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Journal of Magnetism and Magnetic Materials

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Research articles

Magnetic and hydrodynamic torques: Dynamics of superparamagnetic beaddoublets

C. Pease, H.S. Wijesinghe, J. Etheridge, C.J. Pierce, R. Sooryakumar⁎

Department of Physics, The Ohio State University, 191 W Woodruff Ave., Columbus, Ohio 43210, USA

A R T I C L E I N F O

Keywords:Magnetic beadsMagnetic fieldsHydrodynamics

A B S T R A C T

Rotating chains of magnetic microparticles have many applications in lab-on-a-chip technologies. The simplestsuch chain is the fluid-borne doublet, where two beads are in close contact, but remain unattached, allowingeach bead to freely rotate. These beads typically have two components contributing to their net magnetic mo-ment: (i) a superparamagnetic moment and (ii) a field-independent permanent moment. In a rotating magneticfield, there are magnetic torques that separately rotate the doublet and its constituent beads as well as a hy-drodynamic torque from the bead-doublet coupling. This study investigates, through experiments and simula-tions, the dynamics of field-driven doublets. New dynamics were found for the case where the dominant torquestems from the hydrodynamic coupling.

1. Introduction

Actuating magnetic micro- and nano-particles has become importantto the development of several lab-on-a-chip technologies becaue of theirtranslation and rotation in applied external fields, capability for multi-plexing, bio-compatibility, and cellular-relevant length scale. They havebeen investigated in many systems such as ferrofluids [1–3], magneticchains [4–10], magnetotactic bacteria [11,12] and clusters [13,14]. Theensuing applications are broad. They include fluid mixers [2,4,8,9,15–18],micro-swimmers [19,20], sorting [21–25], drug delivery [20,26,27],micro-surgery [20,28], cell detection [29,30], as well as stiffness-, bio- andrheological-sensing [10,31–40]. To optimize these applications, the in-terplay between the hydrodynamic and magnetic forces affecting theirdynamics must be understood. One of the simplest constructs that displaysthis interplay is the detached doublet consisting of two adjacent, but un-attached, fluid-borne magnetic beads rotating in a magnetic field.

Magnetic microbeads that are typically used in lab-on-a-chip devicesare created by embedding magnetic nanoparticles within a polymer matrix[41]. Two components contribute to their net magnetic moment: (i) asuperparamagnetic Néel dipole mH that rapidly aligns with an externalfield H due to the fast stochastic reorientation of the nanoparticle magneticdomains and (ii) a Brownian field-independent permanent moment mP

arising from the slow relaxation of the magnetization of the larger nano-particles within the bead [42,33]. Studies have investigated how magnetictorques arising from the interaction between mp and H (dipole-field in-teraction) affects the rotation of individual beads [33] as well as

chemically attached [43] and detached [44] doublets. Detached doubletspresent an interesting case where the dipole-field torque directly rotatesthe individual beads in the presence of a time varying magnetic field andconsequently affects the doublet’s rotation through the hydrodynamiccoupling of the beads’ and doublet’s rotation.

Detached magnetic particles have previously been studied withoutincluding the hydrodynamic bead-doublet coupling [36,10]. With only themagnetic torque, the doublets rotate at the same rate as the external fieldup to some critical frequency, above which the doublet on average beginsto slow down. Coughlan et. al. [44] found that the hydrodynamic bead-doublet coupling had a small perturbation on the rotation rate of detacheddoublets. The perturbation was small in this case because of the relativeweakness of mP and the dipole-field torque on their beads.

The present study finds new dynamics in the rotation rate of thedoublets when the hydrodynamic bead-doublet coupling is the stron-gest torque. This previously unreported response arises from separatecritical frequencies caused by the moments mH and mP. Simulationsreveal that the hydrodynamic torque τD

Hyd on the doublet is linear withrespect to the rotation rate of the doublet (ωD) and of the individualbeads (ωB), = −τ γ ω γ ωD

HydDD

D DB

B. The coefficients γDD and γD

B represent thestrength of the drag torque on the doublet and the hydrodynamic bead-doublet coupling respectively and are determined from the simulations.Analytical models based on these results compare favorably withmeasurements of the average doublet rotation rate ωD for beads withdifferent strengths of superparamagnetic (mH) and ferromagnetic (mP)moments. This enables the permanent moment (mP) of the beads to be

https://doi.org/10.1016/j.jmmm.2018.07.014Received 23 February 2018; Received in revised form 3 May 2018; Accepted 6 July 2018

⁎ Corresponding author.E-mail address: sooryakumar.1@osu.edu (R. Sooryakumar).

Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

Available online 10 July 20180304-8853/ © 2018 Elsevier B.V. All rights reserved.

T

non-invasively determined from the models by measuring ωD — anadvantage over conventional techniques for measuring mP that requireeither non-symmetric particles [45] or spherical beads attached withsmall particles [31,46], fluorophores [47] or coatings [44,48]. Fur-thermore, these findings provide a framework to leverage both themagnetic and hydrodynamic torques on rotating doublets. For example,our analysis reveals that the hydrodynamic bead-doublet coupling canincrease the maximum torque and rotation rate of a doublet which isbeneficial for many lab-on-a-chip applications.

