Actuation of magnetoelastic membranes in precessingmagnetic fieldsChase Austyn Brisboisa, Mykola Tasinkevycha,b, Pablo Vázquez-Montejoa,c,d, and Monica Olvera de la Cruza,e,f,1

aDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208; bCentro de Fı́sica Teórica e Computacional, Departamentode Fı́sica, Faculdade de Ciências, Universidade de Lisboa, Campo Grande P-1749-016 Lisboa, Portugal; cFacultad de Matemáticas, Universidad Autónoma deYucatán, Mérida, Yucatán, 97110, México; dInstituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad de México, 04510, México;eDepartment of Chemistry, Northwestern University, Evanston, IL 60208; and fDepartment of Physics and Astronomy, Northwestern University, Evanston,IL 60208

Contributed by Monica Olvera de la Cruz, December 12, 2018 (sent for review September 27, 2018; reviewed by Remi Dreyfus and Thomas Halsey)

Superparamagnetic nanoparticles incorporated into elastic mediaoffer the possibility of creating actuators driven by external fieldsin a multitude of environments. Here, magnetoelastic membranesare studied through a combination of continuum mechanics andmolecular dynamics simulations. We show how induced magneticinteractions affect the buckling and the configuration of magne-toelastic membranes in rapidly precessing magnetic fields. Thefield, in competition with the bending and stretching of the mem-brane, transmits forces and torques that drives the membrane toexpand, contract, or twist. We identify critical field values thatinduce spontaneous symmetry breaking as well as field regimeswhere multiple membrane configurations may be observed. Ourinsights into buckling mechanisms provide the bases to developsoft, autonomous robotic systems that can be used at micro- andmacroscopic length scales.

superparamagnetism | molecular dynamics | finite element analysis |membranes | spontaneous symmetry breaking

Superparamagnetic nanoparticles are readily manipulated byexternal magnetic fields, which control the strength and direc-tion of their magnetic dipole moments. Magnetic fields can directthe collective behavior of magnetic colloids to influence static,micro- and mesoscale structures and dynamic particle motion (1–7). Incorporating magnetic particles into elastic media allows forthe precise control of aggregation and locomotion within a mate-rial (8–12). At microscales, superparamagnetic nanoparticlesenable actuating systems that do not require direct connection toan external power source. The lack of a permanent dipole momentprevents unwanted aggregation in the absence of a magnetic field,a property particularly useful in biomedical applications (13–15).

When driven by dynamic, external magnetic fields, linearchains of superparamagnetic particles, or magnetoelastic fila-ments (16), have been incorporated into microswimmers (10,17), active surfaces (11, 18), and gels (12, 19). Actuation of mem-branes of linked superparamagnetic particles is explored here toadd additional functionality, such as switching between open andclosed membrane states for molecular transport or catalysis. Byaligning superparamagnetic particles in a plane, using a rotat-ing magnetic field or at an interface, magnetoelastic membranescould be fabricated using existing techniques (19–21).

We apply a theoretical framework for magnetoelastic mem-branes (22) to examine their actuation under fast precessingfields. We use a combination of analytical description, continuummechanics (CM) solutions, and particle-based, coarse-grainedmolecular dynamics (MD) simulations to study how the preces-sion angle, θ, of the magnetic field (Fig. 1A) affects the config-uration of a square-shaped patch of linked, superparamagneticnanoparticles (Fig. 1B).

