Diamagnetic levitation of water realized with a simple device consisting of ordinary permanent magnets

Techniques for levitating objects in air facilitate the conduct of interesting experiments in physics, chemistry, biology, and medicine, such as the supersaturation of liquids,¹ material synthesis,² and crystal growth under containerless conditions.³ Therefore, several levitation methods have been developed, including electromagnetic,⁴ electrostatic,⁵ acoustic,⁶ and optical trapping⁷ techniques. However, these methods have various limitations; for instance, in electromagnetic levitation, the sample must withstand the heat generated by the electric current flowing through it. In electrostatic levitation, experiments must be conducted in a vacuum to prevent the sample from being neutralized by surrounding gases. Similarly, acoustic levitation and optical trapping are restricted to samples that can resist potential damage from sound pressure or laser light.

Diamagnetic levitation^8–14 utilizes the repulsive force of a magnetic field acting on a diamagnetic body. Because most solids and liquids around us are diamagnetic, diamagnetic levitation has a wide range of applications. It also offers the advantage of achieving a stable levitation state at room temperature without subjecting the sample to potentially damaging control fields, such as high-intensity laser light or sound pressure, with few restrictions on experimental conditions, and no need for an external feedback system to maintain the levitated state. Unfortunately, diamagnetic levitation generally requires exceptionally strong magnets, such as those found in world-class high-field facilities. This is because the magnetic susceptibilities of diamagnetic materials are extremely low, except for bismuth¹⁵ and graphite.^16,17 However, the microstructure of permanent magnets^18–20 fabricated using photolithography generates a large magnetic force in the gap between the magnets, enabling the magnetic levitation of water and other objects. In these methods, two magnets are arranged such that they repel each other. Within a narrow gap of several tens of micrometers between the magnets, their magnetic fields cancel each other, forming a strong $∂B/∂z$⁠. Because the magnetic force is proportional to both B and $∂B/∂z$⁠, a significantly large magnetic force is generated even when a part of the magnetic field is canceled. Historically, diamagnetic levitation has required superconducting magnets, making these methods, which achieve levitation using permanent magnets, remarkable innovations. However, it is worth noting that the size of the levitated objects is often limited to 20–30 μm, constrained by the width of the narrow gap. If magnetic levitation in a larger space could be achieved more easily using permanent magnets, it would pave the way for various studies such as those mentioned at the beginning. Therefore, this study conducted magnetic field distribution simulations to determine a suitable shape or arrangement of permanent magnets for the levitation of larger diamagnetic objects. Consequently, an extremely simple magnetic-levitation system comprising only two pairs of rectangular cuboid magnets, which enabled the levitation of submillimeter-sized water droplets, was developed.

Superconducting magnets capable of generating a magnetic field with | $B∂B/∂z$| exceeding 1400 T²/m are common, and such magnets can be used to enable diamagnetic levitation experiments. In contrast, $|B∂B/∂z|$ produced by rectangular cuboid magnets individually is limited to approximately 10 T²/m near the face center. However, a higher | $B∂B/∂z$| can be generated by exploiting the fact that the magnetic field is concentrated in regions with small radii of curvature at the edges of the magnet. Furthermore, if two such magnets are positioned such that their regions with small radii of curvature attract each other and are brought in close proximity, the magnetic field is expected to become extremely concentrated in the narrow space between them. Consequently, this configuration is expected to generate a large magnetic-field gradient, $∂B/∂z$⁠, while simultaneously producing a high | $B∂B/∂z$|. Unlike previous studies,^18–20 where the magnetic fields generated by two separately placed magnets were canceled out in the gaps between the magnets to create a large $∂B/∂z$⁠, our strategy leverages the superposition of the magnetic field. This approach allows for more efficient utilization of the magnetic field. In this study, we demonstrated that a high | $B∂B/∂z$| can be generated in a narrow valley formed between the edges of two attached permanent rectangular cuboid magnets. This enables the magnetic levitation of diamagnetic materials such as water.

