Template-assisted nano-patterning of magnetic core–shell particles in gradient fields

13 Pages·2014·1.91 MB·English

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Page 1 of 12 Physical Chemistry Chemical Physics
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Controlling the assembly of colloidal magnetic core-shell particles into patterned monolayer structures with
nanoscale feature resolution using soft-magnetic template elements A
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Physical Chemistry Chemical Physics Page 2 of 12
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RSCPublishing t
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ARTICLE
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Template-assisted Nano-patterning of n
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Magnetic Core-shell Particles in Gradient Fields
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Cite this: DOI: 10.1039/x0xx00000x
Xiaozheng Xuea and Edward P. Furlania,b,
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A method is proposed for controlling the assembly of colloidal magnetic core-shell
Received 00th January 2012, t
Accepted 00th January 2012 nanoparticles into patterned monolayer structures with nanoscale feature resolution. The p
method is based on magnetic field-directed self-assembly that is enhanced using soft-magnetic e
DOI: 10.1039/x0xx00000x
template elements. The elements are embedded in a nonmagnetic substrate and magnetized
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www.rsc.org/ using a uniform bias field. A key feature of this approach is the combined use of a uniform
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field with induced gradient-fields produced by the template elements. This enables the
A
customization of a force field with localized regions of attractive and repulsive magnetic force
that provide extraordinary control of particle motion during assembly. The method is
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demonstrated using a computational model that simulates the assembly process taking into
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magnetic and hydrodynamic forces including interparticle interactions, Brownian diffusion,
Van der Waals force and effects of surfactants. The analysis shows that extended geometric i
s
patterns of particles can be assembled with nanoscale resolution, beyond that of the template
y
elements, within milliseconds. This is achieved by tailoring key parameters including the
h
template geometry to produce a force field that focuses the particles into prescribed patterns;
the thickness of the dielectric particle shell to control the magnetic dipole-dipole force upon P
contact and the particle volume fraction to suppress undesired aggregation during assembly.
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The proposed method broadly applies to arbitrary template geometries and multi-layered core-
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shell particles with at least one magnetic component. It can enable the self-assembly of
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complex patterns of nanoparticles and open up opportunities for the scalable fabrication of
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multifunctional nanostructured materials for a broad range of applications. m
Keywords: Field-directed assembly of core-shell nanoparticles, template-assisted self- e
assembly of magnetic-dielectric core-shell nanoparticles, magnetic dipole-dipole interactions. h
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1 Introduction been demonstrated for the assembly of single17-22 and multiple
y
component23,24 colloids into extended patterns. Some methods
The interest in the field-directed manipulation of colloidal rely on directed23,25 or template-assisted26 assembly or r
magnetic particles has grown dramatically over the years due to combinations thereof. However, despite this progress, the t
s
rapid advances in particle synthesis and related enabling assembly of patterned structures with nanoscale feature
technologies1-4, there has been a corresponding proliferation of resolution still remains very challenging. The focus of this work i
m
applications spanning a range of fields that include drug is on a method for achieving this using magnetic-dielectric
delivery5,6, gene transfection7,8,9, microfluidic-based core-shell nanoparticle colloids.
bioseparation10 and sorting11, micro-mixing and bio-chemical In this paper we propose a method for controlling the directed e
sensing. More generally, there has been ongoing emphasis on assembly of colloidal magnetic-dielectric core-shell h
the development methods for controlling the self-assembly of nanoparticles into extended monolayer patterns with nanoscale C
colloidal particles of all kinds into extended patterned resolution. The method is based on magnetic field-directed self-
structures. The motivation for this work comes in part from a assembly that is enhanced using soft-magnetic template
l
desire for a bottom-up approach for the scalable fabrication of elements. The elements are embedded in a nonmagnetic a
integrated nanostructured materials, e.g. for photonic12,13, substrate and magnetized using a uniform bias field. An
magnetic, micro-optical14 and electronic15,16 applications. Such example of a system with hollow cylinder template elements is c
an approach would provide advantages over conventional top- shown in Fig. 1. The template elements can be geometrically i
s
down lithographic-based fabrication and would open up tailored to produce localized regions of attractive and repulsive
opportunities for the low-cost high-throughput production of magnetic force. The ability to customize the force field in this y
functional nanostructured materials. Various methods have way is a key feature of the assembly method as it enables h
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This journal is © The Royal Society of Chemistry 2014 J. Name., 2014, 00, 1-3 | 1
Page 3 of 12 Physical Chemistry Chemical Physics
ARTICLE Journal Name
The proposed method broadly applies to arbitrary template
t
geometries and multi-layered core-shell particles that have at p
least one magnetic component. As such, it can enable the self-
i
assembly of complex patterns of nanoparticles and opens up r
opportunities for the scalable fabrication of multifunctional c
nanostructured materials for a broad range of applications in
s
fields that include photonics, magnetics, electronics, chemical
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and biological sensing, energy storage and harvesting and
catalysis. The computational model enables the rational design n
of novel media for such applications.
