The feasibility of 3D self-assembly of milli-magnetic particles that interact via magnetic dipolar forces is investigated. Typically magnetic particles, such as isotropic spheres, self-organize in stable 2D configurations. By modifying the shape of the particles, 3D self-assembly may be enabled. The design of the particles and the experimental setup are presented. The magnetic configurations of simple particle arrangements are obtained via energy minimization in simulations. The simulations show that a 3D configuration can become energetically favourable over 2D configurations, if the shape of the particle is modified

Abelmann, Leon

Alink, Laurens

Marsman, G.H. (Mathijs)

Woldering, Léon A.

English

Woldering, L.A.

NARCIS 

Simulating three dimensional self-assembly of shape modified particles using magnetic dipolar forces

University of Twente Research Information

SIMULATING THREE DIMENSIONAL SELF-ASSEMBLY OF SHAPE MODIFIED PARTICLESUSING MAGNETIC DIPOLAR FORCESLaurens Alink 1, G.H. (Mathijs) Marsman 1, Léon A. Woldering 1 and Leon Abelmann 11Mesa+ Institute for Nanotechnology, University of Twente, The NetherlandsAbstract — The feasibility of 3D self-assembly ofmilli-magnetic particles that interact via magneticdipolar forces is investigated. Typically magneticparticles, such as isotropic spheres, self-organizein stable 2D configurations. By modifying theshape of the particles, 3D self-assembly maybe enabled. The design of the particles and theexperimental setup are presented. The magneticconfigurations of simple particle arrangements areobtained via energy minimization in simulations.The simulations show that a 3D configurationcan become energetically favourable over 2Dconfigurations, if the shape of the particle ismodified.Keywords: 3D self-assembly, magnetism, dipolarforces, anisotropic particles, energy minimizationI – IntroductionMicrofabrication techniques are currently based ontop-down lithography and therefore inherently two di-mensional (2D), or at best very restricted in the thirddimension. Fabrication by means of three dimensional(3D) self-assembly will open up a wide range of newapplications, such as new types of (smart) supermate-rials with interesting optical, mechanical, electrical andmagnetic properties [1]. On the long term, we envision3D electronics as an answer to atomic limits emergingat the end of Moore’s law progress [2].In this paper, we focus on 3D self-assembly drivenby dipolar magnetic forces. Particles that interact bydipolar forces only, typically assemble in 2D configu-rations preferably [3]. Our approach to overcome thislimitation is to enable 3D magnetic self-assemblies byclever design of the shape of the particles. In order tohave accurate control over the shape, this approach isinvestigated using millimeter sized particles. Some sim-ple configurations of milli-magnetic particles are shownin figure 1. We investigate the stability of such simpleconfigurations by means of simulations and extend intonon-spherically shaped particles.Figure 1: Photographs of simple configurations of toy magnets(’neocubes’) with a diameter of 4 mm. All the configurationsare stable, except for the two rightmost configurations; whenthe magnets are forced in one of these meta-stable configura-tions, they readily rearrange when touched.II – Setup and particle designThere are four distinct elements that play a role inall self-assembly systems: (I) the particles, (II) bindingforces, (III) driving forces and (IV) the environment. Inour case these elements are the following: The particlesare millimeter sized plastic shells, which are fabricatedby 3D-printing. This fabrication technique allows theshape to be to be controlled in an arbitrary fashion.The particles contain an embedded hard magnet. Con-sequently, magnetostatic forces are the main bindingforce. In this case, magnetostatic forces are also drivingthe self-assembly. To enhance the (random) motion ofthe particles, a turbulent flow will be induced in theliquid by means of agitation. All experiments will becarried out in an aqueous environment. We intend tohave the particles hover, i.e., neither sink nor float in theliquid, thus enabling self-assembly without constraintsby the bottom or side walls of the beaker. Therefore,for neutral buoyancy, the density of the liquid will bematched with