IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014 2303507

Deterministic and Stochastic Trajectories of Magnetic Particles:Mapping Energy Landscapes for Technology And Biology

Marci L. Howdyshell1, Michael Prikockis1, Stephanie Lauback1, Gregory B. Vieira1, Kalpesh Mahajan2,Jessica Winter2,3, and Ratnasingham Sooryakumar1

1Department of Physics, Ohio State University, Columbus, OH 43210 USA2Department of Chemical and Biomolecular Engineering, Ohio State University, Columbus, OH 43210 USA

3Department of Biomedical Engineering, Ohio State University, Columbus, OH 43210 USA

Technologies that control matter at the nano- and micro-scale are crucial to realizing engineered systems that can assemble,transport, and manipulate materials at submicron length scales. Two principles: 1) the domain wall structure of patterned magneticstructures and 2) the superparamagnetic properties of nanoparticles, have been previously used to remotely manipulate and transportmagnetic entities to specific sites on a platform. In this paper, changes to the energy landscape during transport as well as thelocal energy profile of individual stationary traps, both of which are central to the functionality of the platform, are evaluatedusing directed forces and stochastic (Brownian) trajectories of trap-confined microparticles. Hybrid magnetic-fluorescent micellenanoconstructs, which are compatible with physiological conditions and safeguard functionality of biomaterials, are shown to beviable markers to label and manipulate individual cells across the platform.

Index Terms— Brownian motion, magnetic domain walls, mobile traps, superparamagnetic (SPM) microparticles.

I. INTRODUCTION

THE continued advancement of nanotechnology will, ingeneral, require the spatial and temporal control to direct

nanoparticles to targeted locations and to place them intoparticular structures and assemblies. Correspondingly, thisrequires femto- to pico-Newton scale forces to be preciselyapplied to a single nanoparticle or to an ensemble of particlesresiding in different environments to which direct access maynot be always possible or convenient. For example, manynext generation smart drug delivery systems that will beresponsive to a patient’s particular needs are being developedbased on transporting nanoshells [1], [2], micelle encapsulatedcargo [3], liposomes [4], [5], or magnetic nanoparticles [6], [7]to specific diseased sites. Concomitantly, miniaturization ofmeasurement or detection devices, such as biomarker recogni-tion of disease conditions, will also be required for augmentedsensitivity to the small signals associated with nano- andmicro-scale biological or inert entities in their native settings.

The transport and detection of nanoscale cargo for technol-ogy or biology face many challenges. Contact-free manipula-tion, overcoming stochastic perturbations, such as Brownianfluctuations [8]–[10], limited throughput, preserving cargointegrity under exposure to large localized optical, electricalor magnetic fields, and selective control and regulation aresome hurdles that need to be addressed. For instance, opticaltweezers [11] and dielectrophoresis [12] utilize optical andelectric fields to exert finely-tuned local forces to manipu-late single tiny objects such as cells, but generally lack thehigh-throughput required for statistically significant results inbiomedical studies. Atomic force microscopy, which offershigh spatial resolution and has led to dip-pen nanolithographyapplications [13] is limited as a viable practical mode for selec-tive transport and detection. Likewise, conventional magnetic

Manuscript received March 7, 2014; revised April 23, 2014; acceptedMay 9, 2014. Date of current version November 18, 2014. Correspondingauthor: R. Sooryakumar (e-mail: sooryakumar.1@osu.edu).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2014.2323959

tweezers have emerged as a useful tool for the study of DNAelasticity [14], but generally remains untapped as an efficientmode for directed selective transport and multiplexed opera-tion. In moving toward lab-on-chip device structures, highlylocalized magnetic fields found at domain walls on patternedwires offer features that mitigate some of the drawbacks ofefficient manipulation that hinder other approaches [15]–[22].

In this paper, we expand upon our previous resultsthat exploit fields generated from stationary domainwalls [23]–[26] to activate mobile traps that mediate themanipulation of targeted magnetic objects. In addition tousing deterministic magnetic forces to map the field-activatedprimary and secondary traps that define the transport dynam-ics, stochastic fluctuations of individual components of high-symmetry ordered micro-particle arrays are utilized as localprobes to chart the energy profiles of individual confinementpotentials. We illustrate that the competition between deter-ministic (confinement and dipolar) and stochastic (Brownian)forces provides a new approach to remotely access and probelocal energy profiles and their response to external magneticfields. The results demonstrate the prospect for a method-ology to isolate, manipulate, and quantify biomarkers thatare emerging as an important approach for environmentalas well as individualized point-of-care testing and diagnosisplatforms. Applications in biotechnology and medicine, wherebiocompatibility is vital and the environmental conditions arefavorable for biological functions, are particularly suited tosuch a hybrid methodology.

