Magnetic levitation of nanoscale materials: the critical role of effective density

The magnetic levitation (MagLev) of diamagnetic materials in a paramagnetic solution is a robust technique for the density-based separation, measurements, and analysis of bulk materials/objects (e.g., beads and plastics). There is a debate in the literature, however, about whether a MagLev technique is reliable for the separation and/or density measurements of nanoscale objects. Here, we show that MagLev can levitate nanoparticles; however, the transition from the bulk to an ‘effective’ density must be acknowledged and considered in density measurements at the nanoscale regime. We performed a proof-of-concept study on MagLev’s capability in measuring the ‘effective density’ of multiscale silver particles (i.e. microparticles, nanopowder, and nanoemulsion). In addition, we probed the effective density of nanoscale biomolecules (e.g. lipoproteins) using a standard MagLev system. Our findings reveal that the MagLev technique has the capability to measure both bulk density (which is independent of the size and dimension of the material) and the effective density (which takes place at the nanoscale regime and is dependent on the size and surrounding paramagnetic solution) of the levitated objects.


Introduction
It is well-documented that magnetic levitation (MagLev) is a robust, low-cost, accurate, and rapid technique for the separation and density measurement of diamagnetic bulk objects/materials (i.e.∼ 1 µm and larger) in (super)paramagnetic liquids [1][2][3][4].However, there has been an extensive debate Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
in the literature regarding the capabilities of MagLev technique for levitation and density measurements of submicron and nanoscale objects (i.e.smaller than ∼100 nm in size) [5][6][7].This ongoing debate can be attributed to two primary factors: (i) the critical role of negligible parameters at the bulk scale which become substantial at the nanoscale, including Brownian motion; and (ii) the lack of consideration of effective density, and not bulk density, in the MagLev measurements.For example, with regard to Brownian motion, it has been experimentally revealed that the density of particles with a size of ∼ ⩽2 µm could not be estimated/measured by MagLev if the particles are uniformly distributed/dissolved in the paramagnetic solution, as Brownian motion prevents the accumulation of particles of the desired height in a reasonable amount of time [7].In addition to Brownian motion, many other negligible forces at microscales become considerable at the submicron and nanoscale regimes [8][9][10][11].More specifically, magnetic and gravitational forces are the two major and dominant forces acting on the object during the levitation of bulk materials, while during the levitation of nanoscale objects, the effects of several other forces such as electrostatic interactions, thermal fluctuations, van der Waals forces, hydrogen bonding, and hydrophilic/hydrophobic interactions are not ignorable, as the magnitude of some of these forces becomes comparable with magnetic and gravitational forces [12][13][14][15].While in lower dimensions, electrostatic interactions become important due to the large surface area of nanoscale objects and their possible surface charges (equation ( 1)), the effect of gravitational forces becomes less considerable due to a significant decrease in the mass of the nanomaterials and consequently the gravitational forces (equation ( 2)).
The relative strength (k/G) of gravitational and electrostatic forces is related to the ratio of Coulomb's constant k (9 × 10 9 Nm 2 C −2 ) to the gravitational constant G (6.674 × 10 −11 Nm 2 kg −2 ) which is many orders of magnitude greater in electrostatic interactions at lower size scales.Therefore, a simple mechanical equilibrium equation including only gravitational and magnetic forces (equation ( 3)) during levitation of bulk materials may not be valid for the levitation of nanoscale materials, and a new equilibrium equation is needed that includes all possible forces present at the lower size scale acting on the object. (3) In fact, the size of the objects has a significant effect on levitation time and (effective) density measurements in MagLev systems, which has not been appreciated so far [7].For instance, when the effect of thermal fluctuations and/or Brownian motion is greater than the net gravitational and magnetic forces of the MagLev, one may expect a longer separation time, particularly for particles smaller than ∼1 µ ms (e.g. for most live organisms and biological species the levitation time is ∼ 2-24 h) [16,17].Slightly larger objects than a few micrometers take less time, but a significant time to separate (e.g.hours for ∼10 µm objects and around 30 min for ∼50 µm objects), mainly because their densities transform back from the effective to the bulk regime [18,19].This limitation may prevent the practical application of MagLev in various areas (e.g.biology and point of care diagnosis) where rapid separation of small biomolecules and less exposure to toxic environments (i.e.high concentrations of the employed paramagnetic salts) are critical.
Prior experimental and theoretical studies have shown that the MagLev technique does not reliably measure the bulk densities of objects smaller than ∼2 µm in radius; instead, such particles/objects form a diffuse cloud in conventional MagLev systems, mainly due to the considerable role of Brownian motion [7].We would like to emphasize once more that, at the sub-micron scale, MagLev technique is expected to provide the effective density of the objects and not their bulk density.This is mainly because the number and type of the forces acting on the objects/nanoparticles are totally different to the forces acting on levitating bulk materials and may be different from one nanoparticle to another due to the various physicochemical properties of nanoscale materials and the consequently different interactions with the surrounding solution [20,21].
Another critical challenge during the levitation of nanoscale objects, which is negligible during the levitation of bulk materials, is their high interaction with the surrounding medium (i.e.paramagnetic solutions) due to their large surface area, chemical composition, shape, and surface chemistry [20,22,23].Any changes and interactions that occur at the particle-liquid interface can significantly affect the levitation profile of the object and, consequently, the density calculations [11,24,25].Such dynamic interactions lead to the wrapping of nanoparticles with water molecules or any other available solvent that greatly alters their density value as compared to their bulk compartments.In addition, the presence of water may cause the formation of hydrogen bindings and/or change the hydrophilicity/hydrophobicity of the nanoparticles, which alters the equilibrium state and levitation profile of the nanoparticles and, consequently, their effective density values.It is reported that hydration shells surrounding nanoparticles in solution considerably affect their physicochemical properties, including electronic and magnetic properties [26].It is also found that hydrated nanoparticles reveal different behaviors compared with non-hydrated nanoparticles in bulk acquired solutions indicating the significant effect of the interactions of nanoscale objects with a surrounding liquid [26][27][28].
In this study, we probe the capability of MagLev technique to levitate submicron objects and identify their effective density by levitating multiscale silver particles, as an example of a non-biological sample, together with high-density lipoproteins (HDL) and very low-density lipoproteins (VLDL), as examples of nanoscale biological samples; and monitor their corresponding levitation profiles versus their sizes.The results highlight the critical effect of size on the levitation profile, and therefore the effective density, of both biological and non-biological samples.

