Analysis of the Dynamics of Magnetic CoreâShell Nanoparticles and

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Analysis of the Dynamics of Magnetic Core-shell Nanoparticles and Self-assembly of Crystalline Superstructures in Gradient Fields Xiaozheng Xue, and Edward P. Furlani J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp513025w • Publication Date (Web): 17 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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The Journal of Physical Chemistry

Analysis of the Dynamics of Magnetic Core-shell Nanoparticles and Self-assembly of Crystalline Superstructures in Gradient Fields Xiaozheng Xue1 and Edward P. Furlani1,2. 1

Dept. of Chemical and Biological Engineering, Univ. at Buffalo SUNY, NY 14260 2

Dept. of Electrical Engineering, Univ. at Buffalo SUNY, NY 14260

Abstract A method is presented for controlling the field-directed self-assembly of colloidal magnetic coreshell nanoparticles into three dimensional (3D) crystalline superstructures with nanoscale feature resolution. This level of resolution is obtained using sub-micron soft-magnetic template elements to guide the assembly in the presence of a uniform bias field. The use of a bias field combined with template-induced gradient fields is a critical feature of this process as it provides highly localized regions of attractive and repulsive magnetic force that enable nanoscale control of particle placement during assembly. We demonstrate proof-of-concept using a computational model that predicts the dynamics of individual particles during assembly as well as the final assembled structure. Our predictions are consistent with reported experimental observations and demonstrate for the first time that 3D crystalline superstructures can be assembled within milliseconds. Uniform hexagonal close packed (hcp), face centered cubic (fcc) and mixed phase hcp-fcc structures can be obtained depending on the template geometry. We further show that the structure and resolution of the assembled particles as well as the rate of assembly can be controlled via careful selection of key parameters including the template geometry, particle constituents, core-shell dimensions and the particle volume fraction. The proposed assembly method is very versatile as it broadly applies to arbitrary template geometries and multilayered core-shell particles that have at least one magnetic component. The magnetic component enables control to drive self-assembly, while the nonmagnetic components can provide desired functionality, e.g. photonic, electronic and biological etc. Moreover, the process is reversible in that the particles can be repeatedly assembled and redisperesed by applying and removing the bias field, thereby enabling the reversible creation of functional media on demand. Alternatively, assembled structures can be transferred intact to a substrate to form thin films. As such, the method opens up opportunities for the scalable bottom-up fabrication of nanostructured materials with unprecedented functionality for a broad range of applications. The computational model is very general and can be applied to many forms of self-assembly to provide a fundamental understanding of underlying mechanisms and enable the rational design of novel media.

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Keywords self-assembly of magnetic core-shell nanoparticles, magnetic template-assisted self1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

assembly of crystalline superstructures, bottom up nanofabrication, Langevin dynamics of magnetic self-assembly, magnetic dipole-dipole interactions.


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1 Introduction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Advances in nanotechnology, especially in nanoparticle synthesis, have opened up opportunities for the bottom-up fabrication of nanstructured materials. Colloidal nanoparticles can be tailored to have different geometries, e.g. spheres, rods, cubes and cage-structures, with unique size- and shapedependent physical properties, e.g. optical, magnetic, electronic etc. The ability to control the assembly of these versatile constituents enables the fabrication of nanostructured materials with extraordinary properties, not found in naturally occurring materials. Bottom-up fabrication has potential advantages in terms of scalability, high-throughput and low-cost over conventional topdown lithographic-based methods, which tend to be costly, time consuming and limited to the development of small-scale planar materials. Self-assembly is considered to be among the most promising means for bottom-up fabrication, i.e. for controlling the organization of colloidal nanoparticles into mesoscopic media1-13. To date, various self-assembly methods2 have been demonstrated for the fabrication of photonic7,14, magnetic7, micro-optical15 and electronic16,17 materials18-25. Some of the methods exploit directed24,26 and template-assisted27 assembly or combinations thereof. However, despite this progress, many fundamental aspects of self-assembly remain unknown and the formation of 3D crystalline superstructures with nanoscale resolution remains very challenging. In this paper we introduce a method to address this challenge. The method is based on magnetic field-directed self-assembly of magnetically-functional core-shell nanoparticles in the presence of soft-magnetic template elements and a bias field. Field-directed transport and assembly of magnetic nanoparticles is currently used for a wide range of applications including drug delivery28,29, gene transfection30-32, microfluidic-based bioseparation33 and sorting34, liquid pressure sealing, levitation35-38, as well as for the fabrication of photonic39-43 and magnetic materials. Here, we extend the use of magnetics to the fabrication of crystalline core-shell particle superstructures. In this paper we demonstrate a method for controlling the self-assembly of colloidal magnetic coreshell nanoparticles into extended 3D crystalline superstructures. In this method, submicron (e.g. lithographically-patterned) template elements are used to achieve nanoscale resolution in particle placement. The templates are made from a soft-magnetic material (e.g. Permalloy, a nickel-iron alloy) and are embedded in a nonmagnetic substrate. A uniform bias field is applied to magnetize the template elements. A key feature of this method is that the template geometry can be tailored to produce highly localized regions of attractive and repulsive magnetic force to enable nanoscale control of particle motion during assembly. Figure 1 shows a Figure 1. A template-assisted self-assembly system showing colloidal core-shell nanoparticles above a nonmagnetic substrate periodic 2D array of that contains an array of embedded softelements. Colloidal magnetic cylindrical template elements. ACS Paragon Plus Environment

portion of a self-assembly system that consists of a nonmagnetic substrate with a embedded cylindrical template