the density of the particles by dissolvingNaCl in the water. The designated experimental setup isschematically depicted in figure 2a.The milli-magnetic particles are neodymium magnetsenclosed in a 3D-printed shape, see figure 2b. The usedmagnets are off-the-shelf neodymium disc magnets.These magnets have their magnetization oriented alongtheir axis. The design specifications the particles arelisted in table 1. The particles are designed such thatthe maximum magnetic force between two particles isslightly larger than the net downward force in water.In this way, the particles can bond magnetically, butweak bonds can still be broken by supplying energy viathe turbulently flowing water. Furthermore, we closelymatch the density of the particles (1.3 kg/L) and theliquid (1.2 kg/L). Therefore, the downward force willbe tuned to a small value. For the first test experiments,the particles have a spherical shape.(a) (b)Figure 2: (a) Schematic drawing of the experimental setup.Shown are the beaker, which contains the liquid with the milli-magnetic particles, and the inlet to which an agitation pumpcan be connected. (b) Exploded view of a milli-magnetic par-ticle. Shown are the neodymium magnet and the 3D-printedplastic encapsulation, which consist of two half-spheres; theirdifference in color allows to identify the ’North’ and ’South’pole of the particle.Page 254Table 1: Design specifications of the particle. In the effec-tive density of the particle, the density and volume of theneodymium magnet and plastic shell are taken into account.The magnetostatic force that is listed here, is the maximummagnetostatic force that two particles can exert on each other[4].property dimensionparticle radius, r 5 mmmagnet radius 2 mmmagnet height 1 mmeffective density particle 1.3 kg/Lsaturation magnetization, Ms 1.2 T/µ0magnetostatic force, Fm 8.3 mNgravitational force 6.8 mNnet downward force in water 1.7 mNIII – Simulation procedureThe preferred magnetic configuration (that is, theorientations of the magnetic moment of the particles)is found by minimizing the magnetostatic energy of theconfiguration dynamically. For simplicity, the particlesare treated as dipoles. Only the torques on the dipolesare considered and not the forces. In other words, thepositions of the particles are fixed in the simulations andthe particles can only lower their energy via rotation.We determined stable positions of the particles by usingthe observed configurations of the neocubes in figure 1as a guide.The magnetostatic energy of a single dipole in amagnetic field is given byU =−Mm ·B, (1)where M = Ms Vm is the magnitude of the dipole mo-ment, with Vm the volume of the particle’s magnet; mis the unit vector in the direction of the dipole momentand B is the magnetic field at the position of the dipole.Figure 3: Force between two cylindrical magnets with acommon axis, as a function of center to center separation.The cylinders have an axial magnetization and dimensions aslisted in table 1. The exact solution [4] (solid line) and dipoleapproximation (dashed line) are shown.This magnetic field is caused by the other dipoles in thesystem. The field of a single dipole is given byB =µ04π1|r|3(3r̂(m · r̂)−m) , (2)where r is the distance between the dipole and the pointwhere the field is evaluated and r̂ is the unit vector fromthe dipole to this point. The magnetostatic force actingon a dipole can be obtained byF =−∇U. (3)In figure 3, the forces between two dipoles are com-pared with the forces between cylindrical shaped mag-nets. At large distances, the dipole approximation isaccurate. However, the approximation is less accurateat distances that are comparable to the size of thecylindrical shaped magnets. Therefore, the accuracy ofthe simulations can be improved by taking the exactfields into account for nearest neighbour particles.The total magnetostatic energy of a configuration ofparticles can be obtained by summing the energies of allthe dipoles in the system.The magnetostatic energy is minimized by allowingthe dipole moments to rotate in the direction of the netfield via∆m∆t=−α m× (m×H) , (4)with H = B/µ0. After each iteration step ∆t, a newconfiguration of the dipole moments is calculated withits resulting field. The damping factor α is manuallyadjusted to optimize simulations speed while maintain-ing