Under a field that precesses at high frequency, the membraneconformation is quasistatic with respect to precession. There-fore, the magnetic energy of the membrane can be obtainedby time-averaging the dipole–dipole interactions over the pre-cession period. The time-averaged interaction potential, 〈Ud〉t ,

between two rapidly rotating dipoles can be approximated asthe magnetic interaction energy with a θ-dependent couplingstrength. In this fast precessing regime, the behavior of anunstretchable membrane patch is characterized by a singledimensionless parameter defined as γ=m(θ)L2o/κ, where m(θ)is the magnetic modulus, Lo is the length of one edge of a squaremembrane, and κ is the bending modulus (22). This “magne-toelastic” parameter describes the relative strength between themagnetic and bending energy of the membrane. The angle ofprecession determines the magnitude of γ via the magnetic mod-ulus, which can take positive or negative values and is zero at acritical precessing angle θ∗. More explicitly, in the case whereθ < θ∗, the magnetic modulus is positive (γ > 0; Fig. 1C) andnegative for θ > θ∗ (γ < 0; Fig. 1F).

The sign and magnitude of the magnetoelastic parameter, γ,indicates how magnetic particles interact with each other (23,24). Therefore, γ influences the configuration of a membrane.In particular, there exist critical γ values which mark significantchanges in membrane behavior. We show how changing the sep-aration between opposite boundaries of the membrane (Fig. 1D)yields a transition from a symmetric to an asymmetric configu-ration at a critical field strength. Similarly, hysteresis present inthe membrane energy as opposite boundaries are twisted by atotal angle, 2α, (Fig. 1E) can either be preserved or eliminatedunder weak or strong fields, respectively, allowing the membraneto resist or promote twisting.

Results and DiscussionWe begin by discussing a magnetoelastic membrane composedof hexagonally close-packed superparamagnetic particles in

Significance

Understanding the properties of elastic membranes withsuperparamagnetic particles advances the design of auton-omous, soft robots. Magnetic fields readily penetrate mostmaterials and can be used remotely to induce rapid and pre-cise changes in membrane shape. Here, we compare analyticaland numerical models to molecular dynamics simulations toexplain how rapidly precessing biaxial magnetic fields canbe used to control the forces on magnetoelastic membranes.These forces are closely linked to its actuating behavior andits potential applications in micromechanics.

Author contributions: C.A.B., M.T., P.V.-M., and M.O.d.l.C. designed research; C.A.B., M.T.,and P.V.-M. performed research; C.A.B., M.T., P.V.-M., and M.O.d.l.C. analyzed data; andC.A.B., M.T., P.V.-M., and M.O.d.l.C. wrote the paper.y

Reviewers: R.D., CNRS; and T.H., ExxonMobil.y

The authors declare no conflict of interest.y

Published under the PNAS license.y1 To whom correspondence should be addressed. Email: m-olvera@northwestern.edu.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplemental.y

Published online January 25, 2019.

2500–2505 | PNAS | February 12, 2019 | vol. 116 | no. 7 www.pnas.org/cgi/doi/10.1073/pnas.1816731116

Dow

nloa

ded

by g

uest

on

July

7, 2

021

https://www.pnas.org/site/aboutpnas/licenses.xhtmlhttp://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttp://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttps://www.pnas.org/cgi/doi/10.1073/pnas.1816731116

PHYS

ICS

Fig. 1. Magnetoelastic membranes under precessing magnetic fields. (A) Magnetic fields precess at an angle θ around the Z axis. (B) A membrane com-posed of hexagonally packed superparamagnetic colloids possessing three bending axes (red arrows) with magnetic moment µ precessing with the field.Time-averaged interaction potential 〈Ud〉t ∝ (3 cos2ϕ− 1)(3 cos2 θ− 1)/r3, where r is the distance between the particles and ϕ is the angle between thecenter-to-center vector and the precession axis. (C) Small-angle precession (θ < θ∗, γ > 0). A schematic cross-section of a membrane shows preferred dipoleorientation which interacts with the potential 〈Ud〉t . (D) Schematic of a membrane minimizing its bending energy by adopting a single “arch.” Each edgehas length Lo where two parallel boundaries are separated by a distance L. (E) A membrane with opposite boundaries at a distance L. Each boundaryis rotated by α in opposite directions for a total angle of 2α. The midpoint of each boundary is stationary and defines a centerline that remains per-pendicular to the magnetic field precession axis. (F) Large-angle precession (θ > θ∗, γ < 0). This leads to a change in the membrane’s magnetic moduluswhich depends on the angle of precession, m(θ)∝ (3 cos2 θ− 1). Cross-section of a membrane shows an alignment of rotating dipoles interacting with thepotential 〈Ud〉t .

