liquid flow and control without solid walls

doi.org/10.26434/chemrxiv.7207001.v1

Liquid Flow and Control Without Solid WallsPeter Dunne, Takuji Adachi, Alessandro Sorrenti, J.M.D. Coey, Bernard Doudin, Thomas Hermans

Submitted date: 15/10/2018 • Posted date: 16/10/2018Licence: CC BY-NC-ND 4.0Citation information: Dunne, Peter; Adachi, Takuji; Sorrenti, Alessandro; Coey, J.M.D.; Doudin, Bernard;Hermans, Thomas (2018): Liquid Flow and Control Without Solid Walls. ChemRxiv. Fileset.

Solid walls become increasingly important when miniaturizing fluidic circuitry to the micron scale. They limitflow-rates due to friction and high pressure drop, and are plagued by fouling. Wall interactions have beenreduced by hydrophobic coatings, porous surfaces, nanoparticles, changing the surface electronic structure,electrowetting, surface tension pinning, and atomically flat channels. We show wall-less aqueous liquidchannels stabilised by a magnetic field that acts on a surrounding immiscible magnetic liquid. This createsself-healing, uncloggable, and near-frictionless liquid-in-liquid microfluidic channels that can be deformed andeven closed without ever touching a solid wall. Basic fluidic operations including valving, mixing, and pumpingcan be achieved by moving permanent magnets. Our approach is compatible with conventional microfluidics,while opening unique prospects for nanofluidics without high pressures.

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Liquid flow and control without solid walls

Peter Dunne1,2†, Takuji Adachi1†, Alessandro Sorrenti1,3, J.M.D. Coey4, Bernard Doudin2, Thomas M.

Hermans1,*

Affiliations:

1 University of Strasbourg, CNRS, ISIS UMR 7006, F-67000 Strasbourg, France 2 University of Strasbourg, CNRS, IPCMS UMR 7504, 23 rue du Loess, F-67034 Strasbourg, France 3 Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zürich, Vladimir Prelog Weg 1, 8093 Zürich, Switzerland 4 School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland * Correspondence to: [email protected] † Equal contribution

Solid walls become increasingly important when miniaturizing fluidic circuitry to the micron

scale or smaller.1 They limit achievable flow-rates due to friction and high pressure drop, and are

plagued by fouling2. Approaches to reduce the wall interactions have been explored using

hydrophobic coatings3,4, liquid-infused porous surfaces4–6, nanoparticle surfactant jamming7,

changing the surface electronic structure8, electrowetting9,10, surface tension pinning11,12, and

atomically flat channels13. An interesting idea is to avoid the solid walls altogether. Droplet

microfluidics achieves this, but requires continuous flow of both the liquid transported inside the

droplets and the outer carrier liquid14. We demonstrate a new approach, where wall-less aqueous

liquid channels are stabilised by a quadrupolar magnetic field that acts on a surrounding

immiscible magnetic liquid. This creates self-healing, uncloggable, and near-frictionless liquid-

in-liquid microfluidic channels that can be deformed and even closed in real time without ever

touching a solid wall. Basic fluidic operations including valving, mixing, and ‘magnetostaltic’

pumping can be achieved by moving permanent magnets having no physical contact with the

channel. This wall-less approach is compatible with conventional microfluidics, while opening

unique prospects for implementing nanofluidics without excessively high pressures.

Magnetic forces have been used to avoid contact with the walls of a device by levitation of particles or

live cells in suspension15, and a first attempt to make wall-less microfluidic channels resulted in

continuous ‘magnetic antitubes’ of water surrounded by an aqueous paramagnetic salt solution16 using

a bulky electromagnet. However, the antitube lifetime was limited by ion interdiffusion between the

two liquids, the salts were toxic, and contact with one stationary wall could not be avoided. Here we

overcome all these limitations, creating entirely wall-less microfluidic channels consisting of

diamagnetic antitubes completely enclosed by an immiscible, non-toxic paramagnetic fluid. The key

magnetic confinement source design is made of a quadrupolar arrangement leading to nearly isotropic

2D confinement with a null magnetic field at the centre (Fig. 1a, 1b). Commercially available 6 x 6 x

50 mm Nd2Fe14B magnets were used for linear channels, or else custom-made bilayers were waterjet

cut to define more complex fluidic circuitry. The magnets were housed in a 3D printed support with

conventional microfluidic inlet and outlet ports (Fig. 1a). The strength of the confinement depends on

the magnetic susceptibility of the paramagnetic host fluid, so commercial ferrofluids were the natural

choice, although their opacity severely limits optical characterisation. Aqueous antitubes were formed

by pumping water into ferrofluid containing quadrupoles, and visualised by top-view X-ray imaging

(Fig. 1c) or in side-view along the channel (Fig. 1d)†. To overcome the ferrofluid opacity, we developed

a new class of rare-earth oil (see Methods), which we call ‘Magoil’ (Fig. 1e), inspired by

diethylenetriaminepentaacetate-based contrast agents used for magnetic resonance imaging17. Antitubes

could then be imaged using standard optical or fluorescent microscopy by adding some tracer (contrast

ink or fluorescent dye, respectively) to the water antitube (Fig. 1f). Antitube extrusion and retraction

can be visualised in real-time (movie S1), and they remain stable for months. Moreover, trapped gas

bubbles that are often problematic in conventional devices can easily be removed, since their buoyancy

in Magoil overcomes the magnetic confinement, and they rise to the oil/air interface whereas the liquid

inside the antitube remains confined (cf. movie S1).

Our liquid-in-liquid design offers advantages of stability and robustness for fluid transport. Fig. 2A

illustrates self-healing after an antitube in the ferrofluid was severed with a spatula. Recovery without

applied external pressure is rapid, and it can even be observed in the magnetically weaker Magoil

† It is possible to stabilize air antitubes using ferrofluids as well.

