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In this paper, we report an innovative MLS design which consists of a gradient-field Maxwell coil placed in the bore of a superconducting (SC) magnet. By optimizing the SC magnet’s field strength and the current in the Maxwell coil, we show that an unprecedented V1% of over 4,000 µL can be achieved in a compact coil of 8 cm in diameter. This optimum V1% increases with the size and the field strength of the MLS. We then discuss how such a MLS can be made using existing high-Tc superconducting materials so that long-time operation with minimal energy consumption can be achieved. To further demonstrate the usefulness of this MLS, we also consider re- ducing its current and the field strength to emulate the gravity on Mars (gM = 0.38g). It turns out that a functional volume over 20,000 µL can be produced, in which the gravity only varies within a few percent of gM. Our design concept may break new ground for exciting applications of MLSs in future low-gravity research.

RESULTS

To aid the discussion of our MLS design, we first introducethe fundamentals of magnetic levitation using a solenoid magnet. Following this discussion, we will present the details of our innovative MLS design concept.

Levitation by a solenoid magnet FIG. 1. (a) Schematic of a solenoid with a diameter of D = 8 cm and a height of √3D/2. (b) Calculated specific potential energy E(r) of The concept of magnetic levitation can be understood by con- a small water sample placed in the magnetic field. The turn-current sidering a small sample (volume ∆V) placed in a static mag- NI of the solenoid is 607.5 kA. The origin of the coordinates is at the netic field B(r). Due to the magnetization of the sample ma- center of the solenoid. The dashed contour denotes the boundary of terial, the energy of the magnetic field increases by [47] the trapping region, and the solid contour shows the low-force region (i.e., acceleration less than 0.01g). (c) The functional volume V1% (i.e., overlapping volume of the two contours) versus the turn-current χB2(r) ∆EB = − ∆V, (1) NI. Representative shapes of the low-force region are shown. 2µ0(1 + χ) where χ is the magnetic susceptibility of the sample material and µ0 is the vacuum permeability. For diamagnetic materials In order to attain a stable levitation, the specific potential with negative χ, ∆EB is positive and therefore it requires en- energy E must have a local minimum at the levitation point so ergy to insert a diamagnetic sample into the B(r) field. Count- the sample cannot stray away. Since E depends on the mate- ing in the gravity effect, the total potential energy associated rial properties besides the B(r) field, we need to specify the with the sample per unit volume can be written as: sample material. Considering the fact that water has been uti- lized in a wide range of low-gravity researches [48–50] and is χB2(r) also the main constituent of living cells and organisms [51], E(r)= − + ρgz, (2) 2µ0(1 + χ) we adopt the water properties at ambient temperature [52] 6 3 3 (i.e., χ = 9.1 10− and ρ = 10 kg/m ) in all subsequent where ρ is the material density and z denotes the vertical co- analyses.− To see× the effect of the B(r) field, we consider an ordinate. This energy leads to a force per unit volume acting solenoid with a diameter of D = 8 cm anda heightof √3D/2, on the sample as: as shown in Fig. 1 (a). These dimensions are chosen to match χ the size of the MLS that we will discuss in later sections. For a F = ∇E(r)= B ∇B ρgeˆz. (3) solenoid with N turns and with an applied current I, B(r) can µ (1 + χ) − 0 · − be calculated using a known integral formula that depends on For an appropriate non-uniform magnetic field, the vertical the product NI (see details in the Method section). E(r) in the component of the field-gradient force (i.e., the first term on full space can then be determined. the right side in Eq. (3)) may balance the gravitational force In Fig. 1 (b), we showthe calculated E(r) near the top open- at a particular location, i.e., the levitation point. Sample sus- ing of the solenoid when a turn-current of NI = 607.5 kA is pension can therefore be achieved at this point. applied. In general, E is high near the solenoid wall due to 3