convergence. The simulation is stopped after 10000steps, or when the change in the total magnetization isbelow a certain tolerance:N∑n=1|∆φn|+ |∆θn|< 10−5 rad, (5)where N is the number of dipoles in the configurationand φ and θ are the angles defining the direction ofdipole moment; ∆φ and ∆θ are the changes of theseangles in a single time step.The energy does not necessarily converge to a globalminimum. Therefore, the simulations are repeatedseveral times, with random initial dipole moments, tofind the magnetic configuration with the lowest energy.IV – Results and discussionThe energies corresponding to the simpleconfigurations in figure 1 have been simulatedfirst. The distance between two neighbouring particlesin these configurations is 2r. Table 2 shows the finalmagnetization state of the simple configurations andthe corresponding energies. The energies scale withM2. The configuration with the lowest energy forN = 3 is the ’line’ configuration in 3.a. For N = 4the ’ring’ configuration in 4.b has the lowest energy.Page 255Table 2: Magnetic configurations and normalized magneto-static energies after energy relaxation; in all configurationsthe dipoles are positioned in plane, except for the 3D configu-ration 4.dconfiguration magnetostatic energyU/M2 J(Am2)−23.a −0.853.b −0.754.a −1.314.b −1.344.c −1.284.d −1.00Configuration 4.c and the 3D configuration 4.d, havehigher energy, as was expected from our experiencewith the neocubes.A. energy barriersThe energy barrier between configuration 3.a and 3.bhas been investigated, see figure 4. The trajectory isparametrized by angle θ , which is defined in the figure.For each θ , the lowest energy state is found via the sim-ulations. Both the 3.a and 3.b configurations correspondto a minimum in the energy landscape. This means thatthese configurations are stable. The maximum energyis attained for θ = 105◦. For the 4-dipole system, twotrajectories are considered. In the first trajectory, an in-plane path from configuration 4.b to 4.c is parametrized.The energy corresponding to this trajectory is given infigure 5. At θ = 120◦, U(θ) has a local maximum.Therefore, the configuration of 4.c is not stable. Thisis in correspondence with the observed meta-stablebehaviour of the neocubes in this configuration.Figure 4: Trajectory in the energy landscape of a 3-dipolesystem; the energy is plotted against θ , which is defined in thelower right. The insets show the configuration of the particlesfor various θ , after minimization of the energy.Figure 5: The in-plane trajectory in the energy landscape ofa 4-dipole system. The energy is plotted against θ , which isdefined in the lower right inset. The magnetic configurationsafter energy minimization for θ = 90◦ and θ = 120◦ are alsoshown.Figure 6: The out-of-plane trajectory in the energy landscapeof a 4-dipole system. The energy is plotted against θ , which isdefined in the lower right inset. The magnetic configurationsafter energy minimization for θ = 0◦ and θ = 90◦ are alsoshown.The second trajectory is a path from configuration4.b to 4.d. The final configuration is ’3D’, since thedipoles move out of the plane. The energy and theparametrization are given in figure 6. The energy attainsa maximum at θ = 90◦, so the 3D configuration is onlymeta-stable. This is again in correspondence with thebehaviour of the neocubes.B. shape modificationsFrom figure 6 we conclude that the 3D configurationwith 4 dipoles is not stable. In this paragraph it isinvestigated whether this 3D configuration can be madestable by modifying the shape of the particles. More-over, if the energy of the 3D configuration can bereduced in this way, the 3D configuration might becomeenergetically favourable over the 2D configurations.Figure 7 illustrates a possible shape modification; theparticles have been indented at three positions. Withsuch a modification, the top particle can be positionedcloser to the 3 bottom particles. This results in a lowerPage 256Figure 7: Illustration of a shape modification. The unmodifiedparticles (left drawing) can be indented to allow a closerpacking of the spheres (right drawing). The exploded view(center drawing) shows the indentations and the verticaltrajectory (line) of the top particle.energy.The effect of this shape modification is investigatedby simulating a trajectory. The 3 bottom