Sonomaglev: combining acoustic and diamagnetic levitation

Acoustic levitation and diamagnetic levitation are experimental methods that both enable the contact-free study of liquid droplets and solid particles. Here we combine techniques of both into a single system that takes advantage of the strengths of each, allowing for the manipulation of levitated spherical water droplets (30 nl–14 µl) under conditions akin to weightlessness, in the laboratory, using a superconducting magnet ﬁtted with two low-power ultrasonic transducers. We show that multiple droplets, arranged horizontallyalong a line, can be stably levitated with this system and demonstrate controlled contactless coalescence of two droplets. Numerical simulation of the magnetogravitational and acoustic potential reproduces the multiple stable equilibrium points observed in our experiments. Contactless manipulation has become an area of increased study over the last 10 years. New experimental techniques have been developed in a wide variety of disciplines, including: analytical chemistry 1–6 , material sciences 7 , pharmacy 8,9 and micro-assembly 10–13 . Many contactless manipulation ex-perimentsrelyuponafamilyoftechniquesclassiﬁedas acoustic levitation. Acoustic levitation uses ultrasonic transducers to create a pressure ﬁeld that applies acoustic radiation forces to suspend objects in a gaseous medium 14 . It has been shown that intricate, readily tuneable pressure ﬁelds can be constructed by using an array of small ultrasonic transducers 15,16 , allowing for manipulation of multiple objects in three dimensions. The discovery that an array of ultrasonic transducers can create customisable acoustic levitation systems has led to a resurgence in the study of non-contact manipulation using acoustic levitation techniques 17–20 . In addition to these and other attractive features of acoustic levitation

Acoustic levitation and diamagnetic levitation are experimental methods that both enable the contact-free study of liquid droplets and solid particles.Here we combine techniques of both into a single system that takes advantage of the strengths of each, allowing for the manipulation of levitated spherical water droplets (30 nl-14 µl) under conditions akin to weightlessness, in the laboratory, using a superconducting magnet fitted with two low-power ultrasonic transducers.We show that multiple droplets, arranged horizontally along a line, can be stably levitated with this system and demonstrate controlled contactless coalescence of two droplets.Numerical simulation of the magnetogravitational and acoustic potential reproduces the multiple stable equilibrium points observed in our experiments.
Contactless manipulation has become an area of increased study over the last 10 years.New experimental techniques have been developed in a wide variety of disciplines, including: analytical chemistry [1][2][3][4][5][6] , material sciences 7 , pharmacy 8,9 and micro-assembly [10][11][12][13] .Many contactless manipulation experiments rely upon a family of techniques classified as acoustic levitation.Acoustic levitation uses ultrasonic transducers to create a pressure field that applies acoustic radiation forces to suspend objects in a gaseous medium 14 .It has been shown that intricate, readily tuneable pressure fields can be constructed by using an array of small ultrasonic transducers 15,16 , allowing for manipulation of multiple objects in three dimensions.The discovery that an array of ultrasonic transducers can create customisable acoustic levitation systems has led to a resurgence in the study of non-contact manipulation using acoustic levitation techniques [17][18][19][20] .
In addition to these and other attractive features of acoustic levitation 21 , the method also has some drawbacks.From the earliest experiments, acoustically-levitated objects were observed to have a tendency to oscillate, attributed to the response of the acoustic field to the presence of the object 22,23 .Objects may also start to rotate spontaneously due to streaming flows in the surrounding gas generated by the high pressure sound waves 24,25 , though techniques to mitigate these effects and control the rotation have been demonstrated recently 26,27 .These same streaming flows may also be problematic in studies of liquid droplets, where the air flow affects heat and mass transfer non-uniformly at the droplet's surface 28 and also sets up flows within the droplet 29 .Acoustically-levitated liquid droplets are often deformed into oblatelike shapes 30,31 ; this typically occurs when the diameter of the droplet is of the same order as the acoustic wavelength.These characteristics, which are usually undesirable (though occasionally exploited 31 ), are avoided in similar experiments using diamagnetic levitation (e.g.Ref. 32).Diamagnetic materials experience a repulsive force when placed in a static, spatially-varying magnetic field.A wide variety of solids and liquids, including water and organic material (including biological), can be levitated in a magnetic field of sufficient strength and gradient (e.g.Refs.33-40); typically a field of order 10 T is required, though levitation of graphite can be achieved using much weaker fields (e.g.Ref. 41).In contrast to the oscillating surface forces applied in acoustic levitation, diamagnetic levitation applies a constant body force that counteracts the force of gravity throughout the levitated object at the molecular level, and so closely mimics the weightless conditions on board an orbital or parabolic flight, or in a drop tower.Diamagnetically-levitated liquid droplets attain a spherical shape, due to the fact that surface tension forces dominate over the residual net body forces (∼ 0.01 g) that stabilize the levitation.