(movie S2). The antitubes cannot be clogged: when glass beads are intentionally jammed into the

antitube (Fig. 2b), they can be flushed out by a minimal applied pressure. Even a bead much larger than

the antitube diameter can be pushed out using less than 10 mbar (Figs. 2c, 2d). The liquid walls of the

antitube stretch to avoid clogging, and return to their original size when the obstruction is expelled. A

change of external pressure alters the antitube size. In Extended Data Fig. 1 we show two extreme cases:

antitubes remain unchanged with externally applied pressure for an open outlet (at atmospheric

pressure), but dilate when the same pressure is applied with the outlet closed off. This can also be seen

in Fig. 2c where the tube dilates behind the bead to accommodate the increased local pressure. In

addition, ferrofluid surfaces have been shown to be anti-scaling18 and resistant to biofilms5.

At equilibrium, stable confinement of an antitube results from the competing magnetic energy of the

confining fluid and the surface energy σ of the magnetic/nonmagnetic interface. These two energy

densities, or effective pressures, inserted into the magnetically augmented version of Bernoulli’s

equation19, give the equilibrium diameter of the antitube:

20 0

42 I I

dMH M

sµ µ

=+

(1)

where HI, MI are the magnetic field and magnetization values at the interface, and M is the field-

averaged magnetization of the confining fluid induced by HI. This simplified expression considers the

magnetic pressure, ½µ0H2, to be significantly larger than any difference in hydrostatic pressure (see

supplementary text, section 1 for the derivation). Equation (1) can be linearized when M = cH, under

the geometrical conditions w ≤ ½ h, and d ≤ ½ w, typical of our devices (see Fig. 1a). This linear model

(LM) gives the minimum equilibrium dimensionless diameter d* = d/w as

( )2

3*2 2 1D

dN

pc c

=+

(2)

where2

0 rD

M wN µs

= is the magnetic confinement number expressing the ratio of magnetic to surface

energies (derivation in supplementary text, section 2). Note that a 1000-fold increase in χ reduces d* by

a factor 100, revealing how important the confining fluid properties are; the detailed dependence is

plotted in Extended Data Fig. 2. At intermediate fields values, typical of our experimental conditions

(Extended Data Fig. 3), significant deviation of the magnetization from linearity appears, and self-

consistent solutions of Eq. (1) are obtained by numerical iteration, in our full model (FM) (see Extended

Data Fig. 4 for computational algorithm).

Fig. 2e shows how the antitube diameter d changes with the dimensions of the magnetic flux source and

the type of confining fluid. Good agreement is found between the experimental points for water

antitubes and the predictions of Eqns. (1) and (2), using measured magnetization and interface energy

(data in Extended Data: Table 1, Figs. 5, and 6). For ferrofluid tubes larger than 150 µm in ferrofluids,

X-ray imaging can be used, while optical imaging of smaller tubes is possible in Magoil.

Miniaturization of the channels is possible thanks to an attractive feature of permanent magnets, namely

that the fields they produce are independent of length scale !. Hence H, M and the magnetic energy

density do not depend on channel size. The interface energy s however scales as !-1; with an oil/water

interface energy of 23 mJ m-2, a field of 100 mT and a susceptibility of 1, the antitube will become

unstable below a diameter of 5 µm (Extended Data Fig. 7). Further miniaturization would require s to

be reduced, which is possible by using a surfactant as illustrated in Fig. 2e (red and orange curves), and

Extended Data Fig. 2 (white circles). By combining a very strong ferrofluid (QK100 with a

magnetisation of 100 kA m–1 in a hydrocarbon medium) and a double surfactant approach (e.g., Span-

8020 in the ferrofluid, and Tween-20 in the aqueous antitube), we estimate that antitube diameters of

~100 nm could ultimately be achieved. Though difficult to image, such nanometer-sized antitubes

would allow practical nanofluidic devices to be realized.

A key advantage of liquid-in-liquid flow is the frictionless transport and negligible pressure drop. A

dramatic illustration is shown in Fig. 3a and movie S3, where flow of a magnetically confined antitube

made of honey is compared to honey flow in a standard tube. Here, we observed an antitube flow of

39.4 ± 0.7 g/hour, 70 times faster than through a conventional plastic tubing of the same diameter d =

1.1 mm, (0.55 ± 0.11 g/hour). This is equivalent to Poiseuille flow of honey through a plastic tube with

a diameter 3 times larger than the antitube. The ferrofluid acts as a lubricating layer, with an effective

slip length at the honey/ferrofluid boundary that can be approximated by21

1hf

f

b t µµ

æ ö= -ç ÷ç ÷

è ø (3)

where tf is the thickness of ferrofluid between the honey and solid wall ( ~1.45 mm), µh and µf are the

kinematic viscosities of honey and ferrofluid respectively, giving a slip length 480 mm. When b/d > 5

the velocity profile is essentially plug-like. Remarkably, the flow rate of honey through the antitube

was 1.5 times faster than when there was no tube, likely due to competition between orifice wetting22

and the higher hydrostatic pressure due to the greater height of the honey column in the antitube design.

Fig 3b shows a 1 m long quadrupole built using 80 magnets, mounted and glued in 20 3D-printed

segments. A water antitube was easily formed and water flows freely through it. Since it is a transparent

cylinder surrounded by ferrofluid, it acts as an optical waveguide (Fig. 3c). Exotic tube cross-sections

can be achieved by further iteration of the initial quadrupole design, leading to novel flow geometries.

For example, four bar magnets pointing in the same direction produce a double lobe antitube (Fig. 3b,

and movie S4) and a circular arrangement of six magnets yields an asymmetric triple antitube (Fig. 3b,

and movie S5), in agreement with calculations. As there are innumerable possible magnet

arrangements23, this result demonstrates a promising aspect of magnetic confinement.

All essential functions required to control microfluidic flow can be implemented with aqueous antitubes

(Fig. 4). Valves can be constructed by adding a magnet whose axis of magnetization is perpendicular

to the quadrupole axis. These valving magnets simply pinch off the antitube by removing the null field

at the centre (Fig. 4a, 4b), thus interrupting the liquid flow (see movies S6 – S8). A single transverse

valving magnet was able to sustain an excess pressure of 125 mbar, whereas a symmetric dual valve

(Fig. 4c) withstood 300 mbar. Pumping arises as an extension of the valving principle; travelling pinch

points can be created by mechanical magnet displacement or by sequential excitation of electro- or

electro-permanent magnets. A proof-of-principle using six transverse valving magnets attached to a

rotor, enables the pinch point to travel along the channel at 10 – 100 mm s-1 (Fig. 4d, movie S9),

resulting in magnetostaltic flow, similar to that produced by a peristaltic pump. A pumping pressure of

20 mbar and flow rates of 500 µL min-1 were achieved (Fig. 4e, 4f) in antitubes with d = 500 µm. This

method has significant advantages compared to traditional external peristaltic pumps as it can be

implemented on-chip and does not create mechanical wear on a tube.