FIG. 2. (a) Schematic of the gradient-field Maxwell coil with a di- ameter D = 8 cm in the presence of an applied uniform field B0. b) Calculated specific potential energy E(r) of a small water sample placed in the magnetic field for I = 112.6 kA and B0 = 24 T. The origin of the coordinates is at the center of the bottom current loop. The black dashed contour denotes the boundary of the trapping re- gion, and the black solid contour shows the low-force region (i.e., acceleration less than 0.01g). the strong B field there. Slightly above the solenoid, there is a trapping region (enclosed by the dashed contour) in which FIG. 3. (a) Calculated V1% versus the loop current I for the coil shown in Fig. 2 with B = 24 T. The largest V is denoted as V . E decreases towards the region center. When a water sam- 0 1% opt (b) The obtained Vopt as a function of B0. The overall maximum Vopt ple is placed in this region, it moves towards the region center is denoted as V , and the corresponding coil current and base field where the net force is zero, i.e., the levitation point. We have ∗ are designated as I∗ and B∗, respectively. also calculated the specific force field using Eq. (3). The solid 0 contour in Fig. 1 (b) denotes the low-force region in which the net force corresponds to an acceleration less than 0.01g. tions can be satisfied approximately. The idea is to combine The overlapping volume of the trapping region and the low- a strong uniform field B0 and a weak field B1(r) that has a force region is defined as our functional volume V1% where fairly constant field gradient ∇B1. In this way, the total field the sample not only experiences weak residue forces but also B = B0 + B1 B0 is approximately uniform and its gradient remain trapped. In Fig. 1 (c), we show the calculated V1% as ≃ ∇B ∇B1 can also remain nearly constant. a function of NI. The trapping region emerges only above a ≃ The uniform field B0 can be produced in the bore of a su- threshold turn-current of about NI = 520 kA. As NI increases, perconducting solenoid magnet. Indeed, for superconduct- V1% first remains small (i.e., a few µL) and has a shape like ing magnets used in magnetic resonance imaging applica- a inverted raindrop. When NI is above about 600 kA, V1% tions, spatial uniformity of the field better than a few parts grows rapidly and peaks at NI = 607.5kA beforeit drops with per million (ppm) in a space large enough to hold a person further increasing NI. In the peak regime, V1% has a highly has became standard [53, 54]. The recently built 32-T all- anisotropic shape due to the non-uniform force field, which superconducting magnet at the National High Magnetic Field makes it unsuitable for practical applications despite the en- Laboratory (NHMFL) further proves the feasibility of produc- hanced V1% value. The required extremely large turn-current ing strong uniform fields using cutting-edge superconducting also presents a great challenge. technology [55]. As for the B1 field, we propose to pro- duce it using a gradient-field Maxwell coil [56]. As shown in Fig. 2 (a), such a coil consists of two identical current Concept and performance of our MLS loops (diameter D) placed coaxially at a separation distance of √3D/2. The current in the top loop is clockwise (viewed To increase V1%, the key is to produce a more uniform field- from the top) while the current in the bottom loop is counter- gradient force to balance the gravitational force such that the clockwise. It was first demonstrated by Maxwell that such a net force remains low in a large volume. Base on Eq. (3), coil configuration could produce a highly uniform field gradi- this can be achieved if we have a nearly uniform B field ent in the region between the two loops [56]. and at the meanwhile the field gradient is almost constant in The B1(r) generated by the gradient-field Maxwell coil can the same volume. These two seemingly irreconcilable condi- be calculated using the Biot-Savart law [47] (see details in the 4

