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Langmuir 2008, 24, 8514-8521
Field Gradients Can Control the Alignment of Nanorods Chinchun Ooi* and Benjamin B. Yellen* Duke UniVersity, Department of Mechanical Engineering and Materials Science, Center for Biologically Inspired Materials and Material Systems, Hudson Hall Room 144, Box 90300, Durham, North Carolina 27708 ReceiVed March 31, 2008. ReVised Manuscript ReceiVed May 13, 2008 This work is motivated by the unexpected experimental observation that field gradients can control the alignment of nonmagnetic nanorods immersed inside magnetic fluids. In the presence of local field gradients, nanorods were observed to align perpendicular to the external field at low field strengths, but parallel to the external field at high field strengths. The switching behavior results from the competition between a preference to align with the external field (orientational potential energy) and preference to move into regions of minimum magnetic field (positional potential energy). A theoretical model is developed to explain this experimental behavior by investigating the statistics of nanorod alignment as a function of both the external uniform magnetic field strength and the local magnetic field variation above a periodic array of micromagnets. Computational phase diagrams are developed which indicate that the relative population of nanorods in parallel and perpendicular states can be adjusted through several control parameters. However, an energy barrier to rotation was discovered to influence the rate kinetics and restrict the utility of this assembly technique to nanorods which are slightly shorter than the micromagnet length. Experimental results concerning the orientation of nanorods inside magnetic fluid are also presented and shown to be in strong agreement with the theoretical work.
I. Introduction Nanowires and other anisotropically shaped colloidal particles exhibit material properties which are uniquely suited to many applications in photonic,1–3 electronic4,5 and biosensor6,7 devices. Much attention has been paid to the synthesis and characterization of a wide variety of nanomaterials.2,8 There is also ongoing work on integrating these nanomaterials into useful systems.9,10 Localized chemical, electric, and magnetic field gradients have demonstrated the ability to control the positions of self-assembled colloidal particles.11–22 Likewise, external fields (e.g., magnetic, electric, hydrodynamic) have been exploited to control the particle’s alignment.9,23–29 In this work, we demonstrate an unexpected result that field gradients can control the alignment of nanorods. Specifically, we discovered that nonmagnetic nanorods suspended inside a highly concentrated solution of magnetic nanoparticles, known as ferrofluid, will align perpendicularly to the direction of the applied external field when exposed to local magnetic field gradients produced by arrays of micromagnets. Reversible * To whom correspondence should be addressed. E-mail:
[email protected] (B.B.Y.);
[email protected] (C.O.). Phone: 919-660-8261(B.B.Y.). Fax: 919-660-8963 (B.B.Y.). (1) Pauzauskie, P. J.; Radenovic, A.; Trepagnier, E.; Shroff, H.; Yang, P.; Liphardt, J. Nat. Mater. 2006, 5(2), 97–101. (2) Xia, Y.; Yang, P.; Sun, Y.; Wu, Y.; Mayers, B.; Gates, B.; Yin, Y.; Kim, F.; Yan, H. AdV. Mater. 2003, 15(5), 353–389. (3) Johnson, J. C.; Choi, H.-J.; Knutsen, K. P.; Schaller, R. D.; Yang, P.; Saykally, R. J. Nat. Mater. 2002, 1(2), 106–110. (4) Bachtold, A.; Hadley, P.; Nakanishi, T.; Dekker, C. Science. 2001, 294, 1317–1320. (5) Law, M.; Goldberger, J.; Yang, P. Annu. ReV. Mater. Res. 2004, 34(1), 83–122. (6) Kneipp, K.; Kneipp, H.; Corio, P.; Brown, S. D. M.; Shafer, K.; Motz, J.; Perelman, L. T.; Hanlon, E. B.; Marucci, A.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. Lett. 2000, 84(15), 3470. (7) Cui, Y.; Wei, Q.; Park, H.; Lieber, C. M. Science. 2001, 293(5533), 1289– 1292. (8) Hu, J.; Odom, T. W.; Lieber, C. M. Acc. Chem. Res. 1999, 32(5), 435–445. (9) Hangarter, C. M.; Rheem, Y.; Yoo, B.; Yang, E.-H.; Myung, N. V. Nanotechnology 2007, 18(1-7), 205305. (10) Whang, D.; Jin, S.; Wu, Y.; Lieber, C. M. Nano Lett. 2003, 3(9), 1255– 1259.
Figure 1. Optical micrographs (a, b) demonstrate the orientation of nanorods in (a) 220 and (b) 26.3 Oe external fields. The light rectangular regions are 8 µm × 3 µm lithographically fabricated cobalt micromagnets. The darker regions depict ferrofluid aggregation between the micromagnets.
switching of nanorod alignment between parallel and perpendicular orientations was accomplished by tuning the external field strength, as demonstrated in Figure 1. (11) Fudouzi, H.; Kobayashi, M.; Shinya, N. Langmuir 2002, 18(20), 7648– 7652. (12) Fudouzi, H.; Kobayashi, M.; Shinya, N. AdV. Mater. 2002, 14(22), 1649– 1652. (13) Velev, O. D.; Kaler, E. W. Langmuir 1999, 15(11), 3693–3698. (14) Lee, C. S.; Lee, H.; Westervelt, R. M. Appl. Phys. Lett. 2001, 79(20), 3308–3310.