We have also demonstrated the applicability of magnetic confinement to more complicated fluidic

circuitry. A quadrupole fluidic device was prepared by first assembling two Nd2Fe14B plates that are

perpendicularly magnetized in opposite directions (Fig. 4g). Any desired shape of channel can be cut,

and the null-field line follows the track centre. In-plane quadrupoles are different (cf. Fig. 1); there the

direction of the channels within the plane should make an angle ≳ 30° to the magnetic axis (Extended

Data Figs. 8–10). Symmetric splitting of the flow was demonstrated in a ferrofluid antitube Y-junction

(movie S10). Merging of the flow at a Y-junction was visualized using antitubes stabilized by Magoil

(Fig. 4h). Remarkably, the mixing occurs immediately after the Y-junction due to a Kelvin-Helmholz

instability16. This is in striking contrast to the laminar flow observed in a 3D printed microfluidic chip

with the same channel size and geometry as the antitube (Fig. 4i).

The near-perfect slip conditions at the liquid wall lead to the unusual fluidic behaviour we have

observed at the sub-mm scale, where the flow in conventional devices would necessarily be laminar.

The magnetic control of basic microfluidic functions that we have implemented clears the path for fully-

integrated antitube fluidics. We envisage that miniaturized fluidic circuits with no solid walls will be

scalable down to submicron sizes, enabling better control over transport of matter at the nanoscale.

References:

1. Tabeling, P. Introduction to Microfluidics. (OUP Oxford, 2005).

2. Mukhopadhyay, R. When Microfluidic Devices Go Bad. Anal. Chem. 77, 429 A-432 A (2005).

3. Zhao, B., Moore, J. S. & Beebe, D. J. Surface-Directed Liquid Flow Inside Microchannels. Science 291, 1023–1026 (2001).

4. Wong, T.-S. et al. Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477, 443–447 (2011).

5. Wang, W. et al. Multifunctional ferrofluid-infused surfaces with reconfigurable multiscale topography. Nature (2018). doi:10.1038/s41586-018-0250-8

6. Leslie, D. C. et al. A bioinspired omniphobic surface coating on medical devices prevents thrombosis and biofouling. Nat. Biotechnol. 32, 1134–1140 (2014).

7. Forth, J. et al. Reconfigurable Printed Liquids. Adv. Mater. 1707603 (2018). doi:10.1002/adma.201707603

8. Secchi, E. et al. Massive radius-dependent flow slippage in carbon nanotubes. Nature 537, 210–213 (2016).

9. Banerjee, A., Kreit, E., Liu, Y., Heikenfeld, J. & Papautsky, I. Reconfigurable virtual electrowetting channels. Lab. Chip 12, 758 (2012).

10. Choi, K., Ng, A. H. C., Fobel, R. & Wheeler, A. R. Digital Microfluidics. Annu. Rev. Anal. Chem. 5, 413–440 (2012).

11. Lee, W. C., Heo, Y. J. & Takeuchi, S. Wall-less liquid pathways formed with three-dimensional microring arrays. Appl. Phys. Lett. 101, 114108 (2012).

12. Walsh, E. J. et al. Microfluidics with fluid walls. Nat. Commun. 8, 816 (2017).

13. Keerthi, A. et al. Ballistic molecular transport through two-dimensional channels. Nature 558, 420–424 (2018).

14. Shang, L., Cheng, Y. & Zhao, Y. Emerging Droplet Microfluidics. Chem. Rev. 117, 7964–8040 (2017).

15. Zhao, W., Cheng, R., Miller, J. R. & Mao, L. Label-Free Microfluidic Manipulation of Particles and Cells in Magnetic Liquids. Adv. Funct. Mater. 26, 3916–3932 (2016).

16. Coey, J. M. D., Aogaki, R., Byrne, F. & Stamenov, P. Magnetic stabilization and vorticity in submillimeter paramagnetic liquid tubes. Proc Natl Acad Sci 106, 8811–8817 (2009).

17. Caravan, P., Ellison, J. J., McMurry, T. J. & Lauffer, R. B. Gadolinium(III) Chelates as MRI Contrast Agents: Structure, Dynamics, and Applications. Chem. Rev. 99, 2293–2352 (1999).

18. Masoudi, A., Irajizad, P., Farokhnia, N., Kashyap, V. & Ghasemi, H. Antiscaling Magnetic Slippery Surfaces. ACS Appl. Mater. Interfaces 9, 21025–21033 (2017).

19. Rosensweig, R. E. Ferrohydrodynamics. (Dover Publications, 2014).

20. Posocco, P. et al. Interfacial tension of oil/water emulsions with mixed non-ionic surfactants: comparison between experiments and molecular simulations. RSC Adv 6, 4723–4729 (2016).

21. Choi, C.-H. & Kim, C.-J. Large Slip of Aqueous Liquid Flow over a Nanoengineered Superhydrophobic Surface. Phys. Rev. Lett. 96, 066001 (2006).

22. Ferrand, J., Favreau, L., Joubaud, S. & Freyssingeas, E. Wetting Effect on Torricelli’s Law. Phys. Rev. Lett. 117, 248002 (2016).

23. Furlani, E. P. Permanent Magnet and Electromechanical Devices. (Academic Press, 2001).

Acknowledgments: We acknowledge the support of the University of Strasbourg Institute for

Advanced Studies (USIAS) Fellowship, The Chaire Gutenberg of the Région Alsace (J.M.D.C.), the

Labex NIE 11-LABX-0058_NIE within the Investissement d’Avenir program ANR-10-IDEX-0002-

02, and SATT Conectus funding. We are grateful to Dr. Hu Boping, of San Huan Corporation for giving

us thin magnetic bilayer sheets.