On the other hand, the options to manipulate multiple objects magnetically are limited.Diamagnetic levitation allows for the levitation of multiple spatially-separated objects simultaneously if the objects have unique ratios of magnetic susceptibility to density.This differs from acoustic levitation where multiple acoustic 'traps' can be created for a particular material by manipulating the acoustic field.The creation of multiple traps for a single material is also possible using diamagnetic levitation, by manipulating the shape of the strong magnetic field, but this is technically more challenging than manipulating an acoustic field and correspondingly more restrictive.
In this letter, we demonstrate combining techniques from both acoustic and diamagnetic levitation to manipulate the position of liquid droplets, drawing on the strengths of each method.The force of gravity is compensated throughout the droplets by applying a vertical diamagnetic body force using a superconducting magnet, providing a simulation of weightlessness.Then, acoustic radiation forces are used to position the droplets.Since diamagnetic rather than acoustic forces provide gravity compensation in this case, the acoustic power required in these experiments is much smaller than that of a purely acoustic levitator.We further show that it is possible to reproduce the locations of the levitated droplets in simulations, calculating the magnetic and acoustic fields.Acoustic levitation has been combined with magnetic fields before, to study active matter levitated in liquids, but in those studies the magnetic field was used to provide propulsion (e.g.Ref. 42) 2 rather than for balancing the force of gravity.
We used a custom-built 18.5 T superconducting solenoid magnet (Cryogenic Ltd., London), with a room temperature, 58 mm-diameter vertical bore, to levitate water droplets diamagnetically in air at room temperature and atmospheric pressure.The existence of a stable equilibrium levitation point for diamagnetic material in a solenoid magnet has been discussed in detail before 35 .This point lies on the central axis, approximately 11 cm above the centre of the solenoid in our magnet, depending on the solenoid current, I.The magnetic field strength at the geometric centre of the solenoid was maintained at a constant B 0 (I) = 17.4 T for all experiments in this letter.The bore was fitted with a custom, 3d-printed PLA ring of inner diameter 39 mm, outer diameter 57 mm and height 20 mm, see figure 1a,b.This ring contained two 10 mm diameter ultrasonic transducers (CamdenBoss CTD40K1007T), of the type used in the multi-emitter acoustic levitator, 'TinyLev' 43 .The transducers were positioned diametrically opposite each other and aligned along a horizontal axis such that their faces were perpendicular to the edge of the ring, as shown in figure 1.The transducers were wired in parallel, driven in-phase, and connected to a function generator (Stanford Research DS345), which was used to drive the transducers with frequencies of 37-40 kHz and up to a maximum peak-to-peak voltage, V pp , of 20 V. The ring was fitted in the bore of the magnet such that the central horizontal plane of the ring intersected with the stable equilibrium levitation position of a diamagnetically levitated water droplet (fig.1a).The sound pressure level close to the bore axis at V pp = 20 V was ∼ 110 dB, obtained from the force balance between the radial magnetic and acoustic forces in equilibrium, and from simulation.The operation of the transducers, despite containing some ferromagnetic material, was not affected by the presence of the strong magnetic field.Images were taken using a camera sited away from the magnet, using a 45 • mirror placed several centimetres above the levitating droplets, at the mouth of the bore, to avoid influencing the acoustic field.