2. Materials and methods

A 3-axis electromagnet provides time-varying magnetic fields for ro-tating the particles. The platform (Fig. 1) includes four orthogonal elec-tromagnets (Magnetec Corp OP-1212) and a custom-made solenoid thatdynamically creates magnetic fields in the xy- and z-directions respec-tively. The fields, which are rotated in the xy-plane at constant frequencies(∼0.1–10Hz), torque the magnetic beads as they are viewed under anoptical microscope (Leica DM2500 MH) with a 40x objective. The beaddynamics are imaged with a high speed camera (Phantom Miro M120) at100 frames per second. The orientation of the doublet in the videos istracked by extracting the doublet from the image through thresholdingand then measuring the orientation of its longest axis.

Invitrogen’s Dynabeads M-270 Carboxylic Acid (2.8 μm diameter)[41] and Bang’s Lab COMPEL Fluorescent Magnetic Carboxylic Acidbeads (2.8 μm diameter) are utilized because of their comparable geo-metries and surface chemistries while displaying definitive differencesin magnetic moments. Larger COMPEL beads (7.9 μm diameter) wereseparately used to visualize individual bead orientation during rotation.

The beads are suspended in a 0.06mm deep fluid well made fromdouble-sided tape sandwiched between a glass slide (Fisher) and acover slip (Thermo-Scientific). The tape binds the device together andprevents leakage. The glass slide is coated with polyethylene glycol(PEG) or casein to reduce non-specific binding of beads to the substrate.The superparamagnetic beads are also diluted in 0.1% Triton solution(Sigma–Aldrich) to decrease the amount of non-specific binding.

The average doublet rotation rate ωD is measured with a givendoublet for the entire field frequency range. Once the field rotations are

initiated and transient bead movement subsides, videos are recorded for60–90 s to include multiple (>6) rotations of the doublet. Doublets wereconsidered permanently attached if an applied z-field could not sepa-rate the beads. The attachment was caused by non-specific bindingbetween the beads’ surfaces. The doublets do not bind to the surfaceand stay oriented in the xy-plane for the entire frequency range.Between measurements, the doublets were magnetically rolled to a newlocation [49] to inhibit non-specific binding to the surface, and a z-fieldis reapplied to confirm the beads remain attached or detached.

To assess the bead-doublet hydrodynamic coupling, steady-stateNavier-Stokes equations were numerically solved. This simulation includesinteractions between the beads as well as near wall effects from the sub-strate. Translational and rotational Brownian fluctuations were neglectedin the simulations because they did not significantly affect the doubletrotation at these bead sizes. The two beads were assumed to have a100 nm gap between their closest points and a 50 nm gap between thebeads and the flat surface [50,51]. The simulation geometry enclosed thebeads in a large cylinder with a diameter ∼ ×40 the bead diameter(Fig. 2a) in order to minimize the wall-induced anisotropic effects on thesteady state solution, and it included a “refinement box” around the beadsthat was more finely discretized to increase the numerical accuracy of thesimulation in that region. A no-slip velocity boundary (Dirichlet condition)and a vanishing pressure gradient (Neumann condition) were imposed onall surfaces including the walls of the cylinder.

A finite volume-based iterative steady-state solver [52] was writtenusing the Semi-Implicit Method for Pressure-Linked Equations [53](SIMPLE) algorithm. The finite volume method is a discretization schemethat is commonly used to solve the incompressible Navier-Stokes equations:

⎜ ⎟⎛⎝

∂→

∂+ → ∇

→→⎞⎠

= −∇→

+ ∇ →ρ vt

v v p μ v· ,2

where μ is the kinematic viscosity, ρ the density, p the pressure, and v thevelocity of the fluid. Together with the incompressible continuity equation

Fig. 1. Diagram of the electromagnet setup. The magnetic particles are placedin the center and are surrounded by four orthogonal electromagnets that applya time-varying magnetic field in the xy-plane and a solenoid (dashed, circularline) that can provide a field along the z-axis. The white arrows depict thedirection of the instantaneous field generated from each electromagnet; theblack arrows show the direction of the magnetic field H encountered by the

Fig. 2. (a) Schematic of the fluid domain and the bead geometry simulated.Navier–Stokes equations were solved at higher resolution in the refinement box nearthe beads. (b) Fluid flow generated by two beads spinning at ωB =5rad s−1

( =ω 0D ) is visualized by the blue trace of several tracer particles, initiated at randomlocations near the beads. Narrow white arrows mark the tracer particles’ direction ofmotion. Dashed arrows show the direction of rotation of the beads, while wide ar-rows indicate the hydrodynamic torque on the doublet due to its coupling to ωB.

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

324

∇→ → =v· 0, the Navier-Stokes equations can be iteratively solved for a dis-cretized fluid volume using the SIMPLE algorithm. Our model did notmake any assumptions about the linearity of the low Reynolds numbersystem. Instead, the full Navier-Stokes equations were solved. These so-lutions are used to determine the net force on a bead by numerically in-tegrating the normal pressure and the shear stress contributions. The re-sulting net forces (Fig. 2b) on the two beads produce the hydrodynamictorque on the doublet and the beads about their centers of mass.

3. Theory

The magnetic moment for our beads is composed of a super-paramagnetic Néel moment mH oriented at θH (along the direction ofexternal field H) and a permanent Brownian moment mP oriented at θP

(Fig. 3a). The net bead magnetic moment ⎯→⎯ = ⎯→⎯ + ⎯→⎯m m mH P is oriented atθM . Both beads are assumed to have the same magnetic moments mH

and mP. When two beads lie adjacent to one another, they form adoublet with a long axis oriented at θD. For detached doublets (Fig. 3b),the individual beads rotate at angular frequency ωB which may differfrom the doublet frequency ωD; for attached doublets the beads mustrotate synchronously such that =ω ωB D.