Method section), from which the specific potential energy E for an inserted water sample can again be determined. As an example, we show in Fig. 2 (b) the calculated E(r) profile for a coil with D = 8 cm and with an applied current of I = 112.6 kA in the presence of a uniformfield B0 = 24 T. Again, we use the dashed contour and the solid contour to show, respectively, the trapping region and the 0.01g low-force region. By eval- uating the overlapping volume of the two regions, we obtain V1% = 4,004 µL. More importantly, this functional volume is much more isotropic as compared to that in Fig. 1 (b), which makes it highly desirable in practical applications. To optimize the coil current I and the base field B0, we have conduct further analyses. First, for a fixed B0, we vary the coil current I. Representative results at B0 = 24 T are shown in Fig. 3 (a). It is clear that V1% peaks at about I = 112.6. We denote this peak value as Vopt. The decrease of V1% at large I is caused by the fact that the field B1 generated by the coil is no longer much smaller than the base field B0, which impairs the uniformity of the field-gradient force. Next, we vary the base field strength B0 and determine the correspond- ing Vopt at each B0. The result is shown in Fig. 3 (b). It turns out that there exists an optimum base field strength of about 24.7 T (denoted as B0∗), where an overall maximum functional FIG. 4. (a) The maximum functional volume V ∗ for coils with dif- volume (denoted as V ∗) of about 4,050 µL can be achieved. ferent diameters D. (b) The required optimum I∗ and B0∗ to achieve This volume is comparable to those of the largest water drops V ∗ versus the coil diameter D. adopted in the past spaceflight experiments [57, 58]. The above analyses assumed a fixed coil diameter D = 8 cm. When D varies, the maximum functional volume V ∗ and NHMFL [60, 61]) is assumed for producing the B0 field. Four the corresponding MLS parameters (i.e., I∗ and B0∗) should sets of gradient-field Maxwell coils made of REBCO pancake also change. To examine the coil size effect, we have repeated rings are placed in the bore of the superconducting magnet. the aforementioned analyses with a number of coil diameters. Each pancake ring is made of 94 turns of the REBCO tape The results are collected in Fig. 4. As D increases from 6 (width: 4 mm; thickness: 0.043 mm) so its cross section is cm to 14 cm, the maximum functional volume V ∗ increases nearly a square (i.e., 4 mm by 4 mm). The pancake rings are from about 1,500 µL to over 21,000 µL, i.e., over 14 times. arranged along the diagonal lines of a standard gradient-field At the meanwhile, the required coil current I∗ and the base Maxwell coil and the averaged diameter of the pancake rings field strength B0∗ increase almost linearly with D by factors of is about8 cm. This coil configurationis foundto producea B1 about 4 and 1.3, respectively. This analysis suggests that it is field with minimal deviations from that of an ideal gradient- advantageous to have a larger coil provided that the desired I∗ field Maxwell coil. While the superconducting magnet at the and B0∗ can be achieved. NHMFL is cooled by immersion in a liquid helium bath, the compact REBCO coils could be cooled conveniently by a 4-K pulse-tube cryocooler inside a shielded vacuum housing. A DISCUSSION room-temperature center bore with a diameter as large as 6 cm can be used for sample loading and observation. When a Practical design considerations current of about 290 A is applied in the REBCO tapes, a total turn-current NI = 4 94 290 A 109 kA can be achieved. × × ≃ The MLS concept that we have presented requires an applied Note that the quenching critical current of the REBCO tape current of the order 102 kA in both loops of the gradient-field can reach 700 A even under an external magnetic field of 30 Maxwell coil. A natural question is whether this is practi- T [62]. Therefore, operating our REBCO coils with a tape cal. One may consider to make the loop using a thin copper current of 290 A should be safe and reliable. wire with 103 turns so that a current of the order 102 A in To prove the performance of the practical MLS design as the wire is sufficient. However, simple estimation reveals that depicted in Fig. 5 (a), we have repeated the previously pre- the Joule heating in the resistive wire can become so large sented optimization analyses. A representative plot of the spe- such that the wire could melt. To solve this issue, we propose cific potential energy E(r) at a total turn-current NI = 108.37 to fabricate the Maxwell coil using REBCO (i.e., rare-earth kA and B0 = 24T is shownin Fig. 5 (b). The overallshapes of barium copper oxide) superconducting tapes similar to those the trapping region and the low-force region are nearly iden- used in the work by Hahn et al. [59]. A schematic of the pro- tical to those of the ideal gradient-field Maxwell coil. The de- posed MLS setup is shown in Fig. 5 (a). A 24-Tsuperconduct- pendance of V1% on the turn-current NI at B0 = 24 T is shown ing magnet with a bore diameter of 120 mm (existing at the in Fig. 5 (c). A peak functional volume Vopt about 3,450 µL 5