Materials
As one of the most common paramagnetic solutions in MagLev systems, MnCl2 (sigma Aldrich) at different concentrations was used to levitate particles of various sizes [12,29].Gadobutrol, also known as gadavist, is used to levitate HDL and VLDL proteins, as we have already shown that proteins get denatured in common paramagnetic solutions such as MnCl 2 or GdCl 3 that have been used for levitation of a wide range of non-biological materials.Fluorescent polyethylene microparticles (see, for example, figures S1 and S2 of the supplementary information) with known densities and standard density solid glass beads were obtained for the calibration of our MagLev systems from Cospheric (www.cospheric.com) and American density materials (www.americandensitymaterials. com), respectively.

MagLev platform
To test the effect of size on the density measurement among all different types of MagLev systems, standard and ring MagLev systems are used in the present work as depicted in figures 1 and 4 [9,30].The standard MagLev system setup includes two rectangular blocks of neodymium permanent magnets (N42grade, NdFeB) (25.4 mm length, 25.4 mm width, and 50.8 mm height) with a 2.5 cm separation distance and the N poles facing each other.Disposable 4 ml plastic cuvettes, were cut to 2.5 cm and used as a levitation container.The ring setup includes two blocks of neodymium ring-shaped permanent magnets (7.62 mm outer dimeter, 25.4 mm inner diameter, and 25.4 mm thickness), with a 1.5 cm distance of separation and N poles facing each other.Disposable glass test tubes with a diameter of 12 mm and a length of 75 mm were used as levitation containers (figure 4).

Characterization
Two blocks of cubic-shaped NdFeB permanent magnets (grade N42, Model # NB044) were provided from mag-net4less.A Gauss meter (vector/magnitude Gauss meter model VGM, Alphalab) was used to measure the magnetic field strength between the magnets (∼0.51 T at the surface of the magnets of the standard MagLev system).Levitation profiles of the particles/objects were recorded by a Nikon D750 digital camera containing a 105 mm Nikkor Microlense.