The next sections summarize the three torques encountered — thedipole-field interaction, the dipole-dipole interaction, and the bead-doublet hydrodynamic coupling — and the doublet’s dynamics whenthese torques combine to rotate detached and attached doublets.

3.1. Dipole-field and dipole-dipole torques

In an external field H m, p experiences the dipole-field torque τBM

given by [45]:

= ⎯→⎯ ×⎯→⎯

= −∘ ∘τ m μ H μ m H θ θsin( ),BM

P H P (1)

where ∘μ is the permeability of free space. For detached doublets, τBM

acts on the individual beads and not on the doublet, but for attacheddoublets, the torque acts on the doublet.

The dipole-dipole torque τDM for two beads each with net magnetic

moments ⎯→⎯ =⎯ →⎯⎯⎯ + ⎯ →⎯⎯m m m( )H P separated approximately by a distance r2(twice the bead radius) is given by [43]:

= −∘τμ m

π rθ θ

34 (2 )

sin(2[ ]),DM

H D

2

3 (2)

which causes a doublet, attached or detached, to rotate around itscenter.

3.2. Bead-doublet hydrodynamic torque

For detached doublets, while the dipole-field torque (Eq. (1)) acts onthe beads and not directly on the doublet, the rotation of the individualbeads creates an additional hydrodynamic torque on the doublet. Thishydrodynamic bead-doublet coupling is the result of a rotating particleexperiencing more drag on the side closer to the adjacent bead than itsother sides, essentially causing the particle to “roll” around the adjacentbead and the doublet to rotate.

In low-Reynolds number dynamics, the forces and torques on a bodyare linearly related to its velocity and angular velocity respectively bydrag coefficients that depend on the physical geometry [44,54]. As aresult, a linear relationship between the hydrodynamic torques τB

Hyd andτD

Hyd and ωB and ωD is expected and confirmed by the simulations(Fig. 4). Thus, for a doublet

= −τ γ ω γ ω ,DHyd

DD

D DB

B

where γDD =0.34 pN μm s rad−1 and = ×γ 6.60 10D

B −2 pN μm s rad−1 arethe doublet drag coefficients from the doublet (D) and bead (B) rotationrespectively. γ ωD

DD is the hydrodynamic torque resisting the rotation of the

doublet; γDB ωB represents an additional hydrodynamic torque on the

doublet arising from the hydrodynamic bead-doublet coupling. Similarlyfor the individual rotating beads within the doublet (Fig. 4b),

= −τ γ ω γ ω ,BHyd

BB

B BD

D

where = ×γ 9.37 10BB −2 pN μm s rad−1 and = ×γ 3.32 10B

D −2 pN μm srad−1 are the corresponding drag coefficients for the bead. Our simula-tions included interactions between the beads as well as near wall effectswith the substrate which are reflected in these values for the drag coef-ficients. Changes in the simulated bead-substrate distance from 50 nm to150 nm yield negligible (<3%) changes in all the drag coefficients exceptγD

D which would decrease by 16%.The drag coefficients resulting from our simulations align with ex-

pectations based on analytical results for beads well separated from thesurface. For two isolated and decoupled beads (radius r = 1.4 μm) inwater (dynamic viscosity = × −η 8.9 10 4 Pa s) where each bead is trans-lating around a fixed Eulerian point at the bead’s edge, γD

D may be ana-lytically approximated by Stokes’ laws as = ≈γ πηr12D

D 3 0.09 pN μm srad−1. Because of the additional drag on the beads from being near thesurface and each other, the actual value of γD

D for non-isolated doublets isexpected to be larger but within the same order of magnitude which isconfirmed by the results of the simulations which yielded γD

D =0.34pN μm s rad−1. Similarly, our simulated value for γB

B ( × −9.37 10 2 pN μm srad−1) is greater than but within an order of magnitude of the analyticalvalue via Stokes’ Law for a rotating isolated bead( = ≈ × −γ πηr8 6.14 10B

B 3 2 pN μm s rad−1) due to the near wall effects.While γD

B (= 6.60×10−2 pN μm s rad−1) is difficult to analytically ap-proximate, the effect of the flow field from the individual bead on therotation of the doublet is expected to be smaller than the drag on the largerdoublet (γD

D = 0.34 pN μm s rad−1) as it moves through the fluid.

3.3. Detached doublets

For detached doublets, the dipole-dipole interaction torques thedoublet into the direction of the field. At low Reynold’s number, the nettorque equals zero, and thus, the magnetic torque is given by

= − = −∘τμ m

π rθ θ γ ω γ ω

34 (2 )

sin(2[ ]) ,DM

H D DD

D DB

B

2

3 (3a)

which yields the rotation rate

= − − +ω c α θ θ c ω(1 ) sin(2[ ]) .D D D H D D B (3b)

where =−

∘αDμ m

V γ γ8 ( )DD

DB

2and the constant = ≈c γ γ/ 0.20D D

BDD is a measure

of the relative strength of the hydrodynamic bead-doublet coupling. Atlow frequencies up to the doublet’s critical frequency ∗ωD, the doublet

Fig. 3. (a) The doublet angle θD is oriented along the direction connecting thecenters of the beads. Induced magnetic moment mH is in direction of the ex-ternal field which is oriented at θH . The permanent magnetic moment mP isfixed along some direction oriented at θP. The net magnetization m is orientedat θM which is between θH and θP. (b) For detached doublets, the beads rotateindependently from the doublet. The doublet rotates at frequency ωD aroundthe center of the two beads. Whereas the individual beads rotate around theircenter at frequency ωB. For attached doublets, the beads and doublet rotate atthe same frequency =ω ωB D.