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particles in suspension are first deposited onto the substrate in the absence of an external magnetic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

field. A uniform external field is then applied that saturates the soft-magnetic template elements to generate a magnetic gradient field that drives the self-assembly process. This method was introduced in our prior work, but was discussed within the limited context of the formation of 1D monolayer particle structures. Here, we extend this work and demonstrate the self-assembly of 3D crystalline particle superstructures. We use a computational model to study the detailed dynamics of the assembly process and to demonstrate proof-of-concept of the approach. The model is based on Langevin dynamics and takes into account magnetic and hydrodynamic forces including interparticle interactions, Brownian diffusion and Van der Waals force. The model predicts the trajectories of the individual particles during assembly and the particle configuration in the final assembled structure. It should be noted that a variety of other modeling approaches are commonly used to predict the field-directed transport and self-assembly of magnetic particles. These include Brownian dynamics46-48, the discrete element method47, the lattice-Boltzmann method48, Monte Carlo49,50 analysis, molecular dynamic simulations51, stochastic dynamics52 and various analytical methods53-55. In this paper we use the computational model to demonstrate template-assisted self-assembly of core-shell particles in the presence of various template geometries including cylindrical, ring-like and cubic geometries. The template elements produce a force field that focuses the particles into compact regions and produce crystalline structures with uniform hcp or fcc packing or mixed phase combinations of the two. Analytical magnetic analysis is used to optimize the dimensions of the elements for the assembly process. We use the computational model to show for the first time that 3D crystalline superstructures can be assembled within milliseconds and with nanoscale precision by tailoring key parameters. Specifically, the template geometry can be designed to produce a force field that provides precise positioning of the particles; the dimensions of the core and shell components are chosen to control the strength of the magnetic and interparticle dipole-dipole forces, and the particle volume fraction is selected to produce a desired number of particle layers while avoiding undesired aggregation during assembly. The main motivation for studying the assembly of core-shell particles is based on a need for a bottom-up method for fabricating functional 3D nanostructured materials. The core-shell structure enables a range of applications and provides a means for controlling the assembly process. With regards to applications, the magnetic component of the particle enables manipulation using an external field that drives the self-assembly process. The shell provides desired functionality: e.g. it could be dielectric or plasmonic for photonic applications, biological for bioapplications or semiconducting for nanoelectronic applications etc. With regards to the assembly process, the existence of a nonmagnetic shell layer provides an important degree of freedom by limiting the magnetic dipole-dipole force between contacting particles. We will show that as the particles assemble, there is a repulsive dipole-dipole force between neighboring particles due to their coACS Paragon Plus Environment

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aligned moments. This acts to separate the particles and reduce the crystalline packing factor. We 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

will demonstrate that the packing factor can be tuned by controlling the ratio of the dimensions of the core and shell components of the particles. To date, various experimental studies have demonstrated self-assembly of nanoparticles with nanoscale56-58 and microscale59-61 resolution using uniform and/or gradient fields as well as the field induced self-assembly of magnetic core-shell particles62. Of particular relevance is the work of Henderson et al.58. This group demonstrated the self-assembly of magnetic particles using a combination of a uniform external field and a spatially alternating gradient field, which was provided by a nano-patterned template in the form of magnetic recording media. This method is similar to ours, the differences being that the gradient field here is provided by submicron (e.g. lithographically-patterned) soft-magnetic elements and the particles have a core-shell structure, which provide advantages. Nevertheless, the prior work of Henderson et al. can serve as a partial proof-of-concept of the proposed assembly method. It should be noted that while we demonstrate the method for two-component core-shell particles and symmetric template geometries, it broadly applies to multi-layered core-shell particles that have at least one magnetic component and arbitrary template geometries. Moreover, the assembly is reversible, i.e. if the bias field is removed the magnetic and dipole-dipole forces become zero and the assembled particles will redisperse. Thus, the method holds potential for the reconfigurable assembly of functional 3D crystalline structures. Alternatively, the assembled structures can potentially be transferred intact to a polymer substrate using techniques similar to those described by Henderson et al.56 As such, the proposed method holds potential for the scalable bottom-up fabrication of functional 3D nanostructured materials at low cost and high throughput for a wide range of applications. The computational model used in this analysis is very general and can be adapted to many forms of self-assembly to provide a fundamental understanding of underlying mechanisms and enable the rational design of novel media. 2 The Computational Model In our previous work we introduced a computational model for predicting the field-directed selfassembly of colloidal magnetic core-shell particles63. The model is based on Langevin dynamics and takes into account magnetic and hydrodynamic forces including interparticle interactions, Brownian diffusion, Van der Waals force and effects of surfactants. We briefly review the model here for convenience. In the model, the motion of each particle is governed by a stochastic ODE of the form, mi

N d 2 xi F F F = + + t + ( ) ( Fdd ,ij + Fvdw,ij + Fsurf ,ij + Fhyd ,ij ) ∑ mag ,i vis ,i B ,i dt 2 j =1

(i =

1,..., N )

(1)

j ≠i

where mi and xi (t ) are the mass and position of the i’th particle. The right-hand-side of Eq.(1) represents the sum of forces on the i’th particle. The first three terms are the magnetic force Fmag ,i due to an external magnetic field (including gradient fields), a viscous (Stokes) drag force Fvis ,i and ACS Paragon Plus Environment


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a stochastic force FB ,i (t ) to account for Brownian motion. The summation term in Eq.(1) accounts for interparticle interactions: Fdd ,ij the magnetic dipole-dipole force; Fvdw,ij Van der Waals force; Fsurf ,ij a repulsive force due to surfactant contact between particles and Fhyd ,ij due to interparticle

hydrodynamic interactions. These interparticle forces give rise to a coupled system of ODEs, one equation for each colloidal particle. We reduce the system of second-order equations to a system of two coupled first-order equations for the velocity and displacement of the particles. We solve these using a dynamic time-stepping approach that greatly accelerates and stabilizes the solution. The discretized coupled equations are as follows,

vi , f = = Dxi

F  +  vi ,0 − sum ,i D D 

Fsum ,i

Fsum ,i D

τ+

Fsum ,i mi   vi ,0 D D

D

 − mi τ e 

(2)

  - mi τ  1-e  D

   