Results and discussions
It has been reported that spherical particles with diameter of ∼2 µm is the lower limit size of objects that have a stable levitation profile and reliable correlation of levitation height with bulk density [7].For all density measurement purposes, MagLev systems need to be initially calibrated using commercially available standard density particles [29,31,32].Figure 2(a) shows the calibration of the standard MagLev system used in this study at different concentrations of MnCl 2 .As is clear from the standard calibration curves there is a linear relationship (equation ( 4)) between the levitation heights of the standard density particles and their corresponding density, which can be used for density measurements, as described in detail in prior reports from our own and other groups [5,9,20,25,33]: The calibration curve presented in figure 2(a) clearly demonstrates the linearity of the magnetic field gradients and dynamic range of density in a standard MagLev system and different concentrations of paramagnetic solution.The dynamic range of density is the difference between the minimum and maximum density of materials/objects that can be levitated in the MagLev system without floating on top or sinking to the bottom of the paramagnetic solution (figures 2(b) and (c)) [32,34,35].The strength of the magnetic fields and concentration of paramagnetic solutions are the two crucial factors associated with the dynamic range of bulk or effective density in the MagLev systems [36,37].It is clear from figure 2(d) that the dynamic range of density (using standard density glass beads) at high concentrations of paramagnetic solution (i.e. 3 M MnCl 2 ) is approximately between 1.05 g cm −3 and 1.57 g cm −3 .This illustrates the fact that it is not possible to levitate objects/materials with densities higher than ∼1.6 g cm −3 using a standard MagLev system and 3 M concentration of paramagnetic solution.Therefore, all materials with densities higher than 1.6 g cm −3 including bulk silver (i.e.∼10.5 g cm −3 ) cannot be levitated, as they stay outside the dynamic range of density and sink to the bottom of the container within the Maglev solution (figure 2(e)).
In order to investigate the effect of the size of the particles on their densities, multiscale silver particles are selected and used to represent high-density materials in various forms (i.e.bulk, micro powder, nanopowder, and colloidal solution).We hypothesize that the density of materials in a non-isolated system is a size-dependent function, as the material density gets shifted from bulk to effective.We test this hypothesis simply by levitating silver with different sizes, as presented in figures 2(f)-(k).Since the density of bulk silver is ∼10.5 g cm −3 , it certainly falls outside the dynamic range of density of the current standard MagLev system even at high concentration of paramagnetic solution (i.e. 3 M MnCl 2 ).Levitating micron size silver powder (∼3 µm) also reveals that it falls outside the dynamic range of the standard MagLev (sinks to the bottom) and is more likely to have densities close to bulk silver, as expected.Interestingly, it is found that nanometer-sized silver nanoparticles, both in the form of powder and colloidal solution, are successfully levitated within the dynamic range of density of the standard MagLev system at 3M concentration of MnCl 2 .Although the levitation profile of silver nanopowder and colloidal nanoparticles of similar sizes (i.e.∼30 nm) are remarkably different within the MagLev system, their effective densities fall within the dynamic range of density of the standard MagLev system.
HDL and VLDL are made of lipids (e.g.cholesterol and triglycerides) and are relevant examples of nanoscale (i.e.∼5-60 nm) biological species in the human body [38].Although high levels of certain types of lipoproteins can be potentially harmful to the body, some lipoproteins can be favorable for the body and prevent heart-related diseases and, therefore, are critical for human health [39,40].It is reported that HDL has a density of 1.06-1.2g cm −3 while the density of VLDL is below 1 g cm −3 (i.e.∼0.95 g cm −3 ) [41,42].Since there is a significant difference between the density of HDL and VLDL, we expect to detect a clear difference in their corresponding levitation heights in the same experimental conditions.This is while levitation heights of both HDL and VLDL are centered around 19 mm within the standard MagLev system, despite their substantial bulk density differences as portrayed in detail in figure 3, indicating the critical role of effective density at the nanometer size.In addition, the levitation height of the objects strongly depends on the concentration of the paramagnetic solution, and this is while no change in the levitation height of the HDL was observed by changing the concentration of the gadobutrol from 0.1 to 1 M, as demonstrated in figure 3.This may be due to the fact that the effective density that nanoscale materials (e.g.HDL and VLDL) experience within the paramagnetic solutions in the MagLev systems are different from their real densities.
It is noteworthy that similar results were observed when a ring MagLev system is used instead of the standard MagLev system, indicating the validity of the obtained results (figure 4).Similar to the standard MagLev system, we observed that the bulk and micron-sized silver particles sedimented at the bottom of the container during levitation in the ring MagLev system containing 3 M concentration of MnCl 2 (figures 4(a) and (b)).Although silver nanopowder and nanocolloids could be successfully levitated in the ring MagLev system, they show different levitation profiles (figures 4(c) to (f)).Interestingly, bulk aluminum with a much lower bulk density (2.7 g cm −3 ) sedimented at the bottom of the container while colloidal gold nanoparticles that have much larger bulk density (i.e.19.3 g cm −3 ), could be successfully levitated within the dynamic range of the MagLev system.These observations reveal that the density of materials in the bulk form (i.e.millimeter and micrometer scales) is different from their effective density at lower scale sizes (i.e.nanometer).In fact, for density calculations of bulk materials, only gravitational and magnetic forces are considered as the main two forces acting on the objects (i.e.equation ( 4)), while at nanoscale, objects/particles experience several forces other than gravitational and magnetic forces, which are not considered in equation ( 4) and, therefore, using this equation for bulk density measurements/calculations may result in inaccurate outcomes.The development of new equations for the measurements/calculation of effective density is a practical way to consider materials' effective density at the nanoscale size.Such equations need to be custom designed based on the physicochemical properties of the nanosized objects and the influential parameters that affect their levitations, as described below.
In other words, at lower size scales (i.e.nanoscale) the density reduces to a large scale due to the emergence of several other physical forces and high interactions of nanoscale objects with the surrounding solution.For example, due to the high surface to volume ratio of nanoparticles, a layer of solvent molecules (for example, water molecules) always covers the surface of the nanoparticles, which substantially alters their effective densities.Moreover, depending on the surface chemistry and hydrophilicity/hydrophobicity of the levitating nanomaterials, some hydrogen bonding may form which affects the density.Another possibility is the effect of surface charges and their movements due Brownian motion which exerts an additional force (i.e.Lorentz force) on the nanoparticles and consequently longer equilibrium times.In fact, the presence of numerous forces at lower size scales causes particles to experience a different environment compared to bulk materials and consequently a different effective density.What makes the density measurements and calculations even more complex at the nanoscale is that the number and type of forces may not be unique for all nanoparticles, as various nanoscale materials have different physicochemical properties and deriving a unique density-levitation height equation for all nanomaterials may not be an easy task in MagLev systems.