Figure 1(a) presents an example of a magnetic arrangement that can levitate water, and a schematic illustration of the magnet configuration is shown in Fig. 1(b). The magnetic levitation system developed in this study comprised four horizontally aligned magnets (two pairs, pair α and pair β) as shown in Figs. 1(a) and 1(b). Pair α comprises the central plate-shaped magnets, while pair β comprises the rectangular cuboid magnets on either side of pair α. Because the edges of the magnets' contact surfaces are not infinitely sharp but instead have a finite, small radii of curvature, a narrow valley is formed between the edges of adjacent magnets of pair α, serving to concentrate the magnetic field, resulting in a very strong upward magnetic force around the uppermost central region of the valley, which is shaded in pink in the enlarged diagram on the right of Fig. 1(b). Figure 1(c) shows a levitated water droplet photographed from the direction along the valley between the magnets in Fig. 1(a). Compared to previous studies,^18–20 its size was approximately 0.3 mm, which was more than 10 times larger in diameter and 1000 times larger in volume.

The roles of the two pairs [pair $α$ and pair $β$⁠, Fig. 1(b)] were different. First, we explain the role of pair α with reference to Fig. 2, followed by an explanation of the role of pair β with reference to Fig. 3. Pair $α$ generated a strong magnetic force field, and the two plate magnets (2 mm wide, 18 mm deep, and 8 mm high) shown in Fig. 1(a) corresponded to these magnets. The magnets comprising pair α were grade N50 neodymium magnets. The magnitude of each magnetic moment was calculated to be 0.337 Am² based on a remnant flux density of 1.47 T. Their directions were antiparallel and oriented vertically upward and downward, respectively. Figure 2(a) shows the distribution of B along the z-axis, which was calculated using a magnetic field simulation software (Simcenter Magnet, Siemens) at a spatial resolution of 5 μm. As expected, the distribution of B should depend on the size of the valley between the magnets, that is, the radius of curvature, R, of the edges. Upon examining the edges of several types of commercially available neodymium magnets, the R values were found to range from 20 to 700 μm. Since the magnet surfaces are typically coated with nickel plating approximately 10 μm thick, the practical minimum R is likely approximately 20 μm. Therefore, B was calculated for seven different magnets with R (=0.05–0.65 mm) similar to that of commercial magnets. The definition of the coordinate system and radius of curvature R used in the calculations are shown in the upper-right inset of Fig. 2(a). The x-axis is the direction perpendicular to the elongated valley between the magnets, the y-axis is the direction parallel to the valley, and the z-axis is the vertical upward direction. The z-coordinate of zero was assigned to the height of the top surface of the magnets. The simulations revealed the following three characteristics. First, in all cases, B reached a maximum slightly below the magnet surface, at a z-coordinate value of approximately −0.5 R. Second, the maximum value of B increased with decreasing R, reaching approximately 2.50 T at R = 0.05 mm [inset in the upper-left of Fig. 2(a)]. Finally, $∂B/∂z$ near the magnet surface was well above 1000 T/m, even at its smallest (for R = 0.65 mm). Because B on the flat surface of the plate magnet was only approximately 0.48 T, the increase in B is due to the concentration of B near the edges, which also contributed to the formation of the steep $∂B/∂z$⁠.

Figure 2(b) shows the distribution of $B∂B/∂z$ along the z-axis, with the inset displaying the maximum value of $B∂B/∂z$ for each R. The dotted lines in the inset indicate $B∂B/∂z$ required to levitate the various materials: $1360 T2/m$ for H₂O (water), $1900 T2/m$ for NaCl (sodium chloride), $2700 T2/m$ for Al₂O₃ (alumina), $3820 T2/m$ for ZnS (zinc sulfide), $5120 T2/m$ for Ag (silver), $6910 T2/m$ for Au (gold), and 11040  $T2/m$ for In (indium). The occurrence of large $B∂B/∂z$ in a small region has also been reported in previous studies,^21,22 where ferromagnetic wires or sheets were placed in an external field generated by an electromagnet or a permanent magnet to produce significant $B∂B/∂z$⁠. For all R values, $B∂B/∂z$ was sufficient to levitate the water. Furthermore, as R decreased, the maximum value of $B∂B/∂z$ increased significantly. It exceeded 40 000  $T2/m$ at R = 0.05 mm. The size of objects that can be levitated is approximately the same as R. For instance, if R is approximately 0.1 mm, metals, such as indium, can also be levitated. This enables experiments on liquid metals that are relevant to flexible or wearable electronics.^23,24 When R is large, it is suitable for the levitation of organic materials. If protein solutions can be levitated, they yield higher-quality protein crystals²⁵ compared with those produced under gravity. Thus, both light organic materials and dense metallic compounds could be levitated diamagnetically. The above-mentioned results show that simply by attaching two such ordinary magnets, we can generate a $B∂B/∂z$ so large that even the world's strongest superconducting magnet cannot reach it.