(3)

where D = 6ph Rhyd , p is the drag coefficient (η is the fluid viscosity and Rhyd,p is the hydrodynamic radius of the particle), τ is the integration time step, vi ,0 , and vi , f are the velocity of the i’th particle at the beginning and end of the time step and Fsum ,i Fmag ,i + FB ,i (t ) + ∑ ( Fdd ,ij + Fvdw,ij + Fsurf ,ij + Fhyd ,ij ) = N

(4)

j =1 j ≠i

In our analysis τ is dynamically adjusted based on the relative velocities and surface-to-surface separations hij of the particles. When

vi , f =

Fsum,i D

 

is large enough τ >>

mi   , Eq.(2) can be simplified to D

. The various force terms in the model are described in detail in our previous work. We

briefly summarize these terms here for convenience. 2.1 Magnetic force The magnetic force is predicted using an “effective” dipole moment method in which the particle is modeled as an “equivalent” point dipole with an effective moment meff . The force on the i’th particle is given by53

= Fmag ,i m f ( mi ,eff ⋅ ∇ ) H a

(5)

where µ f is the permeability of the fluid. H a is the applied magnetic field intensity at the center of particle. Here, H a is a superposition of a uniform bias field and template-induced gradient-fields, i.e. ACS Paragon Plus Environment

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H a = H bias + H template . The moment is given by meff = Vp M p where V p and M p is the volume and magnetization of the particle, respecfully. For magnetic-dielectric core-shell particles, only the core contributes to that magnetic force and consequently, meff = Vcore M p . The moment can be determined using a magnetization model that takes into account self-demagnetization and magnetic saturation of the particles53,54

mi ,eff = Vcore f ( H a ) H a

(6)

where64    f ( Ha ) =    

3( χ p − χ f

(χ

p

)

+ 2χ f ) +3

M sp / H a

 ( χ p + 2χ f ) +3  M Ha <   3 ( χ p − χ f )  sp    ( χ p + 2χ f ) +3  M Ha ≥   3 ( χ p − χ f )  sp  

(7)

In this expression, χ f is the susceptibility of the fluid and χ p is the intrinsic magnetic susceptibility of the particle, i.e. M p = χ p H in where Hin is the field inside the particle. H in differs from H a by the demagnetization field i.e. H= H a − N d M p where N d is the demagnetization factor of the particle, in i.e. N d = 1/ 3 for a spherical particle. The value of χ p can be obtained from a measured M vs. H curve. However, M is often plotted as a function of H a in which case M p = χ a H a , where χ a is the apparent susceptibility. The two values of susceptibility are related as follows= χ p χ a / (1 − N d χ a ) , which reduces = to χ p 3χ a / (3 − χ a ) for a spherical particle65. Thus, the magnetic force can be rewritten as

= Fmag m f Vcore f ( H a )( H a ⋅∇ ) H a

(8)

We use analytical closed-form expressions to predict the fields and force for the template geoemetries. 2.2 Magnetic dipole-dipole interaction The dipole-dipole force in Eq.(1) is obtained from the gradient of a potential U dd ,ij ,

Fdd ,ij = −∇U dd ,ij

(9)

where

µ f  ( µi ,eff ⋅ rij )( µ j ,eff ⋅ rij ) µµ i , eff ⋅ j , eff 3 U dd ,ij = − − 5 3  4π rij rij  ACS Paragon Plus Environment

   

(10)


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Figure 2. Magnetic interaction between two magnetic particles with co-aligned dipoles induced by a uniform applied field: (a) geometry showing relative position of interacting dipoles and direction of applied field, (b) normalized (unit vector) dipole-dipole force field, (c) normalized dipole-dipole force components as a function of relative angular position θ as shown in (a).

and mi ,eff and m j ,eff are the moments of i’th and j’th particle, respectively, and rij is the 6 for identical core shell particles. It is displacement vector between them. Note that Fdd ,ij ∝ Rcore

instructive to consider the nature of the dipole-dipole force between identical paricles that have coalligned moments that are induced by a uniform applied field as shown in Fig. 2a,b. One particle is located at the origin of a reference frame and we study the dipole-dipole force on a second particle as a function of its location in the x-y plane. Figure 2b shows the normalized dipole-dipole force field, i.e. a unit vector for the force on the second particle when it is located at any point in the x-y plane. Note that the dipole-dipole force can be attractrive or repulsive depending on the location of the particle, e.g. it is attractive when the particle is located at any point on y axis and repulsive when it is located on the x-axis. Figure 2c further illustrates the nature of the interparticle force (attractive or repulsive) with the particles at a fixed separation distance as a function of the angular coordinate θ of second particle, taken with respect to the y-axis. 2.3 Van der Waals interaction Van der Waals force is taken into account as an attractive force, which is calculated using66, Fvdw,ij =

di6 A 6 (hij2 + 2di hij ) 2 (hij + di )3

(11)

here A is the Hamaker constant and hij is the surface-to-surface separation distance between the i’th and j’th particle. 2.4 Surfactant force The repulsive force between two particles due to surfactant-surfactant contact is derived from a potential U s ACS Paragon Plus Environment

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Fsurf ,ij = −∇U s

(12)

Where67  rij − 2 R p rij 2 R p + 2δ  = U s 2p R p2 N s k BT 2 − − ln( ) δ δ rij  

(13)

and R p , δ and N s are the radius of particle, the thickness of the surfactant layer and the surface density of surfactant molecules, respectively. 2.5 Viscous drag The drag force on a particle is computed using Stokes’ formula Fvis,i = D

dxi dt

(14)

Where D = 6ph Rhyd , p is described above in Eq.(2). 2.6 Interparticle Hydrodynamics Interactions Hydrodynamic interactions between particles become important at small surface-to-surface separation distances. The force between two neighboring particles is based on lubrication theory and given by66, Fhyd ,ij

6πµ f Vr ,i , j di2 = hij 16

(15)

where hij is the separation bewteen the surfaces and Vr ,i , j is the relative velocity between the particles. When the particles are in contact ( hij ≤ 0 ) this force is considered to be negligible. 2.7 Brownian motion The Brownian force was modeled as a Gaussian white noise process. The magnitde of this force in one dimension is