Fig. 1 | Wall-less magnetic confinement of liquids. Centre: exploded view of the fluidic cell,

made of: a permanent magnets (red, blue) in an in-plane quadrupolar configuration creating a low-field

zone, at the centre where a tube of water (yellow) is stabilized inside a magnetic immiscible liquid; b

contour plot of the magnetic field; c X-ray transmission top-view of a water antitube in ferrofluid; d

optical side-view of a water antitube in ferrofluid; e chemical formula of the transparent paramagnetic

oil called ‘Magoil’ (with Ho3+ unless otherwise specified); and f optical top-view of a water antitube

(dyed black for contrast) in Magoil. The scale bars are 5 mm.

Fig. 2 | Robustness, self-healing, and scaling of water antitubes. a Top-view X-ray images

illustrating the mechanical rupture of a tube using a spatula, self-healing by returning to

equilibrium within minutes; b 600 µm glass beads jammed into a 1.5 mm antitube that can be

expelled with a slight increase in applied pressure; c a 2 mm diameter bead larger than the

antitube diameter (d = 0.5 mm) does not cause clogging. d a small increase in flow rate is

observed after the bead (of panel C) leaves the antitube; e Water antitube diameter d versus

gap width w. Points are experimental data (cf. Extended Data Figs. 12, 13 for method of width

determination), lines are calculated from Equs (1,2): for Magoil (LM), Magoil with 1% Tween-

20 surfactant (cf. Extended Data Fig. 13) in the water (LM), and APG311 (FM). The red curve

shows the FM model outcome with a very strong ferrofluid (QK100, 100 kA m–1) and where

the surface tension s is lowered to 1 mN m–1 using surfactants (e.g., Tween-20 and Span-80).

The inset shows an enlarged area for APG311 data points and error bars from at least three

independent experiments.

Fig. 3 | Special features of magnetic confinement. a Comparative flow of honey under

gravity through an antitube, no tube (i.e., air only), and a normal tube of the same diameter (see

movie S3): b Side-view of a 1 m long water antitube (d = 2 mm), used as a light waveguide,

shown at three ambient light intensities; c unique antitube cross-sections using variations of

the design of Fig. 1 for various magnet arrangements. Top show simulations, bottom

experiments. Scale bar is 3 mm (see movies S4, S5).

Fig. 4 | Magnetically implemented fluidic functionalities. Simulated valving of an antitube

(in yellow) using additional valving magnets (green) a one magnet (Nv = 1, see movie S7), b

two magnets (Nv = 2, see movie S8); c measured flow rate at the exit port upon addition and

removal of 1 or 2 valving magnets, d cross-sectional schematic of a peristaltic pump using a

six-armed wheel rotating at angular frequency w in close proximity to a quadrupole (see movie

S9); e Slow rotation (2 rpm) leads to pulsed flow, while fast rotation (14 rpm) produces a

smoother flow; f the average flow rate and standard deviation vs. rotation rate w; g out of plane

magnetization configuration for a waterjet cut Y-junction in two magnets; h optical image of a

Tb3+–Magoil stabilised aqueous antitube, where blue and pink dye are flowed into the inlets

(300 µL min–1), and mix immediately upon contact before flowing towards the outlet (cf.

magnetic contours in Extended Data Fig. 11). i 3D printed comparison track with solid walls

at the same flow rate (as panel h) exhibiting laminar flow and no convective mixing.

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1

Methods

Synthesis of Rare Earth Oil

Inspired by the well-known gadolinium complexes of DTPA (diethylenetriaminepentacetate), used as

paramagnetic contrast agents in magnetic resonance imaging (MRI), we prepared a novel amphiphilic complex

of DTPA with the paramagnetic rare-earth ion holmium (III), bearing two hydrophobic side chains connected to

the DTPA moiety through ester linkage. In particular, by reaction of DTPA dianhydride with the branched alcohol

2-butyl-1-octanol we obtained a tricarboxylic chelating ligand, which upon complexation with Ho3+ under

alkaline conditions afforded a stable neutral Ho-DTPA complex. The latter, insoluble in water, was mixed with

2-butyl-1-octanol (30% wt of the latter) affording a water-immiscible homogeneous fluid (Magoil) featuring

positive magnetic susceptibility. The magnetic properties of the neutral Ho-DTPA complex (i.e. before mixing

with the alcohol) were evaluated by NMR measurement, using the Evans method1, giving a value 4.40 x10-7 m3

mol-1 for the molar susceptibility, which is comparable with that measured for the inorganic salt Ho(AcO)3 , as

well as of the order of magnitude expected for Ho3+ 2. Our paramagnetic oil is a transparent non-ionic liquid,

slightly pink-yellowish because of the fluorescence of Ho(III), and it is physico-chemically stable in contact with

aqueous solutions. Overall these characteristics make it different from other magnetic fluids, such as: ferrofluids

(i.e. black colloidal suspensions of nanometre-sized magnetic particles), or magnetic room temperature ionic-

liquids (RTIL). Lastly, we stress that the preparation of the oil could be easily scaled up to 25 g, and that we used

also different trivalent rare-earth ions, such as erbium and terbium, obtaining analogous results.

Preparation of the Ho-DTPA-(2-buthyl-1-octanol)2 based paramagnetic oil

DTPA dianhydride (5 g, 14 mmol) was dispersed in dry dimethylformamide (100 mL) under argon

atmosphere, and the obtained suspension was heated to 70-75°C under stirring, until complete solubilization was

observed. After that, 2-butyl-1-octanol (5.5 g, 28 mmol, 2 eq.), dissolved in 10 mL DMF, was added dropwise to

the above solution and the resulting homogeneous mixture was left under stirring at 50°C for 4h. The

2

completeness of the reaction was monitored by liquid chromatography/mass-spectrometry (LC-MS). Afterwards,

the solvent was removed (rotary evaporation + high vacuum pump) to give 2 as a sticky yellowish solid that was

used without further purification.