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2 The Computational Model
Figure 1. A self-assembly system showing soft-magnetic hollow cylinder
template elements embedded in a nonmagnetic substrate. The behaviour of colloidal magnetic particles in the presence of
d
extraordinary control of particle motion during assembly, which an external field is complex function of a number of
results in nanoscale precision in particle placement. We competitive factors including hydrodynamic and magnetic e
demonstrate the method using a computational model that takes forces, Brownian motion, Van der Walls force and the effects t
into account dominant mechanisms of particle transport and can of surfactants. We predict the self-assembly of such particles p
be used to simulate the assembly process. To date, various using a computational model based on Langevin’s equation that e
computational techniques have been successfully applied for takes these effects into account,
c
tdhyen satmudicys 2o7f,2 8m,2a9,g ntehteic dpiasrctirceltee syesletemmesn.t Tmheesteh oindc3l0u, deth Be rolawttniicaen- midd2tx2i =Fmag,i+Fvis,i+FB,i(t)+∑N (Fdd,ij+Fvdw,ij+Fsurf,ij+Fhyd,ij), (1) c
Boltzmann method31, Monte Carlo32,33 and molecular dynamic jj=≠1i A
simulations34, stochastic dynamics35 and various analytical where (cid:1) and (cid:3)(cid:4)(cid:5)(cid:6) are the mass and position of the i’th particle.
methods36,37,38. The model used here is based on Langevin’s The right(cid:2)-hand-s(cid:2)ide of this equation represents the sum of forces on s
equation and takes into account magnetic and hydrodynamic the i’th particle: (cid:7) due to the applied magnetic field, which is a
forces including inter-particle interactions, Brownian diffusion, (cid:8)(cid:9)(cid:10),(cid:2) c
superposition of the bias field and induced gradient-fields; (cid:7) ,
Van der Waals force and the effects of surfactants. A dynamic (cid:13)(cid:2)(cid:14),(cid:2) i
viscous drag due to relative motion between the particles and the s
time-stepping scheme is used to integrate the equations of
surrounding fluid (Stokes drag); (cid:7) (cid:4)(cid:5)(cid:6) a stochastic force to account
motion, which greatly accelerates and stabilizes the (cid:15),(cid:2) y
for Brownian motion; (cid:7) the interparticle magnetic dipole-dipole
simulations. We demonstrate the assembly process for the (cid:16)(cid:16),(cid:2)(cid:17) h
system shown in Fig. 1 using core-shell Fe O -SiO particles. force due to induced dipole moments; (cid:7)(cid:13)(cid:16)(cid:18),(cid:2)(cid:17) Van der Waals force;
The hollow cylinder template elements pro3du4ce a 2force field (cid:7)(cid:14)(cid:19)(cid:20)(cid:21),(cid:2)(cid:17) a repulsive force due to surfactant contact between particles P
that focuses the particles into a ring-like pattern. An analytical and (cid:7) due to interparticle hydrodynamic interactions. We
(cid:22)(cid:23)(cid:16),(cid:2)(cid:17)
force expression is used to optimize the dimensions of the predict the particle dynamics by numerically integrating Eq. (1) l
a
elements for this purpose. Once the template element using an adaptive time stepping method to accelerate and stabilize
dimensions are known, the computational model is used to the computation. The various terms in the model and its c
simulate the assembly process as a function of the particle implementation are described in the following sections. i
volume fraction. The analysis demonstrates that extended m
2.1 Magnetic force
monolayer single particle chain-like patterns can be assembled
within milliseconds with nanoscale precision by tailoring key The magnetic force on a particle due to an external field is computed e
parameters including the template geometry to produce a force using an “effective” dipole moment method in which the particle is h
field that provides precise positioning of the particles; the modeled as an “equivalent” point dipole with an effective moment
thickness of the dielectric shell to control the strength of the (cid:24) . The force on the i’th particle is given by36 C
(cid:25)(cid:21)(cid:21)
interparticle dipole-dipole force upon contact and the particle F =µ (m ⋅∇)H , (2)
volume fraction to suppress undesired aggregation during mag,i f i,eff a y
assembly. To date, various experimental studies have where (cid:26)(cid:21) is the permeability of the fluid and (cid:27)(cid:9) is the applied r
demonstrated self-assembly of nanoparticles with magnetic field intensity at the center of particle. The moment is t
nanoscale39,40,41 and microscale42,43,44 resolution using uniform given by (cid:1) (cid:28)(cid:29)(cid:31) where (cid:29) (cid:28) π#! and (cid:31) are the volume s
and/or gradient fields as well as the field induced self-assembly and magnet(cid:25)iz(cid:21)a(cid:21)tion (cid:30)of (cid:30)the parti(cid:30)cle, !resp(cid:30)ectively.(cid:30) In the case of i
of magnetic core-shell particles45. Of particular relevance is m
magnetic-dielectric core-shell particles, (cid:1) (cid:28)(cid:29) (cid:31) , where
experimental work by Henderson et al.39, wherein a (cid:25)(cid:21)(cid:21) $%(cid:20)(cid:25) (cid:30)
nanoparticle assembly was tailored using a combination of a (cid:29)$%(cid:20)(cid:25) (cid:28) !π#$!%(cid:20)(cid:25) . The moment can be determined using a e
spatially alternating gradient field, which was provided by a magnetization model that takes into account self-demagnetization h
nanoscale patterned template in the form of magnetic recording and magnetic saturation of the particles36,37
C
media, with a uniform external field. This combination of fields m =V f(H )H , (3)
i,eff p a a