zero external field, γ= 0. When opposite, stiff boundaries arebrought together under uniform compression, the adjacent flex-ible boundaries will bend to relieve the relatively high energeticcost of in-plane elastic deformation; the equilibrium state is asingle arch (Fig. 2A). Our MD simulations show that beginningwith a single-arch configuration (L/Lo = 0.5), the total mem-brane energy is dominated by bending (SI Appendix, Fig. S1).As the compression is released by increasing the boundary sep-aration to approach L=Lo , the energy curve decreases linearlydue to the decrease in membrane curvature (Fig. 2B). Therefore,there is constant force acting on the boundaries of a compressedmembrane that drive it to expand flat. If allowed to move freely,the blue boundaries in Fig. 2A would move apart (L→Lo).

MD simulations show that in a precessing magnetic field andunder linear compression, the direction of force on the mem-brane’s boundaries depends on the sign of γ. Under small-angleprecession (γ > 0, θ < θ∗), a membrane, with any initial amountof compression contracts when the boundaries move together(Fig. 3A). Contraction of the boundaries only occurs if thefield strength is sufficient to overcome the penalty of bending.When this requirement is met, the force pulling the boundariestogether increases with field strength. As contraction proceeds,the force on the boundaries decreases and approaches zero. Theenergy curves from MD simulations match closely with thosefrom CM calculations (Fig. 3C). For moderate field strengths,an instability appears on the energy curve (γ= 150 and 300)where the free edges of the single-arch membrane buckle towardthe center due to the favorable alignment of the dipoles, lead-ing to a “double-buckled” configuration. This transition is alsopredicted by our CM calculations for weak-to-moderate fieldstrengths (γ= 100 and 200). These mark the transition point atwhich the energy branches of the single-arch and double-buckledmembranes intersect and are dependent on the boundary sep-aration L/Lo and γ (Fig. 3D). Our MD simulations observethis transition around L/Lo = 0.65− 0.80. Larger values of Lfavor the double-buckled membrane, while smaller values of Lfavor the single-arch membrane. In order for the flexible bound-aries to buckle inward, the stiff boundaries must be sufficientlyspaced so that the increase in bending energy is offset by therecovery of magnetic energy. Increasing γ lowers the value ofL needed to induce the buckling transition. For γ > 300, weobserve only the double-buckled configurations in the rangeof 0.5 0 case, the symmetry across the X −Zplane at the midpoint between the stiff boundaries is brokenunder large-angle precession (γ < 0, θ > θ∗) and large compres-sions. These conditions lead to an asymmetric arch membraneconfiguration shown in Fig. 3D predicted by CM calculationsand MD simulations. Similar transitions have been previouslydescribed in nonmagnetic systems where elastic membranes aredeformed on a solid or liquid substrate (25–27). The asymmetricarch allows the membrane to significantly reduce the force driv-ing expansion (Fig. 3B), with a similar force reduction in CMcalculations (Fig. 3C), and corresponds to the small L branchof the membrane energy. At larger L, we observe the symmet-ric arch configurations which exhibit a larger expansion force.Unlike in the case of the small-angle precession, the membranedoes not reveal a secondary buckling transition. Rather, it pos-sesses a continuous flattening in the X −Y plane as the fieldstrength increases.