An atomiser was used to spray a fine mist of distilled water above the magnet, which then descended into the magnet bore.With the acoustic transducers switched off, the mist of droplets coalesced as one larger droplet at the stable levitation point.When a voltage was applied to the transducers and the experiment repeated, the mist coalesced into several wellseparated droplets, each levitating in stable equilibrium.This image is representative of all experiments we performed using this method: while the sizes of the individual droplets show some variation between repeat experiments, the position and spacing between the droplets is constant for a given frequency and voltage.The technique usually produced droplets that were larger closer to the axis, since the diamagnetic force funnelled the mist toward the axis as it descended, though in principle droplets with uniform size could be levitated.
Figure 2a shows a contour plot of the magnetogravitational potential energy density U mg for water in air 35,44 where χ w = −9.05× 10 −6 , χ a = 5.13 × 10 −7 , ρ w = 998 kg m −3 and ρ a = 1.2 kgm −3 are the volume magnetic susceptibilities (S.I. units) and densities of water (w) and air (a) respectively 45 , B(x, y, z) is the magnetic field strength, g is the acceleration due to gravity, µ 0 is the vacuum permeabil- ity, and z is the vertical coordinate.The terms proportional to ρ a and χ a account for the relatively small buoyancy of the (weakly paramagnetic) air.The magnetic field strength B was calculated from the known current density and geometry of the solenoid by numerical integration of the Biot-Savart integral.The blue circle in the figure marks the location of a local minimum in U mg , i.e. the location of the stable levitation point.We take this point to be the origin of a coordinate system in which the x − y plane is perpendicular to the axis, z, of the solenoid, and in which the x axis intersects the centres of the faces of both transducers (fig.1a).The magnetic field strength at this point is B ≈ 11 T. Figure 2b and c show the total potential energy density where the acoustic potential energy density U acoust of a droplet is given by the Gor'kov potential per unit volume 46 Here, c is the speed of sound in water (w) and air (a) at room temperature and pressure respectively, and p 2 and u 2 are the mean square amplitudes of the pressure and velocity of air, respectively.For the case of water in air The acoustic field inside the magnet bore was calculated using a finite-element method, implemented using the open-source software FreeFem 47,48 , where our computational mesh was generated using the open-source software Gmsh 49 .More information about the method of solution and the geometry of the simulation domain used can be found in the supplementary information.The simulations show that the addition of the acoustic field modulates the potential energy density, U total , giving rise to multiple potential minima, 'wells', along the x-axis (fig.2b, c).Unlike U mg , U total is not cylindrically symmetric about the z (solenoid) axis.Figure 2b shows U total in the y = 0 plane.Blue circles indicate the positions of each well.The two central wells lie on the x-axis, i.e. on the line passing through the centres of the two acoustic transducers.The vertical, z, location of each well increases approximately as x 4 , increasing to z = 3 mm for the wells farthest from the axis, at |x| = 17 mm.This variation in height results from the shape of U mg , in particular an octupole contribution to U mg 50 .Figure 2c shows U total in the z = 0 plane, and includes a photograph of levitating droplets taken from an angle looking down the bore (i.e. down the z axis), superposed on the contours.
By varying the peak-to-peak voltage V pp applied to the transducers we were able to adjust the horizontal (x) positions of the levitating droplets.The montage of photographs in figure 3 shows the horizontal position of two levitated water droplets close to the solenoid axis as a function of V pp .In this particular experiment, the droplet farthest from the axis had a diameter of 1.19 ± 0.05 mm and the droplet closest to the axis was larger, with a diameter of 1.93 ± 0.05 mm (the measurement uncertainty reflects the resolution of our optical set-up).Note that, since the total potential en-This is the author's peer reviewed, accepted manuscript.However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0134297ergy density, U total , is independent of volume, the positions of the stable equilibrium points are independent of the volume of the droplets.When the transducers were driven with V pp = 20.0V, the smaller droplet levitated with its centre at x = 7.21 ± 0.03 mm and the larger levitated with its centre at x = 2.16 ± 0.03 mm.The levitation of both was stable.Decreasing V pp from 20 V to 15 V, the position of the smaller droplet decreased from x = 7.21 mm to 6.85 ± 0.03 mm, while the position of the larger droplet decreased slightly from x = 2.16 mm to 2.06 ± 0.03 mm.On decreasing V pp below 15 V, we observed that the smaller of the two droplets lost equilibrium, as the depth of the local potential well was reduced to zero, and 'fell' toward the larger of the two, coalescing with it.On decreasing V pp further, from 14 V to 8 V, the equilibrium position of the centre of the remaining large droplet decreased from x = 2.06 mm to 1.50 ± 0.03 mm.Below V pp = 8 V, this droplet also lost equilibrium and 'fell' radially, finding a new equilibrium on the solenoid axis.The equilibrium positions obtained from simulation and experiment are shown together in figure 3, for comparison.
