Brownian Dynamics Simulations of Magnetic Nanoparticles Captured

Article pubs.acs.org/JPCC

Brownian Dynamics Simulations of Magnetic Nanoparticles Captured in Strong Magnetic Field Gradients Zhiyuan Zhao,† Isaac Torres-Díaz,†,‡ Camilo Vélez,§ David Arnold,§ and Carlos Rinaldi*,†,‡ †

Department of Chemical Engineering, ‡J. Crayton Pruitt Family Department of Biomedical Engineering, and §Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, United States S Supporting Information *

ABSTRACT: The behavior of spherical single-domain magnetic nanoparticles in strong inhomogeneous magnetic fields is investigated through Brownian dynamics simulations, taking into account magnetic dipole−dipole interactions, repulsive hard-core Yukawa potential, hydrodynamic particle-wall interactions, and the mechanism of magnetic dipole rotation in the presence of a magnetic field. The magnetic capture process of nanoparticles in prototypical magnetic field gradients generated by a sudden reversal in perpendicular magnetization of a flat substrate (defining a “capture line”) is studied as a function of strength of the magnetic field and volume fraction of the magnetic nanoparticles. Capture curves show a regime where capture follows a power law model and suggest that particles with the Brownian relaxation mechanism are captured at a slightly faster rate than particles with the Néel relaxation mechanism under similar conditions of the field gradient. Additionally, evaluation of the shape of the aggregates of captured particles suggests that greater dipole−dipole interactions result in aggregate structures that are flatter/wider than in the case of negligible dipole−dipole interactions. These results can help guide the design of systems for magnetically directed assembly of nanoparticles into complex shapes at a substrate.

1. INTRODUCTION Magnetic nanoparticles, such as particles of iron oxide1 and cobalt ferrite2 with typical diameters in the range of 10−30 nm, can be manipulated by applied magnetic fields, which generate both translational and rotational motion. This property has been recently exploited to drive organization of magnetic and nonmagnetic particles into a variety of structures through directed magnetic assembly at a substrate with patterned magnetic field gradients.3−5 As examples, magnetic nanoparticles have been assembled to generate diffraction gratings,6 to produce geometric patterns such as lines, triangles, squares, and circles,7,8 and to produce structures with more complex shapes that can be cross-linked and released, generating freefloating magnetic microstructures.9 On par with experimental studies of magnetic capture of nanoparticles at a magnetically patterned substrate, several groups have developed computational models to study the mechanism of magnetic particle capture and to provide a theoretical basis for the rational design of magnetic patterns to obtain structures of interest rapidly and reproducibly. The magnetic capture of particles in a variety of applications has motivated computational study of the effects of gravity and buoyancy and of fluid flow on particle trajectories,10,11 distributions12 and capture efficiency.13−15 However, in most of these cases the effect of Brownian motion of the particles has been neglected because the focus has been on the behavior of magnetic micron-sized particles. This is not the case when capturing nanoparticles using magnetically patterned substrates, © 2016 American Chemical Society

where the nanoscale size of the particles can make the effect of translational and rotational Brownian motion significant, especially at large and moderate distances from the magnetic patterns due to spatial decay of the magnetic field gradient. Some prior computational work has been done to study particle assembly in quiescent fluids with an emphasis on studying the shapes adopted by the magnetically captured particles. For example, Xue et al.16 developed a simulation method to study the assembly of magnetic-dielectric core−shell nanoparticles into extended monolayer geometric patterns with nanoscale precision. In their work, the suspension was modeled as containing a low number density of particles, which responded to the action of forces generated by external fields, viscous drag, Brownian motion, magnetic dipole−dipole interactions, and other interparticle interactions. The influence of these interactions on configurations adopted by the captured particles was investigated in situations with varying shapes of the magnetic pattern under different conditions of particle volume fraction and properties. More recently, using a model combining the Langevin equation and the Monte Carlo method, Xue et al.17 provided insight into the self-assembly of mono- and polydisperse magnetic dielectric core−shell nanoparticles into multilayer structures. Again, the focus was on Received: September 17, 2016 Revised: December 7, 2016 Published: December 12, 2016 801

DOI: 10.1021/acs.jpcc.6b09409 J. Phys. Chem. C 2017, 121, 801−810

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The Journal of Physical Chemistry C

case of particles with Brownian relaxation mechanism the algorithm also takes into account the coupling of translational and rotational motion of the nanoparticles, as well as the effects of magnetic torque and thermal rotation, in addition to those already mentioned for particles with Néel relaxation. The magnetic fields generated by the magnetically patterned substrate are calculated using COMSOL Multiphysics and used in the algorithm to calculate the magnetic forces and torques (for the case of particles with Brownian relaxation) on the nanoparticles. For simplicity, here we consider the case of a long “magnetic capture line”, although we point out that the same algorithm can be applied to simulate capture by more complex magnetically-patterned substrates. The effects of mechanism of magnetic relaxation, magnitude of the magnetic field gradients, volume fraction of the nanoparticles, and strength of particle−particle interactions are explored in relation to the rate of magnetic capture, particle capture trajectories, and shape (average width and height) of the collection of captured particles.