Conclusions
The long-standing debate surrounding the suitability of MagLev systems for measuring the density of particles smaller than ∼2 µ ms, primarily influenced by Brownian motion has been addressed in this study.Through comprehensive investigations involving various materials and employing both standard and ring MagLev systems, we have illuminated the critical role of size in the levitation characteristics within MagLev systems.Our findings underscore that, when dealing with sub-micron materials, it is imperative to consider the effective density rather than relying solely on bulk density measurments.The density of many heavy metals such as silver and gold fall outside the dynamic range of density of conventional MagLev systems, even at high concentrations of paramagnetic solutions, while their corresponding nanoscale counterparts (i.e.effective density) can be successfully levitated.Our findings suggest that the effective density of nanoscale materials is different at the bulk scale due to the high interactions of nanomaterials with the surrounding medium and the presence of numerous forces at the nanometer size scale, which become dominant compared to magnetic and gravitational forces as the two main dominant forces at the bulk scale.

Figure 1 .
Figure 1.Schematic showing the standard configuration of the MagLev system used in this research and different possible levitation heights of particles/objects with various size scales.

Figure 2 .
Figure 2. (a) Calibration curve, (b)-(e) representative levitation images of standard density glass beads showing the dynamic range of density of the standard MagLev system, (f) and (g) levitation profiles of various sizes of silver (i.e.bulk and micron-size powder) in 3 M concentration of MnCl 2 within the standard MagLev system, (h) and (i) representative levitation images of silver nanopowder at two different time points, and (j), (k) representative levitation images of silver nanocolloids at two different time points.

Figure 3 .
Figure 3. Representative levitation images of HDL and VLDL at various concentrations of gadobutrol.The yellow dotted ovals show the levitating samples and the corresponding levitation heights.

Figure 4 .
Figure 4. (a), (b) Levitation profiles of various sizes of silver in a 3 M concentration of MnCl 2 in a ring MagLev system, (c), (d) representative levitation images of silver nanopowder at two different time points, (e), (f) representative levitation images of silver nanocolloids at two different time points, (g) schematic representation of a ring MagLev set up, and (h) representative levitation image of bulk aluminum (∼ 1.3 mm in diameter and density of 2.7 g cm −3 ) and citrate stabilized colloidal gold nanoparticles in aqueous solution (∼ 20 nm) in the standard MagLev system (bulk gold has a density of ∼19.3 g cm −3 ).