closely matches the approach that we propose, albeit our where46
gradient field is provided by lithographically-patterned l
slataynedreadlo, nwe hsiocfht- mpargonveidtiec ealdevmanentatgs easn. dN theve eprtahretilcelsess athreis mpurlitoi-r (χ3(χ+p2−χχ)f+)3 Ha<(χ3(pχ+2−χχf)+)3Msp ca
work can serve as particle proof-of-concept of our proposed f(H )= p f  p f  . (4) i
adsifsfeemrebnlty temmeptlhaoted .g eWome etdrieems o(nhsotlrlaotwe ctyhlei ndmeersth, ohdo llfoowr ctuhbreees a  Msp/Ha Ha≥(χ3(pχ+p2−χχf)f+)3Msp ys
and cross-structures) and two different core-shell particle sizes. h
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2 | J. Name., 2014, 00, 1-3 This journal is © The Royal Society of Chemistry 2014
Physical Chemistry Chemical Physics Page 4 of 12
PCCP ARTICLE
In this expression, &(cid:21) is the susceptibility of the fluid and &(cid:30) is the  r −2R r 2R +2δ  t
iwnhtreirnes i(cid:27)c (cid:2)'m iasg tnheet ifci esldu sicnespidtieb itlhitey poafr titchlee. (cid:27)pa(cid:2)'rt idcilfef,e ris. ef.r o(cid:31)m(cid:30) (cid:27)(cid:28)(cid:9)& b(cid:21)y(cid:27) t(cid:2)h'e, Us =2πRp2NskBT2− ij δ p −δijln( prij ), (9) ip
demagnetization field i.e. (cid:27)(cid:2)'(cid:28)(cid:27)(cid:9)()(cid:16)(cid:31)(cid:30) where )(cid:16) is the where #(cid:30), δ and )(cid:14) are, respectively, the radius of particle, the r
demagnetization factor of the particle, i.e. ) (cid:28)1⁄3 for a spherical thickness of the surfactant layer and the surface density of surfactant c
(cid:16)
particle. The value of & can be obtained from a measured (cid:31) vs. (cid:27) molecules. The repulsive force is calculated as the gradient of this s
(cid:30)
potential energy:
curve. However, (cid:31) is often plotted as a function of Ha in which case u
(cid:31)(cid:30)(cid:28)&(cid:9)(cid:27)(cid:9), where &(cid:9) is the apparent susceptibility. The two values F =−∇U =2πRp2NskbTln(2Rp+2δ). (10) n
of susceptibility are related as follows& (cid:28)& ⁄(cid:4)1() & (cid:6), which rep,ij s δ r
(cid:30) (cid:9) (cid:16) (cid:9) ij
reduces to & (cid:28)3& ⁄(cid:4)3(& (cid:6) for a spherical particle47. The a
(cid:30) (cid:9) (cid:9)
magnetic force can be rewritten as 2.5 Viscous drag M
F =µV f(H )(H ⋅∇)H , (5) The drag force on a particle due to the viscosity of the fluid is
mag f p a a a computed using Stokes’ formula
This can be determined once an expression for - is known. For the d
(cid:9) dx
suynsitfeomrm s bhioaws nfi eilnd Fanigd. th1e, tihned uacpepdl ileodc afliiezledd igsr aad iseunpte-fripeoldstsi opnro odfu cthede FD,i =D dti , (11) e
by the magnetized soft-magnetic template elements, i.e. -(cid:9) (cid:28) Where 7(cid:28)69:#(cid:22)(cid:23)(cid:16),(cid:30) is the drag coefficient, : is the fluid viscosity pt
- /- . We use analytical closed-form expressions to and # is the hydrodynamic radius of the particle.
.(cid:2)(cid:9)(cid:14) 0(cid:25)(cid:8)(cid:30)1(cid:9)0(cid:25) (cid:22)(cid:23)(cid:16),(cid:30)
e
predict the fields and force as described in the Appendix. The use of
2.6 Interparticle Hydrodynamics Interactions
analytical analysis reduces the complexity and increases the c
accuracy of the computation as comparred to more commonly used Hydrodynamic interactions between particles become important at c
numerical field analysis. In the latter, the magnetostatic field small surface-to-surface separation distances. The force between two
A
equations are discretized and solved using a computational mesh. neighboring particles is based on lubrication theory and can be
Since the magnetic field has a long range, an extended expressed as follows48,
computational domain is often used to account for for this. 6πµV d2 s
F = f r,i,j i , (12)
Moreover, the field gradient, which defines the force, tends to be lub,ij h 16 c
sensitive to the size of mesh. Mesh size sensitivity studies are ij
usually required to ensure accurate force analysis and while a where 4(cid:2)(cid:17) is the separation bewteen the surfaces and ;(cid:20),(cid:2),(cid:17) is the si
smaller mesh can provide increased accuracy, it also increases relative velocity between the particles. When the particles are in
y
computational cost. An analytical analysis eliminates the need for contact (4(cid:2)(cid:17) <0) this force is considered to be negligible.
h
this complexity. However, it should be noted that the use of
2.7 Brownian diffusion
analytical methods for the magnetic analysis is based on the P
assumption that the bias and template fields are uneffected by other Brownian motion needs to be considerd when predicting the
materials, i.e. the substrate and carrier fluid are assumed to be dynamics of nanoscale particles. We use the following equation to l
nonmagnetic. account for these effects in each dimension: a
ur r r
2k T∆t 2k T∆t c
2.2 Magnetic dipole-dipole interaction x2 = B ∆x =n⋅δ⋅ x2 =n⋅δ⋅ B , (13)
B,i
D D i
In the presence of an applied field, the magnetic core of the m
where > is Boltzmann’s constant, 7 is the Stokes’ drag coefficient
nanoparticles becomes magnetized and aquires an effective moment (cid:15)
as described above, ∆@ is implemented in Eq. (16), ABBC and δ is a
(cid:24)(cid:25)(cid:21)(cid:21) as described above. The potential energy for two dipoles is randomly generated dire(cid:15)c,t(cid:2)ional unit vector and a random distribution e
given by
number between zero and one, which represents the direction and h
µ  (m ⋅r )(m ⋅r ) m ⋅m  magnitude of the displacement due to Brownian motion. These
Udd,ij =−4πf 3 i,eff ijr5 j,eff ij − i,effr3 j,eff , (6) displacements will be applied into the 3 Cartesian directions C
 ij ij  individually.