At a given level of membrane compression, MD simulationspredict a critical field strength above which the membrane adoptsan asymmetric arch shape. The cross-over between asymmet-ric and symmetric configurations is reported in Fig 4A. As

Fig. 2. Effect of boundary separation on an elastic membrane with noexternal field (γ= 0). (A) Under a uniform compression (L/Lo = 0.8) onopposite, stiff boundaries (blue), the strain related to the in-plane defor-mations is released via a buckling transition where the membrane adoptsa symmetric arch shape. The edges that define the boundary separationare colored blue. (B) Total membrane energy obtained from MD simula-tions as the distance, L, between opposite boundaries changes. The totalmembrane energy (in units of kT) is normalized to the total numberof colloids, N. The colloids are modeled as 10-nm particles under roomtemperature.

Brisbois et al. PNAS | February 12, 2019 | vol. 116 | no. 7 | 2501

Dow

nloa

ded

by g

uest

on

July

7, 2

021

https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplemental

Fig. 3. Influence of boundary separation on magnetoelastic membranes. Total membrane energy as a function of L obtained via MD simulations. Theresults show membrane contraction under (A) small-angle magnetic field precession (γ > 0) and expansion under (B) large-angle precession (γ < 0). Curvesare shifted by the value of the energy (in units of kT) at L/Lo = 1, �i , for clarity and normalized by the number of colloids, N. (C) Calculated membranetotal free energy F from CM with γ > 0 and γ < 0. Free-energy curves are shifted by Fi , the values at L/Lo = 1, for clarity and normalized to the bendingmodulus. (D) Snapshots of membrane configurations at boundary separation L/Lo = 0.5, 0.75, and 0.90 (2α= 0) at different values of γ. Field precession atθ < θ∗ causes secondary buckling of the free membrane edges, which is reflected by the brake in the slope of the free-energy curves at γ= 150 and 300.The edges that define the boundary separation are colored blue.

compression increases, the asymmetric configuration becomesmore favorable and the minimum magnetic field strengthrequired to induce the symmetric-to-asymmetric transitiondecreases. Of course, as L approaches Lo , the field strengthneeded to distort the membrane increases rapidly.

We develop analytical solutions that predict this spontaneoussymmetry breaking. The membrane is assumed to be translation-ally invariant in the X direction. This assumption is justified inthe limit of small compression and weak-to-moderate strengthof the magnetic field. In this regime, the membrane shape isdescribed by a profile curve which we parameterize by the angleΘ(l) that the local tangent vector el makes with the precessionaxis, where l is the arc length along the membrane profile curve(SI Appendix, Fig. S7). This analysis yields the total energy ofthe membrane given by H =Lo

∫dlH , where the linear energy

density H coincides with the energy density of a paramag-netic filament (28). The Euler–Lagrange equations arising fromH lead to the profile curves of the ground state. Symmetricsolutions correspond to zero vertical force on the membrane,whereas for a finite vertical force the potential in quadrature Hbecomes asymmetric. This asymmetry is a property inherited bythe corresponding solutions (Fig 4B). Details for the analyticalsolutions can be found in SI Appendix.

In addition to linear forces, a contracting and expanding mag-netoelastic membrane experiences torque on its boundaries.Twisting an elastic membrane at zero external field is disfavored

Fig. 4. Application of magnetic field breaks membrane symmetry. (A)MD simulation data for the transition between symmetric and asym-metric membrane configurations as a function of γ and L/L0. Abovethe critical values of |γ|, only asymmetric configuration exists. (B) Mem-brane profiles found by analytical methods for γ= 100 and −100(L/Lo = 0.75).

2502 | www.pnas.org/cgi/doi/10.1073/pnas.1816731116 Brisbois et al.