10.1021/la801006g CCC: $40.75 2008 American Chemical Society Published on Web 07/17/2008
Field Gradients Can Control the Alignment of Nanorods
The goal of this article is to provide a theoretical explanation for these experimental observations. We discovered that in strong external fields the nanorods align parallel to the external field vector (Figure 1a); however, once the field strength is reduced below a critical threshold the nanorods rotate and align perpendicularly to the external field vector (Figure 1b). The origin of this behavior can be understood as a competition between orientational and positional potential energies. That is, the nanorods prefer to orient parallel to the external field in order to minimize its demagnetizing energy (i.e., orientational potential energy); however, this is in competition with the tendency of nanorods to move toward regions of minimum magnetic field in order to reduce the magnetic field strength inside the nanorod’s volume (i.e., positional potential energy). Through modulation of the external field strength, a convenient switch can thus be implemented for controlling the alignment of nanorods. Here, we develop computational techniques to predict the alignment of nanorods as a function of various system control parameters, including the magnetization and geometry of the micromagnets, strength of the applied uniform field, and relative size of the nanorods with respect to the micromagnets. The nanorod’s alignment is modeled statistically using a Boltzmann distribution to determine the nanorod’s most probable orientation as a function of the experimental system parameters. Through these investigations, computational phase diagrams were constructed to depict the thermal stability of a nanorod’s orientation in different experimental environments. These theoretical investigations are well supported by our experimental work, in which we have studied the average orientation of the nanorod as a function of the applied uniform external field strength. One interesting aspect of this work concerns the dynamic switching between parallel and perpendicular orientations. Specifically, we have found that some equilibrium states predicted by our theoretical model were kinetically inaccessible in our experimental apparatus. Computational results indicate that there is frequently an energy barrier to rotation, which acts similarly to an activation energy in chemical reactions. In the presence of this energy barrier, we discovered that switching between perpendicular and parallel orientations is too slow for observation on our experimental time scale; however for regions in the phase diagram where no energy barrier exists, we confirm reversible switching behavior both theoretically and experimentally. The rest of this manuscript is organized as follows. In Section II, a theoretical model is developed to determine the probabilistic orientation of nanorods as a function of the experimental system (15) Green, N. G.; Morgan, H. J. Phys. D: Appl. Phys. 1997, 30(11), L41–L44. (16) Erb, R. M.; Yellen, B. B. J. Appl. Phys. 2008, 103, 07A312–3. (17) Erb, R. M.; Sebba, D. S.; Lazarides, A. A.; Yellen, B. B. J. Appl. Phys. 2008, 103(6), 063916-5. (18) Yellen, B. B.; Erb, R. M.; Son, H. S.; Hewlin, R., Jr; Shang, H.; Lee, G. U. Lab Chip 2007, 7(12), 1681–1688. (19) Komaee, K.; Yellen, B. B.; Friedman, G.; Dan, N. J. Colloid Interface Sci. 2006, 297(2), 407–411. (20) Yellen, B. B.; Friedman, G. Langmuir 2004, 20(7), 2553–2559. (21) Yellen, B. B.; Friedman, G.; Feinerman, A. J. Appl. Phys. 2003, 93(10), 7331–7333. (22) Yellen, B. B.; Friedman, G. AdV. Mater. 2004, 16(2), 111–115. (23) Ooi, C.; Erb, R. M.; Yellen, B. B. J. Appl. Phys. 2008, 103(7), 07E910–3. (24) Sheparovych, R.; Sahoo, Y.; Motornov, M.; Wang, S.; Luo, H.; Prasad, P. N.; Sokolov, I.; Minko, S. Chem. Mater. 2006, 18(3), 591–593. (25) Tanase, M.; Felton, E. J.; Gray, D. S.; Hultgren, A.; Chen, C. S.; Reich, D. H. Lab Chip 2005, 5(6), 598–605. (26) Smith, P. A.; Nordquist, C. D.; Jackson, T. N.; Mayer, T. S.; Martin, B. R.; Mbindyo, J.; Mallouk, T. E. Appl. Phys. Lett. 2000, 77(9), 1399–1401. (27) Blatt, S.; Hennrich, F.; Lohneysen, H.v.; Kappes, M. M.; Vijayaraghavan, A.; Krupke, R. Nano. Lett. 2007, 7(7), 1960–1966. (28) Donglu, S.; Peng, H.; Jie, L.; Xavier, C.; Sergey, L.B.k.; Eric, B.; Wang, L. M.; Rodney, C. E.; Robert, T. J. Appl. Phys. 2005, 97(6), 064312. (29) Liu, M.; Lagdani, J.; Imrane, H.; Pettiford, C.; Lou, J.; Yoon, S.; Harris, V. G.; Vittoria, C.; Sun, N. X. Appl. Phys. Lett. 2007, 90(10), 103105–3.
Langmuir, Vol. 24, No. 16, 2008 8515
parameters. Section III presents a series of computational phase diagrams for nanorod orientation as a function of different control parameters. Section IV presents the experimental method, followed by a discussion of the experimental results in Section V. A brief conclusion will summarize these results, and discuss possible implications of this assembly technique.