the configurations adopted by the captured particles and not on the dynamics of the capture process. Although the prior computational work cited above has provided important insights into the magnetic particle capture process, these studies assume that the particle responds to the applied magnetic fields through the so-called Néel relaxation mechanism18 of fast internal dipole rotation to align with the local magnetic field. While this is certainly a valid assumption for small spherical magnetic nanoparticles consisting of iron oxide, nanoparticles with other compositions, such as cobalt ferrite, can have their magnetic dipoles “thermally-blocked” in a so-called crystal easy axis due to a large value of anisotropy constant, such that they must physically rotate to align their dipoles with the local magnetic field in what is called Brownian relaxation mechanism.18 For particles with the Brownian relaxation mechanism, the process of magnetic capture could be influenced by both their translational and rotational Brownian motion, as well as by magnetic torques exerted on their thermally blocked magnetic dipoles. The addition of thermally blocked nanoparticles, such as cobalt ferrite, to magnetically assembled structures could yield free-floating magnetic microstructures with mixed magnetic relaxation modes, which in turn could be used in their external manipulation.9 However, we are not aware of any theoretical or computational studies that explicitly compare the magnetic capture of magnetic nanoparticles with Néel and Brownian relaxation mechanisms. In this contribution, we report a computational study of the capture of magnetic nanoparticles with Brownian or Néel relaxation mechanisms in a magnetic field gradient generated by a reversal in perpendicular magnetization of a substrate. This situation, illustrated in Figure 1, is representative of the

2. COMPUTATIONAL SIMULATION METHOD When external magnetic fields are applied to a magnetic particle suspension, the forces and torques acting on each dispersed particle include those due to hydrodynamic drag Fh and Th, those due to external magnetic fields Fm and Tm, those due to magnetic dipole−dipole interactions Fdd and Tdd, those due to other particle−particle interactions, represented here through a repulsive hard-core Yukawa potential for the case of chargestabilized particles FYkw, and those due to thermal agitation resulting from collisions of the particles with solvent molecules FB and TB. For the case of rigid particles, the motion of a magnetic nanoparticle is governed by stochastic linear and angular momentum equations. For nanoparticles, the assumption of negligible inertia is justified19 and the linear and angular momentum balance equations reduce to force and torque balances, expressed as ⎛ 0 ⎞ ⎛ Fh ⎞ ⎛ Fm ⎞ ⎛ Fdd ⎞ ⎛ FYkw ⎞ ⎛ FB ⎞ ⎜ ⎟ = ⎜ ⎟+⎜ ⎟+⎜ ⎟+⎜ ⎟+⎜ ⎟ ⎝ 0 ⎠ ⎝ Th ⎠ ⎝ Tm ⎠ ⎝ Tdd ⎠ ⎝ 0 ⎠ ⎝ TB ⎠

(1)

Because in a dilute suspensions of magnetic nanoparticles long-range magnetic interactions are expected to be more significant than particle−particle hydrodynamic interactions, we explicitly neglect particle−particle hydrodynamic interactions in our simulations.20 Because our interest here is to simulate particle capture during magnetically guided assembly, wherein the goal is to obtain well-defined patterns of nanoparticles at a substrate, our simulations focus on a system where the particles are colloidally stable against aggregation in solution. This implies conditions where repulsive interactions, such as electrostatic repulsion, are dominant over attractive interactions, such as van der Waals interactions. As such, in our simulations we take into account particle−particle interactions through a repulsive Yukawa potential, as an approximation for long-range electrostatic repulsion, and long-range magnetic dipole−dipole interactions. We explicitly neglect the effect of van der Waals interactions as these are of much shorter range and, if dominant, would result in particle aggregation in solution. Furthermore, the strength of the magnetic dipole− dipole interactions is such that particle interactions are nonnegligible, but not high enough to cause significant chain formation. Finally, a hard-sphere repulsion potential is used to prevent particle−particle and particle-wall overlap, which can

Figure 1. (a) Lateral view of device with simulated area and two ideal magnetic poles magnetized with opposite directions. (b) COMSOL simulation of normalized magnetic field in the simulated area.

magnetic field gradients generated for magnetic particle capture by using magnetically patterned perpendicular recording media. The magnetic patterns used here represent those obtained by using selective reversal magnetization.9 Considering that the region with maximum magnetic field gradient lies along the interface of two regions with opposed magnetic poles, we define this as the “magnetic capture line”. A simulation algorithm was developed to account for translational and rotational Brownian motion of the magnetic particles. For the case of particles with Néel relaxation, the algorithm takes into account the dependence of the particle’s magnetic dipole on the magnitude of the local magnetic field, as well as the effects of magnetic force, magnetic dipole−dipole interactions, hydrodynamic drag, hydrodynamic particle-wall interactions, repulsive hard-core Yukawa potential, and thermal motion. For the 802


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mechanism it is assumed that the direction of the dipole instantaneously aligns with the direction of the local magnetic field, whereas instead of a saturated dipole moment the magnitude of the dipole meff is given by the Langevin function L(α) meff 1 = coth α − ≡ L (α ) ms α (6)

be a problem especially close to the magnetic capture line and at high values of the magnetic field gradient. 2.1. Hydrodynamic Force and Torque. If magnetic nanoparticles are placed in a motionless fluid medium, the hydrodynamic force Fh and torque Th exerted on a particle can be related to the particle’s velocity U and angular velocity ω, through the mobility matrix M, in the manner of ⎛ Fh ⎞ ⎛U⎞ ⎜ ⎟ = −M·⎜ ⎟ ⎝ω⎠ ⎝ Th ⎠

where the Langevin parameter is (2)