II. EXPERIMENTAL METHODS

A. Experimental Setup

Co0.5Fe0.5 wires [Fig. 1(a)] were patterned on a siliconwafer using electron beam lithography and sputter deposition.Wires with each segment ∼15 μm long, 1 μm wide, and 12 nmthick were initially magnetized by a momentary 1 T field thatled to either a head-to-head (HH) or tail-to-tail (TT) domainwall at each vertex and generation of a field HDW [23]. Toprobe energy landscapes along the wire length, the wafer con-taining the wires was coated with a 500 nm protective layer of

0018-9464 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2303507 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014

Fig. 1. (a) Zigzag CoFe wires used for vertex-to-vertex transport, witharrows indicating HH and TT domain walls at vertices. (b) Schematicof electromagnet and microscope setup to manipulate and image transportof magnetic particles and labeled cells. (c) View of a 2-D confinementchannel to facilitate cluster formation for tracking Brownian motion. ThePDMS and photoresist are 1 mm and 1.5 μm tall, respectively. The PMMAthickness—and thus the distance of the beads from the wire—may be varied(approximately 0–5 μm).

silicafilm (Emulsitone). A tunable external field Hext(= HXY +HZ ) produced by five orthogonal electromagnets [Fig. 1(b)]modulated the net field in the vicinity of the wires. Fieldsranging from 1 to 150 Oe in each direction were utilized.

Three types of superparamagnetic (SPM) micro-particleswith mean diameters of 1.05 μm (Dynabeads # 65011),2.8 μm (Dynabeads #14305D), and 11 μm (Spherotech#CM-80-10) were utilized. The largest particles were partiallycoated with 100 nm Au (i.e., Janus-type particles) for easyvisualization of their orientation via a high-speed camera(Phantom Miro M120, Vision Research). The contrast betweenthe dark- (Au coated) and light-colored (uncoated polystyrene)surfaces allowed us to distinguish between rolling and trans-lational motion. Furthermore, to probe the local trap profile,individual trajectories of multiple 1.05 μm diameter particleswere tracked as a function of their height Z above the zigzagwires and their response to Hext. For these experiments, thefluid borne particles were constrained to a channel [Fig. 1(c)]bounded by polymethyl methacrylate (PMMA) on the bottomand polydimethylsiloxane (PDMS) on top [22], restrictingthe particles to a quasi 2-D system. The PMMA layer, withthickness dPMMA, on which the particles reside, is directlyatop the zigzag wires, thus providing a convenient means toselectively probe the trap at well-defined heights through theirplanar Brownian trajectories.

The large field gradients (>105 T/m) that occur within tensof nanometers above the vertices offer a unique opportunityto manipulate nanoscale constructs. Sub-100 nm micellesengineered to encapsulate quantum dots and SPM ironoxide nanoparticles (SPIONs), i.e., MagDot structures, weresimultaneously magnetically maneuvered on the wires andfluorescently tracked [24]. The surface of these nanocon-tainers consists of carboxyl-terminated polyethylene glycolthat minimizes non-specific binding. A carboxyl group inconjunction with 1-Ethyl-3-(3-dimethylaminopropyl) carbodi-imide hydrochloride (EDC) chemistry placed biotin on theMagDot surface. Biotinylated anti-human CD45 antibodieswere then attached to the micelles via biotin–avidin–biotinbinding. The micelle-antibody construct attaches to humanleukocytes by antibody-antigen binding, which are imagedusing a 100 W mercury lamp, TRITC filter (604 nm emission),and color charge-coupled device camera (DP70, Olympus,Tokyo).

Fig. 2. Flow chart for algorithm to estimate the trap radius for 1 μmSPM particles as a function of height above a monopole-modeled zigzagwire vertex. The fields are |HZ | = 125 Oe and |HXY | = 0 Oe. The algorithmstarts by: 1) placing an initial particle at height Z at (0, 0, Z). All subsequentsteps occur on the positive x-axis (see sketch on right for 1). 2) Next particle(labeled N = 1) is added, allowed to equilibrate, and then fixed. 3) Themaximum energy gradient (i.e., the force F) as sensed by, and to the right ofthe N = 1 particle is evaluated (see sketch for 3 on right). If F exceeds the10 K B T/μm threshold, then step 2 is repeated with the next particle (N = 2)added. 4) If F does not exceed the threshold, then the x-coordinate of particleN − 1 provides the trap radius. The simulation is re-initialized for a differentvalue of Z and the loop is repeated from step 1.

B. Simulation

To qualitatively evaluate the trap profile with increasingheight above the vertex [system described in Fig. 1(c)],a 1-D simulation governed by the steps below was carried out.Here, the monopole approximation [25] is invoked in which(HDW), the field at location r arising from the domain wall,is given by

HDW = qmr4πr3 (1)

where qm is the effective magnetic charge at the vertex.A trapped SPM bead can be approximated by a point-likemagnetic dipole. When multiple dipoles are present at thetarget site, the field from each dipole (Hdip) is calculatedself-consistently by an iterative algorithm. The sequence ofthe simulation steps carried out in 1-D is given below and isschematically shown in Fig. 2.

1) A point particle (N = 0) of radius R is fixed at position(0, 0, Z ) height Z = R + dPMMA above the vertex. Thesimulation along the x-axis in field |HZ | = 125 Oe and|HXY | = 0 Oe defines the magnetic moment of the firstparticle. (Fig. 2, top-right sketch).

2) A second particle (N = 1) is then introduced on thex-axis and allowed to seek minimum energy. In theabsence of Brownian fluctuations, its attraction towardx = 0 is balanced by dipolar repulsion between particles.