II. Computational Model In previous work, we have studied the behavior of nonmagnetic nanorods in an external magnetic field,23 and we have shown that the equilibrium orientation of the nanorod is well-described by continuum equations for fluid magnetization and an expression for potential energy of a spheroidal particle in an external field, derived by K.C. Kao.30 In the presence of a uniform external magnetic field, H0, the potential energy of a prolate spheroid oriented such that θ is the angle between the external field vector and the major axis of the spheroid is given by
U(θ) )
2 2 2 2πab2 µf(µp - µf) (Gb - Ga)H0 sin (θ) (1) 3 (µf - (µf - µp)Ga)(µf - (µf - µp)Gb)
where µp and µf represent the magnetic permeabilities, with subscripts f and p referring to the fluid and the particle, respectively. The parameters a and b denote the dimensions of the major and minor axes of the spheroid, while Ga and Gb are the demagnetizing factors associated with the shape anisotropy along the major and minor axes, given by
Ga )
ab2 2
∫0∞ (s + a2)ds 3⁄2 (s + b2) Gb )
ab2 2
∫0∞ (s + a2)1⁄2ds(s + b2)2
(2)
In the present work, this expression is no longer useful, due to the highly inhomogeneous magnetic field distribution near the micromagnet array. In this case, the field inside the nanorod’s volume varies strongly both in strength and orientation. Analytical expressions for the potential energy of spheroidal particles in highly nonuniform external fields are unavailable; however, the main features of the field distribution can be adequately described by an equivalent picture of the nanorod as a rigid chain of spherical particles having an aspect ratio and total volume equivalent to that of the actual nanorod. This assumption, often called the discrete dipole approximation, is commonly used to simulate electromagnetic field distribution within complicated geometrical shapes. A. Uniform Fluid Susceptibility. In this section, a model is derived for the potential energy of a rigid chain of nonmagnetic spherical particles inside a ferrofluid having spatially uniform particle concentration, and thus uniform magnetic susceptibility. In practice, the fluid susceptibility is not uniform due to reorganization of the ferrofluid particles in regions of strong magnetic field, as is clearly observed in Figure 1 and in the Supporting Information video, where the darkened regions depict the areas of higher magnetic nanoparticle concentration. However, the assumption of uniform susceptibility will serve as a starting point for determining the stability of the nanorod’s orientation as a function of the external field strength and other control parameters. In Part B, we will relax this assumption by computing the potential energy of the rigid chain in a fluid with locally varying ferrofluid concentration, consistent with prior work by the authors.16,17,31 (30) Kao, K. C. Br. J. Appl. Phys. 1961, 12(11), 629–632.
8516 Langmuir, Vol. 24, No. 16, 2008
Ooi and Yellen
Within a rigid chain of spherical particles, the effective magnetic moment of each particle is well approximated by the field of a point dipole32,33
(
m b) 3
)
χp - χf b ) χVH b VH χp + 2χf + 3
(3)
where V is the volume of a particle, which is assumed to have an effective magnetic susceptibility χj determined from the difference in actual magnetic susceptibility between the particle χp ≈ 0, and a surrounding ferrofluid, which typically has a magnetic susceptibility ranging between 0.1 < χf < 1. In eq 3, b is the total magnetic field at the particle center and contains H b 0, the contributions from the applied uniform external field H b sub, local field produced by the magnetic microarray substrate H as well as fields from other spherical particles in the chain. For an isolated chain of particles, the magnetic moments of each particle are coupled through mutual interactions with other particles in the chain34,35 given by
[
m b i)χV H b iext +
∑
3(m bj · b rij)r bij
j*i
4πrij5
-
m bj 4πrij3
]
(4)
where b rij is the position vector between the ith and jth particle biext ) H b0 + H bisub incorporates all external centers in the chain, and H field contributions from the uniform applied field and the substrate’s field at the location of the ith particle. The linear set of equations for unknowns m b i can be written in the following form:
[] [ m b1 m b2 ) χV ... m bn
I A12 A21 I ... ... An1 An2
... A1n ... A2n ... ... ... I
][ -1
b H ext1 b H
ext2
... b Hextn
]
(5)
where Aij describes the magnetic interaction between the ith and jth particle in the chain and can be written as: χ Aij(θ, φ) ) × 3(2(|j - i|))3
[
(3cos2θ - 1)cos2φ - sin2φ 2
3cosθsinθcos φ 3cosθcosφsinφ
3cosθsinθcos2φ
3cosθcosφsinφ
(3sin θ - 1)cos φ - sin φ 3sinθcosφsinφ 3sinθcosφsinφ 2sin2φ - cos2φ 2
2
2
]
[
]
for the substrate’s field to be approximated by an array of line poles extending infinitely along the Y-direction (illustrated by the dotted lines in Figure 2) and repeating periodically along the X-direction. This arrangement is also practical from the experimental standpoint, because it allows for the magnetization within individual rectangular micromagnets to be controlled using shape anisotropy, while at the same time reducing the sensitivity of an individual micromagnet’s magnetization to the external field strength. With these assumptions, the equivalent magnetic pole density of the substrate can be described in Fourier space using the following expression: ∞
σ(x, z) ) δ(z)
(6)
In this work, we restrict our attention to 1-D orientational motion of a nanorod confined to a flat surface, in which case φ ) 0, allowing simplification of Aij to
3cos2θ - 1 3cosθsinθ 0 χ Aij(θ) ) 3cosθsinθ 3sin2θ - 1 0 3(2(|j - i|))3 0 0 -1
Figure 2. Schematic of the magnetic microarray substrate used to produce local field gradients. The black rectangles represent the lithographically patterned micromagnets, while the dotted lines represent the assumed periodic array of magnetic line poles with pole densityσo. Parameters d and f are lengths corresponding to the spatial periodicity of the micromagnets. The nanorods are represented by a continuous array of spheres linked in a rigid chain.