FIG. 5. (a) Schematic of a practical MLS setup that consists of a 24-T superconducting magnet with four sets of gradient-field Maxwell coils made of REBCO pancake rings. The averaged diameter of the pancake rings is about 8 cm. (b) Calculated specific potential energy E(r) for a small water sample place in this MLS with a total turn-current NI = 108.37 kA. The origin of the coordinates is at the center of the lowest pancake ring. The dashed contour denotes the boundary of the trapping region and the solid contour shows the 0.01g low-force region. (c) Calculated V1% versus the turn-current NI at B0 = 24 T. The peak V1% is denoted as Vopt. (b) The obtained Vopt as a function of B0.

FIG. 6. (a) Contour plot of the specific potential energy E(r) at NI = 66.55 kA and B0 = 24 T in the practical MLS. The black contours denote the boundaries of the regions in which the total force leads to an effective gravitational acceleration within 1% and 5% of gM, respectively. (b) The functional volume VM in which the gravity varies within 5% of gM versus the turn-current NI. (c) The peak volume Vopt versus B0.

is achieved. In Fig. 5 (d), the peak volume Vopt obtained at Emulating reduced gravities in extraterrestrial environment various base field strength B0 is shown. Again, the trend is similar to that in Fig. 3. Therefore, despite the change in the coil geometry as compared to the ideal gradient-field Maxwell Beside levitating samples for near-zero gravity research, our coil, the performance of our practical design does not exhibit MLS can also be tuned to partially cancel the Earth’s grav- any significant degradation. ity so that ground-based emulation of reduced gravities in ex- traterrestrial environment (such as on the Moon or the Mars) can be achieved. To demonstrate this potential, we present further analyses of the practical MLS shown in Fig. 5 with lower applied currents for simulating the Martian gravity gM = 0.38g [43]. In Fig. 6 (a), we show contour plots of the specific potential energy E(r) for water samples in the prac- 6 tical MLS when a turn-current of NI = 66.55 kA is applied where at B0 = 24 T. It is clear that the energy contour lines (red 2 2 2 curves) are evenly spaced in the center region of the MLS, R1 = [r Rcos(φ)] + [Rsin(φ)] + z − suggesting a fairly uniform and downward-pointing force in q , (6) 2 2 2 this region. We then calculate the magnitude of the force us- R2 = [r Rcos(φ)] + [Rsin(φ)] + (z L) − − ing Eq. 3. The two black contours in Fig. 6 (a) represent the q boundaries of the regions in which the total force leads to an L = √3D/2 is the separation distance between the two loops, effective gravitational acceleration within 1% and 5% of gM, and I is the current in each loop. respectively. If we define the volume of the contour in which The magnetic field B1(r) generated by the practical MLS the gravity varies within 5% of gM as our functional volume design as depicted in Fig. 5 (a) can be calculated by super- VM, its dependance on the turn-current at B0 = 24 T is shown imposing the fields produced by the four sets of field-gradient in Fig. 6 (b). This functional volume has a peak value Vopt Maxwell coils. The field of each coil is evaluated in the same of about 22.5 103 µL at NI = 66.55 kA. This peak volume way as outlined above. Counting in the base field B0, the total × (r) (z) is so large such that even small animals or plants can be ac- field is then given by B(r) = [B0 + B1 (r)]eˆr + B1 (r)eˆz commodated inside. We have also calculated the peak volume For a solenoid with a length L and a radius R, if we assume Vopt at different base field strength B0. As shown in Fig. 6 (c), the wire is thin such that the turn number N is large but the initially the peak volume Vopt increases sharply with B0, and total turn-current NI remains finite, an exact expression for then it gradually saturates when B0 is greater than about 24 the generated magnetic field can be derived based on the Biot- T. Operating the MLS at higher B0 gives marginal gain in the Savart law [63, 64]: functional volume. 