^{1,2}, Baptiste Darbois Texier

^{1,2}, Christophe Clanet

^{1,2}and David Quéré

^{1,2}

### Abstract

Liquid oxygen, which is paramagnetic, also undergoes Leidenfrost effect at room temperature. In this article, we first study the deformation of oxygen drops in a magnetic field and show that it can be described via an effective capillary length, which includes the magnetic force. In a second part, we describe how these ultra-mobile drops passing above a magnet significantly slow down and can even be trapped. The critical velocity below which a drop is captured is determined from the deformation induced by the field.

I. INTRODUCTION

II. STATIC SHAPE OF OXYGEN DROPS

III. CAPTURING DROPS

IV. CONCLUSION

### Key Topics

- Magnets
- 39.0
- Magnetic fields
- 20.0
- Fluid drops
- 14.0
- Magnetic liquids
- 12.0
- Liquid surfaces
- 9.0

## Figures

Side views of liquid oxygen drops on a glass plate. (a) Liquid oxygen drop of equatorial radius R = 0.7 mm on a glass plate at room temperature. (b) The same drop in the presence of a magnet (in black, in the picture) 1 mm below it. The drop is deformed and is no longer spherical. (c) Same set-up as before turned upside down. The drop holds against gravity and its shape is almost the same as is (b). The solid line represents 1 mm in each picture.

Side views of liquid oxygen drops on a glass plate. (a) Liquid oxygen drop of equatorial radius R = 0.7 mm on a glass plate at room temperature. (b) The same drop in the presence of a magnet (in black, in the picture) 1 mm below it. The drop is deformed and is no longer spherical. (c) Same set-up as before turned upside down. The drop holds against gravity and its shape is almost the same as is (b). The solid line represents 1 mm in each picture.

Modulus of the magnetic force per unit volume f m (deduced from a measurement of the magnetic field), normalized by the volumic weight of the drop ρg, as a function of z, the distance between the magnet and the bottom of the drop as sketched in the inset.

Modulus of the magnetic force per unit volume f m (deduced from a measurement of the magnetic field), normalized by the volumic weight of the drop ρg, as a function of z, the distance between the magnet and the bottom of the drop as sketched in the inset.

Magneto-capillary length a* as a function of z. The solid line is Eq. (3) , where f m is deduced from the measurement of the magnetic field. Error bars on this line arise from the discrete number of points where B is measured, and are therefore larger where the field rapidly varies. The black dots show half the thickness (a* ≈ h/2) of oxygen puddles. The dashed line represents the capillary length a = 1.1 mm without field.

Magneto-capillary length a* as a function of z. The solid line is Eq. (3) , where f m is deduced from the measurement of the magnetic field. Error bars on this line arise from the discrete number of points where B is measured, and are therefore larger where the field rapidly varies. The black dots show half the thickness (a* ≈ h/2) of oxygen puddles. The dashed line represents the capillary length a = 1.1 mm without field.

Shape of an oxygen drop of volume Ω = 1.2 mm3 in the absence (gray - red online) and presence (black) of a gradient of squared magnetic field of 70 T2/m (obtained at a distance z = 1 mm above a magnet). Solid lines are solutions of Eq. (A3) (derived in the Appendix) and data (circles) are obtained from Figures 1(a) and 1(b) .

Shape of an oxygen drop of volume Ω = 1.2 mm3 in the absence (gray - red online) and presence (black) of a gradient of squared magnetic field of 70 T2/m (obtained at a distance z = 1 mm above a magnet). Solid lines are solutions of Eq. (A3) (derived in the Appendix) and data (circles) are obtained from Figures 1(a) and 1(b) .

(a) Top views of an oxygen drop of radius R = 1 mm passing above a parallelepipedic magnet of width 1 cm indicated by the accompanying Fig. 5 . Time increases from top to bottom (Δt = 25 ms between two images). The drop is deformed above the magnet and it comes out of the magnetic trap significantly slowed down, as seen from the change of slope of the dotted lines (movie 1 is slowed down 20 times). (b) Velocity of the drop as a function of time. The drop arrives at V = 20 cm/s, and it is successively accelerated and decelerated above the magnet, which it leaves at a velocity V′ = 12 cm/s. (c) Measurement of the velocity of a drop passing above a series of three magnets (indicated by their number). The drop loses some speed at each magnet and it is finally captured above the third magnet: the velocity goes to zero and the drop then oscillates in the trap (movie 2 is slowed down 5 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.1] [URL: http://dx.doi.org/10.1063/1.4796133.2]doi: 10.1063/1.4796133.1.

doi: 10.1063/1.4796133.2.

