cm 3 ) solvent was used. In these embodiments, a linear fit of the data gave h=-3.43ρ+12.9 (R2=0.9999). In some embodiments, the reported densities were obtained from the commercial supplier (e.g., McMaster-Carr, Sigma-Aldrich, etc.) from which the material was purchased. N=7 measurements were made.

[0119] In some embodiments, concentrated paramagnetic salts (e.g., DyCl) are prepared by dissolving paramagnetic salts with high magnetic susceptibility (DyCl=5.5×10 -7 m 3 / mol MnCl2 = 1.83 x 10 -7 m 3 / mol) and a steep gradient in magnetic field strength (~43 T / m in axial MagLev vs. 17 T / m in normal MagLev) allows for a wide range of densities (~0 g / cm 3 ~3.7g / cm 3 ) in comparison, previous studies have shown that using aqueous solutions of MnCl2 and conventional MagLev devices, 3 ~3g / cm 3 In some embodiments, tilted MagLev (a variant of conventional MagLev in which the device is tilted with respect to the gravity vector and the sample is levitated along the central axis of the device while resting partially on the wall of the sample container) can achieve the full range of densities observed in materials at room temperature (~0 g / cm). 3 ~23g / cm3 ) can be measured. In some embodiments, the axial MagLev system disclosed herein demonstrates a wider density range than conventional MagLev, is very experimentally convenient, and avoids some of the potential problems of tilted MagLev (e.g., sample resting on the wall of the sample vessel).

[0120] Sample addition and / or removal procedures In some embodiments, an axial MagLev calibration can facilitate adding and / or removing a sample from a vessel. Figures 5A-5C illustrate these steps, according to one or more embodiments, by the floating of "sticky" or viscous samples (a drop of PDMS prepolymer and a plug of Vaseline® gel). This type of sample is difficult to measure with conventional and tilted MagLev devices because they tend to adhere to the vessel walls and the liquid-air interface. The axial MagLev configuration, according to one or more embodiments, also allows for easy removal of a piece of sample of interest (e.g., from a collection of particles).

[0121] In some embodiments, a test tube containing a paramagnetic medium was placed in the axial MagLev device and a "sticky" sample was added to the vessel from above. In these embodiments, the sample entered the medium by gravity and floated within it (near instantaneously for mm-sized samples, Figures 5A-5C).

[0122] 5A-5B show a procedure for adding and / or removing a sample, according to one or more embodiments. FIG. 5A shows a droplet of PDMS prepolymer (doped with graphite powder for visualization) floating in an aqueous solution of 0.5 M MnCl2, according to one or more embodiments. In some embodiments, the droplet appears elliptical because it is visually distorted due to the curvature of the walls of the cylindrical test tube. FIG. 5B shows a plug of Vaseline® gel extruded from a syringe floating in an aqueous solution of 3.0 M MnCl2 with 0.1% (v / v) Tween-20, according to one or more embodiments. FIG. 5C shows the removal of a subpopulation of density standard beads, according to one or more embodiments. In some embodiments, the removed beads floated to the same height as when they were previously removed (see third panel). In these embodiments, the suspension contained 0.5 M MnCl2, 1.4 M NaCl (a diamagnetic co-solute used to match the density of the particles and the liquid), and 0.1% (v / v) Tween-20. The arrows shown in Figure 5C indicate the density (in g / cm) of the particles according to one or more embodiments. 3 ) shows the spread of

[0123] In some embodiments, removing samples from a vessel placed in the MagLev device is easy due to the axial configuration. Density standards (polyethylene particles, diameter ∼200 μm, nominal density 1.13 g / cm 3 This was demonstrated according to one or more embodiments using a 90 degree bent glass Pasteur pipette inserted into the top of a sample container (a test tube with a diameter of ∼25 mm) and a subpopulation of particles floating in an aqueous MnCl2 solution was removed. In certain embodiments, the remaining sample remained floating and was not disturbed. In certain embodiments, the removed fragments floated to the same height in the same medium when placed back in the device (Figure 5C), as expected, but with a ∼5-fold narrower density distribution (thus improving precision). The axial MagLev disclosed herein, in certain embodiments, provides a simple method for preparing high quality density standards.