3) The spatial gradient of the energy profile (i.e., force F)sensed by particle N = 1 is investigated to ensurethat the slope to the right of the energy minimum isgreater than 10 kBT /μm, a threshold (based on thermalenergy and relevant length scale ∼1 μm bead diameter)to guarantee the particle is contained. A sub-thresholdforce implies eventual particle escape from the trap. Ifthe particle is trapped, steps 2 and 3 are repeated, fixingthe previous particles and incrementing N by 1 (Fig. 2,bottom-right sketch).

4) When the N th particle does not meet the thresholdrequirement of step 3, the x-coordinate of particle N −1

HOWDYSHELL et al.: DETERMINISTIC AND STOCHASTIC TRAJECTORIES OF MAGNETIC PARTICLES 2303507

is the trap radius at height Z . The particles are removed,and the simulation is repeated for a different Z .

It is noted that the 1-D simulation provides only a qualitativeestimate of the trap radius. It does not account for thepossibility of an added particle occupying a site not directlybetween trap center and another particle. Moreover, step 2 doesnot allow for existing particles to reach a new equilibriumconfiguration when another enters the trap. Despite theseapproximations, as discussed below, the simulation providesa basis to select the height to probe in the experiment, as willbe discussed in Section III-B.

III. RESULTS AND DISCUSSION

The field-controlled dynamics of the SPM beads are utilizedto characterize the energy landscape and the associated trans-formations that occur to the traps during their transport acrossthe zigzag wire platform. These transformations to the trap-ping sites provide for the deterministic forces that guide andmaneuver the individual particles (Section III-A). In addition,the extent of Brownian excursions of individual componentsof an ordered array of beads held within trapping sites atdifferent elevations are shown to provide a viable means tolocally characterize individual trap profiles (Section III-B).The versatility of these mobile magnetic traps is illustrated bytransporting individual cells labeled with sub-50 nm hybridmagnetic-fluorescent (MagDot) constructs (Section III-C).

A. Energy Landscape and Deterministic Motion

In transporting a SPM bead from one vertex to an adjacentvertex, the fields HXY and HZ are tuned. These weak fieldsdo not modify the general location of the domain walls inthe zigzag wires in any significant way [25] and thus theassociated domain wall fields HDW are determined solelyby the CoFe wire dimensions and initial magnetization. Themoment induced in a SPM particle located at a given heightabove the wires is determined by the net field, Hext + HDW.In the absence of Hext, adjacent vertices and their associatedHDW fields act as primary trapping sites for the bead. Withincreasing HZ , the bead moment is proportionally determinedby Hext and the energy landscapes of adjacent HH and TTvertices, which steadily transform to become attractive andrepulsive, respectively. The introduction of HXY in the pres-ence of HZ , with HXY oriented along the straight segment ofthe zigzag wire, further transforms the character and locationof the traps. In particular, the primary traps weaken and shiftaway from the zigzag vertex to positions that lie betweenvertices. The resulting secondary traps (S) are crucial to thetransport of the beads, which, depending on the depth of thetrapping potential, can be slowed down, momentarily stalled,or completely halted in their movement between vertices.These features are highlighted in Figs. 3 and 4 for beads (diam-eters 2.8 and 11 μm) of different magnetic susceptibilities.The data allow for an experimental determination of in-planeforces exerted by wire traps on particles, found to be as highas 3 pN in this paper.

1) 2.8 μm Particle: Fig. 3 shows the influence of HXY andHZ on the energy landscape during transport of a 2.8 μmdiameter bead along the wire between adjacent vertices.In Fig. 3(a), the out-of-plane field HZ is reversed from+40 to −40 Oe while the in-plane field HXY = 10 Oe. ForHZ = +40 Oe (dashed curve), the particle (dark circle) sitsin a potential minimum (initial trap S0) near the first vertex.

Fig. 3. Potential energy landscapes for a 2.8 μm bead on a wire in(a) HZ = ±40 Oe and HXY = 10 Oe. (b) HZ = ±40 Oe and HXY =80 Oe. (c) HZ = ±10 Oe and HXY = 10 Oe. (d) HZ = ±80 Oe andHXY = 10 Oe. In the presence of a positive HZ field, the initial position ofthe bead (expected position indicated by dark circle) is at the initial trap S0.HZ is then reversed, causing the bead to move to the lower energy at Sf . Themovement of the bead along the energy profile is indicated by arrows. S0,Si, and Sf indicate initial, intermediate, and final traps. Vertical lines (blue)indicate locations of wire vertices. The largest deviation of Si and Sf fromthe vertices occurs at large HXY values (HXY > HZ ).