dipoles areconfigured in a ring, with their centers in the x-y plane.The top particle moves vertically in the −z direction,as illustrated in figure 7. At each position the energy isminimized and U(z) is obtained, where z is the distancebetween the center of the top particle and the x-y plane.At z/r = 2/3√6 the top particle touches the bottomparticles if the particles have their unmodified sphericalshape. To allow a smaller z, the shape of the particlesmust be modified. Figure 8 shows the energies versusz/r that are obtained for two situations. In the firstsituation, all dipole moments are free to rotate. In thesecond situation, the magnetization of the top particle isfixed in the z direction. This is of interest, because themodification of the particles defines a preferred orien-tation of the dipole moment. However, the constraint ofhaving preferred orientations of the dipole moments canbe resolved by designing particles that contain magnetswhich are free to rotate with respect to the shell.The energies of the two cases are compared withthe 4-dipole ring configuration (4.b. in Table 1), sincethis is the 2D configuration with the lowest energy. Incase that the top particle has a fixed magnetization inthe z-direction, the energy is always larger than theenergy of the ring configuration (horizontal line infigure 8). However, in the case where all particles arefree to rotate their dipole moments, the energy is lowerthan the ring configuration if z/r ≤ 1.25. Therefore,for particles with a rotatable magnet and a modifiedshape, the 3D configuration is preferred over the 2Dconfigurations.V – ConclusionsA 3D self-assembly system using milli-magnetic par-ticles has been designed. The design allows the particlesto hover freely in salt water.The magnetostatic energies of simple particle con-figurations are obtained via simulations by dynamicminimization of this energy. The simulated energy bar-riers are in correspondence with the behaviour of theneocubes. Still, the accuracy of the simulations can beimproved by taking into account the cylindrical shape ofthe magnets. In future experiments, the obtained energyFigure 8: Energy versus z position of the top dipole in the3D configuration with 4 dipoles. In the ’free’ case (circles)all dipoles were free rotate their dipole moment during theenergy minimization; in the ’fixed’ case (squares), the momentof the top dipole was fixed the z direction during the energyminimization. The shaded area indicates the region where theshape of the particles needs to be modified. The horizontal lineindicates the energy of the 4-dipole ring configuration.barrier for the 3-dipole system might be validated.Our simulations show that for 4-dipole systems a2D configuration is energetically favoured over a meta-stable 3D structure. However, further simulations in-dicate that by modifying the shape of the particles itis possible to obtain stable 3D configurations. We willnow focus on proving this exciting result experimen-tally. Furthermore, we will investigate the magneticself-assembly of more intricate 3D architectures byclever design of the shape of the constituent particles.References[1] Miko Elwenspoek, Leon Abelmann, Erwin Beren-schot, Joost van Honschoten, Henri Jansen, andNiels Tas. Self-assembly of (sub-)micron parti-cles into supermaterials. J. Micromech. Microeng.,20(6):064001, 2010.[2] L. Abelmann, N. Tas, E. Berenschot, and M. El-wenspoek. Self-assembled three-dimensional non-volatile memories. Micromachines, 1:1–18, 2010.[3] John A. Pelesko. Self assembly - The Science ofThings That Put Themselves Together. Chapman &Hall/CRC, 2007.[4] David Vokoun, Marco Beleggia, Ludek Heller, andPetr Sittner. Magnetostatic interactions and forcesbetween cylindrical permanent magnets. Journal ofMagnetism and Magnetic Materials, 321(22):3758– 3763, 2009.Page 257

Magnetostatic interactions and forces between cylindrical permanent magnets.

Self assembly - The Science of Things That Put Themselves Together. Chapman & Hall/CRC,

Self-assembled three-dimensional nonvolatile memories.

Self-assembly of (sub-)micron particles into supermaterials.

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Simulating three dimensional self-assembly of shape modified particles using magnetic dipolar forces

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