α=

The symmetric and positive-definite mobility matrix can be written as ⎛ MUF MUT ⎞ ⎟⎟ M = ⎜⎜ ⎝ M ωF M ωT ⎠

(3)

(7)

m′ = μ0 mμ̂ ′

(8)

where the prime indicates a vector is in particle coordinates, m can be replaced by ms for Brownian-relaxation particles and by meff for Néel-relaxation particles, and μ̂ is a unit vector specifying the orientation of the magnetic dipole moment. By assuming that all saturated magnetic dipole moments have uniform magnitude and always point along the z-axis of the particle coordinates, μ̂′ can be transformed into the laboratory space through

(4)

where μ is a constant and Ĥ represents a unit vector pointing along the local direction of the magnetic field H. While this simple assumption captures the expectation that the magnetic dipole instantaneously aligns with the local magnetic field, it does not capture the dependence of the strength of the particle’s magnetic dipole on the magnitude of the local magnetic field. For particles that respond to changes of magnetic field through the Néel relaxation mechanism, the magnetic dipoles are continuously changing direction due to thermal energy at characteristic time scales (τN ∼ 10−9 s)22 that are much shorter than the characteristic time scales for particle translation and rotation of interest in magnetic capture. As such, the magnetic dipole is capable of sampling configurations where it is not aligned with the local magnetic field, leading to reduced magnetic dipole strength when one averages over a time longer than the Néel relaxation time but shorter than the characteristic time for particle translation. This reduction in the strength of the magnetic dipole is expected to result in a reduction in the strength of the magnetic forces experienced by the particles, which in turn is expected to influence the rate of magnetic particle capture. In our work, magnetic nanoparticles relaxing by the Brownian mechanism are assumed to always have a saturated dipole moment, of magnitude given by ms = MdVp

kBT

In eq 7 the vacuum permeability is μ0 = 4π × 10−7 N/A2, H represents the magnitude of the magnetic field, kB is the Boltzmann constant, and T represents the absolute temperature. The simulation box is set up in a Cartesian coordinate system and fixed to free space (i.e., the laboratory coordinates). As a result, a magnetic dipole moment in the particle coordinates is given by

where MUF, MUT, MωF and MωT are mobility components that vary with the fluid viscosity and particle position relative to the wall and relate hydrodynamic forces and torques to particle translational and rotational velocity. 2.2. Magnetic Moment and Relaxation Mechanisms. Most work10,12,21 assumes that the magnetic dipole moment of each magnetic nanoparticle is “saturated” along the direction of the local magnetic field, that is, it is commonly assumed that the magnetic dipole moment m of the particles is given by an expression of the form m = μĤ

μ0 msH

μ̂ = A−1·μ′̂

(9)

where A is the transformation matrix in the form ⎡− ζ 2 + η2 − ξ 2 2(ζχ − ηξ) 2(ζη + ξχ ) ⎤⎥ ⎢ 2 ⎥ ⎢ +χ ⎥ ⎢ 2 2 2 − 2(ηξ + ζχ ) − ζ − η + ξ 2(ηχ − ζξ) ⎥ A = ⎢⎢ ⎥ + χ2 ⎥ ⎢ ⎢ 2(ζη − ξχ ) − 2(ζξ + ηχ ) ζ 2 − η2 − ξ 2 ⎥ ⎥ ⎢ + χ2 ⎦ ⎣

(10)

In eq 10, ζ, η, ξ, and χ are the quaternion parameters, which satisfy the condition ζ2 + η2 + ξ2 + χ2 = 1.25 2.3. Magnetic Force and Torque. In our work, the magnetic field results from the reversal perpendicular magnetization in a planar substrate with one region magnetized upward (+z direction) and the other magnetized downward (−z direction). Such a situation is found, for example, in perpendicular recording media, where specific areas of a thin magnetic layer (e.g., films, substrates, tapes, hard drives) are magnetized out of plane, in opposite directions of the original magnetization of the layer. To understand this behavior, a twodimensional (2D) magneto-static finite-element simulation26 was used to estimate the magnetic field at a magnetic pole boundary in a magnetic substrate (Hi8MP video cassette tape).9 This simulation was carried out using the ac/dc module in the COMSOL Multiphysics software (version 5.1), using “magnetic fields, no current” (mfnc) as the physics. The out-ofplane magnetization of the tape material was defined using demag-corrected magnetization curves measured using a vibrating sample magnetometer (VSM-EV9 ADE technologies)

(5)

where Md represents the saturation magnetization of the material and Vp represents the particle volume. The direction of the dipole moment varies due to the balance of rotational Brownian motion and magnetic/hydrodynamic torques on the particle. Our past work23,24 has demonstrated that such a model accurately accounts for the field-dependent ensemble average magnetization of a suspension of magnetic nanoparticles. On the other hand, for magnetic nanoparticles relaxing by the Néel 803


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The Journal of Physical Chemistry C showing a remanence (μ0M) of ∼50 mT and a coercivity (Hci) of ∼65 kA/m. A 2D cross-section simulation was recreated with an air domain (1 μm height and 4 μm length) on top of the magnetic substrate (a 1.75 μm thick and 15 μm length), as illustrated in Figure 1, in which Hmax is the maximum magnetic field in the section. In a medium without time-varying electric fields or currents, magnetic nanoparticles experience a magnetic force due to magnetic field gradients, given by