NN

ON

O

O

O O

OOH

O

NN

ON

O

O O

HO OOHO

R R

OHO

1) DMF, 70°C, 30 min

2) ROH, 5 h

R =

Paramagnetic Oil

DTPA dianhydride

1.5 eq HoCl33 eq NaOHEtOH/H2O 1:1

ROH (10-40 wt%)

1 2

HoN N

O N

O

O

O

O

O

OO

O

OR

R

3

To a stirred 0.2 M solution of the ligand 2 in ethanol (EtOH), an aqueous 1 M solution of NaOH (3 eq.) was

added dropwise at room temperature, followed by the addition of HoCl3(6 H2O) (1.5 eq. with respect to 2),

dissolved in the proper amount of water so to have a final 0.1 M solution of the complex of 3 in ethanol/water

1:1. The resultant homogeneous mixture was left stirring at RT until a viscous pink oil, that responds to magnetic

fields, separated out from the water phase (typically 0.5-1 h). After that, the solution was concentrated by rotary

evaporation to remove EtOH, and the water phase extracted with dichloromethane. The organic phase was

separated, dried over MgSO4 and rotary evaporated to give the complex 3 as a glassy solid. Finally, the latter was

dissolved in the minimum amount of dichloromethane, and the fatty alcohol 2-butyl-1-octanol (30 wt%) was

added to the resultant solution. Afterwards, the mixture was rotary evaporated under high vacuum to completely

eliminate dichloromethane, resulting in the target homogeneous transparent paramagnetic oil.

3

Measurement of the magnetic susceptibility of the paramagnetic oil

The magnetic susceptibility of the neutral complexes 3 was measured by the Evans method, using NMR

spectroscopy1. This method relies on the fact that the chemical shift of the 1H NMR signals of a molecule depend

on the bulk susceptibility of the medium. In particular, we used the resonance line of the residual proton of CDCl3

(i.e. the NMR solvent) as a reference, by comparing its chemical shift in pure CDCl3 and in a CDCl3 solution

containing the complex 3. From the difference in the chemical shifts we could calculate the mass and molar

susceptibilities of 3 (cmass = 4.88 x10-7 m3 kg-1 ; cmol = 4.40 x10-7 m3 mol-1).

Magnetic Properties of the Commercial Ferrofluids

Magnetization measurements were performed using vibrating sample magnetometer with a compact variable

permanent magnet source3. Samples were mounted into 3D printed sample holders with cylindrical chambers of

diameter 3 mm and length 3.88 or 5.88 mm, ensuring a well-defined demagnetization factor, and maximal use of

the uniform region of the pick-up coils.

The magnetic field dependence of the magnetization was assumed to be due to dispersed non-interacting

spherical particles in a non-magnetic medium, and the data was fitted with either a single component, or two

component Langevin function, which are characterized by:

( )LBM Mf a= (1)

where MB is the bulk magnetization (480 kA m-1 for magnetite), f is the volume fraction of magnetic material,

and L is the Langevin function

( ) ( )L cotha aa1

= - (2)

where 3

0

B6kBM d HT

a µ p= , µ0 is the permeability of free space, d is the particle diameter, kB is Boltzmann’s

constant, and T is the temperature.

4

The linearized initial susceptibilities then become, a) for one component

2

130

B18kBM dT

µ p fc = (3)

and b) for two components

( )2

3 301 1 2 2

B2 18k

BM d dTpc µ f f+= (4)

where the subscripts denote components one and two respectively, and the saturation magnetization for one and

two components are Ms = f1MB and Ms = (f1 + f2)MB respectively.

After correcting for non-zero magnetic field offsets, the applied field H was corrected for the appropriate

demagnetization factor N to get the internal field H’ using:

'H H NM= - (5)

For a cylinder of length L, and radius R with its symmetry axis along z, the z-averaged demagnetization factor in

xy for a transverse magnetization is4

2 212x yN N L R RLé ù= = + -ë û

(6)

Thus, N was 0.39 and 0.51 for the sample holders containing APG 311 and EMG 900 respectively.

A very strong ferrofluid QK100 with Ms of 100 kA m–1 is available from Qfluidics (see www.qfluidics.com), but

has not been used in experiments in the current work. Simulations based on its magnetic properties are included

in Fig. 2e in the main text.

Remnant Magnetization of Permanent Magnets

The remnant magnetization Mr of the permanent magnets used was measured by profiling the external field

with a Gauss-meter in the z direction along the central axis (Extended Data Fig. 4 below), and fitting the resultant

profile to analytical expressions5. For a cylindrical magnet magnetized along its symmetry axis this expression is

5

( )

02 2 222

rz

M z L zBz Rz L R

µ é ù+ê ú= -ê ú++ +ë û

(7)

while for a cuboid it is

( ) ( )22 2 2 2 2

1 10 tan tanrz

z L a b z LM z a b zBab ab

µp

- -é ùæ ö æ ö+ + + + + +ê úç ÷= - ç ÷ç ÷ê úç ÷ è øê úè øë û

(8)

The results are shown in Extended Data Table 2.

Surface Tension of Magnetic Liquids

A simple home-build pendent drop setup was used to measure the surface tension of the magnetic liquids.

The solutions were gradually dispensed in a step-wise fashion through a polished 74 µm diameter glass capillary

using a syringe pump into a glass cuvette in a 3D printed support (see Extended Data Fig. 5). The pendent drops

were imaged using a Canon EOS 50D and Tamron superzoom 18-270 mm lens in RAW mode. RAW images

were converted to monochrome TIFF files, with image processing being carried out in ImageJ 6, and the surface

tension calculated using “Pendent_Drop”, an ImageJ plugin 7.

The drops were gradually enlarged, with multiple images taken at each size until the drops broke off, and

this was repeated at least 6 times. The values reported in table S1 is pooled mean and variance, calculated from

at least 6 different droplets, for at least 15 droplet sizes, and 3 images per size. Overall, between 30 – 100 images

of droplets of various sizes were used per surface tension value.

Method for Self-Consistent Diameter Calculation using the Full Model (FM)

For a given set of conditions, (i.e. permanent magnet remnant magnetization, quadrupole spacing, ferrofluid

nanoparticle diameter and volume fraction, surface tension), d can be calculated in a self-consistent manner as

shown in Extended Data Fig. 7 by calculating the magnetic field and magnetization at a position d/2, next

6

generating a new d using eq.(1), use this as a new input d, and looping repeatedly until the difference between

input and output is negligible.