When HZ is reversed to −40 Oe (solid curve), this vertexbecomes repulsive and, since no intermediate traps are stabi-lized between the vertices, the particle moves steadily fromthis unfavorable energy state toward the neighboring final trapSf located at the other vertex. When |HXY | > |HZ |, as inFig. 3(b) (HXY = 80 Oe, |HZ | = 40 Oe), two secondarytraps of different energy depths occur. The intermediate trap(Si) nearer to the initial vertex is weakened by the repulsivecontribution of HZ to the potential energy while the con-structive superposition of HZ and HDW at the second vertexrenders a deeper secondary trap (Sf). For a given |HZ |, theintermediate trap Si steadily becomes more pronounced withincreasing HXY and Si transforms from a weak shoulder toa distinct trap that slows the particle’s motion. For weakplanar fields (|HXY | ≤ |HZ |), HXY is not strong enough toeffectively influence the orientation of the bead’s inducedmagnetic moment to generate a clear intermediate trap, asshown in Fig. 3(c) and (d), where HXY = 10 Oe and|HZ | = 10 and 80 Oe, respectively.

As shown in Fig. 4, the energy profiles presented in Fig. 3are consistent with the measured speeds of the bead. Fig. 4(a)confirms that for low HXY (10 Oe) and HZ (40 Oe), thebead initially accelerates reaching speeds ∼60 μm/s as itmoves away from the initial trap S0, which is transformedto a repulsive site by reversing HZ . The motion is thenslowed as the bead encounters a flatter energy landscapebefore emerging and gaining speed as it moves toward thedeeper final trap Sf where it is rapidly brought to rest.In Fig. 4(b), HXY = 70 Oe (|HXY |>| HZ |) and an intermediatetrap causes the particle to be temporarily localized beforeeventually escaping from Si to reach target destination Sf .For HXY = 80 Oe [Fig. 4(c)], however, the intermediatetrap is sufficiently deep that the particle is, as expected,permanently halted well before reaching the adjacent vertex.Fig. 4(d)–(f) demonstrates experimental confirmation thatwhen |HXY | < |HZ |, intermediate traps do not form therebyenabling the particle to reach the next vertex.

2303507 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014

Fig. 4. Experimentally measured speed of 2.8 μm bead moving alongthe wire (solid lines) and corresponding potential energies (dashed lines)calculated from the model. Plots are for HZ = −40 Oe and HXY = (a) 10,(b) 70, and (c) 80 Oe. As HXY increases, an intermediate secondary trapSi emerges, causing the bead (b) to slow or (c) come to rest. Experimentson the same bead with HXY = 10 Oe and HZ = (d) −10, (e) −70, and(f) −80 Oe do not result in intermediate traps and the bead reaches thedestination vertex.

2) 11 μm Particle: The 11 μm magnetic Janus-typeparticles offer different responses compared with their 2.8 μmcounterpart. These changes can be traced to the effective beadmoment above the vertex. Despite having larger expected sus-ceptibilities (which increases the magnetic potential energy),the 11 μm beads experience weaker effective fields andbroader primary traps due to the large field gradients asso-ciated with HDW. The initial (S01, S02) and final (Sf1 andSf2) traps are located along the wire a few micrometers fromthe vertex center; this distance increases with increasing HXY .On 14.5 μm long wires, the broadened primary traps approacheach other with no intermediate traps evident [Fig. 5 (a)–(d)].According to the model, for wires ∼40 μm and longer, theinitial and final traps are more separated (not shown), enablingan intermediate trap to emerge for the 11 μm sized particles.

The experimental results of Fig. 5(e) and (f) for the 11 μmparticles confirm that the corresponding translational speedsare smaller than those on the 2.8 μm particles. The smallermeasured speeds and reductions in the distance traveled alongthe wire with increasing HXY are in line with the model.Confirmation of theoretical predictions (Figs. 3–5) of themeasured starting and ending locations, as well as recordedchanges in particle speed with applied fields for the differentparticles thus validates the models related to: 1) domain-wall-generated fields (HDW); 2) response of the energy landscapeto HXY + HZ ; and 3) the magnetic properties of the beads.

We have also observed (Fig. 6) that the 11 μm Janusparticles exhibit an initial rolling motion prior to sliding alongthe wire. Upon reversing HZ , [Fig. 6(a), i–viii], the entirebead is observed to rotate to align with the net field prior tosliding along the wire to reach the neighboring vertex whilemaintaining its orientation [Fig. 6(b), i–vii]. These findingsreveal a field-induced rotational torque on the micro-particleimmediately after the field is reversed. This in turn suggeststhe presence of a small ferromagnetic character for the bead,which is likely due to a size distribution in the embedded mag-netic nanoparticles. Despite the weak ferromagnetic character,the overall field response of the beads is largely in agreementwith that of a SPM microsphere.

Fig. 5. Experimentally measured speed of the 11 μm bead along the wireand corresponding potential energy landscape calculated from the model.(a) and (b) Potential energy in fields HZ = ±40 Oe and HXY = 60 and150 Oe, respectively. (c) and (d) Potential energy for HXY =10 Oe and HZ =±10 and ±80 Oe, respectively. Vertical lines (blue) indicate locations of wirevertices. (e) Experimentally determined particle speed in HZ = −40 Oe andHXY = 60 and 150 Oe. (f) Measured particle speed for HXY = 10 Oe andHZ = −10 and −80 Oe. As HZ increases relative to HXY secondary trapsshift closer to wire vertex and, as predicted in (a)–(d), the particle travelsa larger distance. S01, S02, Sf1, and Sf2 are the initial and final traps fordifferent field values.