y
where (cid:24) and (cid:24) are the moments of i’th and j’th particle,
(cid:2),(cid:25)(cid:21)(cid:21) (cid:17),(cid:25)(cid:21)(cid:21) 2.6 Equations of Motion
respectively, and 2 is the displacement vector between them. The r
(cid:2)(cid:17) Particle motion during assembly is predicted by solving Langevin’s t
dipole-dipole force in Eq. (1) is obtained as the gradient of the
equation, which can be rewritten as s
potential,
Fdd,ij =−∇Udd,ij. (7) midd2tx2i +Dddxti =Fsum,i + FB,i(t), (14) mi
2.3 Van der Waals interaction where e
Van der Waals force is taken into account as an attractive force, F =F +∑N (F +F +F +F ), (15) h
which is calculated using48, sum,i mag,i dd,ij vdw,ij rep,ij lub,ij
j=1 C
A d6 j≠i
F = i , (8)
vdw,ij 6 (h2+2dh )2(h +d)3 Eq.(14) can be solved by first reducing it to a pair of coupled first-
ij i ij ij i order equations and then integrating these equations using a l
where 3 is the Hamaker constant and 4 is the surface-to-surface a
(cid:2)(cid:17) numerical time stepping scheme. The discretized first-order
separation distance between the i’th and j’th particle. c
equations are as follows:
2.4 Surfactant force F m  F  −Dτ si
∆x = sum,iτ+ iv − sum,i1-e mi +∆x , (16)
The repulsive force caused by the surfactant-surfactant contact is i D D i,0 D   B,i y
taken into consideration in our model. The potential energy 5 of
this interaction is given by49, (cid:14) and h
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Page 5 of 12 Physical Chemistry Chemical Physics
ARTICLE Journal Name
F  F  −Dτ #%(cid:19)0 (cid:28)#(cid:20)(cid:2)'(cid:10),%(cid:19)0 (cid:28)500 M(cid:1). These are chosen so that the particles t
vi,f = sDum,i +vi,0− sDum,ie mi , (17) assemble over the annulus of the element. The spacing of the p
elements is taken to be 2 µm center-to-center so that there is
i
where F is the integartion time step, G(cid:2),H and G(cid:2),(cid:21) are the velocity of negnigible overlap of their fields, which we verify below. Thus, it r
the i’th particle at the beginning and end of the time step and ∆@(cid:15),(cid:2) is sufficies to perform the c
the displacement due to Brownian motion. In our analysis, the time template design for a s
step F is dynamically adjusted based on the relative velocities and single isolated element.
u
surface-to-surface separations 4 of the particles. When the time Analytical closed-form
(cid:2)(cid:17)
step F is large enough (F≫(cid:8)J ), Eq. (17) can be simplified to expressions for the n
K field and force of a
F a
v = sum,i . (18) single hollow cylinder
i,f D soft-magnetic element M
in a uniform bias field
3 Analysis of Self-assembly are presented in the d
Appendix. These are
We demonstrate the proposed assembly method for the system used to optimize the e
shown in Fig. 1. Here, the templates are hollow soft-magnetic force field as a t
cylinders that are embedded in a non-magnetic substrate. A uniform function of the element Figure 2. Axial magnetic force D(cid:8)(cid:9)(cid:10),E p
along a horizontal line 100 nm above the
bias field -.(cid:2)(cid:9)(cid:14) is applied upward, perpendicular to the substrate, to dimensions. These template element as a function of the height e
magnetize the elements along their axis. When a colloid of core-shell formulas can be easily h. Rout is outer radius of the element. c
nanoparticles are introduced onto the substrate they assemble into adapted for the more
structures, the geometry of which depends on parameters that general case of closely spaced elements as described in the Appendix. c
include the strength of the bias field, the dimensions of the template We determine a template height h that provides a viable force field A
elements, the core and shell dimensions and the volume fraction of for assembly. For the force analysis, a reference frame is chosen
the nanoparticles. The computational model can be used to predict with the x-y plane coincident with the surface of the substrate and
s
the assembled structure given these parameters, or alternatively, to with the z axis aligned with the axis of the element. Thus, the top
determine a specific mix of parameters that produce a pre-defined surfaces of both the substrate and the template element are at _(cid:28)0. c
assembled structure. In this section, we demonstrate the latter and We use Eqs. (A3) and (A4) in the Appendix to compute the radial i
s
determine parameters that produce a prescribed monolayer ring-like and axial force components D and D across a horizontal
(cid:8)(cid:9)(cid:10),(cid:20) (cid:8)(cid:9)(cid:10),E
particle pattern. For the purpose of analysis, and without loss of line that spans a unit cell of the system and falls along the diameter y