[0124] Swelling ratio of crosslinked polymers in solvents In some embodiments, PDMS was chosen as a model cross-linked polymer to demonstrate the use of axial MagLev to characterize the swelling behavior of cross-linked polymers in solvents. In certain embodiments, a small piece of PDMS (1.5 mm diameter, ~1 mm thickness, prepared by a 1.5 mm biopsy punch) was soaked in the solvent for 24 hours. In certain embodiments, the sample was blotted dry and added to the MagLev device, and density measurements were performed using a 1.5 M MnCl2 aqueous solution containing 0.1% (v / v) cetyltrimethylammonium bromide (a surfactant to remove air bubbles). In some embodiments, MagLev can directly measure the density of a sample, independent of the volume (or shape) of the sample. In some embodiments, the measured density was converted to a volumetric swelling ratio (see Table 1) using Equation 1: TIFF0007612570000001.tif1690

[0125] In some embodiments, in Formula 1, V sp is the volume of the swollen sample, V p is the volume of the dry sample, and ρ p is the density of the sample, and ρ sp is the density of the swollen sample, and ρ s is the density of the solvent.

[0126] In some embodiments, the diameters of dry and swollen samples from the same image were measured using the flying height to estimate the swelling ratio. In some embodiments, the agreement of the results (within 10% deviation) demonstrated the performance of the MagLev technique. In some embodiments, deviations from literature values ​​may be due to differences in sample preparation. In some embodiments, this demonstration also highlights the simplicity and compatibility of the MagLev technique, which does not require sophisticated tools (e.g., microscopes) to measure small samples. Table 1. Swelling ratios of PDMS samples in specific solvents TIFF0007612570000002.tif39164a.N=3, average of 3 measurements b. The volume swelling ratio is calculated using the formula f = (D / D0) 3 where D is the diameter of the swollen PDMS disk and D0 is the diameter of the dry PDMS disk. The PDMS sample was assumed to swell isotropically in the solvent. N=3. Swelling ratios of PDMS pieces reported in c.Lee et al. "Solvent Compatibility of Poly(dimethylsiloxane)-Based Microfluidic Devices" 2003, Vol.75, pp.6544-6554. The following ratios of the length of the swollen PDMS sample to the length of the dry sample were used: chloroform 1.39, chlorobenzene 1.22, toluene 1.31. These values ​​were converted to volumetric swelling ratios.

[0127] In some embodiments, axial MagLev utilizes an axially symmetric magnetic field generated by two ring magnets with opposing like poles to perform density-based analysis, separation, and manipulation. In some embodiments, this configuration of MagLev removes the physical barrier along the central axis of conventional MagLev devices, where the sample vessel (e.g., a rectangular cuvette or capillary tube) is physically sandwiched between two block magnets. In some embodiments, axial MagLev has four useful features: (i) Axial MagLev allows easy access to the levitating sample and paramagnetic medium, allowing easy addition or removal of sample or suspending medium; (ii) Axial MagLev provides complete clearance (360°) around the sample vessel, allowing the levitating sample to be observed, and further allowing easy observation of the sample from above and below. (iii) Axial MagLev does not limit the height of the sample container and further accommodates a wide variety of container types (e.g., cuvettes, vials, test tubes, graduated cylinders) that fit within the inner diameter of the ring magnet, and (iv) Axial MagLev focuses small and / or dilute particles along a common vertical axis to aid in their visualization and manipulation.

[0128] In some embodiments, the axial MagLev generates a linear magnetic field between the magnets. In certain embodiments, with appropriate configuration, a non-linear magnetic field may be used to perform density-based measurements and separations. In certain embodiments, the working distance between the two magnets can be extended to at least ∼38 mm while the magnetic field gradient (and therefore density) is monotonically varied.