Fig. 6. 11 μm Janus particle exhibits (a) rolling and (b) sliding motionduring vertex-to-vertex transport. Schematic images (i–iv) are paired with(v–viii) illustrating the orientation of dark- and light (translucent)-coloredregions during rolling and translational motion.

B. Trap Profile and Stochastic Motion

To further investigate the primary trap, as discussed inSection II-B, a PMMA layer provides an expedient means tocontrol the height above the wires where several interactingparticles, confined to two dimensions, reside and interact withthe trapping fields. Their Brownian movements selectivelyprobe the trapping potential at each specified height. Forinstance, at dPMMA = 1.65 μm, the primary contribution toeach particle’s magnetic moment arises from the applied fieldHZ = 125 Oe), which yields a much larger contribution thanthe domain wall field (HDW = 17 Oe) at this height. Since the

HOWDYSHELL et al.: DETERMINISTIC AND STOCHASTIC TRAJECTORIES OF MAGNETIC PARTICLES 2303507

Fig. 7. Theory and simulation results for 1 μm diameter (2R) SPM beads inan external field HZ = 125 Oe. (a) Computationally estimated trap radius asa function of PMMA thickness for 1.05 μm beads. (b) Relative contributionof the domain wall field (solid line) and external field (dashed line) to thetotal field sensed by a single bead as a function of its height directly abovethe vertex. (c) Potential energy profile for a bead whose center is at heightZ = 0.5 μm (dotted curve) and Z = 3.5 μm (solid curve) as a function ofx position (y = 0). (d) Trajectories simulated for a monopole-modeled domainwall with initial random particle positions. The concentric circles from insideto outside are constant energy contours of (−10, −8, −6, −4, and −2) kBT .

channel is engineered to restrict particles to horizontal motion,the resulting out-of-plane moments give rise to repulsivedipolar forces that are balanced by the trap-related confinementforces, yielding a spatially ordered planar array of confinedparticles. Since all particles in the array fluctuate about theirmean positions, the trap profile can be probed.

Fig. 7(a) shows results derived from the 1-D simulationsdescribed in Section II-B, which estimate the trap radius asa function of dPMMA for the 1.05 μm beads. As shown, atdPMMA = 0, the trap radius is equal to the bead diameter,and it grows rapidly in size as dPMMA increases to 0.25 μm.Beyond this, the trap size decreases until a height of 0.75 μmbefore expanding again. This trend is accounted for by changesto the dipole moments of the particles and correspondingmodifications to the energy contours that occur with increasingheight. The dipole moment depends on the total magneticfield, Hext + HDW, and close to the wire (dPMMA < 1.0 μm),the primary contribution to the moment arises from HDW[Fig. 7(b)]. For instance, a 1.05 μm diameter bead locateddirectly above the vertex has a moment of 2.6 × 10−14 Am2,but when the center is at a height Z = 1.5 μm, its moment isreduced to 10−14 Am2. The reduction represents a significantchange to the dipolar repulsion. When close to the vertex,dipolar effects determine the trap radius. At dPMMA = 0 (in the1-D simulation), a maximum of two particles can be trappedbecause their dipolar forces prohibit additional particles fromapproaching (0, 0, Z), i.e., a third particle cannot attain stableequilibrium within the confines of the trap. At dPMMA =0.25 μm, weakened dipolar forces allow a third particle to betrapped, thereby growing the trap radius. From dPMMA = 0.25to 0.75 μm, the equilibrium position of the third particle drawscloser to (0, 0, Z) as their repulsion weakens. At dPMMA =1.0 μm, the domain wall field accounts for only 20% of thedipole moment (reduced from ∼80% at a distance of one beadradius above vertex). As dPMMA increases beyond 1.0 μm,

the contribution of HDW to the moment diminishes, anddipolar repulsion is no longer dominant in determiningtrap size.

The second factor that establishes the trap radius is theshape of the potential energy contours. With increasing Z,the potential energy profile widens (but becomes shallower),a feature shown in Fig. 7(c) where the solid (dotted) curvecorresponds to the bead center located at 3.5 μm (0.5 μm)above the wire vertex. This gradual change in shape allows forincreased separation between trapped particles and thus alsoaccounts for the rising trap radius as dPMMA exceeds 1.0 μm.The trap radius also depends on the magnetic moment m ofthe particles. Dipolar forces (∼m2) increase faster than trapconfinement forces (∼m) at a given height. Increasing m willnot result in trapping additional beads; rather, it will expandthe trap radius until dipolar repulsion expels one of the outerbeads from the trap, shrinking the trap radius.

In contrast to the 1-D case in Fig. 7(a), Fig. 7(d) showsmore relevant 2-D simulated trajectories of eight particles neara wire vertex with HZ = 125 Oe; the origin is located at adistance equal to a bead radius directly above the center of thevertex and corresponds to dPMMA = 0. The concentric circlesindicate constant energy contours of −10, −8, −6, −4, and −2kBT . The beads initially occupy a random configuration, and atthe end of the 85 s simulation are located at positions indicatedby the black outlined circles in Fig. 7(d). The associatedtrajectory of each bead is determined according to [9] and [22].Simulations confirm that the four particles lying on the outersections of the trap undergo extended Brownian trajectories,while the remaining four stay tightly confined near the origin.The results reveal that only four of eight particles will betrapped for the duration of the simulation.