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

325

will rotate synchronously with the field ( =ω ωD H). At frequenciesabove ∗ωD, the doublet will rotate slower than H because drag overcomesthe magnetic torque. Likewise, the beads will rotate synchronously withωH up to the bead’s critical frequency ∗ωB and at higher frequencies willalso desynchronize with ωH . ∗ωD is dependent on the strength of thedipole-dipole torque and the hydrodynamic bead-doublet coupling andreaches its maximum value αD when there is maximal torque from thebeads’ rotation (i.e. =ω ωB H) at ∗ωD which occurs when ⩾∗ ∗ω ωB D. Whenthe bead-doublet coupling is strong, the maximum average doublet

rotation rate ωDMAX may be larger than αD even though the doublet is not

rotating synchronously with H.Similarly, the magnetic torque on the individual beads is given by

= − = −∘τ μ m H θ θ γ ω γ ωsin( ) ,BM

P H P BB

B BD

D (4a)

and the bead rotation rate

= − − +ω c α θ θ c ω(1 ) sin( )B B B H P B D (4b)

where =−

∘αBμ m Hγ γ

P

BB

BD and = ≈c γ γ/ 0.35B B

DBB . ∗ωB is dependent on the dipole-

Fig. 4. (a) Simulations measured the hydrodynamic torque on the doublet τDHyd while ωB or ωD varied and the other rate is fixed at 5 rad s−1. These results apply to

both attached and detached doublets. The fixed frequency rate was chosen to allow data points from both the >ω ωD B and <ω ωD B cases. The hydrodynamic torqueon the doublet was found to be linearly proportional to the rotation rate of the beads and the doublet and to obey = −τ γ ω γ ωD

HydDD

D DB

B where = × −γ 6.60 10DB 2 pN μm s

rad−1 and γDD = 0.34 pN μm s rad−1. (b) Hydrodynamic torque on the beads within a doublet is linear = −τ γ ω γ ωB

HydBB

B BD

D with = × −γ 9.37 10BB 2 pN μm s rad−1 and

= × −γ 3.32 10BD 2 pN μm s rad−1.

Fig. 5. (a) Frequency-dependent average doublet frequency ωD falls into three regions for detached doublets when >∗ ∗ω ωB D. The corresponding experimental data isseen in Fig. 8. (b) Average bead (dashed line) and doublet (solid line) rotation rate when >∗ ∗ω ωB D. (c) Frequency-dependent average doublet frequency ωD fordetached doublets when <∗ ∗ω ωB D. The corresponding experimental data is seen in Fig. 9. The same critical frequency ∗ωD may be reached at different values for ∗ωB and

∗ωD but with slight differences in the frequency response. (d) Average bead (dashed line) and doublet (solid line) rotation rate when <∗ ∗ω ωB D.

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

326

field torque and the hydrodynamic bead-doublet coupling from therotating doublet and reaches its maximum frequency αB when the hy-drodynamic bead-doublet coupling is largest ( =ω ωD H) at ∗ωB whichoccurs when ⩽∗ ∗ω ωB D.

Because of the interdependence of the bead and doublet dynamics,ωD and ωB fall into two cases depending on their relative critical fre-quency values ( ∗ωB and ∗ωD) or equivalently on whether the dimension-

less parameter = −

−αα

mm

HV γ γ

γ γ

8 ( )

( )BD

P DD

DB

BB

BD2 is greater than or less than one. When

≈ = ∼m m χVH,Hαα

mχ HV

BD

P2 .

3.3.1. Case I ( >∗ ∗ω ωB D):As illustrated in Figs. 5a and b, Eqs. (3) and (4) lead to

In this case, = + −∗ ∗ ∗α ω k ω ω( ) ( )B B B B D2 2 where = −

−kBc c

c(1 )

(1 )B D

Band

= ∗α ωD D. The values for m and mP for the beads can be determined fromthe two critical frequencies ∗ωD and ∗ωB. The doublet and individual beadrotations fall into three distinct regions: (1) < ∗ω ωH D, (2)

< <∗ ∗ω ω ωD H B, (3) and > ∗ω ωH B.When < ∗ω ωH D, both the individual beads and the doublet rotate at

ωH . The doublet does not align with H but trails it by a fixed angle thatincreases with ωH until the dipole-dipole torque reaches its maximum at

= ∗ω ωH D.When < <∗ ∗ω ω ωD H B, the individual beads continue to rotate at ωH .

Whereas, ωD slows when ωH just exceeds ∗ωD. The doublet cannot rotatefast enough to keep up with H so it increasingly trails further behind it.When H is too far ahead of the doublet, the doublet will slow down androtate backwards to realign with it. This process repeats creating aperiodic oscillatory motion (Supplementary Video 2). While the dipole-dipole torque τD

M decreases with ωH on average, the hydrodynamicbead-doublet coupling continues to increase linearly because the beadsare still rotating at ωH as evident in Fig. 5b. Once the hydrodynamicbead-doublet coupling dominates, ωD increases linearly with ωH suchthat ≈ ≈ω c ω ω0.2D D H H . If αB is sufficiently large, the hydrodynamicbead-doublet coupling can rotate the doublet faster than the dipole-dipole interaction τD

M alone.When > ∗ω ωH B, the beads can no longer rotate at ωH decreasing the

hydrodynamic bead-doublet coupling and causing ωB and ωD to de-crease (Fig. 5a) as both the beads and doublet start to exhibit periodicoscillatory motion.