FB = ξ

2 Dk BT Dt

(16)

Where k B is Boltzmann’s constant, D is is the Stokes’ drag coefficient as described above and ξ is a random number with a Gaussian distribution. The 3D Brownian force is obatined by applying Eq. (16) for each force component.. 3 Analysis of Self-assembly



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We use the computational model to study the detailed dynamics of the 3D self-assembly of core1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

shell particles. Three different template geometries are considered: cylindrical, ring-like and cubic structures. To demonstrate proof-of-concept we choose Fe3O4-SiO2 core-shell particles, but note that the method broadly applies to multi-layered core-shell particles that have at least one magnetic component. Fe3O4 has a density rcore = 5000 kg/m3 and a saturation magnetization M sp = 4.78×105 A/m. The SiO2 shell has a density ρ shell = 2648 kg/m3. The template elements are chosen to be permalloy (78% Ni, 22% Fe), which has a saturation magnetization M e, s = 8.6 × 105 A/m. The bias field is taken to be H bias = 3.9 × 105 A/m ( Bbias = 5000 Gauss), which is sufficiently strong to saturate both the nanoparticles and the template elements as discussed below. This field can be obtained by positioning a commercial rare-earth NdFeB magnet immediately beneath the substrate54. The effective dipole moment of the saturated particles is meff = Vcore M sp . To simplify the analysis, we assume that the hydrodynamic radius of the particles is the same as their physical radius. However, in general, it is larger because of the presence of surfactants on the surface. The carrier fluid is assumed to be nonmagnetic ( χ f = 0 ), with a viscosity and density equal to that of water,

η = 0.001 N⋅s/m2 and ρ f = 1000 kg/m3⋅ 3.1 Cylindrical Template Elements We first consider self-assembly using a soft-magnetic cylindrical template element. To demonstrate the analysis, we choose a cylinder with a radius Rm = 200 nm, let h denote its height. The spacing of the elements is taken to be 1 µm center-to-center so that there is negnigible overlap of their fields, which we verify below. Thus, it suffices to perform the template design for a single isolated element. Analytical closed-form expressions for the field and force of a soft-magnetic cylindrical element in a uniform bias field can be found in in our previous work63. It should be noted that in all of this work we choose the bias field that is sufficiently strong to completely saturate the soft-magnetic elements. 3.1.1. Magnetic Field Analysis We first determine a template height h that provides a viable force field for assembly. For the force analysis, a reference frame is chosen with the x-y plane coincident with the surface of the substrate and with the z axis

Figure 3. Axial magnetic force along a aligned with the axis of the element. Thus, the top horizontal line above the template element as a function of the height h. Rm is radius of the surfaces of both the substrate and the template element element. ACS Paragon Plus Environment

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are at z = 0. We use field and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

force equations from our previous work63 to compute the radial and axial force components Fmag ,r and

Fmag , z across a horizontal line that spans a unit cell of the system and falls along the diameter of the element, i.e. -500mm ≤ x ≤ 500nm. The

force

components

are

computed at a distance z = 100 nm above the element for a range of element heights: h = 100, 200, 300 and 400 nm. The force profiles for Fmag , z are plotted in Fig. 3 and show relatively little change for h ≥ 300nm.

Figure 4. Axisymmetric magnetic field and force components at z=100 nm above the template element: (a) Br, (b) Bz, (c) Fmag,r, and (d) Fmag,z

Thus, we choose the

element height to be h = 300 nm. Note that the force profiles are axisymmetric because of the cylindrical symmetry of the template geometry as shown in Fig. 4. Also, recall that this analysis is based on the assumption that the template element is saturated. This occurs when H bias ≥ N d M es , where Nd is the demagnetization factor of the element63. For the chosen element dimensions (Rm = 200 nm, h = 300 nm), Nd = 0.46 as described in the literature68 and it follows that the element is saturated because

H bias > 0.4126 M es =3.96 × 105 A/m ( Bbias ≥0.5T). Next, the magnetic force provided by a template element with dimensions: h = 300 nm and Rm = 200 nm is analyzed in more detail. The x and z components of the force Fmag , x and Fmag , z are computed along the same horizontal line as above, i.e. -500 nm ≤ x ≤ 500 nm, for a range of distances Figure 5. Magnetic force along horizontal lines 100, 200 and 300 nm above the template (arrows indicate direction of force): (a) Fmag,x and (b) Fmag,z. ACS Paragon Plus Environment

above the element, z = 100, 200 and 300 nm. The


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force profiles are plotted in Fig. 5 and the analysis shows that there is a relatively strong attractive 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(downward-directed) axial force Fmag , z over the center of the element, x/Rm ≤ 1 as indicated by the red arrows, which promotes assembly in this region. Significantly, there is also a relatively weak repulsive (upward-directed) axial force above near the exterior edge of cylinder (r = Rm) as indicated by the blue arrows, which prevents particles from assembling there. The repulsive force is due to the superposition of the uniform bias field and the gradient-field of the element. Specifically, the bias field induces an upward-directed moment meff zˆ , whereas the magnetized element produces a field gradient ∇H e that changes sign throughout the computational domain. Since the force is proportional to the product of these two terms as shown in Eq.(5), it is attractive (negative) in regions where these terms have opposite signs and repulsive (positive) when they have the same sign. The radial field component Fmag ,r also plays a critical role in focusing the particles during the assembly process. Note from Fig. 4c that this force component peaks (in a negative sense) above the radial edge of the element and therefore acts to move the particles inward towards the center of the element. This is also reflected in the plot of Fx, which shows that the force is directed inward to the center of element as indicated by the blue arrows in Fig. 5a. Thus, the magnetic force directs particles towards the center of the template during assembly, which promotes the formation of a tapered 3D pyramid structure. It is important to note that the ability to produce regions of attractive and repulsive magnetic force is a key feature of the proposed assembly method that enables nanoscale precision of particle placement. It is instructive to investigate the impact of neighboring template elements on the magnetic force. Heretofore, we have assumed that there is negligble overlap of the magnetic force gnerated by neighboring elements, but it remains to verify this. To this end, we compute the axial force field Fmag,z(x,y,z) for a 3 by 3 array of elements (1 mm center-to-center spacing) at a distance z = 100 nm above the elelemnts. The force distribution, which is shown in Fig. 6, is computed by first obtaining

the

total

field

via

superposition of the individual element fields and then computing the force using the total field. The analysis shows that the force falls off

dramatically

elements

and

between that

there

the is

Figure 6. Magnetic force field Fmag,z(x,y,z) at z = 100 nm above a 3 by 3 array of template elements.

negligible overlap of their effects.