Fm = μ0 m·∇H

⎛ F ⎞ ⎛ F ⎞ ⎜ ∑ dd, ji ⎟ ⎛ ⎞ ̂ μ μ · ∇ H ( ) m ⎛ Ui ⎞ 0 i i ⎜∑ Ykw, ji ⎟ ⎟ ⎜ j≠i ⎜ ⎟ ⎜ ⎟ = Mi ·⎜ ⎟⎟ ⎟ + Mi ·⎜⎜ j ≠ i ⎟ + Mi ·⎜ ⎝ ωi ⎠ ⎝ μ0 mi(μ̂i × H)⎠ ⎜∑ Tdd, ji ⎟ ⎝ ⎠ 0 ⎠ ⎝ j≠i ⎛ FB, i ⎞ + Mi ·⎜⎜ ⎟⎟ ⎝ TB, i ⎠ (17)

Dimensionless variables are introduced according to Ũi =

(11)

(12)

Fdd, ji 3μ0 mjmi 4πrji 4

(13)

Tdd, ji =

4πrji 3

(14)

κ ̃ = aκ

(19-1)

MiUT ⎛ 4 1 ⎞ UT =⎜ ⎟m i aDr ⎝ 3 kBT ⎠

(19-2)

M iωF ⎛ 4 a ⎞ ωF =⎜ ⎟m i Dr ⎝ 3 kBT ⎠

(19-3)

miUT,

(19-4)

miωF,

miωT

⎛ dx̃ i ⎞ ⎛ Ũ ⎞ ⎜ ⎟ = ⎜⎜ i ⎟⎟dt ̃ ⎜ ̃⎟ ⎝ dΦi ⎠ ⎝ ω̃ i ⎠

(20)

where t ̃ is nondimensionalized according to t ̃ = Drt. Integrating from t ̃ to t ̃ + Δt ̃ based on the first-order forward Euler method, and applying the fluctuation−dissipation theorem30 to the Brownian terms, the motion equation results in ⎛ ⎞ 4 UF UT ⎛ dx̃i ⎞ ⎜ αmax 3 (mi ·e Fm, i + mi ·e Tm, i)⎟ ⎟Δt ̃ ⎜ ⎟=⎜ ⎜ ̃⎟ ⎜ ⎝ dΦi ⎠ ⎜ α 4 (m ωF ·e + m ωT·e )⎟⎟ i Fm, i Tm, i ⎝ max 3 i ⎠

exp[−κ(rji − σ )] rji

(15)

⎛ ⎛ ⎜ β ⎜m UF· ⎜ dd ⎜ i ∑ j≠i ⎜ ⎝ +⎜ ⎛ ⎜ ωF ⎜ βdd ⎜⎜mi ·∑ ⎜ j≠i ⎝ ⎝

where σ is the hard-core diameter, λρ−1/3 denotes the cutoff distance, λ is a prefactor to modulate the cutoff distance, ρ−1/3 is proportional to the average interparticle distance, ε represents the pair potential, and κ represents the inverse Debye screening length. The force due to the repulsive hardcore Yukawa potential is obtained from FYkw, ji = −∇uYkw, ji

H , Hmax

where and are the dimensionless components of the mobility matrix corresponding to MUF i , ωF ωT ̃ MUT i , Mi , and Mi , respectively. By setting dxĩ and dΦi as the infinitesimal translation and rotation vectors of the particle i, their relationship with velocity and angular velocity are given by

[3(m̂ j·rjî )(m̂ i × rjî ) + (m̂ j × m̂ i)]

σ

H̃ =

MiUF ⎛ 4 a ⎞ UF =⎜ ⎟m i aDr ⎝ 3 kBT ⎠

miUF,

where mj and mi represent the magnitudes of magnetic dipole moment j and i, respectively, rji represents the center distance between particle j and i, r̂ji represents the unit vector of center distance, and m̂ j and m̂ i represent the unit vectors of dipole moment j and i, respectively. The algorithm also takes into account repulsion between the particles due to electrostatic interactions, modeled using a repulsive hard-core Yukawa potential, truncated at σ ≤ rji ≤ λρ−1/3 and which has the form29 uYkw, ji = ε

∇̃ = a∇,

M iωT ⎛ 4 1 ⎞ ωT =⎜ ⎟m i Dr ⎝ 3 kBT ⎠

[rjî (m̂ j·m̂ i) + m̂ j(rjî ·m̂ i) + m̂ i(rjî ·m̂ j)− 5rjî (rjî ·m̂ j)(rjî ·m̂ i)]

μ0 mjmi

ωi , Dr

where a is the radius of the uniform-size particles, Dr = kBT(8πηa3)−1 is the rotational diffusivity for a sphere and η is the viscosity of the carrier fluid. The mobility matrix components are nondimensionalized according to

The magnetic torque is zero for particles relaxing by the Néel mechanism because we consider their magnetization is always collinear with the local magnetic field. 2.4. Particle−Particle Interactions. The simulation algorithm included the magnetic forces and torques resulting from magnetic dipole−dipole interactions between particles. The magnetic dipole−dipole force and torque exerted by particle j on particle i are given by27,28

=

ω̃ i =

(18)

Particles that relax by the Brownian mechanism also experience a magnetic torque when their dipole moment is not aligned with the local magnetic field, given by

Tm = μ0 m × H

Ui , aDr

⎛ ⎜β 8 ⎜ Ykw 3 ⎜ +⎜ ⎜ 8 ⎜ βYkw ⎜ 3 ⎝

(16)