3.3.2. Case II ( <∗ ∗ω ωB D):As illustrated in Figs. 5c and d, from Eqs. (3) and (4)

where = + −∗ ∗ ∗α ω k ω ω( ) ( )D D D D B2 2 for = −

−kDc c

c(1 )

1D B

Dand = ∗α ωB B.

When < ∗ω ωH D, the doublet rotates at ωH . At ∗ωD, the individualbeads are already rotating slower than ωH and contribute less hydro-dynamic torque. As such, the critical frequency ∗ωD is up to 14% belowαD. The lower limit of the critical frequency ( ≈∗ω α0.86D D) occurs ifthere is no dipole-field torque (i.e. =∗ω 0B ) and the upper limit( =∗ω αD D) occurs if the beads rotate at ωH when ∗ωD is reached (i.e.

=∗ ∗ω ωB D).When > ∗ω ωH D, both ωB and ωD decrease (Fig. 5d) as both the beads

and doublet exhibit periodic oscillatory motion.

3.4. Attached doublet

Attached doublets are similar to the detached case except the beadsare rigidly attached =ω ω( )B D . Hence, they are affected by both themagnetic torques τD

M and τBM and the hydrodynamic torques τD

Hyd andτB

Hyd [43] yielding the hydrodynamic drag coefficient= − + −γ γ γ γ γ2( )AD D

DDB

BB

BD . The doublet rotation rate becomes

= − − + − −ωγ

γ γ α θ θ γ γ α θ θ1 [( ) sin(2[ ]) 2( ) sin( )].DAD

DD

DB

D H D BB

BD

B H P(7)

showing that ωD not only depends on αB and αD, but also on the angle ϕ(= −θ θD P) between the permanent moment and the doublet axis.

3.5. Maximum rotation rate

Many lab-on-a-chip applications would benefit from an increasedmaximum torque or rotation rate. The maximum average doublet ro-tation rate ωD

MAX in turn is dependent on the ratio of the critical fre-quencies and whether the beads are attached or detached (Fig. 6).

For a detached doublet, based on Eqs. (5) and (6),

=

⎧

⎨

⎪

⎩⎪

<= < ⪅

− − ×

− ⪆

∗ ∗ ∗

∗ ∗ ∗ ∗

∗

∗ ∗ ∗ ∗

ω

ω ω ωω α ω ω ωω c

ω ω ω ω

whenwhen 4.63

(1 )

( ) ( ) when 4.63

DMAX

D B D

D D D B D

B D

B D B D2 2

where ∗ω4.63 D approximates + −− −

∗ωcc D

1 (1 )1 (1 )

DD

22 .

For an attached doublet, the maximum rotation rate is also depen-

=⎧

⎨⎪

⎩⎪

<

− − − < <− − + − − − − − >

= ⎧⎨⎩

<− − + − − − − − >

∗

∗ ∗

∗

∗

∗

ω

ω ω ω

ω c ω α ω ω ω

ω c ω c ω c ω c α c α ω ω

ωω ω ωω c ω c ω c ω c α c α ω ω

if

(1 ) if

({(1 ) [( ) (1 ) ] } [(1 ) ] ) ifand

if({(1 ) [( ) (1 ) ] } [(1 ) ] ) if

.

D

H H D

H D H D D H B

H D H D H B D B B D D H B

BH H B

H B H B H D B D D B B H B

2 2

2 2 2 1/2 2 2 1/2

2 2 2 1/2 2 2 1/2 (5)

= ⎧⎨⎩

<− − + − − − − − >

=⎧

⎨⎪

⎩⎪

<

− − − < <− − + − − − − − >

∗

∗

∗

∗ ∗

∗

ωω ω ωω c ω c ω c ω c α c α ω ω

ω

ω ω ω

ω c ω α ω ω ω

ω c ω c ω c ω c α c α ω ω

if({(1 ) [( ) (1 ) ] } [(1 ) ] ) if

andif

(1 ) if

({(1 ) [( ) (1 ) ] } [(1 ) ] ) if

,

DH H D

H D H D H B D B B D D H D

B

H H B

H B H B B H D

H B H B H D B D D B B H D

2 2 2 1/2 2 2 1/2

2 2

2 2 2 1/2 2 2 1/2 (6)

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

327

dent on the fixed angle = −ϕ θ θD P (Eq. (7)). Both magnetic and hy-drodynamic torques are maximized when =ϕ π/4 which yields

=− + −

− + −ω

γ γ α γ γ αγ γ γ γ

( ) 2( )2 2

.DMAX D

DDB

D BB

BD

B

DD

DB

BB

BD

If the beads attach while in an external magnetic field, =ϕ 0.Whether doublets move faster while attached or detached depends

on whether drag or magnetic torque increases more with attachment.When doublets become attached, the magnetic torque increases fromthe addition of the dipole-field torque τB

M on the doublet, but the dragalso increases from the addition of the hydrodynamic torque τB

Hyd fromthe beads. The faster state is generally dependent on the ratio of ∗ωB to

∗ωD (Fig. 6), except when ≈∗ ∗ω ωB D, then it is highly dependent on ϕ.Otherwise, when >∗ ∗ω ωB D, attached doublets have a faster maximumrotation rate than their detached counterparts; for <∗ ∗ω ωB D, the doub-lets can rotate faster when detached.