[0129] 6A-6F show the selection of the ring magnet shape and the separation between the ring magnets using Comsol® simulations, according to one or more embodiments. FIG. 6A is a schematic diagram showing four independent physical parameters (od: outer diameter, id: inner diameter, h: magnet height, d: separation) used to define the configuration of the device, according to one or more embodiments. The parameter combinations considered using Comsol® simulations in these embodiments are shown in Table 2. FIG. 6B shows an example of determining a linear gradient of maximum magnetic field strength of ∼0.4 T along the central axis between the magnets (dashed line in inset), according to one or more embodiments. FIG. 6C shows an example of a non-linear gradient of maximum magnetic field strength of ∼0.5 T along the central axis between the magnets (dashed line in inset), according to one or more embodiments. FIG. 6D shows fine tuning of the separation between two magnets (od: 3, id: 1, h: 1), according to one or more embodiments. FIG 6E shows the magnetic field strength Bz along the central axis between two magnets for separations ranging from 0.2 to 2 in increments of 0.2, according to one or more embodiments. FIG 6F shows a plot of the curve from FIG 6E at d=0.6. In some embodiments, the parameters are all unitless, since the aspect ratio of these parameters (d=1) was focused on optimizing the magnetic field shape.

[0130] Density is a universal property of all materials, and simple, inexpensive, and useful MagLev devices, such as the axial MagLev device described herein, may expand the utility of MagLev-based density analysis in chemistry, biochemistry, and materials science. In some embodiments, the compact design, portability, affordability, and ease of use of the axial MagLev device allow for characterization of materials (e.g., swelling behavior of cross-linked polymeric materials in solvents), separation of samples (e.g., especially those with small amounts, such as crystals), and manipulation of samples (e.g., hard, soft, and sticky, such as gels) without physical contact. Table 2. Parameters examined using Comsol® simulation TIFF0007612570000003.tif45164

[0131] material In some embodiments, polyamideimide (Torlon®) and polytetrafluoroethylene (Teflon®) were obtained from McMaster-Carr. In some embodiments, aluminum foil was obtained from Sigma-Aldrich. In some embodiments, zirconium silicate was obtained from Cospheric LLC. In some embodiments, the density of air was obtained from the CRC Handbook of Chemistry and Physics, 98th Edition.

[0132] Correlation between swelling ratio and density In some embodiments, Equation S1 represents the density of a crosslinked polymer sample in a dry state, Equation S2 represents the density in a swollen state, and Equation S3 represents the swelling ratio of the sample. TIFF0007612570000004.tif57110

[0133] In formulas S1 to S3, m p is the mass of the sample, m s is the mass of solvent present in the swollen sample, and V sp is the volume of the swollen sample, V p is the volume of the dry sample, and ρ p is the density of the sample, and ρ sp is the density of the swollen sample, and ρ s is the density of the solvent.

[0134] In some embodiments, for swollen samples, the mass of solvent is represented by Equation S4. TIFF0007612570000005.tif1297

[0135] In some embodiments, equations S1-S4 are solved to obtain equation S5, which describes the relationship between swelling ratio and density. TIFF0007612570000006.tif17104

[0136] Although one or more specific materials or steps have been shown and described for purposes of illustration, it will be understood that the materials or steps can be varied in specific respects or the materials or steps can be combined to produce desired results. Furthermore, modifications of the disclosed embodiments and claimed invention are possible and are within the scope of this disclosed invention.

Claims

1. 1. A magnetic levitation system for levitating a diamagnetic or paramagnetic sample, comprising: A first cylindrical magnet; A second cylindrical magnet arranged coaxially with the first cylindrical magnet; a first cavity formed within the first cylindrical magnet and coaxially aligned with the first cylindrical magnet; a container configured to hold a paramagnetic solution; The surface of the first cylindrical magnet has a polarity and faces the surface of the second cylindrical magnet having the same polarity as the polarity in a parallel manner, and the Bz component of the magnetic field, which is a magnetic field component in a direction parallel to the central axis, is changed linearly along the central axis between the first cylindrical magnet and the second cylindrical magnet. Magnetic levitation system.

2. The system of claim 1 , wherein the first cavity is cylindrical.

3. The system of claim 1 , wherein the first cavity spans the entire height of the first cylindrical magnet.

4. The system of any one of claims 1 to 3, wherein the linear magnetic field extends into the first cavity.

5. The system of claim 1 , wherein the first cylindrical magnet and the second cylindrical magnet have different radii or heights.