2.5. Motion Equation. After substituting the different forces and torques acting on the nanoparticles the motion equation of particle i becomes 804

e Fdd, i rjĩ 4 e Fdd, i rjĩ 4

+ miUT·∑ j≠i

+ miωT·∑ j≠i

⎞ e Tdd, i ⎞ ⎟ ⎟ rjĩ 3 ⎟⎠ ⎟ ⎟ ⎟Δt ̃ ⎞ e Tdd, i ⎟ ⎟⎟ rjĩ 3 ⎟⎠⎟ ⎠

⎤⎞ ⎞ 1 κ̃ + ⎟⎟(miUF·rjî )⎥ ⎟⎟ 2 ⎥⎦ rjĩ ⎠ ⎝ rjĩ j≠i ⎣ ⎟ ⎟Δt ̃ + ⎤⎟ ⎡ ⎞ ⎛ κ ̃ ⎟ ωF ⎥ −κ(̃ rjĩ − 2)⎜ 1 ⎢ ∑ ⎢e ⎜ r 2 + r ̃ ⎟(mi ·rjî )⎥⎟⎟ ji ⎠ ⎝ jĩ ⎦⎠ j≠i ⎣ ⎡

⎛

∑ ⎢⎢e−κ(̃ rjĩ − 2)⎜⎜

⎛ X̃ (Δt )̃ ⎞ ⎟ ⎜ i ⎟ ⎜ ̃ ⎝ Wi (Δt )̃ ⎠

(21)


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Figure 2. Zoomed-in 3D configuration snapshots of particles close to the capture line at various times for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, and particle volume fraction ϕ = 0.05% for magnetic nanoparticles relaxing by the (a) Brownian relaxation mechanism and (b) Néel relaxation mechanism. The north pole of magnetic dipole moment is specified in red color whereas the south pole in white. Coordinate dimensions are scaled by particle radius.

As the time step Δt is assumed longer than the relaxation time for particle momentum, the particle has null acceleration. The first term on the right side of eq 21 calculates the translational and rotational variance of particles due to the external magnetic field at each time step, where we have defined e Fm, i = μ̂i ·∇̃ H̃

volume interactions between particles and between particles and the wall are taken into account by applying a hard sphere interaction. The calculation for far-field hydrodynamic interactions between single particles and the wall is included in the normalized mobility terms31

(22)

1 −1 −3 −5 (9hĩ − 2hĩ + hĩ )(δpq − δp3δq3) 16 1 −1 −3 −5 − (9hĩ − 4hĩ + hĩ )δp3δq3 8

miUF , pq = −

e Tm, i = μ̂i × H̃

(23)

and the maximum Langevin parameter amax = μ0mHmax/(kBT) . Similarly, the second term and the third term on the right side of eq 21 calculates the one-step variance in both position and orientation of particles due to magnetic dipole−dipole interactions and repulsive hard-core Yukawa potential, respectively, by defining e Fdd, i = rjî (m̂ j· m̂ i) + m̂ j(rjî · m̂ i) + m̂ i(rjî · m̂ j) − 5rjî (rjî · m̂ j)(rjî · m̂ i)

1 (m̂ j × m̂ i) 3

(25)

and by setting the parameter of magnetic dipole−dipole interaction βdd = μ0m2/(πa3kBT) and the parameter of hardcore Yukawa repulsion βYkw = ε/(kBT) . In eq 22, X̃ i(Δt )̃ and ̃ i Δt )̃ are random vectors characterized by a Gaussian W( distribution with mean and covariance, respectively, as ⟨X̃ i(Δt )̃ ⟩ = 0 ⟨X̃ i(Δt )̃ ·X̃ i(Δt )̃ ⟩ =

8 UF (m i + m iUT)Δt ̃ 3

⟨W̃ i (Δt )̃ ⟩ = 0 8 ⟨W̃ i (Δt )̃ ·W̃ i (Δt )̃ ⟩ = (m iωF + m iωT)Δt ̃ 3

miω, pqF = −

3 ̃ −4 hi ε3pq 32

(27-2)

miω, pqT = −

15 ̃ −3 3 ̃ −3 hi (δpq − δp3δq3) − hi δp3δq3 64 32

(27-3)

where h̃i denotes the radius-scaled distance between particle i and the wall, δpq is the Kronecker delta function, and ε3pq is the Levi-Civita symbol. Because of the symmetry of the mobility matrix, the matrix component miUT can be obtained by transposing mωF i . 2.6. Methods To Identify and Quantify Magnetically Captured Nanoparticles. An aggregate-box method was used to count the number of magnetic nanoparticles that are captured by the patterned magnetic field. In this method, a box with size 0.3 μm × 4 μm × 0.2 μm immediately above the capture line was considered. The number of magnetically captured particles at a given time point was calculated by subtracting the initial number of particles from the current number of particles in the aggregate box. Although this method is appropriate for arbitrary strength of external magnetic fields, it cannot evaluate the configuration of the captured particles and is unable to track particle position. In order to track changes in the shape of the aggregates of captured particles, we applied another method where particles that reach the capture line are treated as being magnetically captured. This approach is supported by observations of particle trajectories in simulations,

(24)

e Tdd, i = (m̂ j·rjî )(m̂ i × rjî ) +

(27-1)

(26)

To preclude the overlap of particles with neighbors and take into account confinement due to the boundary wall, excluded 805