Side-View Optical Imaging and Pressure/Flow Measurements

Images were acquired with either i) a Leica MZ16 stereo microscope with a Lecia IC80HD digital camera,

or ii) a Nikon SMZ745T stereo microscope and a SONY 1/1.8” 20im/sec 1600x1200 pix color CCD camera,

while an Elveflow OB1 pressure driven pump and Elveflow FS4 flow sensors were used to set the applied pressure

and measure the resulting flow rates in and out of the antitubes. All image processing and analysis was carried

out in ImageJ 6. Image thresholding was performed using Otsu’s clustering method, with the resulting image

converted to a binary mask. Particle analysis and circle fitting was performed on each image, resulting in cross-

sectional area, and roundness/circularity. Example cross-sections and the measurement setup are shown in

Extended Data Fig. 1.

X-Ray Transmission Imaging

A MyRay dental x-ray imaging system was adapted for imaging through ferrofluids, consisting of an X-ray

source (MyRay RX-DC eXtend) and a detector (MyRay Zen X T1) were purchased from Castebel (Waver,

Belgium), and confined inside an in-house constructed box covered by lead plates. In general, X-ray emission at

65 kV and 6 mA with the exposure time of 0.1 s was used to obtain a good contrast image of water antitubes

confined in ferrofluids.

To measure the tube diameter, first a background measurement with a microfluidic device (Extended Data Fig.

12a, 12b) fully filled with ferrofluid was taken. The image processing loop (also in ImageJ) consisted of taking

the average of 10 images, rotated to have the tube fully vertical, with the intensity values inverted (Extended Data

Fig. 12c). Next a plot profile is generated by taking column averages (Extended Data Fig. 12e, red line). Then a

freshly cleaned device is filled with water, and progressively ferrofluid was added in 50 µl steps at a flow rate of

100 µl/min, with 10 images taken at each step. The same imaging process carried out as with the background,

followed by subtraction of the background plot profile, leading to a sharp peak due to the lower absorption of

7

water (Extended Data Fig. 12e). This is fitted with a Gaussian peak function, whose full width half max

corresponded to tube diameters measured from the optical side view technique. The procedure of adding ferrofluid

was continued until a continuous tube is no longer observed.

Top-View Optical Imaging

A Nikon SMZ745T stereo microscope and a SONY 1/1.8” 20im/sec 1600x1200 pixel colour CCD camera were

used to image water tubes inside Magoil. Similar to the x-ray transmission images, the optical images were

rotated, inverted (see Extended Data Fig. 13), and profiles of column averages were extracted for data fitting. The

background was subtracted, and the resulting peaks fitted with a gaussian function.

8

Supplementary Information

Section 1. Derivation of Equilibrium Diameter Equation (1)

In a fluid, pressure can be considered not only as a force per unit area, but also as an energy density,

remembering that F Fd WPA Ad V

= = = . See Extended Data Fig. 14 for geometry. In fact, the famous Bernoulli

equation is a statement of the conservation of energy, which relates fluid pressure to kinetic and potential energy.

The magnetostatic energy is2 0mU µ= - ×M H , which in a magnetically soft system can be replaced by the vector

magnitudes 0mU MHµ= - as M and H are collinear. For a linear medium such as a concentrated paramagnetic

salt solution, or dilute ferrofluid, this can be further simplified to ( ) 201/ 2mU Bµ c= - . This energy density also

represents a magnetic pressure ( ) 201/ 2mp Bµ c= , acting throughout the fluid.

To stabilize a liquid anti-tube inside a magnetic liquid, a simple expression can be derived using the

augmented Bernoulli equation, defined as8

* 21/ 2 g constmp v h pr r+ + - = (9)

where p* is the composite pressure

* ( , ) s mp p T p pr= + + (10)

where ps is the magnetostrictive pressure, negligible in fluids, and pm is the fluid-magnetic pressure

0 00d

H

mp M H MHµ µ= =ò (11)

Here, the defining boundary condition is

0* n cp p p p+ = + (12)

where p0 is the ambient pressure, pn is the magnetic normal traction, and pc

20

12n Ip Mµ= (13)

9

2cp ks= (14)

where σ is the surface tension, the mean curvature, κ, is defined as

1 2

1 1 12 r r

kæ ö

= +ç ÷è ø

(15)

Applying this boundary condition at the centre of the quadrupolar field, P1, and the interface between water and

the immiscible ferrofluid, P2, (see Extended Data Fig. 13), while noting that at P1, H1 = 0, and σ = 0, results in

*1 0

* 22 0 0

122 I

p p

p p Mks µ

=

= + - (16)

By inserting these pressures into the FHD Bernoulli equation, while ignoring gravity, which acts in the

perpendicular direction, the following expression for the equilibrium mean curvature κ can be derived

20 2 0 ,2

1 12 2 IMH Mk µ µsæ ö= +ç ÷è ø

(17)

For a circular cross-section tube, where, r1 = r, and r2 is infinite i.e. there is no curvature along the tube axis, the

equation simplifies to

20 0

22 I

rMH M

sµ µ

=+

(18)

Or for the diameter:

20 0

42 I

dMH M

sµ µ

=+

(19)

If the tube symmetry axis lies parallel to gravity, then including gravity, the expression for the curvature

becomes

( )201 2 ( )2 2 IMH M glµk rsé ù= + + Dê úë û

(20)

where l is the tube length, and 1 1r r rD = - is the density difference between water and the surrounding ferrofluid.

Section 2. Derivation of Linear (LM) and Saturation Models (SM) for the Equilibrium Tube Diameter

10

The magnetic field due a 2D rectangle of width 2a, height, 2b, with remnant polarization 0r rJ Mµ= in y

can be described by5

( )( )

( )( )2 22 2 2 2

2 2atan atan

2r

x

a b y a b yMHa x b y a x b yp

æ öæ ö æ ö- - -ç ÷= - - -ç ÷ ç ÷

ç ÷ ç ÷ç ÷- + + - - - + + -è ø è øè ø (21)

( ) ( )( ) ( )

( ) ( )( ) ( )

2 2 2 2

2 2 2 2log log4yr a x b y a x b yM

a x b y aH

x b yp

æ öæ ö æ ö- + + - - - + + -ç ÷-ç ÷ ç ÷

ç ÷ ç ÷ç ÷+ + - - + + -è ø øè ø=

è (22)

The field due to the quadrupolar configuration with gap 2w then becomes

( )( )