6. The system of any one of claims 1 to 4, wherein the first cylindrical magnet and the second cylindrical magnet have the same radius, height, or both.

7. The system of any one of claims 1 to 6, further comprising a second cavity formed in the second cylindrical magnet and coaxially aligned with the second cylindrical magnet.

8. The system of claim 7 , wherein the second cavity is cylindrical.

9. the distance between the surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is d; The height of the first cylindrical magnet and the second cylindrical magnet is h １ and ２ and h １ and h ２ The system of any one of claims 1 to 8, wherein the ratio of at least one of: to d is from about 0.2:1 to about 10:

1.

10. h １ and h ２ 10. The system of claim 9, wherein a ratio of at least one of is equal to d is about 1.67:

1.

11. The inner diameter of the first cylindrical magnet is i.d. １ and id １ and d is from about 0.2:1 to about 10:1; A system according to claim 9 or claim 10.

12. id １ 12. The system of claim 11, wherein the ratio of d to is about 1.67:

1.

13. The outer diameter of the first cylindrical magnet is od １ and od １ and d is from about 0.3:1 to about 100:1; A system according to any one of claims 9 to 12.

14. od １ 14. The system of claim 13, wherein the ratio of n to d is about 5:

1.

15. a second cylindrical cavity formed in the second cylindrical magnet and coaxially aligned with the second cylindrical magnet; The inner diameter of the second cylindrical magnet is i.d. ２ and id ２ and d is from about 0.2:1 to about 10:1; A system according to any one of claims 9 to 14.

16. id ２ 16. The system of claim 15, wherein the ratio of d to n is about 1.67:

1.

17. a second cavity formed in the second cylindrical magnet and coaxially aligned with the second cylindrical magnet; The outer diameter of the second cylindrical magnet is od ２ and od ２ and d is from about 0.3:1 to about 100:1; A system according to any one of claims 9 to 16.

18. od ２ 20. The system of claim 17, wherein the ratio of n to d is about 5:

1.

19. The first cavity is cylindrical, and the inner diameter (i.d. １ ): Outer diameter (od １ ): Height (h １ 20. The system of claim 1, wherein the ratio of the distance (d) between the surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is approximately 1.67:5:1.67:

1.

20. a second cylindrical cavity formed in the second cylindrical magnet and coaxially aligned with the second cylindrical magnet; The inner diameter (id) of the second cylindrical magnet ２ ): Outer diameter (od ２ ): Height (h ２ ): the ratio of the distance (d) between the surfaces of the same pole of the first cylindrical magnet and the second cylindrical magnet is about 1.67:5:1.67:1; A system according to any one of claims 1 to 19.

21. The system of any one of claims 1 to 20, wherein the container configured to hold a paramagnetic medium is at least partially disposed between the first cylindrical magnet and the second cylindrical magnet.

22. 22. The system of claim 21, wherein the vessel further comprises an inlet configured to allow the addition or removal of the paramagnetic medium or paramagnetic or diamagnetic sample.

23. 23. The system of claim 21 or claim 22, wherein the container comprises two ends, at least one of which extends into or through the first cavity.

24. a second cavity formed in the second cylindrical magnet and coaxially aligned with the second cylindrical magnet; The other of the two ends of the container extends into or through the second cylindrical cavity.

24. The system of claim 23.

25. The system according to any one of claims 21 to 24, wherein the container is a cuvette.

26. The system of any one of claims 21 to 25, wherein the paramagnetic medium comprises an aqueous solution of a paramagnetic compound.

27. The paramagnetic compound is MnCl ２ , MnBr ２ , CuSO ４ , GdCl ３ , DyCl ３ , HoCl ３ 27. The system of claim 26, wherein the anti-inflammatory agent is selected from the group consisting of a Gd chelate compound, and combinations thereof.

28. 28. The system of claim 27, wherein the Gd chelate is gadolinium(III) diethylenetriaminepentaacetic acid.