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Brownian mechanism and the Néel mechanism. Videos S1 and S2 contain a projection of particle positions into a 2D space with magnetically captured particles highlighted in red and noncaptured particles highlighted in black. In Figure 2, the Brownian-relaxation particles are arranged such that their dipole moments are aligned relative to each other, whereas particles with Néel relaxation have their dipoles aligned in the local direction of the magnetic field. This is easier to observe in Videos S3 (particles with Brownian mechanism) and S4 (particles with Néel mechanism) of the Supporting Information. Combining this difference with a comparison of the aggregate size and structure between the Brownian-relaxation particles and the Néel-relaxation particles, one can observe that starting from the same initial configuration the particles with Brownian relaxation mechanism are captured at a faster rate while also forming distinct particle/dipole moment chains aligning along the direction of the external magnetic field. However, these particle chains are not observed for the Néelrelaxation particles. This suggests that magnetic dipole−dipole interactions exert greater influence on the capture behavior of Brownian-relaxation particles, as compared to particles undergoing Néel relaxation. Figure 3 shows trajectories of four representative magnetic nanoparticles relaxing by the Brownian and Néel mechanisms,

where we observed that captured particles can continue to move within the aggregates. We found that the capture-line method was only suitable for cases with strong applied fields (αmax ≥ 70) but it is more suitable to observe the shape of aggregates based on the positions of captured particles. 2.7. Simulation Parameters and Conditions. Simulations were made for spherical magnetic nanoparticles with uniform magnetic radius of 10 nm, dispersed in a solvent with the viscosity of water and in a simulation box with size of 4 μm × 4 μm × 1 μm. The bottom surface of the simulation box corresponds to an impermeable wall, whereas the vertical surfaces are modeled as periodic boundaries. Particles that leave the simulation box through the top surface are reintroduced at random positions in the x- and y-directions. The temperature in the simulations corresponds to 300 K. Runs were executed starting from random particle configurations and using a minimum time interval of Δt ̃ = 0.01. The cutoff distance corresponding to a prefactor of λ = 1 was taken into account for both magnetic dipole−dipole interactions and repulsive hardcore Yukawa potential. In addition, for the repulsive hardcore Yukawa potential we used a radius-scaled inverse Debye screening length of κ̃ = 3, and a scaled interaction energy of 5 ≤ βYkw ≤ 20. The scaled dipole−dipole interaction was βdd = 100. The maximum Langevin parameter in the simulation box was varied in the range of 10 ≤ αmax ≤ 500, which for a nanoparticle with diameter of 10 nm and a temperature of 300 K corresponds to maximum magnetic field 17.63 kA/m ≤ Hmax ≤ 881.73 kA/m. Nanoparticle volume fractions of 0.001% ≤ ϕ ≤ 0.1% were considered.

3. RESULTS AND DISCUSSION Simulation runs for magnetic nanoparticle capture under a wide range of conditions were analyzed with respect to particle configurations and representative particle trajectories over time, magnetic capture rates by evaluating the number of captured magnetic nanoparticles as a function of time, and temporal variation of the width and height of the collection of captured particles. These results are presented and discussed below. 3.1. Particle Motion. Figure 2 shows representative zoomed-in 3D configuration snapshots of particle close to the capture line at different time steps during the magnetic capture process for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter of αmax = 100, particle volume fraction ϕ = 0.05%, and for particles relaxing by the Brownian and Néel relaxation mechanism. These snapshots were plotted through POV-Ray, using a top-view camera perspective, setting the left side and right side of the capture line as the south pole and north pole of the magnetic substrate, and specifying the north pole of the magnetic dipole moment in red and the south pole in white. In such 3D configurations, all dimensions are scaled by the uniform particle radius. Particles that appear larger in Figure 2 are closer to the camera plane in constructing the image. As seen in the figure, magnetic nanoparticles, which are initially dispersed in random positions and orientations, are attracted by the strong magnetic field gradients and translate toward the capture line (x = z = 0), where the magnetic field strength is highest. With the increase in the number of captured particles and due to the balance of excluded volume interactions, electrostatic repulsion, and magnetic interactions, the size of the particle aggregates along the capture line grows over time. These behaviors are better observed in Supporting Information Videos S1 and S2, which are at the same conditions as Figure 2 and for particles relaxing by the

Figure 3. Trajectories of representative magnetic nanoparticles that respond to the applied magnetic field by the (a) Brownian relaxation mechanism and (b) Néel relaxation mechanism for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, and particle volume fraction ϕ = 0.05%.

starting from the same initial positions. The simulations were executed at a particle volume fraction of ϕ = 0.05% and with the same magnetic field and particle interaction conditions. Although all the particles shown in Figure 3 are eventually captured by the magnetic field gradient, it is observed that the particles relaxing by the Néel relaxation mechanism undergo more complex paths as compared with the particles relaxing by the Brownian mechanism. These differences are also observed in Supporting Information Videos S1 and S2, which suggest that particles relaxing by the Néel mechanism, which experience relatively weaker magnetic forces and torques due to their reduced effective dipole moments, experience greater translational Brownian motion than particles relaxing by the Brownian mechanism. Together, Figures 2 and 3 suggest that particles relaxing by the Brownian mechanism are captured at a faster rate than particles relaxing by the Néel mechanism. 3.2. Magnetic Capture Rates for Different Relaxation Mechanisms. Figure 4 shows the number of captured particles as a function of capture time (i.e., the magnetic capture curves) for particles that relax by the Brownian and Néel mechanisms under identical conditions of particle−particle interactions, magnetic field gradient, and particle volume fraction. It should 806