4. Experimental results and discussion

The following sections will compare the measured ωD for detachedDynabead and COMPEL doublets with the models predicted above. It ispreceded with a discussion justifying that the two types of beads in-vestigated have a permanent moment mP, and is concluded with acomparison of ωD

MAX for attached and detached doublets.

4.1. Identification of permanent moment mP

If the magnetic moment of a bead does not immediately align withH, it may be the result of a permanent moment [13,16,29–31,35] or thepresence an easy axis [46]. An easy axis is caused by the magneticanisotropy resulting from either non-uniformly distributed magneticnanoparticles or nanoparticles that exhibit shape anisotropy, and ischaracterized by a magnetization that is linearly dependent on H andcan flip orientation along its axis [33,46].

A permanent moment results from large nanoparticles with a Néelrelaxation time that is long compared with π ω2 / H and has a fixedmagnitude. The orientation of the permanent moment is fixed[33,44,45] at low fields but can be re-magnetized [47] at high fieldswhere H can overcome the coercive energy and flip the magnetization’sdirection. Single bead studies [33,47] have demonstrated that 2.8 μmDynabeads have a permanent moment and not an easy axis.

In order to identify the type of magnetization for our beads, at-tached doublets were studied, because both the dipole-field and dipole-dipole torque would directly affect the doublet rotation. When ωH is

above the critical frequency, ωD will start to slow down and exhibitoscillatory motion. The doublet will oscillate at a frequency that isdependent on the type of magnetization that underlies its rotation[33,46,47]. For a permanent moment mP fixed along a specific direc-tion, the period of the oscillation is the time required for the field tomove completely around the doublet yielding the oscillation frequency

= −ω ωΩ H D1 . Whereas for the dipole-dipole interaction τDM (Eq. (2)), an

easy axis, or a permanent moment at high H that can flip directions, theoscillation period is the time required for the field to go halfway aroundthe doublet yielding the oscillation frequency = −ω ωΩ 2( )H D2 .

A doublet’s oscillation frequency can be directly determined fromthe power spectrum of θ t( )D (Fig. 7). Experimentally, the power spec-trum of the COMPEL beads display much stronger frequency peaks atΩ1 than Ω2 (Fig. 7a) whereas the Dynabeads have pronounced peaks atboth Ω1 and Ω2 (Fig. 7b). The presence of Ω1 in both COMPEL andDynabeads confirms a fixed direction, permanent moment in bothcases. Whereas, the relative weakness in Ω2 for COMPEL beads com-pared with Dynabeads, suggests that the dipole-dipole interaction andconsequently m is weaker for COMPEL beads than Dynabeads at =H 20Oe.

4.2. Detached COMPEL beads

As shown in Fig. 8a, the typical frequency response of a detachedCOMPEL doublet agrees with Eq. (5) allowing COMPEL beads to beused to explore bead dynamics when >∗ ∗ω ωB D.

The time dependence of θD is also measured (Fig. 8b). As expected,the doublet rotates at ωH when < ∗ω ωH D, and undergoes periodic os-cillatory motion when < <∗ ∗ω ω ωD H B. These oscillations were found todecrease in magnitude at higher field frequencies when the bead-doublet coupling provides greater torque than the dipole-dipole inter-action. In the case where > ∗ω ωH B, the doublet was often found to rotateconsistently at a frequency below ωH but not oscillate. The bead-doublecoupling does not appear to cause any periodic oscillations in θD eventhough the individual beads are expected to oscillate.

When > ∗ω ωH B, we expect the decrease in ωD is due to ωB slowing.However, a direct measure of ωB could not be achieved because the

Fig. 6. The maximum rotation rate ωDMAX in multiples of αD at various values of

∗ ∗ω ω/B D. For detached doublets, ωDMAX increases to =ω αD

MAXD where it stays

constant until ⪆∗ ∗ω ω4.63B D when the hydrodynamic bead-doublet coupling islarge enough to increase ωD

MAX (Fig. 5a). For attached doublets, ωDMAX increases

monotonically and is greatest when = − =ϕ θ θ π/4D P .

Fig. 7. Power spectrum of linear fit removed doublet angle of attached (a)COMPEL beads and (b) Dynabeads at 5 Hz with H=20 Oe. The dashed linesrepresent Ω1 and Ω2. Both beads have peaks at Ω1 indicating the presence of apermanent moment.

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

328

2.8 μm beads are spherically symmetric with any defects too small to beoptically resolved. On the other hand, 7.9 μm COMPEL beads exhibit asimilar ωD as the 2.8 μm beads and their orientation can be observeddue to the presence of minor surface defects (Supplementary Videos 1and 2). Between 1.4 and 1.8 Hz, ω π/2H increases past ∗ω π/2B , i.e. theindividual 7.9 μm beads slow down and rotate asynchronously, whichcoincides with ω π/2D starting to decrease from 0.30 Hz to 0.19 Hz. Thisresponse for the 7.9 μm beads agrees with our model that the decreasein ωD is due to ωB slowing when > ∗ω ωH B.

The model (Eq. (5)) also quantitatively agrees with experimentalresults on the relative strength of the hydrodynamic bead-doubletcoupling to the hydrodynamic drag as measured by =c γ γ/D D

BDD (Eqs.