generality, the core shell particles are taken to be 60 nm Fe3O4-SiO2 of the element, i.e. (1 (cid:26)(cid:1)<(cid:3)<1 (cid:26)(cid:1). The force components are h
with a core diameter of 30 nm (#$%(cid:20)(cid:25) (cid:28)15 M(cid:1)) and a shell thickness computed at a distance _(cid:28)100 M(cid:1) above the element for a range of P
of 15 nm. In principle, any multi-layered core-shell particle can be element heights: h = 100, 200 and 300 nm. The force profiles for

used as long as it has at least one magnetic component. FeO has a
3 4 D are plotted in Fig. 2 and show relatively little change for
density N$%(cid:20)(cid:25) (cid:28)5000 >O⁄(cid:1)! and a saturation magnetization (cid:31)(cid:14)(cid:30)(cid:28) 4(cid:8)`(cid:9)(cid:10)2,E00 M(cid:1). Thus, we choose the element height to be 4(cid:28) al
4.78T10U 3⁄(cid:1) . The SiO2 shell has a density N(cid:14)(cid:22)(cid:25)11 (cid:28) 200 M(cid:1). Note that the force profiles are axisymmetric because of c
2648 >O⁄(cid:1)!. The template elements are chosen to be permalloy the cylindrical symmetry of the template geometry. The 3D force
(78% Ni, 22% Fe), which has a saturation magnetization (cid:31)(cid:25),(cid:14)(cid:28) and corresopnging field profiles are shown in Fig. 3. Also, recall that mi
8.6T10U 3⁄(cid:1) . The bias field is taken to be(cid:27) (cid:28)3.9T
.(cid:2)(cid:9)(cid:14)
10U 3⁄(cid:1) (cid:4)X.(cid:2)(cid:9)(cid:14)(cid:28)5000 YZ[\\(cid:6), which is sufficiently strong to e
saturate both the nanoparticles and the template elements as
h
discussed below. This field can be obtained by positioning a rare-
earth NdFeB magnet immediately beneath the substrate37. The C
effective dipole moment of the saturated particles is(cid:1) (cid:28)
(cid:25)(cid:21)(cid:21)
(cid:29)$%(cid:20)(cid:25)(cid:31)(cid:14)(cid:30) . To simplify the analysis, we assume that the y
hydrodynamic radius of the particles is the same as their physical
r
radius #(cid:22)(cid:23)(cid:16),(cid:30) (cid:28)30 M(cid:1). However, it should be noted that in general, t
the hydrodynamic radius is larger because of the presence of s
surfactants. Similarly, we assume that the colloid is monodispersed, i
whereas in reality there will be a particle size distribution. The m
carrier fluid is assumed to be nonmagnetic (& (cid:28)0), with a viscosity
(cid:21)
and density equal to that of water, :(cid:28)0.001 )∙\⁄(cid:1)^ and N (cid:28) e
(cid:21)
1000 >O⁄(cid:1)!. h
3.1 Magnet Force Analysis C
We first analyze the magnetc force. The goal is to determine
template dimensions that produce a force field that focuses the l
a
particles into a prescibed pattern, in this case a ring-like structure. To
demonstrate the analysis, we choose the inner and outer radial c
dimensions of the ring assembly to be # (cid:28)400 M(cid:1) and i
(cid:20)(cid:2)'(cid:10),(cid:2)' s
#(cid:20)(cid:2)'(cid:10),%(cid:19)0 (cid:28)500 M(cid:1), respectively. Let #(cid:2)' and #%(cid:19)0 denote the inner
and outer radii of the template element and let h denote its height. Figure 3. Axisymmetric magnetic field and force components at z=100 y

To obtain the desired assembly, set #(cid:2)'(cid:28)#(cid:20)(cid:2)'(cid:10),(cid:2)' (cid:28)400 M(cid:1) and nm above the template element: (a) Br (b) Bz, (c) D(cid:8)(cid:9)(cid:10),(cid:20), and (d) D(cid:8)(cid:9)(cid:10),E h

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this analysis is based on the assumption that the template element is
t
saturated. It remains to verify this. From Eq. (A8) in the Appendix, p
this occurs when - `) c , where ) is the demagnetization
.(cid:2)(cid:9)(cid:14) (cid:16) (cid:25)(cid:14) (cid:16) i
factor of the element. For the chosen element dimensions (#(cid:2)'(cid:28) r
400 M(cid:1), # (cid:28)500 M(cid:1), 4(cid:28)200 M(cid:1)), ) d0.450 and it follows c
%(cid:19)0 (cid:16)
that the element is saturated because -.(cid:2)(cid:9)(cid:14)`0.4c(cid:25)(cid:14). s
Next, the magnetic force provided by a template element with
u
dimensions: # (cid:28)400 M(cid:1), # (cid:28)500 M(cid:1), 4(cid:28)200 M(cid:1) is
(cid:2)' %(cid:19)0
analyzed in more detail. The x and z components of the force D n
(cid:8)(cid:9)(cid:10),b
and D(cid:8)(cid:9)(cid:10),E are computed along the same horizontal line as above, i.e. a
(1 (cid:26)(cid:1)<(cid:3)<1 (cid:26)(cid:1), for a range of distances above the element, z = M
100, 150 and 300 nm. The force profiles are plotted in Fig. 4 and the Figure 5. Magnetic force field D (cid:4)(cid:3),a,_(cid:6) at _(cid:28)100 M(cid:1) above a 3
(cid:8)(cid:9)(cid:10),E
analysis shows that by 3 array of template elements.