(7)
Figure 2 provides an illustration of a chain of spherical particles located above an array of uniformly magnetized micromagnets arranged in a 2-D pattern. This arrangement was chosen as a compromise between theoretical and experimental considerations. This arrangement simplifies our theoretical analysis by allowing (31) Hovorka, O.; Yellen, B. B.; Dan, N.; Friedman, G. J. Appl. Phys. 2005, 97(10), 10Q306–3. (32) Panofsky, W. K. H. and Phillips, M. Classical Electricity and Magnetism, 2nd ed.; Addison Wesley Pub: New York, 1955. (33) Stratton, J. A., Electromagnetic Theory, 1st ed.; McGraw Hill Book Company: New York, 1941. (34) Yellen, B. B.; Friedman, G. J. Appl. Phys. 2003, 93(10), 8447–8449. (35) Yellen, B.; Friedman, G.; Feinerman, A. J. Appl. Phys. 2002, 91(10), 8552–8554.
∑[
nπ nπ 1 d d cos x+ - cos xd n)1 d 2 d 2
( (
))
( (
))] (8)
where d is the characteristic length of the micromagnet. After applying the boundary conditions, the magnetostatic potential produced by the micromagnet array in the z > 0 region is given by
φ(x, z) ) -
σod ∞ 1 nπ sin(knx)e-kn|z| sin π n)1 n 2
∑
( )
(9)
where σo is the magnitude of the pole density and has the units of A/m, and kn ) nπ/d are the spatial harmonics in the periodicity of the micromagnet array. The substrate’s magnetic field is determined from the negative gradient of the potential, given by
[ ]
[
∞ Hx nπ -kn|z| cos(knx) b Hsub ) ) σ0 sin e Hz -sin(knx) 2 n)1
∑
( )
]
(10)
Here, we restrict our attention to the situation where the external uniform magnetic field is aligned in the same direction as the magnetization of the micromagnets. We also consider the case where the particle chain lies flat on the substrate, such that the vertical position of each particle in the chain is exactly one particle radius above the substrate. These assumptions allow for the system to be described by two nondimensional parameters, β ) a/d, where a is the radius of a spherical particle in the chain, and ζ ) |H0/σ0|, which reflects the relative strength of the external field
Field Gradients Can Control the Alignment of Nanorods
Langmuir, Vol. 24, No. 16, 2008 8517
with respect to the substrate’s field. With this notation, the total external field at the location of the ith particle can be uniquely determined from two degrees of freedom: (1) the orientation of the chain with respect to the external field direction, θ, and (2) the position of the central Cth particle in the chain.
[
bi ) H ext σ0
ζ+
cos(2nπβcos(θ)(i - C))exp(-nπβ)] ∑ [sin( nπ 2) n
0-
sin(2nπβcos(θ)(i - C))exp(-nπβ)] ∑ [sin( nπ 2) n
]
(11)
By combining eq 11 with eqs 5 and 7, the magnetic moment of each particle can be determined and the potential energy of the composite chain computed as a function of its position and orientation, as given by
U(θ, C) ) -
µ0 2
∑ mbi(θ, C) · Hbi(θ, C)
(12)
i
Equation 12 can provide a qualitative explanation for the general assembly process. The effective magnetic moment of each particle in the chain is a negative quantity due to the fact that the ferrofluid has stronger magnetic susceptibility than the particle (i.e., χp < χf). Thus, the potential energy of the nanorod is strictly positive and is minimized when the nanorod is positioned in regions of weakest magnetic field. For the system under consideration, the weakest field is located directly on top of the micromagnets, where the micromagnet’s field partially cancels the external field. If particle-particle interactions within the chain were negligible, then this analysis indicates that each particle within the chain will move to the same X-position on top of the micromagnets where the field is weakest, implying alignment of the particle along the Y-direction (perpendicular to the external field direction). On the other hand, if particle-particle interactions within the chain are important, then the system can reduce its demagnetizing energy by aligning in the same direction as the external field. Understanding the competition between positional and orientational energy of a nanorod is at the heart of these computational studies. The mean statistical orientation of the nanorod is computed from the Boltzmann distribution function, which gives the probability of finding the chain with orientation θ through the magnitude of its potential energy U(θ) with respect to randomizing thermal energy kBT. The simulation procedure is able to compute the expected orientation of the nanorod by fixing the position of the central particle in the chain and studying the potential energy solely as a function of its orientation, θ. This assumption is reasonably consistent with the experimental results, where the nanorod is observed to rotate around a more or less fixed pivot point when the external field is modulated. Under this assumption, the Boltzmann distribution is solely a function of θ, making it possible to determine the probability of finding a nanorod whose orientation is within a defined interval, Θ1 > θ > Θ2 using the following expression:
P(Θ1 > θ > Θ2) )
∫ ∫Θ
π -U(θ,ζ)⁄kT dθ 2 e 0 Θ2 -U(θ,ζ)⁄kT 1
e
computational standpoint, it is not entirely accurate as ferrofluid has been shown to concentrate near regions of stronger magnetic field and deplete from regions of weaker magnetic field16,17 (See Figure 1 or the Supporting Information video, for example). Due to the spatial variations in particle concentration and concomitant variations in ferrofluid susceptibility, the calculation of potential energy of the spherical chain becomes a more complicated matter and necessitates the use of numerical techniques to calculate the magnetic field self-consistently. In spite of these complications, it is possible to develop a first order estimate for the contribution of ferrofluid concentration gradients to the local magnetic susceptibility of the fluid. In previous work, an approximate theoretical model was derived to study the local ferrofluid concentration C(r b) near patterned micromagnets. For details on the derivation, the reader should refer to prior works.16,17,31 The final expression is given by
