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  • S1

    Magnetic Levitation in Chemistry, Materials Science, and Biochemistry

    Supporting Information

    Shencheng Ge,1 Alex Nemiroski, 1 Katherine A. Mirica, 1 Charles R. Mace, 1 Jonathan W.

    Hennek,1 Ashok A. Kumar, 1 and George M. Whitesides1,2,3*

    1 Department of Chemistry & Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA

    02138, USA

    2 Wyss Institute for Biologically Inspired Engineering, Harvard University, 60 Oxford Street, Cambridge,

    MA 02138, USA

    3 Kavli Institute for Bionano Science & Technology, Harvard University, 29 Oxford Street Cambridge,

    MA 02138, USA

    * Corresponding author: George M. Whitesides (gwhitesides@gmwgroup.harvard.edu)

  • S2

    Table of Contents

    S1. Magnetism Relevant to MagLev........................................................................................................... S3

    S1.1 Diamagnetism ............................................................................................................................. S3

    S1.2 Paramagnetism ............................................................................................................................ S4

    S1.3 Ferromagnetism .......................................................................................................................... S5

    S1.4 Superparamagnetism ................................................................................................................... S5

    S2. Qualitative Characteristics of MagLev ................................................................................................. S6

    S2.1 Basic Principles of MagLev ........................................................................................................ S6

    S2.2 What is Important? ρ, Δρ, χ, Δχ, B, ∇B, g, V, … ........................................................................ S8

    S3. The “Standard” MagLev System and its Quantitative Description ...................................................... S8

    S3.1 Relating Density to the Height of Levitation .............................................................................. S9

    S3.2 Approximate and Exact Solutions for a Spherical Object ........................................................ S11

    S4. Error Analysis ..................................................................................................................................... S12

    S5. Theoretical Guide to Adjust Sensitivity and Dynamic Range ............................................................ S14

    S6. Understanding and Controlling Orientation of Levitated Objects ...................................................... S15

    S7. Quality Control: Heterogeneity in Density in Injection-Molded Parts ............................................... S17

    Figure S1. .................................................................................................................................................. S19

    Figure S2. .................................................................................................................................................. S20

    Figure S3. .................................................................................................................................................. S22

    Table S1. ................................................................................................................................................... S24

  • S3

    S1. Magnetism Relevant to MagLev

    S1.1 Diamagnetism

    Diamagnetism is a characteristic of all matter, but is only apparent in materials with no

    unpaired electrons. If, however, a molecule or material contains unpaired electrons, then its

    interactions with an applied magnetic field may dominate diamagnetism. To understand the

    origin of diamagnetism, consider a simplified classical picture of an electron orbiting the nucleus

    of an atom. As a charge in motion, this electron generates a magnetic field. When an external

    magnetic field is applied, the electron alters its motion to oppose the change of field (Lenz’s

    law).[1] The consequence of this effect is induced magnetization in a substance that opposes the

    applied field; this molecular-level response to an applied magnetic field is called diamagnetism.

    The effect of diamagnetism is, thus, universal for all matter. Most organic materials are

    diamagnetic. Examples of organic diamagnetic materials include water, most organic liquids,

    typical biological polymers such as proteins (those that do not contain transition metals), DNA,

    and carbohydrates, and most synthetic or semi-synthetic polymers. A few representative

    exceptions include stable free radicals (e.g., trityl and nitroxyls), O2, many organometallic

    compounds and chelates of transition metals; these may be paramagnetic (see Section S1.2).

    A measure of the type and magnitude of magnetization of a material in response to an

    applied magnetic field is the magnetic susceptibility,  𝑣 (volume susceptibility, unitless), as

    defined in eq S1, where M⃗⃗⃗ (A m-1) is the magnetization of the material, H⃗⃗ (A m-1) is the applied

    external magnetic field. Eq S2 describes the closely related magnetic field B⃗⃗ (T) present in a

    material, where 𝜇0 is the magnetic permeability in vacuum.

     𝑣 = M⃗⃗⃗ /H⃗⃗ (S1)

  • S4

    B⃗⃗ = 𝜇0(H⃗⃗ + M⃗⃗⃗ ) = 𝜇0(1 + 𝑣)H⃗⃗ (S2)

    The magnetic susceptibility of typical diamagnetic materials is around -10-5 (in SI unit, the

    negative sign indicates that the induced magnetization opposes the applied field), and is

    essentially indistinguishable for many materials, including some of the metals and the majority

    of the organic materials (Figure 2). Bismuth and pyrolytic graphite are notable exceptions, and

    are up to two orders of magnitude more diamagnetic than common diamagnetic materials.

    Because the induced magnetization of diamagnetic materials is negative and small, they are

    slightly repelled from regions of magnetic field. This repulsion can be sufficient to suspend the

    most diamagnetic materials (e.g., pyrolytic graphite) against gravity in air (and even in vacuum)

    using permanent magnets, while it is not noticeable when common diamagnetic materials

    interact with the modest field of the permanent magnets in air (although they can be important

    with the much higher fields of superconducting magnets). The magnetic susceptibility of most

    diamagnetic materials is independent of temperature over commonly encountered ranges.[2]

    S1.2 Paramagnetism

    In one view of paramagnetism, it originates from the unpaired electrons present in the

    material that produces permanent magnetic moments. In the absence of an applied magnetic

    field, the spins of the unpaired electrons are randomly oriented in space and time due to thermal

    energy: background magnetic fields—a field gradient—(e.g., from the earth) introduce

    interactions far weaker that those from thermal motions. The presence of an external magnetic

    field thus tends to align (weakly) the magnetic moments in the direction of the applied field. The

    response of paramagnets to an applied field at room temperature is typically three orders of

    magnitude greater

  • S5

    than that of diamagnets. They are attracted to, rather than repelled by, the applied field. Typical

    magnetic susceptibilities of paramagnets are ~ 10−3−10−5 (in SI unit). Paramagnetic species

    relevant to the type of MagLev we discuss in this review include simple paramagnetic salts, such

    as MnCl2, GdCl3, HoCl3, DyCl3, CuCl2, and FeCl2, and the chelates of some of these ions (e.g.,

    Gd3+). See Section 3.3 for a more detailed discussions.

    S1.3 Ferromagnetism

    Ferromagnetism is the property of a material that exhibits spontaneous (and permanent)

    magnetic moments; that is, the material has a high magnetic moment even in the absence of an

    external field.[1] Ferromagnetism only occurs in materials that contain strongly interacting

    unpaired spins. These spins in the material interact in such a way that they align with each other

    in the same aligned direction in localized regions—termed magnetic domains. In the apparently

    “unmagnetized” ferromagnetic materials, the magnetic moments of the magnetic domains are

    disordered and thus, effectively cancel. An external magnetic field can align the magnetic

    domains in the materials, and the collective alignment of spins in magnetic domains produces a

    net magnetic moment. Strong magnetic moments remains in ferromagnetic materials when the

    applied magnetic field is removed. Common ferromagnetic materials are iron, iron oxides (e.g.,

    magnetite), cobalt, nickel, their alloys (e.g., Alnico), and, importantly, alloys containing rare-

    earth metals (e.g., NdFeB and SmCo). Permanent NdFeB magnets that enable the type of

    MagLev we describe in this review are ferromagnetic.[3,4]

    S1.4 Superparamagnetism

    Superparamagnetic materials behave qualitatively similarly to paramagnetic materials in

    an applied magnetic field