there is a relatively d
strong attractive total field as described in the Appendix. The analysis shows that the
e
(downward-directed) force falls off dramatically between the elements and that there is
axial force D over negligible overlap of their effects. t
(cid:8)(cid:9)(cid:10),E p
the annulus of the Lastly, we estimate the effective range of the magnetic force, i.e. the
element, # <e< distance z beyond which directed assembly is thwarted by Brownian e
(cid:2)'
#%(cid:19)0 as indicated by motion. The particles within this distance contribute to a rapid c
the red arrows, which assembly (on the order of milliseconds), while those beyond this
c
promotes assembly in range have a negligible contribution. We use the following criterion
this region. to estimate this distance, A
Significantly, there is F (z) D ~k T , (19)
mag,z p B
also a relatively weak s
Here, g(cid:7) (cid:4)_(cid:6)g is the magnitude of the magnetic force acting on a
repulsive (upward- (cid:8)(cid:9)(cid:10),E
directed) axial force particle of diameter 7(cid:30) at a distance z above the element and >(cid:15) and c
T are as defined in Eq. (13). This relation is basically a comparison i
above a substantial s
between thermal energy and the energy expended by the magnetic
protion of its interior
region (0<e<#(cid:2)') force in moving a particle a distance 7h. The particles within a y
as indicated by the distance z from the element such that g(cid:7)(cid:8)(cid:9)(cid:10),E(cid:4)_(cid:6)g7(cid:30)i>Xj will h
blue arrows, which predominantly contribute to the assembly, much less so for those P
Figure 4. Magnetic force along horizontal prevents particles beyond this height will not. We evaluate (cid:7)(cid:8)(cid:9)(cid:10),E(cid:4)_(cid:6) over the center of
lines 100, 150 and 300 nm above the from assembling there. the annulus # (cid:28)(cid:4)# /# (cid:6)⁄2 where the attractive force is
(cid:9)(cid:13)(cid:25) (cid:2)' %(cid:19)0 l
template (arrows indicate direction of The repulsive force is maximum. A plot of the ratio (cid:7)(cid:8)(cid:9)(cid:10),E(cid:4)#Zkl,_(cid:6)7(cid:30)⁄>Xj vs. z is shown a
force): (a)D(cid:8)(cid:9)(cid:10),b and (b)D(cid:8)(cid:9)(cid:10),E due to the in Fig. 6. From this plot we see that the magnetic force is dominant c
superposition of the below 0.75 µm and the Brownian force is dominant above this height.
uniform bias field and the gradient-field of the element. Specifically, Thus, we estimate that the effective range of directed assembly for mi
the bias field induces an upward-directed moment (cid:1) _C, whereas this template is approximately _<1 (cid:26)(cid:1). We study the dynamics of
(cid:25)(cid:21)(cid:21)
the magnetized element produces a spatially varying field gradient assembly next. e
f(cid:27) that changes in sign depending on the location relative to the
(cid:9) h
element. Since the force is proportional to the product of these two
terms as shown in Eq. (2), it is attractive (negative) in regions where C
these terms have opposite signs and repulsive (positive) when they
have the same sign. The radial field component D(cid:8)(cid:9)(cid:10),(cid:20) also plays a y
critical role in focusing the particles over the annulus. Note from Fig.
r
3a that it is directed outward over the interior of the annulus0<e< t
# , and inward over the exterior of the annulus e`# . This is s
(cid:2)' %(cid:19)0
also reflected in the plot of D(cid:8)(cid:9)(cid:10),b, which is directed outward over the i
interior of the annulus 0<(cid:3)<# , and inward over the exterior of m
(cid:2)'
the annulus (cid:3)`# . as indicated by the arrows Thus, the magetic
%(cid:19)0
force field directs the particles over the annulus, which promotes e
assembly of the ring structure. It is imporatnt to note that the ability h
to produce regions of attractive and repulsive magnetic force is a key
C
feature of the propsed assembly method as it enables nanoscale
precsion of particle placement.
It is instructive to investigate the impact of neighboring template l
a
elements on the magnetic force. It is assumed that there is negiligble
overlap of the magnetic force. It remains to verify this. To this end, c
we compute the axial force field D (cid:4)(cid:3),a,_(cid:6) for a 3 by 3 array of Figure 6. Ratio of magnetic to thermal energy along a vertical line over i
(cid:8)(cid:9)(cid:10),E
elements (2 µm center-to-center spacing) at a distance _(cid:28)100 M(cid:1) the annulus of a template element. s
above the elelemnts. The force, which is shown in Fig. 5, is y
computed by first forming the total field via superposition of the h
individual elelemnt fields and then computing the force using the
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ARTICLE Journal Name
3.2 Dynamics of the Assembly Process information.
t
In this section we use the computational model to study the In this analysis the particles are initially randomly distributed and p
their final configuration is a single particle ring pattern that forms
dynamics of the assembly process. The bias field and template i
dimensions are as above. We use a computational domain centered over the annulus of the template element. The assembly was r
with respect to a single element. The domain spans a unit cell, i.e. 2 completed in less than 30 ms for all the cases studied. The final c
particle assembly has many important features. For example, it has a
µm along both the x and y axes and 1 µm in the z-direction as shown s
line width of 60 nm, i.e. the diameter of the nanoparticles. This is
in Fig. 7a and 8a. The base of the domain is at _(cid:28)0, which u
smaller than the line width of the template element, i.e. the width of
coincides with the top surface of both the substrate and the template
the annulus which is 100 nm. Another interesting feature is that the n
element. Periodic boundary conditions for particle transport are
particles are nearly uniformly spaced. This occurs because
imposed at the lateral sides of the domain to account for a 2D array a
neighbouring particles are pushed apart by a mutually repulsive
of template elements. We simulate the assembly of 60 nm FeO-
3 4 dipole-dipole force, which is due to the alignment of all dipole M
SiO particles for a range of different percent volume fractions: φ =
2 moments upward, parallel to the applied field. The spacing between
0.0655, 0.0873, 0.11, 0.131, 0.153 and 0.175%. These values
correlate to an integer number of particles in the computational the particles m(cid:30) in the first layer (i.e. monolayer) varies somewhat d
because of the effects of Brownian motion. However, as the particle
domain, e.g. φ = 0.0655 and 0.175% correspond to 15 and 40 volume fraction increases, both this spacing and the variation in e
particles, respectively. The modeling shows that the final assembly
spacing decrease as shown in Fig. 9. In this plot, the nearest t
for all of the φ values is chain-like ring pattern. Representative plots neighbouring spacing is normalized with respect to the particle p
of the initial and final particle distributions for φ = 0.0873 and
diameter 7 and the error bars reflect the range of spacing in the e
0.1750% are shown in Figs. 7 and 8, respectively. Animations of the (cid:30)
final assembly as computed by the model. The decrease in spacing
particle dynamics during assembly are provided in the supporting c
and its variation is due to an increase in strength of the repulsive
c
dipole-dipole force, which ultimately dominates the Brownian force
and produces a denser particle packing. As the volume fraction A
s
c
i
s
y
h
P
l
a
c
i
m
Figure 9. Normalized particle spacing m and variation in spacing vs.