[ ( )
C(r b) ) 1 +
1 - Cb sinh(ξ0) ξ(r b) Cb sinh(ξ(r b)) ξ0
]
-1
(14)
b (r where ξ(r b) ) γMsV|H b)|/kBT is the dimensionless ratio between magnetic and thermal energy, where Ms and V are the saturation magnetization and the volume of the iron-oxide nanoparticles, and Cb is the bulk volume fraction of magnetic nanoparticles in the suspension. In this expression, the multiplicative factor γ is included to account for the minor degree of particle aggregation in the fluid. Small aggregates affect the formation of ferrofluid concentration gradients through cooperative interactions with a field gradient, as discussed in previous work. For this particular ferrofluid, the multiplicative factor γ ) 8 was found to match previous experimental results. Equation 14 qualitatively predicts the correct physical behavior of magnetic nanoparticle suspensions as a function of the magnitude of the local field strength. When the local field strength greatly exceeds that of the external field, the ferrofluid concentration approaches 100%. On the other hand, when the local field is weaker than the external field (such as the field directly on top of one of the micromagnets, where the substrate’s field subtracts from the external field), the local ferrofluid concentration is lower than the bulk value. With these assumptions, an expression can be derived for the local magnetic susceptibility as a function of the local particle concentration:
χf(r b) ) C(r b)χB
(15)
where χB is the extrapolated value for the susceptibility of bulk iron oxide. This linear relationship for ferrofluid susceptibility can be justified based on Onsager’s previous studies of polar liquids where a linear relationship was obtained between electrical permittivity and molecular number density.36 The potential energy of the chain of spherical particles can now be computed from eq 12 by inserting eqs 14 and 15 into eq 3. As will be shown in Section III, inclusion of local ferrofluid concentration gradients into our model does affect the general shape of the phase diagrams; however it does not have a dramatic effect on the regions of the phase diagram where reversible switching of the nanorod between parallel and perpendicular orientations was experimentally observed.
III. Simulation Results (13)
dθ
B. Variation in Fluid Susceptibility. While the assumption of uniform ferrofluid concentration is convenient from the
Simulations were conducted using Matlab software from Mathworks (Natwick, MA). In the following simulations, the ferrofluid concentration and fluid susceptibility were chosen to be uniform, except in one simulation where the effect of ferrofluid (36) Onsager, L. J. Am. Chem. Soc. 1936, 58(8), 1486–1493.
8518 Langmuir, Vol. 24, No. 16, 2008
Figure 3. This phase diagram represents the probability of finding the nanorod with equilibrium orientation between 85° and 90° with respect to the external field vector. A color map is used to present the probability distribution, with blue representing low probability and red high probability. A value of σ0 ) 50 Oe was used in these simulations.
Figure 4. This phase diagram presents the preferred orientations for the nanorod as a function of ζ and 2εβ. The color map is used to present the equilibrium orientation, with blue regions denoting 0°, and dark red regions denoting 90°. Dark blue regions indicate areas where no particular orientation predominates. A value of σ0 ) 50 Oe was used in these simulations as well.
concentration gradients was investigated. To allow for direct comparison with experimental work, the aspect ratio, ε, of the rods was held constant at ε ) 15. Details concerning the ferrofluid and nanorods are provided in Section IV. For these simulations, we assume the magnetic susceptibility of undiluted ferrofluid (supplied at 3.9% vol. fraction) was 0.56, which is consistent with our prior investigations23 and the material specifications from the manufacturer. The system was also assumed to be at a room temperature of 24 °C. Simulation results for a particular setup are presented in two characteristic phase diagrams, shown in Figures 3 and 4. Figure 3 depicts the probability of finding a nanorod with orthogonal orientation with respect to the external field direction. Some basic trends are clearly observed. At high values of ζ (i.e., external field is much larger than the substrate’s field), the probability of finding a nanorod aligned perpendicularly to the external field vector is very low, as indicated by the blue regions, due to the fact that the nanorod’s orientation-dependent potential energy in the external uniform magnetic field dominates over its position-dependent potential energy in the magnetic field gradient. As the external field strength is reduced, a transition occurs and the perpendicular orientation becomes more energetically stable. The “pockets of stability”, represented by the dark red regions in Figure 3, indicate the conditions for which the probability of observing perpendicular orientations is greater than 90%. The size of the pocket is clearly a function of the relative size of the nanorod with respect to the micromagnet. A large range of ζ values will lead to a stable perpendicular orientation when the nanorod length is approximately 5-10 times the micromagnet’s length (i.e., 5 < 2εβ < 10). For these values of 2εβ, the nanorod’s
Ooi and Yellen
Figure 5. This diagram illustrates the effect of the system’s scaling trends on the size of the pockets. A positive correlation is observed between the size of the system and the size of the pocket. σ0 for this simulation was 50 Oe.