( ) ( )

( )( )

( ) ( )

( )

2 2 22 2 2

2 2 22 2 2

22 2

2 22atan atan2 2

2 22atan atan2 2

2atan at2

r

r

r

x

a b yM aya y a w x a b y a w x

a b

H

M

yM aya y a w x a b y a w x

aya y a w x

p

p

p

æ öæ ö æ ö-ç ÷- - -ç ÷ ç ÷

ç ÷ ç ÷ç ÷- + + - - + - + - + - - +è ø è øè øæ öæ ö æ ö- -ç ÷+ - -ç ÷ ç ÷

ç ÷ ç ÷ç ÷- + + - - + - + - - + - - +è ø è øè ø

æ ö- - -ç ÷

ç ÷- + + + +è ø

=

( )( ) ( )

( )( )

( ) ( )

2 22

2 2 22 2 2

2 2an

2

2 22atan atan2 2r

a b ya b y a w x

a b yaya y a w x a b y a w x

Mp

æ öæ ö-ç ÷ç ÷

ç ÷ç ÷- + - + + +è øè øæ öæ ö æ ö- -ç ÷+ - -ç ÷ ç ÷

ç ÷ ç ÷ç ÷- + + + + - + - - + + +è ø è øè ø

(23)

and

( )( )

( ) ( )( ) ( )

( )( )

( ) ( )( ) ( )

( )( )

2 2 22

2 2 22

2 2 22

2 2 22

22

22

2 2 2log log

4 2

2 2 2log log

4 2

log4 2

r

r

y

r

y a w x b y a w xMy w x b y w x

y a w x b y a w xMy w x b y w x

y w xMy a w x

Hp

p

p

æ öæ ö æ ö+ - - + - - + - - +ç ÷- +ç ÷ ç ÷

ç ÷ ç ÷ç ÷+ - + - - + - +è ø è øè øæ öæ ö æ ö+ - - + - + - - +ç ÷- -ç ÷ ç ÷

ç ÷ ç ÷ç ÷+ - + - + - +è ø è øè ø

æ ö+ ++ - +ç ÷

ç ÷+

=

+ +è ø

( ) ( )( ) ( )

( )( )

( ) ( )( ) ( )

2 2

2 2

2 2 22

2 2 22

2log

2 2

2log log

4 2 2 2r

b y w xb y a w x

y w x b y w xMy a w x b y a w xp

æ öæ ö- - + +ç ÷ç ÷

ç ÷ç ÷- - + + +è øè øæ öæ ö æ ö+ + - + +ç ÷- -ç ÷ ç ÷

ç ÷ ç ÷ç ÷+ + + - + + +è ø è øè ø

(24)

11

By using a Maclarin expansion, the magnetic field magnitude Hg in the gap, along x for y = 0, or along y for x =

0, assuming a,b > w, is

22 r

wM xHwp

= (25)

while differences in the fields along x or y only become apparent for higher order terms. For a gap w, this becomes

4 r

wM xHwp

= (26)

At saturation, M and MI are Ms, and substituting these, Hw for H, and x/2 for d into (19) results in a quadratic

equation

2 200

2 2 0r ss

M M x M xwµ µ s

p+ - = (27)

Solving this for x, and remembering 2d x= , and gives this following expression

2

02

68

48

s s

r s rr

M M www M MMM

d pp s pµ

+ -= (28)

Using the Buckingham pi-theory, three dimensionless variables can be deduced, the dimensionless diameter

* ddw

= , the source factor sm

r

MM

a = , the confinement number 2

0 rD

M wN µs

= , and equation (28) is simplified to

64*8 8m m

D m

dN

p pa p aa

= + - (29)

For linear magnetic media far from saturation, M and MI become χH/2, and χH respectively, substituting these,

Hw for H, and x/2 for d into (19) results in a cubic equation

2 2 3 2 3 2 20 02 2 0r rM x M x wc µ cµ p s+ - = (30)

Solving this for x, remembering 2d x= , gives the following expression

12

2 2

32 2 2

0 0r r

wM M

d p sc µ cµ+

= (31)

Or in dimensionless form

( )2

3*1I

dN

pc c

=+

(32)

Section 3. Angular stability dependence of tracks cut into magnets for different magnetization directions

Magnetic field distributions for tracks cut into two N42 stacked magnets (µ0M = 1.26 T) of size 50 ´ 50 ´ 5

mm (x ´ y ´ z), oppositely magnetized along one of three axes Mx, My, and Mz (Extended Data Figs. 8, 9, 10,

respectively). The track widths were 14 mm, of total length 50 mm, with an angled section length of 20 mm.

References

1. Evans, D. F. 400. The determination of the paramagnetic susceptibility of substances in solution by nuclear

magnetic resonance. J. Chem. Soc. 0, 2003–2005 (1959).

2. Coey, J. M. D. Magnetism and Magnetic Materials. (Cambridge University Press, 2010).

3. Cugat, O., Byrne, R., McCaulay, J. & Coey, J. M. D. A compact vibrating-sample magnetometer with variable

permanent magnet flux source. Rev. Sci. Instrum. 65, 3570–3573 (1994).

4. Wysin, G. M. Demagnetization fields. (Kansas State University, 2012).

5. Furlani, E. P. Permanent Magnet and Electromechanical Devices. (Academic Press, 2001).

6. Schindelin, J. et al. Fiji: an open-source platform for biological-image analysis. Nat. Methods 9, 676–682

(2012).

7. Daerr, A. & Mogne, A. Pendent_Drop: An ImageJ Plugin to Measure the Surface Tension from an Image of a

Pendent Drop. J. Open Res. Softw. 4, (2016).

8. Rosensweig, R. E. Ferrohydrodynamics. (Dover Publications, 2014).

13

Extended Data Fig. 1

a Relative dilation of an antitube in APG311 ferrofluid with a quadrupolar gap, w, of 10 mm under flow, and with

the outlet closed; b side view through the antitubes; c experimental setup using a stereomicroscope to image light

travelling through a quadrupole containing ferrofluid.