29. The paramagnetic medium is MnCl ２ The system of any one of claims 21 to 27, comprising an aqueous solution.

30. The system of any one of claims 21 to 24, wherein the paramagnetic medium comprises a hydrophobic paramagnetic medium or a paramagnetic ionic liquid.

31. The system of any one of claims 1 to 30, wherein a maximum magnetic field along a common axis of the first cylindrical magnet and the second cylindrical magnet is about 0.20 to 0.50 T.

32. The system of any one of claims 1 to 31, wherein the first cylindrical magnet and the second cylindrical magnet are NdFeB magnets.

33. A second cylindrical cavity may be provided coaxially aligned with the second cylindrical magnet over the entire height of the second cylindrical magnet, The inner diameter of the first cylindrical magnet or the second cylindrical magnet is about 15 mm to about 40 mm. A system according to any one of claims 1 to 32.

34. 34. The system of claim 33, wherein an inner diameter of the first cylindrical magnet or the second cylindrical magnet is about 25 mm.

35. A second cylindrical cavity may be provided coaxially aligned with the second cylindrical magnet over the entire height of the second cylindrical magnet, The outer diameter of the first cylindrical magnet or the second cylindrical magnet is about 50 mm to about 100 mm. A system according to any one of claims 1 to 34.

36. 36. The system of claim 35, wherein an outer diameter of the first cylindrical magnet or the second cylindrical magnet is about 76 mm.

37. The system of any one of claims 1 to 36, wherein the height of the first cylindrical magnet or the second cylindrical magnet is between about 15 mm and about 50 mm.

38. 38. The system of claim 37, wherein the height of the first cylindrical magnet or the second cylindrical magnet is about 25 mm.

39. 39. The system of claim 1, wherein the distance between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet is between about 5 mm and about 50 mm.

40. 40. The system of claim 39, wherein the distance between like-polarized surfaces of the first cylindrical magnet and the second cylindrical magnet is about 15 mm.

41. 1. A method for analyzing a diamagnetic or paramagnetic sample, comprising: Providing a system according to any one of claims 1 to 40; placing the container configured to hold a paramagnetic medium between the first cylindrical magnet and the second cylindrical magnet; adding a paramagnetic medium and a diamagnetic or paramagnetic sample, separately or together, to the vessel; levitating the diamagnetic or paramagnetic sample under the linear magnetic field between the first cylindrical magnet and the second cylindrical magnet; A method comprising:

42. 42. The method of claim 41, wherein the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is added via a first cylindrical cavity.

43. 43. The method of claim 41 or claim 42, further comprising the step of removing the paramagnetic medium or the paramagnetic sample or the diamagnetic sample.

44. 44. The method of claim 43, wherein the paramagnetic medium or the paramagnetic sample or the diamagnetic sample is removed through a first cylindrical cavity.

45. The method of any one of claims 41 to 44, wherein the paramagnetic medium comprises an aqueous solution of a paramagnetic compound, a hydrophobic paramagnetic medium, or a paramagnetic ionic liquid.

46. The paramagnetic compound is MnCl ２ , MnBr ２ , CuSO ４ , GdCl ３ , DyCl ３ , HoCl ３ 46. ​​The method of claim 45, wherein the anti-inflammatory agent is selected from the group consisting of a Gd chelate compound, and a combination thereof.

47. 47. The method of claim 46, wherein the Gd chelate is gadolinium(III) diethylenetriaminepentaacetic acid.

48. The paramagnetic medium is MnCl ２ The method of any one of claims 41 to 46, comprising an aqueous solution.

49. The method of any one of claims 41 to 48, wherein the paramagnetic medium further comprises cetyltrimethylammonium bromide.

50. The method of any one of claims 41 to 48, wherein the paramagnetic or diamagnetic sample comprises a cross-linked polymer.

51. 51. The method of claim 50, further comprising determining a density of the cross-linked polymer.

52. generating a standard curve correlating the flying height of a sample in the system with its density; determining the density of the cross-linked polymer using the standard curve and the flying height in the system; 52. The method of claim 51 further comprising:

53. The method of any one of claims 50 to 52, wherein the cross-linked polymer comprises polydimethylsiloxane.

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