Figures 3(a) and 3(b) show the distribution of B and $B∂B/∂z$ along the z-axis for pair $α$ alone and pairs $α$ and $β$ combined, respectively. Pair $β$ is the blue magnets (6.5 mm wide, 6.5 mm deep, and 8 mm high) depicted in the lower-left inside Fig. 3(a). The magnets comprising pair β were grade N52 neodymium magnets. The magnitude of each magnetic moment was calculated to be 0.387 Am² based on a remanent flux density of 1.44 T. Their directions were opposite to the magnetic moment of the adjacent pair α, being either vertically upward or downward. Here, the radius of curvature, $R$⁠, of the edges was set to 0.45 mm. Compared to the one-pair system, B and $B∂B/∂z$ were weakened in the two-pair system. However, in both cases, the maximum value of the magnetic force field was still larger than $1360 T2/m$ (dotted line). The magnetic and gravitational forces on the water were balanced at the intersection points [◆ (filled diamond) and ◇ (open diamond)] of the dotted and solid lines shown in Fig. 3(b). Figure 3(c) shows the distribution of the sum of the magnetic and gravitational potential energies along the z-axis, also indicating the points where the magnetic and gravitational forces are balanced. For both one- and two-pair systems, stable levitation was not realized at the points ◆ (filled diamond) because the energy curves exhibited a maximum point. However, the energy curves reached a minimum at the point ◇ (open diamond); thus, the points ◇ (open diamond) were energetically stable at least along the z-axis. To achieve stable non-contact levitation, the energy at the point ◇ (open diamond) must be a local minimum point along all axes. The insets of Fig. 3(c) show the energy density distribution along the x- and y-axes at the height z corresponding to the points ◇ (open diamond) both for one- and two-pair system. The gravitational potential energy was set to zero. The energy curves for the one-pair system exhibited a maximum point for both along the x- and y-axes. This implied that it was “unstable” along the horizontal direction. However, the curves for the two-pair system exhibited a minimum point, facilitating stable non-contact levitation. A stable non-contact levitation point did not appear in a one-pair system; however, it consistently appeared in a two-pair system. As shown here, pair β was responsible for creating a local potential minimum by partially attenuating the magnetic field generated by pair α. To ensure the creation of the potential minimum, the depth of the magnets along the y-axis in pair β should be sufficiently shorter than that of the magnets in pair α. Figure 3(d) shows a two-dimensional map of the potential energy density in the xz- and xy-plane for the two-pair system with R = 0.45 mm. The direction and strength of the magnetic field at each point on the plane are indicated by the direction and size of the white arrows in the map of the xz-plane. The sector-shaped areas at the bottom of the left and right sides indicate the edges of the magnets, and the levitation point of the water is represented by a cross mark on the z-axis (z = 0.067 mm). The levitation point is based on calculations that neglect the volume of the levitating object. However, for a water droplet of finite size, the larger the diameter, the lower the levitation position. In this system, a water droplet with a diameter of approximately 450 μm would come into contact with the edge of the magnet.

Figure 4 shows a schematic of the experimental setup. The magnets were placed in a transparent plastic chamber on a 6-axis (⁠ $xyzθϕψ$⁠) control stage, thereby facilitating precise control of tilt and position. Tiny water droplets with a diameter of a few micrometers generated by an ultrasonic humidifier were introduced through an 8 mm diameter silicone tube into the chamber. They drifted through the inner space and fell gradually, with some gathering at the levitation point. The formation of a relatively large levitating water droplet was captured using a charge-coupled device (CCD) camera equipped with a microscope. By changing the position of the microscope, it was possible to obtain videos from all x, y, and z directions (video clips in supplementary material S1).