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3.1.2 Self-Assembly Dynamics In

this

section

we

use

the

computational model to study the dynamics of the assembly process. The

bias

field

and

template

dimensions are as above. We use a computational

domain

centered

with respect to a single element. The domain spans a unit cell, i.e. 1 mm along both the x and y axes and 1 mm in the z-direction as shown in Fig. 7. The base of the domain is at z = 0, which coincides with the top surface of the substrate and the template

element.

Periodic

Figure 7. Initial and final particle distributions for a cylindrical template element: (a) initial random particle distribution, (b) perspective of final assembled particle structure, (c) lateral view of assembled particle structure, (d) field-directed assembly of nonmagnetic beads in a ferrofluid using cylindrical traps20 (adapted with permission).

boundary conditions for particle transport are imposed at the lateral sides of the domain to account for a 2D array of template elements. We simulate the assembly of 40 nm Fe3O4-SiO2 particles with an 18 nm Fe3O4 core. Note that the assembled structure shown in Fig 7c has a uniform hcp crystalline structure that is tapered due to focusing by the magnetic force as described above. This structure is very similar to those observed by Yellen et al. as shown in Fig. 7d20. This group studied the assembly of non-magnetic submicron (500nm diameter) beads that were suspended in a ferrofluid

(10

nm

nanoparticles).

These

beads

acquire

an

effective

dipole

moment

meff ,bead = − M ff Vbead due to the volume of ferrofluid they displace, where M ff is the net

magnetization of ferrofluid. Note that the effective moment is negative, which means that the nonmagnetic beads experience a magnetic force that is opposite to that of magnetic beads. They also experience an interparticle dipole-dipole force and self-assemble towards regions of a weaker field, again opposite to that of magnetic particles. Yellen et al. observed that the beads assembled into a tapered crystalline structure (Fig. 7d) over cylindrical magnetic traps (similar to our template elements albeit larger in diameter and thinner) that were designed to provide a weaker field than their surrounding medium. A comparison of Fig. 7d with Figs 7c, and 9 shows that the observed structures are similar to the ones that we predict. This is expected because the equations that govern the assembly for the two types of particles are essentially the same.



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Figure 8. Self-assembly dynamics: sequence of simulation images showing the positions of the particles at various times during the assembly process.

The model that we use predicts the trajectories of each particle during the assembly process. Fig. 8 consists of a time sequence of simulation images showing the positions of the particles at various stages of assembly. Note that in this case the final assembly was completed within 20 ms. An animation video of the assembly process can be found in the supporting documentation. The individual layers in the assembled structure are shown in Fig. 9. The particles in the various layers are color coded for clarity. Note that these first four layers are well-formed with no defects (i.e. interparticle gaps). However, when the particle layers are sufficiently far away from the template element, defects in crystalline structure can occur, which are due to a decrease in effect of the gradient field force that tends to pack the particles. Specifically, as the distance from the template element increases, the force due to the gradient field becomes weaker while the

magnetic

dipole-dipole

force,

which is repulsive for co-aligned neighboring

particles,

remains

relatively constant because the bias

field is sufficient to saturate the particles. Thus, template experience

farther from

element, a

the

more

the

particles dominant

interparticle repulsion and assemble with more space between them, which

Figure 9. Multi-layer hexagonal packed structure. (a), (b), (c) and (d) are indicated as 1st, 2nd, 3rd and 4th layer of the particles’ structure after assembly. The particle core is shown as a solid sphere and the shell is shown as transparent.


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Figure 10. Orientation of magnetic moments in the nanoparticles (red) in the assembled structure and the external field distribution (blue).

can produces gap defects in the crystalline structure. The orientations of the dipole moments of the particles in the assembled structure are shown in Fig. 10. Note that the moments are co-alligned due to the applied field. which is also shown. Thus, the dipole-dipole force between neighboring particles in the same layer is repulsive, which would prohibit the assembly if it were not for the dominant gradient field force provided by the magnetic template element. A carefull analysis of the magnetic forces is required in order to ensure the formation and stability of a desired crystaliine superstructure. The crystalline structure that forms during assembly depends on various factors including the template geometry, the core-shell properties and dimensions, and the particle volume fraction. These parameters can be tuned to achieve crystalline superstructures with desired features, i.e. geometry, resolution and packing. For example, we studied the packing factor PF for the hcp structure of Fig. 9 as a function of the ratio Rcp of the radius of the Fe3O4 core Rcore to that of the whole particle Rp. A plot of this relationship is shown in Fig. 11. This plot shows that the PF is maximum when Rcp is approximately 0.45. As Rcp decreases below this value, the spacing between particles increases due to a weaker magnetic force provided by the template element, which is 3 proportional to Rcore . As Rcp increases above the

optimal value, the repulsive dipole-dipole force, 6 core

which is proportional to R

, dominates the

Figure 11. The packing factor of the assembled structure vs. the ratio of Fe3O4 core radius to the particle radius (the blue line is a least squares fit to the computed red dot values).