The use of acoustic levitation in areas such as analytical chemistry and biochemistry is growing, where it allows analysis of small volumes of liquid without the problems associated with the presence of container walls, such as adsorption and contamination of the analyte, or measurement interference (see, e.g.Ref. 51 and references therein).Recent further strides have been taken to extend the technique to enable contactless transport and processing of multiple levitated objects, which opens up many more applications including in biochemistry, pharmaceuticals and materials science (e.g.Refs.52 and 53).Here we have demonstrated that, by combining techniques from diamagnetic and acoustic levitation, we can trap quiescent liquid droplets with volumes ∼ 10 −8 -10 −5 L in a series of potential wells arrayed along a horizontal line, and that the location of these traps, and thus the droplets, can be varied precisely.By lifting the drops diamagnetically, the sound pressure levels were much reduced compared to those of an acoustic levitator: ∼ 110 dB compared to 150-165 dB 28 , a greater than 100-fold reduction in the amplitude of the pressure.Consistent with this, we did not observe phenomena such as vibration of the droplets, deformation of their equilibrium shape, or instabilities in their position, which are characteristic of acoustic levitators 22,26,30 ; further experiments are planned to quantify the reduction in fluid flows within and surrounding the droplets using t his set-up.We suggest that this technique may be useful in areas where particularly delicate handling of material is required, where control over the internal droplet flow is important or where evaporation and cooling by the oscillating air flow are problematic.We note that many common techniques used in biochemical analysis are compatible with a strong magnetic field, such as fluorescence microscopy 54 , Raman spectroscopy 55 , and light scattering 56 .
The combination of acoustic and magnetic forces creates additional possibilities to manipulate the droplets: here, we demonstrated controlled contactless coalescence of two droplets by varying the ratchet-like potential U total , using a relatively simple acoustic set-up consisting of two low-power transducers, a feat that otherwise requires large transducer arrays 52,53 .Droplets can also be moved vertically by adjusting the electric current in the solenoid.With the addition of more acoustic transducers, we anticipate that this flexibility will allow the creation of a system that can provide spatial positioning in three directions within the bore of the magnet.

SUPPLEMENTARY MATERIAL
See supplementary material for additional information about numerical simulations.
FIG. 1. a) Diagram of the experimental set-up inside the superconducting magnet bore.The two ultrasonic transducers are aligned along an axis, x, perpendicular to the vertical bore axis, z.Dashed lines represent the magnetic field lines.b) Image showing the 3dprinted PLA ring used to mount the transducers in the magnet bore.c) Line of eight droplets of diameters 0.4-3.0mm (30 nl-14 µl) levitating in the bore of the magnet, with position controlled by the acoustic-transducers.
Fig-This is the author's peer reviewed, accepted manuscript.However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0134297ure 1c is a photograph, taken from an angle looking down the axis of the magnet bore, showing the formation of eight separate droplets from the mist.The droplets in this experiment have diameters in the range 0.4-3.0mm (volume 30 nl-14 µl), and are separated by approximately 5 mm.The two centre droplets lie approximately 2 mm from the axis of the bore.
FIG. 3. a) Montage of images of two levitating droplets showing the variation in their horizontal position, x, as Vpp was reduced from 20 V to 8 V. Down-pointing arrows indicate the voltage below which droplets lost equilibrium.b) Measured x-coordinates of the centres of the droplets.In both plots, red circles show the calculated xcoordinates of local minima in U total .The blue unbroken lines indicate the x-coordinates of the two minima at Vpp = 20 V.