Dow

nloa

ded

by g

uest

on

July

7, 2

021

https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttps://www.pnas.org/cgi/doi/10.1073/pnas.1816731116

PHYS

ICS

due to the buildup of elastic stress. Rather than changing com-pression, we rotate the two stiff boundaries in opposite directionsby an angle α for a total angle of 2α. The rotation occurs alonga centerline drawn along the Y axis that extends through thecenter of each stiff boundary. Starting from 2α= 0, we twistthe membrane by rotating the boundaries to 2α= 90◦ and thenreverse the direction of rotation back to the parallel state. Weobserve hysteresis in the �(α) curve (Fig. 5B). Twisting mem-brane edges beyond 2α' 70◦ from lower angles induces a con-figurational transformation to relieve elastic and bending energylocalized in two opposite corners of the membrane (Fig. 5A).Further twisting concentrates elastic stress at a single point nearthe center of the membrane. This conformation persists untilthe boundaries are rotated close to a parallel position (18◦)where the membrane transitions back to a symmetric arch. Thistransformation also leads to a drop in membrane stretching andbending energies (SI Appendix, Fig. S1B).

In the case of membrane twisting under a precessing magneticfield, the conformation depends, to a great extent, on the mag-nitude and sign of γ. MD simulations highlight different regimesfor magnetically induced changes in the membrane configurationthat correspond to changes in the torque on the stiff membraneboundaries. Under small-angle precession (γ > 0) and relativelyweak fields, the �(α) curves reveal a hysteresis in the mem-brane energy (see blue and red curves in Fig. 6A). Twisting theboundaries from 0◦ to 90◦ causes the membrane to undergotwo configurational transitions. The reverse path results in asingle transition back to the original, single-arch state. From0◦ to 90◦, the change in the slope at low angles (25◦− 33◦)

Fig. 5. Effect of boundary rotation on an elastic membrane with noexternal field (γ= 0). (A) MD simulated membranes show the conforma-tional transition as the angle between stiff, separated boundaries (blue)increases (images represent 2α= 0, 45◦, 90◦, respectively). (B) Total mem-brane energy (in units of kT) normalized by the number of colloids, N,obtained from MD simulations as the total angle, 2α, between boundarieschanges with constant separation (L/Lo = 0.5). The membrane undergoesa conformational transition at some threshold value of 2α' 70◦ to relieveaccumulated stretching strain causing the formation of a hysteresis.

indicates a buckling transition driven by the magnetic field wherethe center of the single arch buckles inward (Fig. 6D). Afterthis point, there is a buildup of elastic stress along a V-shapedcrease in the membrane. This leads to a second transition atlarger angles (57◦ for γ= 150 and 87◦ for γ= 300) back toa similar configuration as the single arch where the elasticstress is more distributed or, in the case of strong fields, con-centrated along a line. We refer to this as an “unbuckling”transition, where the membrane bending and stretching energyrecovers from a magnetically induced, buckled state. Unlikein the γ= 0 case, the weak field energy curves possess localminima at intermediate angles, around 80◦. Above a thresh-old value of γ > 300, the free edges of the membrane buckleinward (Fig. 6D) and adopt an S-shaped configuration, whicheliminates the membrane energy hysteresis. In this regime, themembrane configuration is dominated by magnetic interactionsand, above γ > 600, it has a global energy minimum at the mosttwisted state.

For large-angle precession (γ < 0), we also observe a hys-teresis in the �(α) curves under weak fields (Fig. 6B). Small|γ| (< | − 24|) results in a transition similar to the purely elas-tic case with differences in the angle at which the membraneunbuckles, 78◦, before transitioning back to a single arch.MD simulations show that the combination of larger |γ| andboundary rotation exacerbates the bending energy penalty whilethe membrane tries to flatten against the boundaries (Fig. 6D)and locks in a “loop” on one side of the membrane formedfrom the asymmetric arch. The hysteresis loop is not presentfor γ 0, a flat membrane with rigid, parallelboundaries contracts. If the sign of γ is flipped (γ < 0), themembrane expands to its original flat configuration. However,applying a field such that γ� 1, a flat membrane twists as itcontracts and remains in the twisted state until γ decreases.Thus far, switching between a flat and twisted structure hasbeen observed by altering the chemical environment (29) orintrinsic structure (30, 31) of polypeptide beta-sheets. We showthat magnetoelastic membranes can switch between these twomorphologies rapidly using external magnetic fields without thechallenge of changing a membrane’s mechanical or chemicalenvironment.