Figure 7. Initial and final particle distributions for φ = 0.0873%: (a) (cid:30)
volume fraction. e
initial random particle distribution, (b) perspective of final assembled
particle ring, (c) lateral view of assembled particle ring, (d) magnified increases further, the particles will eventually begin to form an h
top view of assembled ring.
addition layer of the ring. It is useful to determine a first-order
C
estimate of the volume fraction at which this occurs. To this end, we
study the energetics of competing assembly configurations. Let N be
y
the number of particles in the assembly. In the first configuration, all
N particles are arranged in a ring pattern with uniform spacing r
between them. In the second configuration, N-1 particles are in a t
s
ring pattern and the remaining particle is vertically stacked on one of
these particles, thereby forming a vertical two particle chain. The i
m
second configuration represents the transition from a single layer of
particles to the beginning of a second layer. We compare the
e
magnetostatic energy of the two configurations, n and n ,
(cid:20) $
respectively. The magnetostatic energy of each assembly is h
computed using n(cid:8)(cid:9)(cid:10)(cid:28)∑p(cid:2)qs∑p(cid:17)q(cid:2)rs5(cid:16)(cid:16),(cid:2)(cid:17)/∑p(cid:2)qs(cid:1)(cid:2)∙X(cid:9),(cid:2), where C
5 is the potential energy due to dipole-dipole interaction
(cid:16)(cid:16),(cid:2)(cid:17)
between the i’th and j’th particles as given in Eq. (6) and (cid:1)(cid:2)∙X(cid:9),(cid:2) is l
the energy of the i’th dipole due to its interaction with the applied a
field X , which is evaluated at the center of the particle. It is c
(cid:9),(cid:2)
instructive to compare the relative energy difference ∆n(cid:28)n$( n(cid:20) i
Figure 8. Initial and final particle distributions for φ = 0.1750%: (a) between the two configurations at different heights above the s
initial random particle distribution, (b) perspective of final assembled elements as a function of the number of particles, which translates y
particle ring, (c) lateral view of assembled particle ring, (d) magnified top into a volume fraction. When ∆ni0, the energetics favor a single
h
view of assembled ring. layer ring-like pattern because n is higher than n . However, when
$ (cid:20)
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PCCP ARTICLE
t
p
i
r
c
s
u
n
a
M
d

Figure 10. Normalized relative energy ∆E as a function of volume e
fraction and distance z above the template element.
Figure 11. Initial and final particle distributions for a hollow cube t
∆nd0 the two particle chain configuration is favored, which template element for a volume fraction of φ = 0.2185%: (a) initial p
corresponds to the initial formation of an additional layer of particles. random particle distribution, (b) perspective of final assembled particle e
pattern, (c) lateral view of assembled particle pattern, (d) magnified top
Parametric plots of normalized ∆n are shown in Fig. 10. This figure
view of assembled rectangular pattern. c
can be used to obtain a first-order estimate of the onset of a
subsequent layer of particles (assembled upon the current layer) as a c
function of an initial volume fraction of the particles and the height A
of the current layer above the substrate. From this data we find that
at z = 50 and 100 nm ∆ni0 for all volume fractions up to 0.2%. s
Thus, a monolayer ring pattern is favoured at these heights, which is
c
consistent with the simulations above. Partial vertical stacking
occurs at z = 150, 200 and 250 nm at volume fractions above 0.18%, i
s
0.13% and 0.055%, respectively, which implies that an additional
layer will form under these conditions. It is important to emphasize y
that the analysis above is simply a first order estimate of the impact h
of the volume fraction on the formation of an additional particle
P
layer. It is based on magnetostatic analysis alone, i.e. ignoring Van
der Waals force and the effects of surfactants. It does not account for
the possibilities that the particles assemble into multiple rings or zip- l
a
chain patterns etc. A full simulation needs to be performed to obtain
c
a rigorous understanding of the formation of multilayerd systems.
Finally, in the analysis above we have demonstrated the assembly i
m
process for a hollow cylinder template element. However, the Figure 12. Initial and final particle distributions for a cross template
approach broadly applies to arbitrary template geometries. To element for a volume fraction of φ = 0.1750%: (a) initial random
confirm this, we now model self-assembly for two other template particle distribution, (b) perspective of final assembled particle pattern, e
geometries, a hollow cube and a cross structure. Both geometries are (c) lateral view of assembled particle pattern, (d) magnified top view of h
assembled cross pattern.
200 nm deep and have a 100 nm wall thickness. The cube is 1000
C
nm on a side and each segment of the cross is also 1000 nm long.
The particles are as above i.e. 60 nm FeO-SiO, as is the
3 4 2
y
computational domain, which spans 2 µm along both the x and y
axes and 1 µm in the z-direction. The initial and final particle r
distributions for these geometries are shown in Figs. 11 and 12, t
s
respectively. The particle volume fractions for these simulations are
0.2185% and 0.1750%, respectively. We used semi-numerical i
m
models to compute the 3D field distribution and force of the
template elements as described by Furlani51. A similar analysis is
e
performed with smaller particles, i.e. with a 30 nm core and 10 nm
thick shell. The assembly of these particles for the same cross h
structure as above is shown in Fig. 13 assuming a volume fraction of C
0.0650%. The particles form a zip-chain like patterns at the ends of
the cross and a multilayer structure at its centre due to the smaller
l
size of the particles. a

c

Figure 13. Initial and final particle distributions for a cross template i
s
element for a volume fraction of φ = 0.0650%: (a) initial random
particle distribution, (b) perspective of final assembled particle pattern, y
(c) lateral view of assembled particle pattern, (d) magnified top view of
h
assembled cross pattern.