2 ζ+ µ0NI 2 R k 2 2 B(r)(r,z)= − K(k2)+ E(k2) 4π Lr r  k k ζ − (7) Summary ζ+ µ0NI 1 R r B(z)(r,z)= ζk K(k2)+ − Π(h2,k2) 4π L√Rr   R + r ζ In conclusion, our analyses have clearly demonstrated the su- − periority of the proposed MSL concept in comparison with where conventional solenoid MSLs. An unprecedentedly large and 2 4Rr isotropic functional volume, i.e., about three orders of magni- k = 2 2 tude larger than that for a conventional solenoid MSL, can be (R + r) + ζ 4Rr , (8) achieved. The implementation of the superconducting magnet h2 = technology will also ensure stable operation of this MLS with (R + r)2 a minimal energy consumption rate, which is ideal for future ζ = z L/2 low-gravity research and applications. ± ± and the functions K(k2), E(k2), and Π(h2,k2) are given by: π/2 dθ METHOD K(k2)= Z0 1 k2 sin2 θ Magnetic field calculation π/2 p − E(k2)= dθ 1 k2 sin2 θ . (9) Z0 p − The magnetic field B(r) generated at r by a current loop π/2 dθ in three-dimensional space can be calculated using the Biot- Π(h2,k2)= Z 2 2 2 2 Savart law [47]: 0 (1 h sin θ) 1 k sin θ − p − µ0I dl (r l) B(r)= × − , (4) 4π I r 3 Numerical method | | where dl the elementary length vector along the current loop. The magnetic fields produced by the solenoid, the ideal For a field-gradient Maxwell coil with a radius R = D/2, the gradient-field Maxwell coil, and the practical MLS design are generated magnetic field B1(r) can be decomposed into an all calculated using MATLAB. Considering the axial symme- axial componentand a radial componentdue to the axial sym- try, we only evaluate the fields in the r-z plane. The sizes of metry. If we set the z-axis along the co-axial line of the two the computational domains for different types of designs are loops and place the coordinate origin at the center of the bot- essentially shown in Fig. 1 (b), Fig. 2 (b), and Fig. 5 (b). Typ- tom loop, the two components can be evaluated as: ically, the computational domain is discretized using a square r z 2π grid with spatial resolutions ∆ = 10 µm and ∆ = 10 µm, (r) µ0I Rzcos(φ) R(L z)cos(φ) which gives good convergence of the numerical results. The B1 (r,z)= 3 + − 3 dφ 4π Z0  R R  1 2 (5) calculations assumed water properties at the ambient tempera- 2π 2 2 (z) µ0I R Rr cos(φ) Rr cos(φ) R ture, but the same procedurescan be applied to other materials B1 (r,z)= − 3 + 3 − dφ 4π Z0  R1 R2  with different magnetic susceptibilities and densities. 7

DATA AVAILABILITY Florida.

The computer codes and the data supporting the findings of AUTHOR CONTRIBUTIONS this study are available from the corresponding author upon request. W.G. designed the research; H.S. conducted the simulations; H.S. and W.G. analyzed the results and wrote the paper.

ACKNOWLEDGMENTS COMPETING INTERESTS

This work is supported by National Science Foundation under The authors declare no competing interests Grant No. CBET-1801780. The work was conducted at the National High Magnetic Field Laboratory, Florida State Uni- versity, which is supported by National Science Foundation REFERENCES Cooperative Agreement No. DMR-1644779 and the state of

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