Based on these simulations, a PMMA height of 1.65 μmwas selected for the trap probing experiments described below,allowing many particles to be reliably trapped and be spatiallyseparated to enable their distinct Brownian excursions bemonitored. The fluctuations were then used to confirm aparticular model for the magnetic trap, thereby mapping theenergy landscape at a given height. Two approaches are usedto determine HDW: 1) the monopole model (1) and 2) the2-D version of the Object Oriented MicroMagnetic Frame-work (OOMMF). The latter considers properties of the CoFewire (such as saturation magnetization) and its shape. It alsoprovides for the spatial distribution of the wire magnetization,enabling HDW to be extracted by integrating the divergence ofthe magnetization over all space [26].

Fig. 8(a) shows the trajectories tracked over 85 s of an arrayof eight 525 nm radius particles with HZ = 125 Oe. Therecorded paths are overlaid on an image of the initial cluster.The center of each bead is 2.15 μm above the wire (dPMMA =1.65 μm). The related radial rms fluctuations as a functionof the average in-plane displacement (rc) from each particleto the geometric center of the vertex are shown in Fig. 8(b)along with the simulation results (triangles). The error barslargely arise from pixelation in the image and accuracy of theLabVIEW protocols. As shown [Fig. 8(c)], the particles areobserved to lie on concentric rings, with those on a given ringhaving similar in-plane displacements from the vertex.

To map the trap energy profile, the real-time particletracking results are compared with simulations based on themonopole model and protocols using OOMMF to obtain HDW.The former predicts a symmetric potential energy profilethat allows for net rotation of the entire cluster about the

2303507 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 11, NOVEMBER 2014

Fig. 8. Experimental results for 1 μm particles in field HZ = 125 Oe at beadcenter height 2.15 μm. (a) Brownian trajectories experimentally tracked over85 s. Image background shows particles at their initial positions. (b) Radialrms fluctuations as a function of average in-plane distance from the vertex forthe eight particles (filled circles) of part (a). Results based on monopole modelfor HDW are identified by triangles. The simulation yields very similar resultsfor particle pairs 1, 2 and 3, 4, as well as for the four particles 5–8, such thatsome corresponding triangles overlap on the plot. (c) Brownian trajectoriesfrom (a) as plotted over constant energy contours given by HDW derived fromOOMMF simulations. The + identifies the geometric center of the Ro/Ri =1.3/0.3 μm wire. Trajectories are unlabeled for clarity, but correspond to thosein (a).

central axis, and similar Brownian excursions for all particleson the same ring. However, in experiment, particles 5 and 8are observed to undergo significantly different trajectories andreveal a trap that is weaker over the trajectory of particle 5 thanit is for particle 8. Furthermore, the absence of cluster rotationover the 85 s tracking period indicates that the monopolemodel does not fully describe the trapping fields.

The OOMMF simulations yield wire magnetization profilesand their response to the vertex geometry and external fields.The curvature of the rounded segments of the vertex can,for example, shift the potential energy minimum away fromthe geometric center of the wire. The related constant energycontours [Fig. 8(c)] manifest from a wire with outer/inner radiiof Ro/Ri = 1.3/0.3 μm and are consistent with the recordedexperimental tracks. The associated forces vary by as muchas 5% along different radial directions in agreement with theabsence of rotations of the entire cluster around the vertex,revealing that each step is not energetically equivalent due toasymmetries in the trapping potential.

C. Manipulating Biomarkers: MagDotNanocontainers and Cell Transport

The ability to characterize and isolate cells based on abiomarker signature is a vital tool for the implementation ofsingle-cell-based personalized diagnostics. An indication ofthe population of cells that contain a given biomarker canspecify disease progression [27]. Furthermore, quantificationof biomarker concentrations on a cell surface provides infor-mation that can guide therapeutic intervention [28].

Here, we demonstrate a new mechanism for isolation,manipulation, and quantification of biomarkers on cells.Leukocytes were labeled with MagDots, synthesized as

Fig. 9. Bright field image of leukocytes conjugated to micelles on zigzagwire array (4 μm nearest vertex-to-vertex distance) in the presence ofHZ = 100 Oe. Micelles encapsulate quantum dots and SPIONs. (a) Cellsare trapped at wire vertices, indicating the presence of SPIONs in micelles.(b) and (c) Upon several reversals of HZ , cells are released, guided by fluidflow (approximate direction indicated by white arrows), and trapped at newvertices. (d)–(g) Single leukocyte transported and trapped at four differentvertices is imaged using a fluorescence microscope confirming the presenceof both SPIONs and quantum dots (emission wavelength = 604 nm) on thecell surface.