(3b) and (5)). In the region < <∗ ∗ω ω ωD H B, a near-linear response≈ω c ωD D H is observed at high frequencies (Fig. 8) when the hydro-

dynamic bead-doublet coupling is the dominant doublet torque. Fromthe response of ten doublets, it is experimentally found that

= ±c 0.20 0.04D which is in good agreement with the simulation-de-termined value of ≈c 0.20D (as found in Section 3.3). Because γD

D issensitive to the height above the substrate (decreasing 16% when thegap changes from 50 nm to 150 nm), the agreement in cD validates ourchoice of a 50 nm bead-substrate gap in the simulations.

The model described by Eq. (5) includes assumptions that influencethe fit to the data around the critical frequencies. For instance bothbeads are taken to have the same critical frequency ∗ωB that leads to a

sharp peak in the model at ∗ωB. When ∗ωB for the beads is different, theresponse of ωD at ∗ωB becomes smoother as seen in the measured data inFig. 8. In addition, the current model ignores effects from thermallydriven Brownian fluctuations [47] which causes the fits to over-esti-mate ωD around the critical frequencies. For example, even if < ∗ω ωH Dthe addition of Brownian torque can cause the doublet to oscillate and,on average, slow down if ωH is sufficiently close to the critical fre-quency. Consequently, not including Brownian motion increases thepredicted peak in ωD at the critical frequencies. Based on measurementsof seven COMPEL doublets, the values of ωD at the critical frequenciesbased on the fits to Eq. (5), which do not include Brownian fluctuations,are ±31 7% and ±26 7% larger than the highest measured values of ωD

at ∗ωD and ∗ωB respectively. Numerical simulations reported by Reenenet al. [47] for the simpler case of individually rotating 2.8 μm Dyna-beads found a similar result where the exclusion of Brownian motioncaused an increase of ±16 5% in the critical frequency.

From the critical frequencies measured using the doublet’s fre-quency response, both m and mP can be determined for individualbeads. Compared with single bead techniques that require coatings orfluorophores to identify the bead’s orientation, the present approach isa non-invasive means for determining mP. Based on a sampling of 13doublets of 2.8 μm COMPEL beads at =H 20 Oe, the critical frequenciesare = ±∗ω π/2 0.40 0.02D Hz and = ±∗ω π/2 3.60 0.16B Hz. As such, themaximum critical frequencies are = ±α π/2 0.40 0.02D Hz and

= ±α π/2 5.17 0.17B Hz where =−

∘αBμ m Hγ γ

P

BB

BD and =

−∘αD

μ mV γ γ8 ( )D

DDB

2. The re-

sulting magnetic moments are = ± × −m (1.05 0.04) 10P15 Am2 and

= ± × −m (7.23 0.18) 10 15 Am2. The magnetic susceptibility≈ =χ 0.37m

VH if ⎯→⎯ ≈ ⎯ →⎯⎯⎯m mH , which is justified because mH is− × m(5.9 6.8) P depending on the angle between mH and mP.

4.3. Detached dynabeads

A typical frequency response (shown in Fig. 9a) of ωD for detachedDynabeads at H = 20 Oe agrees with a fit based on Eq. (6), therebyallowing Dynabeads to be used to explore bead dynamics when

<∗ ∗ω ωB D.As depicted in Fig. 9b, the doublet rotates at ωH when < ∗ω ωH D, and

undergoes periodic oscillatory motion when > ∗ω ωH D. Unlike with theCOMPEL beads, these oscillations continue for all the measured highfrequencies, since the dipole-dipole torque, which causes the oscilla-tions, remains the dominant torque.

Detached Dynabeads partially coated with a non-magnetic filmwere previously studied by Coughlan et al. [44] who found that thehydrodynamic bead-doublet coupling affected ωD, but did not calculatethe bead magnetic moment from ωD. Based on six doublets at H = 20Oe, the measured maximum critical frequencies are = ±α π/2 3.15 0.23D

Hz and = ±α π/2 2.21 1.79B Hz. The resulting magnetic moments are= ± × −m (4.49 3.67) 10P

16 Am2 and = ± × −m (2.03 0.07) 10 14 Am2. Thelower accuracy for mP is likely a result of the permanent moment beingrelatively weak for Dynabeads and not having a significant effect ondoublet rotation. The magnetic susceptibility can be approximated inthis case as ≈ =χ 1.11m

VH because mH is roughly 45× greater than mP.As expected, the Dynabeads display a smaller value of mP and a larger mthan COMPEL beads. In comparison, Coughlan et al. [44] found com-parable values for the magnetic moments of the same type of Dyna-beads with =χ 1.33 at H = 10 Oe and = × −m 3.0 10P

16 Am2 through abulk particle property measurement system and single bead analysisrespectively. The magnetization among the beads vary considerably[47], which is the likely cause for the variations in the measuredmagnetic moments.

4.4. Maximum rotation frequency

ωDMAX for attached and detached COMPEL and Dynabeads are

summarized in Fig. 10. ωDMAX is not directly dependent on the ratio of

Fig. 8. (a) Frequency-dependent ω π/2D for a representative 2.8 μm COMPELdoublet at H= 20 Oe and fits to Eq. (5). (b) The doublet orientation θD over timefor the three different regimes. The measured orientation (blue points) is com-pared with a guide line rotating at frequency ωH (red, dashed line). When

< ∗ω ωH D, the doublet rotates at ωH . When < <∗ ∗ω ω ωD H B, the doublet undergoesperiodic oscillatory motion where it alternates between periods of rotating andstalling. To illustrate the periodicity of the oscillations, each period is containedwithin separate guide lines rotating at ωH but separated by π/2. Each period thuslasts until = −ϕ θ θH D increases by π/2. These oscillations decrease in magnitudeat higher field frequencies as the bead-doublet coupling provides more torquethan the dipole–dipole interaction. When > ∗ω ωH B, the doublet will often rotateconsistently at a frequency below ωH but not oscillate. In this case, the bead-doublet coupling does not appear to cause any periodic oscillations in θD.