R of the magnets used in the experiment was approximately 0.4–0.5 mm, implying a maximum B of approximately 1.5 T was formed in the valley between the magnets. Figure 5(a) depicts the situation immediately after the introduction of the water droplets. Water droplets of a few micrometers diameter drifted into the valley between the magnets (black band in the background). After 60 s, the water droplets aligned parallel to the valley. They formed medium-sized droplets of several tens of micrometers in size [Fig. 5(b)]. According to the simulation results, the force acting on the droplet along the y-axis, corresponding to the slope of the potential curve (⁠ $∂E/∂y$⁠), was 100 times smaller than the force along the x-axis (⁠ $∂E/∂x$⁠) [inset of Fig. 3(c)]; therefore, the water droplets along the y-axis require a long time to gather at the center, i.e., the levitation point. The water droplet at the levitation point remained in its position. Consequently, the droplet at the center continued to grow preferentially over the others [Fig. 5(c), after 120 s] because of the collisions with other water droplets that moved toward the equilibrium point. Even after the humidifier was turned off, the water droplets continued to collide and the number of droplets in the valley decreased [Fig. 5(d), after 300 s]. After waiting further, a relatively large water droplet remained, as shown in Fig. 1(b). Moreover, when the sample chamber was sealed, the water droplet did not change in size and remained levitated (further details in supplementary material S1).

Thus, this study conducted detailed computer simulations of the magnetic field to develop a simple method to achieve the diamagnetic levitation of water using only ordinary permanent magnets and experimentally validated its effectiveness. The proposed method exhibited five outstanding features. First, the device structure was extremely simple. It was assembled using only the force of attraction between the magnets, without any special microfabrication, intricate assembly techniques, or jigs to hold the magnets in place. Second, the initial and operating costs were exceptionally low. Typically, superconducting magnets cost more than $100 000; however, the magnets used in this device cost approximately $1 each. Furthermore, owing to the lack of the need to run a large current or cool the coils, the operating costs were negligible. This rendered the proposed device notably energy efficient. Third, the device was remarkably stable. The depth of the potential well corresponds to approximately 30 μm in the gravitational field. The force required to extract an object from the well ranges from one-tenth to one-third of the gravitational force. In practice, even light tapping of the experimental table to induce vibrations does not allow the object to escape from the potential well. In addition, the region that satisfied the condition of noncontact levitation of the object, that is, the region where the energy curve exhibited a minimum point along all axes, was extended by several tens of percent around the levitation point in $B∂B/∂z$⁠. Consequently, this device was highly resistant to disturbances. This spatial stability also guaranteed temporal stability, facilitating the conduct of experiments for as long as required. Fourth, the system was highly flexible. By simply adjusting the strength and dimensions of the magnets, they could be applied to levitate not only water, but also other solids and liquids (further details in supplementary material S2). Fifth, the system was highly scalable. Its small size and simple structure render it applicable to various experiments, and it is easy to set up an optical system and control experimental parameters such as temperature and humidity. In addition, if a medium that generates magnetic buoyancy is filled around the levitating object, the range of materials that can be levitated is significantly expanded (the magneto-Archimedes effect^26–30). As described above, the system proposed in this study offers various advantages. Moreover, it is expected to have applications in materials science,^31,32 fluid dynamics,^13,33 crystal growth,³⁴ surface chemistry,³⁵ biochemistry,^36,37 and preliminary ground-based research for experiments conducted under microgravity in space.

See the supplementary material for additional information on the magnetic field calculations and video clips of the formation process of levitated droplets captured along the x-, y-, and z-axes.

We would like to express our sincere gratitude to Ms. Marin Motoyanagi for her support in executing the simulations and to Mr. Masayuki Sugaya for his assistance in fabricating the experimental equipment. This study was financially supported by JSPS KAKENHI (Grant No. JP23K04697).

The authors have no conflicts to disclose.

Tomoya Naito: Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Writing – original draft (equal). Tomoaki Suzuki: Methodology (equal). Yasuhiro Ikezoe: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon request.