The CM calculations agree with the predictions of MD sim-ulations for the configurations and configurational transitionsdriven by membrane compression at 2α= 0. In addition, theanalytical solutions to the continuum model constructed herefor unstretchable membranes are similarly valid under weakmagnetic fields. Now that we have validated these MD simula-tions with theoretical predictions, we set the stage for addressingdynamical aspects of magnetoelastic membrane behavior in theregime of slow precessing fields.

MethodsMD Simulations. The magnetoelastic membrane at room temperature issimulated as a hexagonally close-packed monolayer of soft spheres pos-sessing a dipole moment. Each sphere interacts with its nearest neigh-bors through stiff harmonic springs and angular harmonic potentials

Brisbois et al. PNAS | February 12, 2019 | vol. 116 | no. 7 | 2503

Dow

nloa

ded

by g

uest

on

July

7, 2

021

https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplemental

Fig. 6. Influence of boundary rotation on magnetoelastic membranes. MD simulations results for membrane energy as a function of 2α, the twistingangle of two opposite membrane edges, at L/L0 = 0.5, and for several values of (A) γ > 0 and (B) γ < 0. Membranes resist twisting at small 2α undersmall-angle magnetic field precession (γ > 0) and for all rotation angles under large-angle precession (γ < 0). The solid lines represent rotation from 2α= 0to 2α= 90◦ and the dashed lines represent the reverse path back to 2α= 0. There exists a threshold value of γ above which the hysteresis from rotationdisappears (green and magenta curves). Curves are shifted by the value of the energy (in units of kT) at 2α= 0, �i , for clarity and normalized by thenumber of colloids, N. Energy is given in the Lennard-Jones units. (C) Membrane free energies from CM calculations as a function of 2α for γ > 0 andγ < 0. Free-energy curves are shifted by Fi , the free-energy values at 2α= 0, for clarity and normalized to the bending modulus. (D) Snapshots of themembrane configurations at 2α= 15◦, 50◦, 75◦, respectively, under different values of γ (L/Lo = 0.5). The edges that define the boundary separation arecolored blue.

representing the stretching and the bending of the membrane, respec-tively. Spheres along the boundaries parallel to the X axis are bound totheir initial position with an additional harmonic spring potential for apredefined distance, L/Lo, and angle, α. The bending of the membrane isdetermined along three directions along the membrane (Fig. 1B, red arrows)

and uses the harmonic angle potential, Ubend =κ

2(θNN −π)2, where θNN is

the angle three spheres make along one bending direction, and κ is thebending constant. To prevent the membrane from moving through itselfwe add the repulsive Weeks–Chandler–Andersen potential to all spheres.For the magnetic interactions, each sphere possesses a point dipole with

the potential Udipole =µ04π

(1r3ij

(⇀µi ·

⇀µj)− 3r5ij

(⇀µi ·

⇀rij )(

⇀µj ·

⇀rij )

), where µ0 is the

magnetic permeability of free space,⇀µ is the magnetic moment, and

⇀rij is

the displacement vector between spheres. The total energy of the mem-brane is determined as the sum of each potential and the kinetic energy.Assuming the spheres represent 10-nm particles, the magnitude of thedipole moment represents typical values for superparamagnetic materi-als (20 to 140 Am2/kg). The required precession frequency to meet thequasistatic condition for the time-averaged interaction potential scalesas ω∼σ−3 (12). For particles in water at 25◦C, the required frequencyis 1 MHz.