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Figure 4. Number of captured magnetic nanoparticles as a function of capture time for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and with magnetic nanoparticles relaxing by the Brownian relaxation mechanism and Néel relaxation mechanism.

be noted that the y-intercepts of the magnetic capture curves vary with simulation conditions because the capture curves are plotted starting from the second time step. In Figure 4, the curves initially have almost constant slopes. After some time, the slope of the capture curve decays as the number of captured particles saturates. We thus separate the capture process into two periods. In the first period, the dispersed particles surge toward the capture line due to the magnetic force and assemble along the capture line. The slope of the capture curve during this period is considered to follow a power law between the number of magnetically captured particles NC and the dimensionless capture time t ̃ ln NC = γ ln t ̃ + ln C

(28)

where γ is the power law exponent (i.e., the slope of the capture curve in log−log coordinates), a measure of the magnetic capture rate, and C is a constant depending on the simulation conditions. During the second period, the rate of capture slows down due to the decreasing number of free particles in the simulation box, eventually becoming zero when all the particles are captured. Making a comparison between the two curves in Figure 4, it is again evident that particles with Brownian relaxation mechanism are more rapidly captured than particles with the Néel relaxation mechanism, but apparently the difference stems from rapid capture in the initial time steps. As a result, the particles with Brownian magnetic relaxation mechanism are more rapidly depleted from the simulation box, hence the number of captured particles begins to asymptote earlier. Similar results are obtained for other conditions of the magnetic field gradient and volume fraction of the nanoparticles. 3.3. Magnetic Capture Rates as a Function of the Strength of the External Magnetic Field Gradient. Figure 5a,b compares magnetic capture curves for various maximum Langevin parameters for particles that relax by the Brownian and Néel mechanisms. As shown in the figure, when αmax ≥ 30 for both Brownian-relaxation particles and Néel-relaxation particles, the capture curves look smooth with a wide range of power-law behavior. On the other hand, for smaller values of the maximum Langevin parameter, the magnetic capture curves fluctuate significantly. These distinguishing behaviors are explained by the Brownian agitation of the particles, which becomes significant as the magnetic force and torque exerted on the particles decreases due to decreasing magnitude of the

Figure 5. Number of captured magnetic nanoparticles as a function of capture time for interaction parameters βdd = 100 and βYkw = 5, particle volume fraction ϕ = 0.05%, various maximum Langevin parameters, and with magnetic nanoparticles relaxing by the (a) Brownian relaxation mechanism and (b) Néel relaxation mechanism, respectively. (c) Magnetic capture rates fitted based on the data in (a,b).

magnetic field gradient. Focusing on the capture period that follows the power law model, we fitted the magnetic capture rates in order to compare the slopes as a function of the maximum Langevin parameters, as shown in Figure 5c. This figure illustrates that with the same maximum Langevin parameter, the capture rates for particles relaxing by the Brownian mechanism are slightly higher than the rates for particles relaxing by the Néel mechanism. Additionally, as the maximum Langevin parameter αmax ≥ 30 (i.e., the maximum magnitude of applied magnetic field 52.90 kA/m) for both Brownian-relaxation particles and Néel-relaxation particles, the magnetic capture rates change very little with respect to the variation of field strength. 807


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This result suggests that, when ϕ ≥ 0.005% for Brownianrelaxation particles and ϕ ≥ 0.01% for Néel-relaxation particles, the magnetic capture rate is not a function of particle volume fraction any more. These results appear to be at odds with experimental and simulation work by Faraudo et al.,32 who studied rates of particle capture for separation applications and observed that the time to capture particles decreased with increasing particle concentration. However, in their work Faraudo et al.32 considered particles with a diameter of 410 nm (compared to 10 nm considered here) and were interested in so-called cooperative magnetophoresis, where a uniform applied field is used to cause chain formation, which leads to faster separation of the particles in a given magnetic field gradient. In our study, we were interested in the capture of individual particles so as to generate a pattern on a substrate. In that case, chain formation in solution would lead to low fidelity pattern formation and as such is undesirable. Thus, our simulations were performed under conditions that do not lead to significant chain formation and we do not see the enhancement in particle separation with increasing concentration reported by Faraudo et al.32 3.5. Shape of Magnetic Particle Aggregates. The size evolution of the magnetic nanoparticle aggregates on the capture line with different relaxation mechanisms is shown in Figure 7 by plotting the average height and width of aggregates as a function of capture time based on the capture-line method. From the figure, one can observe that the aggregates assembled with two different relaxation mechanisms grow in height at roughly equal rates. However, aggregates assembled by particles relaxing by the Brownian mechanism grow faster in width than aggregates assembled by particles relaxing by the Néel mechanism. That is because Brownian-relaxation particles aggregating on the capture line preferentially form particle strings due to magnetic dipole−dipole interactions. This phenomenon is also observed in Figure 2. Figure 8 presents a comparison of the size evolution of particle aggregates for different ratios of magnetic interaction parameter and Yukawa repulsion parameter with other conditions and parameters uniform. For particles relaxing by the Brownian mechanism, the aggregates assembled with magnetic dipole−dipole interactions and hard-sphere Yukawa repulsion undergo greater growth in width than aggregates assembled without such interactions. At βdd = 100 and βYkw = 5, the aggregates have a lower height and a larger width than aggregates obtained using other parameters. This indicates that by generating plentiful particle strings, dominant magnetic