14


Linear model phase space of equilibrium tube diameter for the dimensionless susceptibility, and gap number (ND)

for equation (32). White circles correspond to four experimentally accessible conditions for a gap width of 6 mm,

using i) APG 311 ferrofluid + water (χ = 0.083 or log(χ ) = 0, and σ = 23 mN m-1); ii) APG 311 ferrofluid + 1%

surfactant (χ = 0.083 or log(χ ) = 0, and σ = 6 mN m-1), iii) EMG900 ferrofluid + water (χ = 1.67 or log(χ ) = 0.2,

and σ = 29 mN m-1); iv) Magoil and water (χ = 0.47 10-3 or log(χ ) = -3.3, and σ = 6 mN m-1. log(d*) = 0

corresponds to a tube diameter equal to the gap width, values greater than zero are nonphysical as they correspond

to tube diameters greater than the gap width.

15


Demag corrected magnetisation loops of the two ferrofluids used, APG311, and EMG900.

16


Simplified algorithmic loop to perform the self-consistent equilibrium diameter calculation; note: it does not show

external loops used for convergence stability tests.

17


Schematic of a a cylindrical magnet, b a cuboidal magnet (quad), and c the measured (points) and fitted (lines)

field profiles for a series of magnets.

18


Pendent drop of APG 311 ferrofluid in a air, b water, and c water + 1% tween-20.

19


Dimensionless antitube diameter for APG311 in water compared to three models: FM: Full Model; SM:

Saturation Model; LM: Linear Model

20


XY Cross-section contour plots of a quadrupole for junctions with an angle of 15 – 90° for an in-plane (Mx)

magnetisation.

21


XY Cross-section contour plots of a quadrupole for junctions with an angle of 15 – 90° for a parallel (My)

magnetisation.

22


XY Cross-section contour plots of a quadrupole for junctions with an angle of 15 – 90° for an out of plane (Mz)

magnetisation.

23


a A waterjet cut Y-junction in two magnets, with a Magoil stabilised aqueous antitube, as shown in (Fig. 4h). b

Calculated magnetic field contours along the central xy plane for the Y-junction.

24


a & b Typical quadrupole assembles used for x-ray measurements; c inverted transmission x-ray image; d

averaged transmission; e background corrected transmission through the water anti-tube fitted with a Gaussian

peak function.

25


Optical micrographs of a water tube in Magoil for a gap width w = 220 µm a, and w = 307 µm b, c & d are the

greyscale, rotated and inverted images, e & f are the column average profiles of each gap, and g & h are the

Gaussian function fits to the background subtracted profiles. Note: the 94 µm antitube is thermodynamically

stable, as the image was taken during the extrusion of the water antitube, while the 30 µm tube is

thermodynamically unstable. After injection, water was then extracted, resulting in a thinning of the tube, which

at this diameter collapses into droplets in a matter of minutes.

26


2D geometry of 4 bar magnets in a quadrupolar configuration, the hatched region denotes the ferrofluids, while

the white region in the centre is the contained liquid tube.

27

Extended Data Table 1.

Measured physical properties of the two commercial ferrofluids (Ferrotec Corporation, www.ferrotec.com) and

paramagnetic oil (Magoil) used: density ρ, nanoparticle diameters d1 and d2 (for bimodal size distributions),

volume fractions φ1 and φ1 of magnetite in the suspension (corresponding to particles of size d1 and d2), initial

susceptibility χ , surface tension σ with air, surface tension with water, and surface tension with 1% tween-20 in

water.

Sample ρ d1 d2 φ1 φ2 χ σair σwater σtween

kg m-3 nm nm % mN m-1 mN m-1 mN m-1

APG

311

950

±50

8.2

±

0.03

– 1.3 ±

4.10-3 –

0.085

±0.001

24.7

±0.8

22.8

±0.8

5.9

±0.6

EMG

900

1720

±90

11.3

±0.1

5.0

±0.1

9.2

±0.1

3.3

±0.1

1.67

±0.04

22.0

±0.5

29.2

±1.4

7.0

±0.3

Magoil 970

±50 – – – –

4.7 x10-4

±0.5x10-

4

– 5.8

±0.4 –

28

Extended Data Table 2.

Measured remnant magnetization in units of tesla T for a variety of permanent magnets used, compared to typical

values for the magnet grade, including the 6x50x6 mm used for the quadrupolar confinement.

Type Grade Typical Br Br meas. Error B @ surface

T T T mT

Cube 10x10x10 N42 1.28 – 1.32 1.35 0.03 486

Cylinder 14x10 N38 1.22 – 1.25 1.18 0.02 478

Quad 9x50x9 N42 1.28 – 1.32 1.26 0.02 409

Quad 6x50x6 N42 1.28 – 1.32 1.24 0.02 392

Quad 50x50x5 N42 1.28 – 1.32 1.28 0.00 112

29

Movie S1

Extrusion and extraction of water into Holmium based Magoil in a quadrupolar field using two in/outlets. The

water is dyed with ink to enhance visibility. Bubbles are removed from the top Magoil–air interface manually.

Movie S2

Rupture of an anti-tube in Magoil by a plastic stick, which self-heals without any flow from an external pump.

Movie S3

Honey flowing from reservoirs under gravity through three different configurations: an antitube, no tube (i.e., air

only), and a normal tube of the same diameter.

Movie S4

Injection of APG311 ferrofluid into a water-containing thin cell surrounded by four 10x10x10 mm cube magnets

with the field orientation shown in Fig. 3C (left).

Movie S5

Injection of APG311 ferrofluid into a water-containing thin cell surrounded by six 10x10x10 mm cube magnets

with the field orientation shown in Fig. 3C (right).

Movie S6

Valving an antitube in ferrofluid using a fifth magnet; upon addition of a magnet the flow stops, upon removal,

the flow recommences.

Movie S7

Animation of the 0.15 T isovolume of the magnetic field in a quadrupolar field upon valving with one magnet.

Movie S8

Animation of the 0.15 T isovolume of the magnetic field in a quadrupolar field upon valving with two magnets.

30

Movie S9

Non-contact peristaltic pumping using moving valve-points (occlusions) generated by a rotating wheel of valving

magnets.

Movie S10

Splitting of water through a free-hanging Y-junction in a ferrofluid. The ferrofluid is held in place by the magnetic

field gradient force supplied by the magnets, there is no physical support below it. Water is injected from one

input on the left and splits in two on the right.

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