Continuum Elastic Model. The total free energy of a magnetoelastic mem-brane is a summation of the stretching, Fs, bending, Fb, and magnetic,Fm, energies and is minimized using a finite element method (SI Appendix).The continuum model assumes a “quasistatic” regime, where the preces-sion period is short compared with the characteristic time scale for colloidaltranslation. In this regime, the time-averaged magnetic energy due to dipo-lar interaction between nearest neighbors, induced by a magnetic fieldprecessing about ẑ that is quadratic in the projections of the membranetangent vectors ei ≡ ∂iX onto the precession axis:

Fm =1

2

∫Sref

dS m(θ)(

2

3− gij(ei · ẑ)(ej · ẑ)

), [1]

where X is the position vector and gij are components of the inverse of themetric tensor gij = (ei · ej). The magnetic modulus m(θ) (with units of energyper area) depends on the precession angle θ:

m(θ) =3µ0

4π∆l

(µ

∆l2

)2(3 cos2 θ− 1). [2]

Here, µ is the magnitude of the induced magnetic dipoles and ∆l is theseparation between their centers.

2504 | www.pnas.org/cgi/doi/10.1073/pnas.1816731116 Brisbois et al.

Dow

nloa

ded

by g

uest

on

July

7, 2

021

https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816731116/-/DCSupplementalhttps://www.pnas.org/cgi/doi/10.1073/pnas.1816731116

PHYS

ICS

This representation shows how membrane configurations depend on theprecession angle θ. As the precession angle changes, the magnetic inter-action, 〈Ud〉t , passes through zero. The magnetic modulus vanishes at thecritical angle θ∗ = arccos(1/

√3). From Eq. 1, we see that to minimize Fm

membranes will tend to develop regions aligned parallel or orthogonalto the precession axis ẑ to maximize or minimize (ei · ẑ) (Fig. 1 C and F,respectively).

ACKNOWLEDGMENTS. This work was supported as part of the Center forBio-Inspired Energy Science, an Energy Frontier Research Center fundedby the US Department of Energy, Office of Science, Basic Energy Sciencesunder Award DE-SC0000989. M.T. acknowledges financial support from Por-tuguese Foundation for Science and Technology Contract IF/00322/2015.P.A.V.-M. acknowledges support from Consejo Nacional de Ciencia y Tec-nologı́a Grant Fondo Institucional de Fomento Regional para el DesarrolloCientı́fico, Tecnológico y de Innovación (FORDECYT) 265667.

1. Driscoll M, et al. (2017) Unstable fronts and motile structures formed by microrollers.Nat Phys 13:375–379.

2. Kaiser A, Snezhko A, Aranson IS (2017) Flocking ferromagnetic colloids. Sci Adv3:e1601469.

3. Liu Q, Ackerman PJ, Lubensky TC, Smalyukh II (2016) Biaxial ferromagnetic liquidcrystal colloids. Proc Natl Acad Sci USA 113:10479–10484.

4. Spiteri L, Messina R, Gonzalez-Rodriguez D, Bécu L (2018) Ordering of sedimentingparamagnetic colloids in a monolayer. Phys Rev E 98:020601.

5. Kim E, Stratford K, Cates ME (2010) Bijels containing magnetic particles: A simulationstudy. Langmuir 26:7928–7936.

6. Li H, Halsey TC, Lobkovsky A (1994) Singular shape of a fluid drop in an electric ormagnetic field. Europhys Lett 27:575–580.

7. Kamien RD, Nelson DR (1993) Directed polymer melts and quantum criticalphenomena. J Stat Phys 71:23–50.

8. Hu W, Lum GZ, Mastrangeli M, Sitti M (2018) Small-scale soft-bodied robot withmultimodal locomotion. Nature 554:81–85.

9. Zrinyi M, Barsi L, Buki A (1997) Ferrogel: A new magneto-controlled elastic medium.Polym Gels Networks 5:415–427.

10. Dreyfus R, et al. (2005) Microscopic artificial swimmers. Nature 437:862–865.11. Wei J, Song F, Dobnikar J (2016) Assembly of superparamagnetic filaments in external

field. Langmuir 32:9321–9328.12. Dempster JM, Vázquez-Montejo P, Olvera de la Cruz M (2017) Contractile actuation

and dynamical gel assembly of paramagnetic filaments in fast precessing fields. PhysRev E 95:052606.