3.4. Magnetic Capture Rates for Different Particle Volume Fractions. Figure 6 shows the magnetic capture

Figure 6. Number percentage of captured magnetic nanoparticles as a function of capture time for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameters αmax = 100, various particle volume fractions, and with magnetic nanoparticles relaxing by the (a) Brownian relaxation mechanism and (b) Néel relaxation mechanism, respectively.

curves for various particle volume fractions, for particles that relax by the Brownian mechanism and Néel mechanism. For all the capture curves, the interaction parameters were βdd = 100 and βYkw = 5, and the maximum Langevin parameter was αmax = 100. As seen in the figure, besides the curves at particle volume fraction ϕ = 0.001% for Brownian-relaxation particles and ϕ = 0.001% and ϕ = 0.005% for Néel-relaxation particles, all other curves under the same relaxation mechanism have similar shape and slopes. The abnormal behavior of the curves can be explained by the discontinuous supply of particles during the magnetic capture process at very low particle volume fraction.

Figure 7. Average (a) width and (b) height of particle aggregates as a function of capture time for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and with magnetic nanoparticles relaxing by the Brownian relaxation mechanism and Néel relaxation mechanism. 808


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Figure 8. Average (a) height and (b) width of aggregates assembled by Brownian-relaxation particles and average (c) height and (d) width of aggregates assembled by Néel-relaxation particles as a function of capture time for maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and various ratio of magnetic interaction parameter and hard-core Yukawa repulsion parameter.

100 and particle volume fraction ϕ ≥ 0.005% for Brownianrelaxation particles and ϕ ≥ 0.01% for Néel-relaxation particles, the magnetic capture rate is not a function of particle volume fractions. Additionally, strong magnetic dipole−dipole interactions are verified to contribute tight aggregate structures and result aggregates assembled by particles relaxing by the Brownian mechanism in wider width than aggregates assembled by particles relaxing by the Néel mechanism. In summary, this study provides a theoretical understanding of magnetic capture mechanisms and the behaviors of the dispersed magnetic nanoparticles, relaxing by the Brownian relaxation mechanism and Néel relaxation mechanism, respectively. These theoretical predictions of magnetic capture in a quiescent fluid can help in controlling the size of particle aggregations formed during magnetic assembly and in designing better devices for magnetic nanoparticle separation capture.

dipole−dipole interactions (βdd/βYkw = 20) result in particle aggregates with flat shapes. When βYkw increases to 10, we observed that the particle aggregates increase in height and have narrower width. This behavior is explained by the enhanced Yukawa repulsion, which inhibits the formation of particle strings by increasing the interparticle distance at equilibrium. As a result, captured magnetic nanoparticles have to accumulate on aggregates in the height dimension. At βYkw = 20 for βdd/βYkw = 5, the repulsive Yukawa interaction becomes so strong that the aggregates exhibit a looser structure than those at smaller βYkw. In this case, both the height and width of the aggregates rises, as shown in Figure 8a,b. For particles relaxing by the Néel mechanism, the introduction of magnetic dipole−dipole interactions generates little influence on the aggregate size. However, an increase in repulsive Yukawa interactions still results in an increase in the height and width of particle aggregates, resulting in loose structures.

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4. CONCLUSIONS In the present work, magnetic capture rates and evolution of aggregate sizes at a capture line generated by a magnetic pole reversal are investigated by using Brownian dynamics simulations of a magnetic nanoparticle suspension in strong external magnetic field gradients generated at a solid substrate. The simulations suggest that under identical conditions of particle size, volume fraction, and magnetic fields particles with Brownian relaxation mechanism are captured at a faster rate than particles with Néel relaxation mechanism. We also observed that when ϕ = 0.05% and the maximum Langevin parameter αmax ≥ 30 for both Brownian-relaxation particles and Néel-relaxation particles, the strength of magnetic fields have little effect on the power-law dependence of number of captured particles with capture time. Similar observations were made for the influence of particle volume fraction. When αmax =

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b09409. Evolution of magnetic nanoparticle configurations during the magnetic capture process, for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and for particles relaxing by the Brownian relaxation mechanism for view of captured particles (AVI) Evolution of magnetic nanoparticle configurations during the magnetic capture process, for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and for particles relaxing by the Néel relaxation mechanism for view of captured particles (AVI) 809


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Evolution of magnetic nanoparticle configurations during the magnetic capture process, for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and for particles relaxing by the Brownian relaxation mechanism for 3D view including the orientation of magnetic dipole moments (AVI) Evolution of magnetic nanoparticle configurations during the magnetic capture process, for interaction parameters βdd = 100 and βYkw = 5, maximum Langevin parameter αmax = 100, particle volume fraction ϕ = 0.05%, and for particles relaxing by the Néel relaxation mechanism for 3D view including the orientation of magnetic dipole moments (AVI)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]fl.edu. Phone: 1-352-294-5588. ORCID

Carlos Rinaldi: 0000-0001-8886-5612 Present Address

(I.T.-D.) Department of Chemical and Biomolecular Engineering, Johns Hopkins University. Baltimore, Maryland 21218, United States. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the U.S. National Science Foundation, Grant CBET1511113. REFERENCES

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Brownian Dynamics Simulations of Magnetic Nanoparticles Captured

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