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329

the critical frequencies. In our case, the maximum frequency forCOMPEL and Dynabead doublets are comparable, which is possiblebecause they are each characterized by different magnetic moments: mis 181% larger for Dynabeads and mP is 134% larger for COMPEL beads.However, the effect of attachment on ωD

MAX is dependent on the relativestrength of ∗ωB to ∗ωD as predicted in Fig. 6.

When >∗ ∗ω ωB D, attached doublets generally rotate faster than de-tached doublets, which is also observed with COMPEL beads. The at-tached doublets rotate over 4× faster ( ±2.73 0.15 Hz) than detacheddoublets ( ±0.62 0.04 Hz). The hydrodynamic bead-doublet couplingallows detached doublets to rotate faster than when only under amagnetic dipole-dipole torque τD

M , but it is a smaller effect on ωDMAX

than if the beads are attached. The hydrodynamic bead-doublet cou-pling allows ωD

MAX to be 50% greater than αD ( ±0.62 0.04 Hz versus±0.40 0.02 Hz) but occurs when ωH is 9× greater ( ±3.6 0.16 Hz versus±0.40 0.02 Hz).When <∗ ∗ω ωB D, detached doublets generally rotate faster than their

attached counterparts as seen (Fig. 10) with Dynabeads where detacheddoublets rotated 20% faster ( ±2.66 0.08 Hz) than attached doublets( ±2.21 0.11 Hz).

5. Conclusion

This study has addressed the role of hydrodynamic and magnetictorques on the rotational dynamics of a pair of magnetic beads.Simulations confirm a linear dependence of the hydrodynamic torqueon ωD and ωB, i.e. = −τ γ ω γ ωD

HydDD

D DB

B and = −τ γ ω γ ωBHyd

BB

B BD

D which wasexperimentally confirmed by the doublet response (Figs. 8 and 9).Models for the cases where >∗ ∗ω ωB D and <∗ ∗ω ωB D agreed with ωD

measured for COMPEL and Dynabeads respectively which allowed the

beads to be used as examples of the two regimes. Moreover, our si-mulation results were validated when the value for = ≈c γ γ/D D

BDD 0.20,

which reflects the drag-reduced bead-doublet coupling, matched theexperimentally determined = ±c 0.20 0.04D from COMPEL beads.

For detached doublets, χ and mP were determined from the fre-quency-dependent response of ωD. This represents a non-invasivemethod of determining the permanent bead moment mP of individualbeads which is central to the dynamics of single and chain rotations. Incontrast to the more precise value of = ± × −m (1.05 0.04) 10P

15 Am2 forCOMPEL beads, our value of = ± × −m (4.49 3.67) 10P

16 Am2 forDynabeads had more uncertainty due to the relative weakness of mP

and its modest effect on ωD.It is also noted that while doublets have been assumed to rotate

faster when detached [44], this is true only when <∗ ∗ω ωB D (Figs. 6 and10). When >∗ ∗ω ω ω,B D D

MAX is greater for attached doublets.In general, micromagnetic actuation is hindered by the effects of

viscosity which places a limit on the maximum rotation rate.Attachment is one way to overcome that barrier by exchanging torquebetween a body and the its constituent parts. For the doublets, when theDynabeads were detached or the COMPEL beads attached, ωD

MAX wasable to increase by decreasing the rotation rate of the individual beads.Likewise, this effect could be extended to more complex many-bodysystems.

The findings presented in this study have broad applications. Forinstance, simple, non-invasive bead calibration can be performed bymeasuring the frequency-dependent ωD to determine the relativestrengths of permanent and induced moments. For drug delivery pro-tocols, the calibrated, bio-compatible beads may be administered alongwith drugs. Because of the permanent moment, individual beads wouldbe able to roll independently through small pores to reach the targetsite before chaining into larger structures for on-site micro-mixing [26].The mixing rate would be increased for beads with a large permanentmoment and could potentially lead to increased drug effectiveness. Theresults thus provide a framework to engineer emergent behaviors ofmagnetic microbeads in complex and crowded environments.

Acknowledgment

The authors gratefully acknowledge the financial support providedby the U.S. Army Research Office under contract W911NF-14-0289.

Fig. 9. Frequency-dependent ω π/2D for a representative 2.8 μm Dynabeaddoublet at H=20 Oe and fits to Eq. (6). (b) The doublet orientation θD overtime for the two different regimes. The measured orientation (blue points) iscompared with a guide line rotating a ωH (dashed, red line). When < ∗ω ωH D, thedoublet rotates at ωH . When > ∗ω ωH D, the doublet undergoes periodic oscilla-tory motion where it alternates between periods of rotating and stalling. Eachperiod is contained within guide lines separated by π/2 indicating that eachperiod lasts until = −ϕ θ θH D increases by π/2.

Fig. 10. Maximum doublet frequency for attached and detached Dynabeadsand COMPEL beads at 20 Oe, based on 8 attached and 13 detached COMPELdoublets and 6 attached and detached Dynabead doublets.

C. Pease et al. Journal of Magnetism and Magnetic Materials 466 (2018) 323–332

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Appendix A. Supplementary data

Supplementary data associated with this article can be found, in theonline version, at https://doi.org/10.1016/j.jmmm.2018.07.014.

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