13. Faraudo J, Andreu JS, Calero C, Camacho J (2016) Predicting the self-assembly ofsuperparamagnetic colloids under magnetic fields. Adv Funct Mater 26:3837–3858.

14. Mahmoudi M, Sant S, Wang B, Laurent S, Sen T (2011) Superparamagnetic ironoxide nanoparticles (SPIONs): Development, surface modification and applications inchemotherapy. Adv Drug Deliv Rev 63:24–46.

15. Sabale S, et al. (2017) Recent developments in the synthesis, properties, and biomed-ical applications of core/shell superparamagnetic iron oxide nanoparticles with gold.Biomater Sci 5:2212–2225.

16. Furst EM, Suzuki C, Fermigier M, Gast AP (1998) Permanently linked monodisperseparamagnetic chains. Langmuir 14:7334–7336.

17. Ido Y, Li Y, Tsutsumi H, Sumiyoshi H, Chen C (2016) Magnetic microchains andmicroswimmers in an oscillating magnetic field. Biomicrofluidics 10:011902.

18. Vilfan M, et al. (2010) Self-assembled artificial cilia. Proc Natl Acad Sci USA 107:1844–1847.

19. Bannwarth MB, et al. (2015) Colloidal polymers with controlled sequence andbranching constructed from magnetic field assembled nanoparticles. ACS Nano9:2720–2728.

20. Kralj S, Makovec D (2015) Magnetic assembly of superparamagnetic iron oxidenanoparticle clusters into nanochains and nanobundles. ACS Nano 9:9700–9707.

21. Geng BY, et al. (2007) Hydrophilic polymer assisted synthesis of room-temperatureferromagnetic fe3o4 nanochains. Appl Phys Lett 90:043120.

22. Vázquez-Montejo P, Olvera de la Cruz M (2018) Flexible paramagnetic membranes infast precessing fields. Phys Rev E 98:032603.

23. Smallenburg F, Dijkstra M (2010) Phase diagram of colloidal spheres in a biaxialelectric or magnetic field. J Chem Phys 132:204508.

24. Osterman N, et al. (2009) Field-induced self-assembly of suspended colloidalmembranes. Phys Rev Lett 103:228301.

25. Diamant H, Witten TA (2011) Compression induced folding of a sheet: An integrablesystem. Phys Rev Lett 107:164302.

26. Zhu L, Chen X (2013) Mechanical analysis of eyelid morphology. Acta Biomater9:7968–7976.

27. Stoop N, Müller MM (2015) Non-linear buckling and symmetry breaking of a softelastic sheet sliding on a cylindrical substrate. Int J Non Linear Mech 75:115–122.

28. Vázquez-Montejo P, Dempster JM, Olvera de la Cruz M (2017) Paramagnetic fila-ments in a fast precessing field: Planar versus helical conformations. Phys Rev Mater 1:064402.

29. Flynn JD, McGlinchey RP, Walker RL, III, Lee JC (2018) Structural features ofα-synuclein amyloid fibrils revealed by Raman spectroscopy. J Biol Chem 293:767–776.

30. Zhang S, et al. (2013) Coexistence of ribbon and helical fibrils originating fromhIAPP20–29 revealed by quantitative nanomechanical atomic force microscopy. ProcNatl Acad Sci USA 110:2798–2803.

31. Moyer TJ, Cui H, Stupp SI (2013) Tuning nanostructure dimensions with supramolecu-lar twisting. J Phys Chem B 117:4604–4610.

Brisbois et al. PNAS | February 12, 2019 | vol. 116 | no. 7 | 2505

Dow

nloa

ded

by g

uest

on

July

7, 2

021