width is commensurate with the micromagnet’s length. The range of stable perpendicular configurations is dramatically reduced for smaller nanorods (i.e., 2εβ < 5), an effect that can be understood as follows. When the nanorods are much smaller than the micromagnets, the field variation across the nanorod becomes smaller, and thus the nanorod’s orientation-dependent potential energy begins to dominate at lower values of the external field strength. At the other end of the spectrum, when the nanorods are much larger than the magnets (i.e., 2εβ > 20), the substrate’s field at the center of each particle in the chain becomes weaker, due to the exponentially decaying field distribution produced by the micromagnet array. With further increases in 2εβ, the orientation-dependent potential energy again dominates at lower values of the external field strength. It is worth noting that the jagged characteristic shape of the “pockets of stability” is caused by the discretization of the relative dimensions of the nanorod with respect to the periodicity of the array. Figure 4 presents the most probable orientation of the nanorod as a function of the control parameters, 2εβ and ζ. As clearly seen in Figure 4, nanorod alignment along the external field direction occurs for large ζ and large 2εβ. For fixed ζ in the range of 2-4, a continuous transition between parallel and perpendicular orientation occurs as 2εβ is allowed to increase. This transition is further evidence that the field variation across the nanorod becomes more influential to its total potential energy as the nanorod size increases with respect to the fixed size of the micromagnets. A third distinct region was observed for low values of 2εβ and ζ, for which no particular orientation dominates. This behavior results from the dominating influence of thermal fluctuation energy for small nanorods exposed to weak external fields. In Figure 5, we study the scalability of the manipulation technique by analyzing the shape of the pockets as a function of the system size. From this analysis, it is clear that a large range of perpendicular orientations is possible for 330 nm diameter nanorods, such as those used in the experimental work; however the stable region shrinks with the nanorod’s size. For rods of about 80 nm diameter and aspect ratio 15:1, virtually no pocket exists. This general trend results from the increasing influence of thermal fluctuation energy as the entire system is scaled down, since the potential energy of the nanorod is proportional to its volume. In Figure 6, a simulation is presented for a constant nanorod volume but with variable values of σ0. As expected, the size of the pocket decreased with decreasing σ0. A stable perpendicular orientation is barely possible at 5 Oe and below, whereas large pockets exist at values of 50 Oe and above. The rapid increase in pocket size with respect to σ0 results both from an increasing
Field Gradients Can Control the Alignment of Nanorods
Figure 6. This phase diagram presents the relationship between pocket size and the substrate’s pole density σ0. The trends are similar to those presented in Figure 5. The pocket size increases with increasing σ0. The pockets represent regions of the phase diagram where there is >90% probability of observing nanorods in an orientation between 85° to 90° relative to the direction of the external field vector.
Figure 7. This diagram illustrates the effect of magnetic susceptibility variation on the pocket size as a result of different values for the aggregation parameterγ. The pockets represents the condition for which there is a >90% probability of observing nanorods with an orientation between 85° and 90° relative to the external field vector. The value of σ0 is taken to be 50 Oe to enable direct comparison with the other simulations.
substrate’s field and a proportional increase in the external field magnitude due to its relationship with the control parameter ζ. In Figure 7, the relationship between pocket size and the degree of magnetic susceptibility variation is presented as a function of the aggregation parameter, γ. The diagram for γ ) 8 is cut off at the top due to the scale of the diagram, but it is clear that an increase in the aggregation parameter causes a dramatic increase in the pocket’s height; however there is very little change in the pocket’s width as compared to the uniform susceptibility model. In essence, the effect of susceptibility variation is to extend the range of external field strengths for which a perpendicular orientation is stable. One possible explanation for this effect is that the ferrofluid avoids the regions on top of the micromagnets. Thus, the orientation-dependent potential energy tends to be smaller in this region, since the local fluid susceptibility will be much weaker than in the bulk fluid.
IV. Experimental Methods EMG 705 ferrofluid, consisting of nanometer sized iron-oxide grains suspended in aqueous fluid via proprietary surfactants, was purchased from Ferrotec (Nashua, NH). Cylindrical nanorods of alternating gold and silver segments were a kind gift of Oxonica Inc. (Mountain View, CA). These nanorods were found to have mean diameter of 332 nm ( 19.8% and mean length of 4.92 µm ( 30.9% (corresponding to 15:1 aspect ratio) as measured by scanning electron microscopy. These nanorods were chosen because they have been used with EMG 705 in previous work by the authors without any significant degradation in colloidal stability of the nanorod or ferrofluid mixture.
Langmuir, Vol. 24, No. 16, 2008 8519
Figure 8. Reversible switching behavior of nanorods is validated through simulations and experiments. The experimental data (solid line) depict the range of field strengths for which the reversible switching was observed as a function of 2εβ. The dotted lines present data from simulations, using values of σ0 ) 30, 43.75, and 70 Oe for the dashed, dotted, and dash-dotted lines, respectively. The value of σ0 ) 43.75 Oe appears to be a good fit with the experimental work.