(a) Top views of an oxygen drop of radius R = 1 mm passing above a parallelepipedic magnet of width 1 cm indicated by the accompanying Fig. 5 . Time increases from top to bottom (Δt = 25 ms between two images). The drop is deformed above the magnet and it comes out of the magnetic trap significantly slowed down, as seen from the change of slope of the dotted lines (movie 1 is slowed down 20 times). (b) Velocity of the drop as a function of time. The drop arrives at V = 20 cm/s, and it is successively accelerated and decelerated above the magnet, which it leaves at a velocity V′ = 12 cm/s. (c) Measurement of the velocity of a drop passing above a series of three magnets (indicated by their number). The drop loses some speed at each magnet and it is finally captured above the third magnet: the velocity goes to zero and the drop then oscillates in the trap (movie 2 is slowed down 5 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.1] [URL: http://dx.doi.org/10.1063/1.4796133.2]doi: 10.1063/1.4796133.1.

doi: 10.1063/1.4796133.2.

Exit velocity V′ of an oxygen drop of radius R = 1 mm escaping a magnetic trap, as a function of its initial velocity V. We observe a critical velocity V* = 13.5 cm/s below which the drop is captured by the magnet (V′ = 0). The solid line represents Eq. (5) , and the gray thin line is V′ = V.

Exit velocity V′ of an oxygen drop of radius R = 1 mm escaping a magnetic trap, as a function of its initial velocity V. We observe a critical velocity V* = 13.5 cm/s below which the drop is captured by the magnet (V′ = 0). The solid line represents Eq. (5) , and the gray thin line is V′ = V.

Side view chronophotograph of an oxygen drop of radius R = 1 mm passing above the magnet (black rectangle below the glass plate) at a velocity V = 60 cm/s. Time interval between successive photos: 8 ms (movie is slowed down 50 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.3]doi: 10.1063/1.4796133.3.

Side view chronophotograph of an oxygen drop of radius R = 1 mm passing above the magnet (black rectangle below the glass plate) at a velocity V = 60 cm/s. Time interval between successive photos: 8 ms (movie is slowed down 50 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.3]doi: 10.1063/1.4796133.3.

Critical velocity V* as a function of the deformation R max /R. The dashed line represents Eq. (6) with (6γ/ρR)1/2 = 19.5 cm/s, adjusted to fit the data.

Critical velocity V* as a function of the deformation R max /R. The dashed line represents Eq. (6) with (6γ/ρR)1/2 = 19.5 cm/s, adjusted to fit the data.

Top view chronophotograph of an oxygen drop of radius R = 1 mm passing above a magnet located at z = 0.5 mm below the liquid, and indicated by the white lines. Initial velocity of the drop is V = 48 cm/s. The variation of R in the three first pictures is due to small initial vibration of the drop. Interval between images: 10 ms (movie is slowed down 60 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.4]doi: 10.1063/1.4796133.4.

Top view chronophotograph of an oxygen drop of radius R = 1 mm passing above a magnet located at z = 0.5 mm below the liquid, and indicated by the white lines. Initial velocity of the drop is V = 48 cm/s. The variation of R in the three first pictures is due to small initial vibration of the drop. Interval between images: 10 ms (movie is slowed down 60 times) (enhanced online). [URL: http://dx.doi.org/10.1063/1.4796133.4]doi: 10.1063/1.4796133.4.

Diagram of the drop. s is the curvilinear coordinate along the profile of equation r(ζ) defining the shape of the drop.

Diagram of the drop. s is the curvilinear coordinate along the profile of equation r(ζ) defining the shape of the drop.

(a) Shape of an oxygen drop of volume Ω = 1.2 mm3 in the absence of magnetic field. Solid line is the solution of Eq. (A3) and circles are obtained from Figure 1(a) . (b) The same drop placed at z = 1 mm above the magnet. Similarly, solid line comes from computation and circles are obtained from Figure 1(b) . (c) Magnetic field B as a function of height z above a centimetric cylindrical magnet. We do not have data below z = 0.5 mm, which corresponds to half the thickness of the probe used for the measurement. (d) Computed shape of a drop of radius Ω = 1.2 mm3 in the absence of magnetic field. (e) Computed shape of the same drop placed in a hypothetical magnetic field with a very strong gradient. The magnetic force is much higher at the bottom of the drop that at the top, yielding a shape with an enhanced curvature at the bottom, which is very different from the one presented in (b). (f) Hypothetical magnetic field with a very strong gradient. The corresponding equation is B(z) = 0.5 exp (−z/0.3) (with z expressed in mm).

(a) Shape of an oxygen drop of volume Ω = 1.2 mm3 in the absence of magnetic field. Solid line is the solution of Eq. (A3) and circles are obtained from Figure 1(a) . (b) The same drop placed at z = 1 mm above the magnet. Similarly, solid line comes from computation and circles are obtained from Figure 1(b) . (c) Magnetic field B as a function of height z above a centimetric cylindrical magnet. We do not have data below z = 0.5 mm, which corresponds to half the thickness of the probe used for the measurement. (d) Computed shape of a drop of radius Ω = 1.2 mm3 in the absence of magnetic field. (e) Computed shape of the same drop placed in a hypothetical magnetic field with a very strong gradient. The magnetic force is much higher at the bottom of the drop that at the top, yielding a shape with an enhanced curvature at the bottom, which is very different from the one presented in (b). (f) Hypothetical magnetic field with a very strong gradient. The corresponding equation is B(z) = 0.5 exp (−z/0.3) (with z expressed in mm).

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