attractive magnetic force, which is proportional



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3 to Rcore . Thus, the repulsive force tends to increase the separation between the particles, which results

in a larger interparticle gap and lower PF. 3.2 Other Templates Elements We have demonstrated the assembly process for a cylindrical template element. However, as noted above, the approach broadly applies to arbitrary template geometries. To demonstrate the flexibility of the assembly process we analyze two other template geometries, a hollow cylindrical ring-like element and a cubic structure. 3.2.1 Cylindrical Ring Element Consider a ring-like cylindrical element with an inner radius Rin = 100 nm, an outer radius Rout = 200 nm and a height h = 200nm (Fig. 12a). This element has the properties of permalloy and will be saturated in the 0.5T bias field that we used above63. We study self-assembly for this system using a 30 nm Fe3O4-SiO2 (Dcore = 16 nm) particle for the simulation. The smaller particles are chosen here to provide nanoscale resolution in the assembly, which occurs over the 100 nm wide annulus of the ring structure. The z-component of the magnetic force Fmag,z for this system as seen by the first layer of particles (i.e. at z = 15 nm above the element) is plotted in Fig. 12b. The field and force for this structure are obtained using analytical closed-form expressions63. Note that the force is downward (i.e. attractive) towards the surface of the template (i.e. negative, blue) above the annulus of the template and upward (i.e. repulsive, positive, red) in the remainder of the computational domain. Fig. 12e shows details of the progression of the formation of three full layers and a partial fourth layer of particles. Note that the particles assemble into a mixture of hcp and fcc crystalline structures over the annulus as shown in Figs. 12d, e. On explanation for this mixed phase assembly is that the hcp and fcc patterns are geometrically and energetically similar and may be randomly seeded during the

Figure 12. Initial and final particle distributions for a ring-like cylindrical template element: (a) cylindrical ring geometry and magnetization, (b) axial magnetic force (blue is downward/attractive, red is upward/repulsive), (c) initial random particle distribution, (d) lateral view of assembled particle structure, (e) formation of multilayered crystalline structure where the particles are color coded by layer.


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formation of a superstructure due to a weaker gradient of magnetic field for a ring-link cylindrical 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

structure and Brownian motion of the particles as they seat themselves within the structure. 3.2.2 Cubic Structure We now consider a cubic template element as shown in Fig. 13a. This element is 300 nm on a side and assumed to have the properties of permalloy. The particles are 40 nm Fe3O4-SiO2 (Dcore = 15 nm) and the computational domain spans 1 μm along both the x and y axes and 1 μm in the zdirection. The magnetic force for this structure as seen by the first layer of particles (at z = 20 nm) is shown in Fig. 13b. The field and force for this structure are obtained using analytical closed-form expressions, which are given in the Appendix (in supporting information). Note that the force is downward over the surface of the template with more pronounced attractions near the edges (i.e. negative, yellow to blue), and upward (positive, red) in the remainder of the computational domain.

Figure 13. Initial and final particle distributions for a cubic template element: (a) template geometry and magnetization, (b) axial magnetic force (blue is downward/attractive, red is upward/repulsive), (c) initial random particle distribution, (d) perspective of assembled particle structure, (e) formation of multi-layered crystalline structure where the particles are color coded by layer.

The initial and final particle distribution for this template element are shown in Figs. 13c,d. Fig. 13e shows details of the progression of the formation of four layers of particles. Note in this case that the particles assemble into a uniform fcc pattern. 4 Discussion

The analysis above demonstrates the feasibility of using template-assisted self-assembly of magnetic core-shell nanoparticles to create 3D crystalline superstructures that have nanoscale feature resolution. The predicted results are consistent with experimental data both in terms of the assembled geometry and crystalline structure (see Fig. 9d and related discussion20). The computational model used for the analysis enables the rational design of such structures from first principles via determination of key parameters that control the assembly process: notably, the template geometry, the particle properties (especially the core-shell dimensions), and the particle ACS Paragon Plus Environment


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volume fraction. The template geometry defines the extent and overall geometry of the particle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

assembly. Sub-micron templates (e.g. lithographically patterned) can be used to provide nanoscale feature resolution. Specifically, the template geometry can be tailored to produce a gradient field that provides highly localized regions of attractive and repulsive magnetic force. The particle trajectories become focused in the force field for precise placement during assembly. It is important to note that the ability to create nanoscale regions of attractive and repulsive magnetic force is a key feature of the assembly method and is due to the combination of the uniform bias field with the localized gradient fields. If the template elements were permanent magnets and there were no bias field, the magnetic force would be purely attractive and the ability to focus the particles during assembly would be degraded along with the spatial resolution of the assembled structure. The bias field should be strong enough to induce magnetic saturation in both the template elements and the particles. The saturation field is typically on the order of 0.5-1T and can be readily obtained using standard rare earth NdFeB permanent magnets. The choice of template material is also 2

important because the magnetic force is a nonlinear function of its magnetization Me, i.e. Fmag ~M e . Thus, a higher saturation magnetization produces a stronger magnetic force, which enables more precise control of the assembly. With regards to the particle properties, both the core and shell properties are important. The magnetization Mp and especially the radius of the core Rcore directly 3

2

6

impact the magnetic and dipole-dipole forces, i.e. Fmag ~M p Rcore and Fdd ~M p Rcore . These parameters can be tuned to control the particle motion and interparticle coupling during assembly. The shell thickness also impacts the dipole-dipole force, but only when the particles are in contact. It can be increased to reduce Fdd in order to suppress undesired chaining and reduce the prevalence of defects in the crystalline structure. The overall particle size also impacts the hydrodynamic forces, e.g. larger particles exhibit greater viscous drag and assemble more slowly. Since we observed mixed phase fcc and hcp crystalline structures in the ring structure assembly, we hypothesized that these structures must represent similar energy states. To check this hypothesis, we calculated the energy of two representative particle configurations, each of consisted of 13 particles. We computed the magnetostatic energy of both configurations assuming that they were positioned above the center of a cylindrical soft-magnetic template element. The magnetostatic energy of each assembly is computed using = Emag

∑ ∑

U + ∑ i= m ⋅ Ba ,i , where U dd ,ij is the potential i= 1 j= i +1 dd ,ij 1 i N

N

N

energy due to dipole-dipole interaction between the i’th and j’th particles as given in Eq. (10) and mi ⋅ Ba ,i is the energy of the i’th dipole due to its interaction with the applied field Ba ,i , which is

evaluated at the centre of the particle. The difference in energy between the two configurations is approximately 0.2% of the total magnetostatc energy. Significantly, the difference in energy is on the same order as the thermal kinetic energy of a particle, i.e. 3/2kBT. Thus, we conclude that the hcp ACS Paragon Plus Environment