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Page 9 of 12 Physical Chemistry Chemical Physics
ARTICLE Journal Name
4 Discussion one magnetic component. Furthermore, once particle patterns
t
have been formed, they can be transferred to a different p
The analysis above demonstrates the viability of controlling the substrate using techniques similar to those described by
assembly of the core-shell nanoparticles into extended monolayer Henderson et al.39 In this previous experimental study, field- ri
geometric patterns with nanoscale precision. To achieve this, various directed patterning of magnetic nanoparticles was achieved c
parameters need to be carefully chosen and computational modeling using the magnetic field gradients at the surface of commercial
s
is invaluable for determining these. There are three groups of disk drive media. The particle patterns were successfully
parameters that are especially important: the particle properties, the transferred to the surface of a polymer film by spin-coating and u
template properties and the particle volume fraction. With regards to peeling. The assembled particle patterns were preserved after n
the particles, both the core properties, magnetization (cid:31)(cid:30) and peeling because the particles were immobilized during the spin- a
especially radius # and the shell thickness are important. The coating process by the large field gradients provided by the
$%(cid:20)(cid:25)
first two parameters directly impact the magnetic and dipole-dipole recording media. However, the lower limit gradient threshold M
forces, i.e. D ~(cid:31) #! and D ~(cid:31)^#u . These parameters can required for viable pattern transfer is not known. It seems
(cid:8)(cid:9)(cid:10) (cid:30) $%(cid:20)(cid:25) (cid:16)(cid:16) (cid:30) $%(cid:20)(cid:25)
be tuned to control the particle motion and interparticle coupling reasonable to assume that a similar pattern transfer process may d
during assembly. The shell thickness also impacts the dipole-dipole apply to our approach, albeit this may require the fabrication of
force, but only when the particles are in contact. It can be increased nanoscale template elements to provide sufficient particle e
to reduce D in order to suppress undesired chaining and promote immobilization, which could be challenging. In this case, a t
(cid:16)(cid:16)
the formation of a monolayer. The overall particle size also impacts single template substrate could be used to reproduce numerous p
the hydrodynamic forces, e.g. larger particles exhibit greater viscous nano-patterned materials. In summary, we have used modelling e
drag and assemble more slowly. As for the template elements, the to demonstrate the feasibility of new and interesting phenomena
c
most important properties are the dimensions and to a lesser extent in the form of template-assisted assembly of single aligned
the level of magnetization (cid:31) . The template dimensions are magnetic core-shell particle structures at the millisecond time c
(cid:25)
especially critical as they can be tailored to produce localized scale and the use of a nonmagnetic (e.g. silica) shells to further A
regions of attractive and repulsive force that enable nanoscale control interparticle forces. Prior experimental work39
precision in particle placement. On the other hand, (cid:31)(cid:25) depends on reinforces the viability of our approach. The ability to produce s
the template material and the strength of the bias field and has a such nanostructured materials opens up opportunities for the
nonlinear contribution to the magnetic force D ~(cid:31)^. Finally, the scalable high-throughput fabrication of multifunctional c
(cid:8)(cid:9)(cid:10) (cid:25)
particle volume fraction can be adjusted to control undesired nanostructured materials for a broad range of applications and i
s
our computational model enables the rational design of such
aggregation during the assembly process. Specifically, as the volume
media. y
fraction increases the particles are closer together and tend to
aggregate during assembly, which interferes with the formation of h
the desired monolayer pattern. Thus, the volume faction must be Acknowledgements
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kept low enough to avoid this. All the aforementioned parameters
The authors acknowledge financial support from the U.S. National
can be determined for a given application using a combination of
Science Foundation, through award number CBET-1337860. l
magnetic field modeling and particle transport modeling as a
demonstrated above.
c
i
5 Conclusions m
We have presented a method for controlling the assembly of
e
colloidal magnetic-dielectric core-shell nanoparticles into
extended geometric patterns with nanoscale precision. This is h
achieved using soft-magnetic template elements with nanoscale C
line-widths to guide the assembly in the presence of a uniform
bias field. The combination of a uniform field and localized
y
high gradient fields produced by the template elements enables
nanoscale precision of particle placement. We demonstrate r
proof-of-concept using a computational model that takes into t
s
account dominant mechanisms that govern the assembly
dynamics. We show the first time that prescribed geometric i
m
patterns of particles can be assembled within milliseconds and
with a line width resolution substantially greater than that of the
e
template geometry. The increased resolution is due to the
nanoscale precision in particle placement, which is achieved by h
tailoring key parameters including the template geometry to C
produce a force field that focuses the particles into prescribed
patterns; the thickness of the dielectric particle shell to control
l
the magnetic dipole-dipole force upon contact by providing a a
separation between the magnetic cores; and the particle volume
c
fraction to suppress undesired aggregation during assembly. We
have demonstrated the model using hollow cylinder, hollow i
s
cube and cross-shaped templates and dual-layer core-shell
y
particles. However, it broadly applies to templates with an
arbitrary geometry and multiple-layer particles that have at least h
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8 | J. Name., 2014, 00, 1-3 This journal is © The Royal Society of Chemistry 2014