Cobalt micromagnets were fabricated on Si wafers using e-beam lithography and the conventional metal lift-off technique. A layer of PMMA 950K (Microchem Inc., Santa Clara, CA) was spin coated onto clean Si wafers at 4000 rpm for 40s and then exposed to 100pA of electron beam current for 1.075s using Elionix ELS-7500 EX Lithography System obtained from STS-Elionix (Billerica, Ma). The substrate was then developed with methyl isobutyl ketone (MIBK) solution diluted with isopropyl alcohol (IPA) at a 1:3 ratio for 30s. A trilayer of Au/Co/Cr was deposited onto the chip wafers via e-beam metal evaporation using a CHA Industries Solution e-beam metal evaporator (Fremont, CA). The bottom layer of 5 nm Cr was included in order to improve metal adhesion to the Si substrate. The middle layer of Co was adjusted over a range of thicknesses, and the top layer of 5 nm Au was used to minimize oxidation of the Co film. The thickness of the middle Co layer was intentionally varied in order to simulate experiments with a constant value of σ0 with respect to the changing length of the micromagnets. The effective magnetic line pole density of the substrate can be approximated as σ0 ) Msτ/d, where Ms, τ, and d are the remnant magnetization of Cobalt, the thickness of the magnetic film and the length of the micromagnets, respectively. The goal of experimental work was to test the accuracy of the computational phase diagrams presented in Figures 3–7; however, we discovered that reversible switching between perpendicular and parallel states occurred only for values of 2εβ < 1 (i.e., for nanorods shorter than the micromagnet length). The data presented in Figure 8 were obtained using five different micromagnet lengths of 8, 10, 12, 14, and 16 µm. In order to simulate a constant value of σ0, the Cobalt layer’s thicknesses were selected such that the ratio τ/d ) 0.0875 was maintained throughout all experiments. Specifically, for the five micromagnet lengths tested in these studies, the thickness of the Co layer was adjusted to 70, 87.5, 105, 122.5, and 140 nm, respectively. Experiments on reversible alignment of nanorods were conducted by first mixing 5 µL of nanorods from the stock solution with 50 µL of EMG 705 ferrofluid. Since the dilution factor was relatively insignificant, the stabilizing surfactants inside the ferrofluid were not significantly degraded. An aliquot of 0.75 µL of the final mixture was placed between the chip and a glass coverslip, and then exposed to uniform magnetic field applied through a pair of solenoids with iron cores while under microscopic observation. The experimental protocol was conducted as follows. A 200 Oe magnetic field was initially applied to the system, while the micromagnet array was viewed via bright field reflective microscopy under 40× magnification using a Leica DMLM microscope. At this field strength, nearly all the single nanorods were aligned parallel to the external field vector and were located directly on top of one of the micromagnets. The orientations of individual nanorods were then observed as the magnetic field was slowly reduced in order to simulate quasi-equilibrium conditions. At certain critical field strengths, the nanorods were observed to rotate orthogonally to the
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This does not imply that a transition between parallel and perpendicular orientations is impossible for 2εβ > 1, but instead that the relevant time scale for switching is much larger than the timescales used in our experimental investigations. For example, consider the expected time required for a nanorod to overcome a potential energy barrier by random fluctuations. Assuming a 1-D random walk model, the mean square Brownian displacement of a nanorod’s orientation in the absence of an energy barrier is described by38,39
〈θ2 〉 ) 2Drt
Figure 9. In the above diagram, (A) illustrates the conditions for which the nanorods would align perpendicular to the uniform external field vector, while (C) represents the magnitude of the energy barrier to rotation from a parallel to a perpendicular orientation, for the points on (A) along the pocket border denoted by the dotted lines. The magnitude of the energy barrier is given in terms of kBT. (B) and (D) are expanded views of the black box regions in (A) and (C). Image (D) especially, illustrates that the energy barrier has a zero value only within a very narrow range of values for ζ and 2εβ.
external field vector. Nearby this critical field strength, the field was oscillated over 1-2 Oe intervals, while providing periodic pauses between one-half to one minute, in order to provide sufficient time for the nanorods to settle into their equilibrium orientation. The field at which the nanorods had rotated approximately perpendicular to the external field vector was recorded with a handheld gaussmeter (Lakeshore Cyrotonics, Westerville, OH). Measurements were conducted on single nanorods, and 50 data points were obtained for each setup. By this method, an average switching field was obtained and used for comparison with the expected switching fields predicted by the computational model.
V. Results and Discussion The experimental data for each micromagnet array layout is presented in Figure 8. Assuming a saturation magnetization of 0.5 T, which is consistent with the magnetization of thin Cobalt film,37 we determined the value of σ0 to be 43.75 Oe for the experimental setup. This value was used as an input for computing the expected switching field from the computational model. The theoretically expected switching field is compared with that obtained from experiments as shown in Figure 8. For comparison, we also provide the theoretically expected switching fields using σ0 ) 70 Oe and σ0 ) 30 Oe, respectively. The agreement between theory and experiment was found to be reasonably accurate, and in strong support of the computational models. When conducting these experiments, it was discovered that reversible switching was only observed for values of 2εβ < 1, contrary to theoretical simulations where switching between parallel and perpendicular states is expected to occur over a much larger range of 2εβ values. This discrepancy can be explained by analyzing the potential energy landscape of the nanorod as it rotates from parallel to perpendicular orientations. In this system, a potential energy barrier to rotation was observed for nearly all regions of the phase diagram except for very small values of 2εβ. For nanorods of this size, the energy barrier can be quite massive (e.g., greater than 100 kBT as shown in Figure 9), thus the majority of pathways into the pockets appeared to be kinetically inaccessible in our experimental system. This is clearly illustrated in Figure 9, in which the energy barrier is plotted as a function of 2εβ. (37) Parker, G. J.; Cerjan, C. J. J. Appl. Phys. 2002, 87(9), 5514–5516.