described previously [24], [29], containing both quantum dotsand SPIONs. On average, each micelle contains one quantumdot and 4–10 SPIONs. The presence of SPIONs on the leuko-cyte surfaces is verified because of the localization of cells atmagnetic wire vertices [Fig. 9(a)]. Furthermore, as discussedabove, by varying external magnetic fields, the traps are maderepulsive, allowing the magnetically-labeled cells to be guidedby fluid flow and trapped at nearby vertices [Fig. 9(b)–(c)].The presence of quantum dots on leukocytes was verified byfluorescence microscopy. A single cell trapped at differentvertices shows a fluorescent signature, indicating simultane-ous magnetic and fluorescent labeling [Fig. 9(d)–(g)]. Thefluorescence intensity provides a quantitative indication ofthe number of fluorescent micelles, and in turn the relativenumber of labeled cell surface receptors. By imaging usinglaser scanning confocal microscopy, stacks of fluorescenceintensity from multiple focal planes can be reconstructed intoa 3-D cell image. By tracking several fluorescing MagDotsundergoing Brownian motion, the size of each was determined.The fluorescent intensity was calculated by averaging theintensity over the pixels tracked as a function of time. Theresulting average intensity for several micelles thus providedan estimate of the total number of micelles bound to a cellsurface. This yields an estimation of the number of surfacereceptors because we expect the two to mainly interact ona one-to-one basis; however, receptor dimerization and otherfactors prevent a direct measurement of receptor numbers andthus this indication provides only a relative measure.

If the magnetic trap profile is appropriately characterized asdiscussed earlier in this paper, the strength at which the cellis bound to the magnetic trap can indicate the total volume ofmagnetic material on the cell, providing another quantitativemeasure of surface bioreceptors.

IV. CONCLUSION

Emerging advances in nanotechnology will require capa-bilities to capture, transport, and spatially localize targetedmultifunctional nanoparticles. In this paper, we have demon-strated that a combination of interacting magnetic particlesand externally applied magnetic fields can map the energyprofile of mobile conveyor traps that originate from stationary

HOWDYSHELL et al.: DETERMINISTIC AND STOCHASTIC TRAJECTORIES OF MAGNETIC PARTICLES 2303507

domain walls formed at the vertices of zigzag wires. Unlikeapproaches in which the particle transport is mediated bymoving domain walls [15], the present methodology is notprone to limitations such as domain wall pinning that couldrestrain particle maneuverability. Furthermore, transport path-ways are not restricted to trajectories that lie directly abovethe wire conduits. We have also demonstrated that the interac-tions between a collection of individual fluid borne particlesconfined to the trap and their stochastic (Brownian) motionsprovide for a means to characterize the trap profile and itsvariation with height above the wire vertex. Moreover, the veryhigh magnetic field gradients near the surface of the conveyorplatform has offered promising opportunities to manipulatefunctional nanoparticles that are less than 50 nm in diameter.Hybrid magnetic-fluorescent quantum dot (MagDot) nanoma-terials, which are especially useful for detecting analytes andbiomolecules of comparable size, have enabled individual cellsbe labeled and transported on the same platform. In contrastto the large micrometer-sized magnetic beads often used incell separation, the MagDot materials provide opportunitiesto target specific cell surface biomarkers. We have shownthat the integration of nano-materials with the versatility ofthe magnetic conveyor platform has enabled the transport ofindividual cells to specific sites while providing fluorescenttracking signatures. This combination of technologies offersexciting opportunities to isolate, quantify, and manipulate cellsand biomolecules.

ACKNOWLEDGMENT

This work was supported in part by the U.S. Army ResearchOffice under Contract W911NF-10-1-0353, in part by theNational Science Foundation under Grant EEC-0914790 andGrant CMMI-0900377, and in part by the Ohio State Univer-sity Institute for Materials Research. The authors would liketo thank Fengyuan Yang for his assistance with fabricating thewires and useful discussions.

REFERENCES

[1] A. R. Lowery, A. M. Gobin, E. S. Day, N. J. Halas, and J. L. West,“Immunonanoshells for targeted photothermal ablation of tumor cells,”Int. J. Nanomed., vol. 1, no. 2, pp. 149–154, 2006.

[2] K. Fu et al., “Measurement of immunotargeted plasmonic nanoparti-cles’ cellular binding: A key factor in optimizing diagnostic efficacy,”Nanotechnology, vol. 19, no. 4, p. 045103, 2008.

[3] G. Ruan, D. Thakur, S. Deng, S. Hawkins, and J. O. Winter,“Fluorescent—Magnetic nanoparticles for imaging and cell manipula-tion,” J. Nanoeng. Nanosyst., vol. 223, nos. 3–4, pp. 81–86, 2009.

[4] D. Sutton, N. Nasongkla, E. Blanco, and J. M. Gao, “Functionalizedmicellar systems for cancer targeted drug delivery,” PharmaceuticalRes., vol. 24, no. 6, pp. 1029–1046, 2007.

[5] N. Rapoport, Z. G. Gao, and A. Kennedy, “Multifunctional nanoparticlesfor combining ultrasonic tumor imaging and targeted chemotherapy,”J. Nat. Cancer Inst., vol. 99, no. 14, pp. 1095–1106, 2007.

[6] G. F. Goya, V. Grazu, and M. R. Ibarra, “Magnetic nanoparticles forcancer therapy,” Current Nanosci., vol. 4, no. 1, pp. 1–16, 2008.