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and fcc structures have similar energy and can be randomly seeded as the particles form the third 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

layer of the assembly. Finally, the particle volume fraction can be adjusted to control the number of layers in the assembled structure and aggregation during the assembly process. Specifically, as the volume fraction increases the number of layers in the structure increases. However, the dispersed particles are closer together and can aggregate during assembly, which interferes with the formation of the crystalline structure. Thus, the volume faction must be high enough to achieve a desired multilayered structure, but low enough to avoid aggregation. All the aforementioned parameters can be determined for a given application using a combination of magnetic field modeling and particle transport modeling as demonstrated above. 5 Conclusions We have presented a method for directing the assembly of colloidal magnetic-dielectric core-shell nanoparticles into extended 3D crystalline structures with nanoscale precision. Sub-micron softmagnetic template elements are used to guide the assembly in the presence of a uniform bias field. The use of a uniform bias field combined with localized high gradient fields produced by the template elements is a key feature of the method that enables nanoscale precision of particle placement, which translates to nanoscale feature resolution in the assembled structure. We demonstrate proof-of-concept of the method for various template geometries using a computational model that takes into account the dominant assembly mechanisms. The model can be used to study critical details of the assembly process including the trajectories of individual particles, the structure of the final assembly and the time required to complete it. Our predicted results are consistent with experimental observations20 and we have shown for the first time that 3D crystalline core-shell particle structures with controllable nanoscale features can be assembled within milliseconds. We have further shown that during this time, only the particles within a distance of one micron from the substrate effectively contribute to the assembly process for the template geometries that we studied. The particles at distances greater than one micron experience a relatively weak magnetic force. Their motion is dominated by Brownian dynamics and they do not reach the substrate during the assembly process. Moreover, we have theoretically demonstrated for the first time that the core-shell particles preferentially self-assemble into hcp, fcc and mixed phase crystalline structures in the presence of high gradient magnetic fields, which act to focus the particles into compact regions, e. g. over the surface of the template elements. We attribute this to the fact that hcp and fcc structures have a similar configuration of particles and hence a similar magnetic energy, which is lower than that of less densely packed crystalline structures such as body centered cubic (bcc) etc. The mixed hcp-fcc phase assembly that we predict is due to that fact that the particle arrangement and hence magnetic energy of hcp and fcc structures are very similar and can be randomly seeded during the assembly process due to the stochastic motion of the particles as they become seated in the assembly. Lastly, it ACS Paragon Plus Environment


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should be noted that the magnetic force that focuses the particles is countered by the magnetic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

dipole-dipole force, which is repulsive for particles near the surface of the template due to their coaligned dipole moments. We have found that the dipole-dipole force can be reduced by increasing the thickness of the nonmagnetic shell of the particle, which enables a more densely packed structure. Other experimental and theoretical studies have demonstrated template-directed self-assembly of magnetic particles20,56,59-61, but ours is the first study to demonstrate the feasibility of assembling 3D crystalline core-shell particle superstructures with nanoscale resolution using sub-micron (e.g. lithographically patterned) soft-magnetic templates. The nanoscale resolution is achieved by tailoring the template geometry to produce a force field that focuses the particles into prescribed structures during assembly. The packing factor and crystalline defects can be controlled by tuning the size of the magnetic core to control the interplay between the magnetic and dipole-dipole forces. The particle volume fraction can be adjusted to achieve a desired number of particle layers while avoiding aggregation during assembly. While we have demonstrated the assembly method using cylindrical, ring-shaped and cubic template geometries and dual-layer core-shell particle, it is very versatile and broadly applies to arbitrary template geometries and multiple-layer core-shell particles with at least one magnetic component. Moreover, the assembly process is reversible, i.e. the particles can be repeatedly be assembled and redispersed by applying and removing the bias field. This enables the reversible creation of functional media on demand. Alternatively, the assembled structures can potentially be transferred intact to a substrate to form functional thin films using techniques similar to those described by Henderson et al.56 as described in our previous work.63 Thus, a single template substrate could be used to reproduce numerous nanostructured materials. The ability to produce such materials using bottom up self-assembly opens up opportunities for the bottom-up scalable high-throughput fabrication of functional nanostructured materials with unprecedented properties for a broad range of applications. In addition, the computational model that we present is very general and can be adapted to many other self-assembly processes to provide a fundamental understanding of underlying mechanisms and enable the rational design of novel media.

Acknowledgement The authors acknowledge financial support from the U.S. National Science Foundation, through award number CBET-1337860. Supporting Information Available This information is available free of charge via the Internet at http://pubs.acs.org


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AUTHOR INFORMATION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Corresponding Author Edward Furlani Dept. of Chemical and Biological Engineering Dept. of Electrical Engineering 113B Davis Hall University at Buffalo (SUNY) Buffalo, New York 14260-4200 Email: [email protected]; phone: (716) 645-1194; Fax: (716) 645-3822. Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. These authors contributed equally: Xiaozheng Xue1 and Edward P. Furlani1,2. 1

Dept. of Chemical and Biological Engineering, University at Buffalo SUNY, NY 14260

2

Dept. of Electrical Engineering, University at Buffalo SUNY, NY 14260

Funding Sources The authors acknowledge financial support from the U.S. National Science Foundation, through award number CBET-1337860.