(16)
where Dr is the rotational diffusion coefficient of the nanorod and is determined as a function of the nanorod’s geometry, assumed to be a circular cylinder with length L and radius r, and in a fluid with viscosity, η
Dr )
kT γd
(17)
The rotational drag coefficient, γd, is given by
γd )
πη 3
L3
( 2rL ) - 0.66
(18)
ln
Assuming values of L and r that match the nanorods used in our experiments, temperature of 297 K, and ferrofluid viscosity of η ) 0.005 Pa s, as reported by the manufacturer, the rotational diffusion coefficient is calculated to be Dr ) 0.135 rad2/s. In the absence of a potential energy barrier, the characteristic frequency for rotation by π/2 radians can be determined by solving eq 16, and is found to be approximately 0.01 Hz for this experimental system. In the presence of an energy barrier to rotation, this characteristic time scale increases dramatically. In essence, the kinetics of nanorod rotation behaves similarly to the kinetics of a chemical reaction, given in the Arrhenius form: k ) Ae-UA/kT where the prefactor A is assumed to be the characteristic Brownian rotation frequency. Thus, even for small energy barriers of approximately 1 kBT, the characteristic time required to switch between parallel and perpendicular states is on the order of 5 min. Considering that most of the pocket displayed an energy barrier greater than 100-200 kBT, our inability to observe reversible switching for 2εβ > 1 makes physical sense. Although the experimental results and theoretical predictions are in strong agreement, caution must be taken when applying this model as some of the assumptions did not always hold true. This is clearly observed in Figure 10, where the nanorods were observed to rotate about an off-axis position with respect to the center of the micromagnets. This trend was more pronounced for longer micromagnets (i.e., regions of smaller β), indicating that the location of minimum field can change as a function of the micromagnet’s size and the external field strength. This effect was neglected in our computational phase diagrams, where we assumed that the nanorods would remain centered directly on top of the micromagnets. In fact, the good fit between theory and experiment shown in Figure 8 required the assumption that the nanorod’s center was positioned slightly off axis from the micromagnet’s center, more specifically, at the midpoint between the center and the edge of the micromagnets. This assumption appears to be quite satisfactory, as can be seen from the position of the nanorods on the 12 µm micromagnets, shown in Figure 10b. In each instance, the (38) Shelton, W. A.; Bonin, K. D.; Walker, T. G. Phys. ReV. E 2005, 71(3), 036204-8. (39) Tirado, M. M.; de la Torre, J. G. J. Chem. Phys. 1980, 73(4), 1986–1993.
Field Gradients Can Control the Alignment of Nanorods
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Figure 10. Microscopy images obtained for two of the experimental setups, with micromagnet sizes of 8 (A) and 12 µm (B). The switching fields of (A, B) were 18.3 and 15.6 Oe, respectively. In both cases, the external field is applied along the long axis of the micromagnets, but the rods are aligning perpendicular to the external field vector. As can be seen from the second image especially, the nanorods are off-center, which contradicts an assumption made in the computational model. The black dotted lines indicate the actual center of the micromagnets, and the location where the nanorods were assumed to be located in the computational results presented in Figure 3–7.
deviation between the expected switching field, using a value of σ0 ) 43.75 G, was within a few Oe of the experimentally obtained values. These results suggest that the computational model adequately describes the physics of this phenomenon, and are reasonable representations for predicting the controllability of the nanorod’s switching behavior.
VI. Conclusion Here we developed a computational model to explain our interesting experimental observations concerning the effect of field gradients on the alignment of nanorods. The controllability of nanorod alignment was studied as a function of various control parameters, including the substrate’s field strength, the nanorod’s volume, and the size ratio between the nanorod and micromagnets. Although these simulations indicated that large pockets of stability existed for perpendicular orientations, most of these regions were discovered to be kinetically inaccessible due to the presence of an energy barrier to rotation. Only in the regions of the phase diagram where the nanorod was shorter than the micromagnets, in which case the potential energy barrier to rotation is absent, was reversible switching between parallel and perpendicular orientations observed experimentally. The ability to control nanorod alignment with field gradients provides yet another route to guiding self-assembly processes
for integrating nanoscale devices into useful engineering systems. This technique is expected to be of particular value for the assembly of nanowire cross junctions, which have applications in lasers and electronics devices.40–42 Similar control over nanorod alignment in dielectrophoresis systems is also expected to occur when combined with ferroelectric surface elements; however this type of behavior has not been reported previously. Acknowledgment. This work was supported in part by the National Science Foundation under NER Grant No. 0608819 and CMMI Grant No. 0625480. We gratefully acknowledge the kind gift of bar-coded nanorods from Scott Norton, Oxonica, Inc. Supporting Information Available: Supplementary movie 1 is provided to illustrate reversible switching of the nanorod’s orientation in oscillating external magnetic field. This material is available free of charge via the Internet at http://pubs.acs.org. LA801006G (40) Huang, Y.; Duan, X.; Cui, Y.; Lauhon, L.; Kim, K.; Lieber, C. M. Science 2001, 294, 1313–1317. (41) Cui, Y.; Lieber, C. M. Science 2001, 291, 851–853. (42) Chen, Y.; Jung, G. Y.; Ohlberg, D. A. A.; Li, X.; Steward, D. R.; Jeppesen, J. O.; Nielsen, K. A.; Stoddart, J. F.; Williams, R. S. Nanotechnology 2003, 14, 462–468. (43) Snider, G. S. Appl. Phys. A: Mater. Sci. Process. 2005, 80, 1165–1172.