[7] Y. Piao et al., “Wrap-bake-peel process for nanostructural transforma-tions from β-FeOOH nanorods to biocompatible iron oxide nanocap-sules,” Nature Mater., vol. 7, pp. 242–247, Feb. 2008.

[8] A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion ofindividual biomolecules in solution,” Proc. Nat. Acad. Sci. United StatesAmer., vol. 103, no. 12, pp. 4362–4365, 2006.

[9] A. Chen et al., “Regulating Brownian fluctuations with tunable micro-scopic magnetic traps,” Phys. Rev. Lett., vol. 107, no. 8, p. 087206, 2011.

[10] W. R. Browne and B. L. Feringa, “Making molecular machines work,”Nature Nanotechnol., vol. 1, no. 1, pp. 25–35, 2006.

[11] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observationof a single-beam gradient force optical trap for dielectric particles,”Opt. Lett., vol. 11, no. 5, pp. 288–290, 1986.

[12] B. Edwards, N. Engheta, and S. Evoy, “Electric tweezers: Experimentalstudy of positive dielectrophoresis-based positioning and orientationof a nanorod,” J. Appl. Phys., vol. 102, no. 2, p. 024913,2007.

[13] R. D. Piner, J. Zhu, F. Xu, S. Hong, and C. A. Mirkin, “‘Dip-Pen’nanolithography,” Science, vol. 283, no. 5402, pp. 661–663, 1999.

[14] I. D. Vlaminck et al., “Highly parallel magnetic tweezers by targetedDNA tethering,” Nano Lett., vol. 11, no. 12, pp. 5489–5493, 2011.

[15] E. Rapoport, D. Montana, and G. S. D. Beach, “Integratedcapture, transport, and magneto-mechanical resonant sensing ofsuperparamagnetic microbeads using magnetic domain walls,” LabChip, vol. 12, no. 21, pp. 4433–4440, 2012.

[16] P. Vavassori et al., “Domain wall displacement in Py square ring forsingle nanometric magnetic bead detection,” Appl. Phys. Lett., vol. 93,no. 20, p. 203502, 2008.

[17] M. Donolato et al., “Magnetic domain wall conduits for single cellapplications,” Lab Chip, vol. 11, no. 17, pp. 2976–2983, 2011.

[18] M. Donolato et al., “Nanosized corners for trapping and detecting mag-netic nanoparticles,” Nanotechnology, vol. 20, no. 38, p. 385501, 2009.

[19] M. Donolato et al., “On-chip manipulation of protein-coated magneticbeads via domain-wall conduits,” Adv. Mater., vol. 22, no. 24,pp. 2706–2710, 2010.

[20] A. Beguivin et al., “Simultaneous magnetoresistance and magneto-optical measurements of domain wall properties in nanodevices,” J.Appl. Phys., vol. 115, no. 17, p. 17C718, 2014.

[21] L. A. Rodríguez et al., “Optimized cobalt nanowires for domain wallmanipulation imaged by in situ Lorentz microscopy,” Appl. Phys. Lett.,vol. 102, no. 2, p. 022418, 2013.

[22] M. Prikockis et al., “Programmable self-assembly, disassembly, transportand reconstruction of ordered planar magnetic micro-constructs,” IEEETrans. Magn., vol. 50, no. 5, pp. 1–6, May 2014.

[23] G. Vieira et al., “Magnetic wire traps and programmable manipulationof biological cells,” Phys. Rev. Lett., vol. 103, p. 128101,Sep. 2009.

[24] G. Ruan et al., “Simultaneous magnetic manipulation and fluorescenttracking of multiple individual hybrid nanostructures,” Nano Lett.,vol. 10, no. 6, pp. 2220–2224, 2010.

[25] G. Vieira, A. Chen, T. Henighan, J. Lucy, F. Y. Yang, andR. Sooryakumar, “Transport of magnetic microparticles via tunablestationary magnetic traps in patterned wires,” Phys. Rev. B, vol. 85,no. 17, p. 174440, 2012.

[26] A. Chen, T. Byvank, G. B. Vieira, and R. Sooryakumar, “Magneticmicrostructures for control of Brownian motion and microparticletransport,” IEEE Trans Magn., vol. 49, no. 1, pp. 300–308, Jan. 2013.

[27] S. Stott et al., “Isolation of circulating tumor cells using a microvortex-generating herringbone-chip,” Proc. Nat. Acad. Sci. United States Amer.,vol. 107, no. 43, pp. 18392–18397, 2010.

[28] Z. Lui, W. G. Cumberland, L. E. Hultin, H. E. Prince, R. Detels, andJ. V. Giorgi, “Elevated CD38 antigen expression on CD8+ T cells isa stronger marker for the risk of chronic HIV disease progression toAIDS and death in the multicenter AIDS cohort study than CD4+ cellcount, soluble immune activation markers, or combinations of HLA-DRand CD38 expression,” J. Acquired Immune Deficiency SyndromesHuman Retrovirol., vol. 16, no. 2, pp. 83–92, 1997.

[29] K. D. Mahajan et al., “MagDot-nanoconveyer assay for detection andisolation of molecular biomarkers,” Chem. Eng. Prog., vol. 108, no. 12,pp. 41–46, 2012.