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60. Yellen, B. B.; Friedman, G., Analysis of Repulsive Interactions in Chains of Superparamagnetic Colloidal Particles for Magnetic Template-based Self-assembly. Journal of Applied Physics 2003, 93, 8447-8449. 61. Yellen, B. B.; Friedman, G., Programmable Assembly of Colloidal Particles Using Magnetic Microwell Templates. Langmuir 2004, 20, 2553-2559. 62. Malik, V.; Petukhov, A. V.; He, L.; Yin, Y.; Schmidt, M., Colloidal Crystallization and Structural Changes in Suspensions of Silica/Magnetite Core-Shell Nanoparticles. Langmuir 2012, 28, 14777-14783. 63. Xue, X.; Furlani, E. P., Template-assisted Nano-patterning of Magnetic Core-shell Particles in Gradient Fields. Physical Chemistry Chemical Physics 2014, 16, 13306-13317. 64. Furlani, E. P.; Ng, K. C., Analytical Model of Magnetic Nanoparticle Transport and Capture in the Microvasculature. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 2006, 73, 061919/1-061919/10. 65. Furlani, E. P.; Sahoo, Y.; Ng, K. C.; Wortman, J. C.; Monk, T. E., A Model for Predicting Magnetic Particle Capture in A Microfluidic Bioseparator. Biomedical Microdevices 2007, 9, 451-463. 66. Russel, W. B.; Saville, D. A..; Schowalter, W. R., Colloidal Dispersions. Cambridge University Press: Cambridge UK, 1989. 67. Rosensweig, R. E., Fluid Dynamics and Science of Magnetic Liquids. Academic Press, New York (In Advances in Electronics and Electron Physics) 1979, 48, 103-199. 68. Beleggia, M.; Vokoun, D.; De Graef, M., Demagnetization Factors for Cylindrical Shells and Related Shapes. Journal of Magnetism and Magnetic Materials 2009, 321, 1306-1315. 69. Furlani, E. P., Permanent Magnet and Electromechanical Devices; Materials, Analysis and Applications. Academic Press: New York, 2001. 70. Furlani, E. P.; Xue, X., A Model for Predicting Field-Directed Particle Transport in the Magnetofection Process. Pharmaceutical Research 2012, 29, 1366-1379. 71. Furlani, E. P., A Method for Predicting the Field in Permanent Magnet Axial-field Motors. IEEE Transactions on Magnetics 1992, 28, 2061-2066. 72. Craik, D., Magnetism: Pinciples and Applications. John Wiley & Sons: New York, 1995; pp 294-376. 73. Furlani, E. P., Computing the Field in Permanent-magnet Axial-field Motors. IEEE Transactions 1994, 30, 3660-3663. 74. Moon, F. C., Magnetosolid Mechanics. John Wiley & Sons: New York, 1984. 75. Furlani, E. P., Field Analysis and Optimization of NdFeB Axial Field Permanent Magnet Motors. IEEE Transactions on Magnetics 1997, 33, 3883-3885. 76. Furlani, E. P., A Formula for the Levitation Force between Magnetic Disks. IEEE Transactions on Magnetics 1993, 29, 4165-4169.



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203x203mm (300 x 300 DPI)


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Figure 1. A template-assisted self-assembly system showing colloidal core-shell nanoparticles above a nonmagnetic substrate that contains an array of embedded soft-magnetic cylindrical template elements. 95x71mm (300 x 300 DPI)



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Figure 2. Magnetic interaction between two magnetic particles with co-aligned dipoles induced by a uniform applied field: (a) geometry showing relative position of interacting dipoles and direction of applied field, (b) normalized (unit vector) dipole-dipole force field, (c) normalized dipole-dipole force components as a function of relative angular position θ as shown in (a). 127x42mm (300 x 300 DPI)


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Figure 3. Axial magnetic force along a horizontal line above the template element as a function of the height h. Rm is radius of the element. 137x149mm (300 x 300 DPI)



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Figure 4. Axisymmetric magnetic field and force components at z=100 nm above the template element: (a) Br, (b) Bz, (c) Fmag,r, and (d) Fmag,z 127x127mm (300 x 300 DPI)


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Figure 5. Magnetic force along horizontal lines 100, 200 and 300 nm above the template (arrows indicate direction of force): (a) Fmag,x and (b) Fmag,z. 232x426mm (300 x 300 DPI)



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Figure 6. Magnetic force field Fmag,z(x,y,z) at z = 100 nm above a 3 by 3 array of template elements. 63x31mm (300 x 300 DPI)


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Figure 7. Initial and final particle distributions for a cylindrical template element: (a) initial random particle distribution, (b) perspective of final assembled particle structure, (c) lateral view of assembled particle structure, (d) field-directed assembly of non-magnetic beads in a ferrofluid using cylindrical traps20 (adapted with permission). 101x81mm (300 x 300 DPI)



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Figure 8. Self-assembly dynamics: sequence of simulation images showing the positions of the particles at various times during the assembly process. 152x76mm (300 x 300 DPI)


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Figure 9. Multi-layer hexagonal packed structure. (a), (b), (c) and (d) are indicated as 1st, 2nd, 3rd and 4th layer of the particles’ structure after assembly. The particle core is shown as a solid sphere and the shell is shown as transparent. 127x127mm (300 x 300 DPI)



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Figure 10. Orientation of magnetic moments in the nanoparticles (red) in the assembled structure and the external field distribution (blue). 846x396mm (96 x 96 DPI)


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Figure 11. The packing factor of the assembled structure vs. the ratio of Fe3O4 core radius to the particle radius (the blue line is a least squares fit to the computed red dot values). 95x71mm (300 x 300 DPI)



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Figure 12. Initial and final particle distributions for a ring-like cylindrical template element: (a) cylindrical ring geometry and magnetization, (b) axial magnetic force (blue is downward/attractive, red is upward/repulsive), (c) initial random particle distribution, (d) lateral view of assembled particle structure, (e) formation of multi-layered crystalline structure where the particles are color coded by layer. 50x20mm (300 x 300 DPI)


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Figure 13. Initial and final particle distributions for a cubic template element: (a) template geometry and magnetization, (b) axial magnetic force (blue is downward/attractive, red is upward/repulsive), (c) initial random particle distribution, (d) perspective of assembled particle structure, (e) formation of multi-layered crystalline structure where the particles are color coded by layer. 50x20mm (300 x 300 DPI)


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