A gradient field defeats the inherent repulsion between magnetic nanorods

When controlling the assembly of magnetic nanorods and chains of magnetic nanoparticles, it is extremely challenging to bring them together side by side while keeping a desired spacing between their axes. We show that this challenge can be successfully resolved by using a non-uniform magnetic field that defeats an inherent repulsion between nanorods. Nickel nanorods were suspended in a viscous film and a non-uniform field was used to control their placement. The in-plane movement of nanorods was tracked with a high-speed camera and a detailed image analysis was conducted to quantitatively characterize the behaviour of the nanorods. The analysis focused on the behaviour of a pair of neighbour nanorods, and a corresponding dynamic model was formulated and investigated. The complex two-dimensional dynamics of a nanorod pair was analysed analytically and numerically, and a phase portrait was constructed. Using this phase portrait, we classified the nanorod behaviour and revealed the experimental conditions in which nanorods could be placed side by side. Dependence of the distance between a pair of neighbour nanorods on physical parameters was analysed. With the aid of the proposed theory, one can build different lattices and control their spacing by applying different field gradients.

2. Introduction

In the past decade, one-dimensional magnetic nanostructures, such as magnetic nanorods, chains of magnetic nanoparticles and nanotubes filled with magnetic nanoparticles, have offered great opportunities for the design of multi-functional devices and for the manufacturing of anisotropic nano- and microstructures [1–4]. These applications include, for example, optofluidics [5–9], microrheology [10–14], magnetic swimming [15–20], photonics [21], drug delivery [22] and electromagnetic shielding [23]. Particularly, in the manufacturing of composite materials, different configurations of magnetic fields are usually applied to obtain the desired pattern of magnetic rods or chains [2,4,24–27]. A uniform magnetic field is usually used to align the nanorods in one direction or to form self-assembled chains from magnetic nanoparticles. Recently, the strategies for aligning an assembly of non-interacting magnetic nanorods in both Newtonian and non-Newtonian fluids under a uniform magnetic field have been proposed and developed [28,29]. However, in many cases, one needs to deal with a concentrated colloid of magnetic nanorods where the interactions between nanorods are crucial for the patterning of the microstructures [30–36].

The main challenge when attempting to control the assembly of magnetic nanorods is bringing them together and placing them side by side [7,35,36]. Indeed, when two identical magnetic nanorods come together side by side, they are prone to move away owing to their inherent repulsion. A uniform magnetic field keeps them parallel to each other but they tend to form a tandem with a head-to-tail ordering. Phase diagrams for the long nanorods demonstrate a significant enlargement of the region of repulsion compared with the point dipoles [7,37]. Therefore, one needs to develop a new strategy in order to defeat this inherent repulsion.

One possible strategy is to use a non-uniform magnetic field with a field gradient [30,33,35,38]. In a non-uniform magnetic field generated by a magnet, the magnetic force acting on a nanorod with magnetization vector M is written as F=V (M⋅∇)B, where B is the magnetic field vector and V is the volume of the nanorod. As follows from this formula, magnetic nanorods can be pushed towards each other by generating a special field gradient. For example, in the region close to the axis of a cylindrical magnet, the radial component of the magnetic field B_r is much weaker than the axial component B_y, B_r≪B_y, where the y-axis is taken along the axis of the cylinder. The magnetic force can be approximately calculated as F=V (M∂/∂y)B, where M is the magnetization of the nanorod. For a cylindrical magnet like the one shown in figure 1a,b, the radial component of the magnetic field is positive (B_r>0) and its gradient is negative along the y-axis (dB_r/dy<0). Thus, the radial component of magnetic force F_r is negative (F_r<0) and the nanorods tend to cluster at the central axis. These arguments show that the placement of magnetic nanorods next to each other can be accomplished by applying a non-uniform magnetic field with a strong axial gradient of the radial component of a magnetic field [39].

However, such strong gradients are difficult to produce at the micrometre scale. On the other hand, the gradient dB_y/dy of the axial component of a magnetic field is much easier to create and control. Taking advantage of this type of field non-uniformity, various microstructures have already been successfully fabricated [30,33,35], and interest in using this type of field gradient is growing [3,9,30,33,34,40–42]. However, owing to the lack of complete understanding about the behaviour of nanorods in a strong dB_y/dy gradient, the inter-rod spacing is still difficult to control [2,9,43].

In this paper, we overcome this challenge by modelling the behaviour of a pair of magnetic nanorods in a non-uniform magnetic field with a dB_y/dy gradient. We analyse the dynamics of the nanorods suspended in a two-dimensional Newtonian film. The nanorods are pulled together by the dB_y/dy gradient. They also interact with each other through their magnetic poles. In experiments, we use nickel nanorods; their dynamics is filmed using dark field microscopy. We develop a particle-tracking algorithm to follow the nanorod movement and analyse the nanorod trajectories. The analysis of experiments shows that the model adequately describes the behaviour of interacting magnetic nanorods. The conditions for controlled placement of magnetic nanorods were theoretically studied and experimentally examined.

3. Experiments

3.1 Preparation of a dispersion of nickel nanorods

To make nickel nanorods, we used electrochemical template synthesis [44,45]. Nanorods were synthesized inside 200 nm pores of alumina membranes (Watman Ltd.). We exactly followed the procedure described in the supporting information of Tokarev et al. [11]. This experimental protocol enables one to produce nanorods of approximately 6 μm in length and less than 200 nm in diameter. Following the protocol as described in [28,40], we stabilized the nanorods with a layer of polyvinylpyrrolidone (PVP). The PVP-coated nickel nanorods were dispersed in a 76 wt% water–glycerol mixture (76% glycerol, 24% water). In experiments, a dispersion of nickel nanorods of low concentration (0.005 wt%) was used. A 1 μl droplet of this dispersion was placed on a glass slide (VWR International, LLC) and immediately covered by another glass cover slide (VWR International, LLC). Two 26 μm thick Nylon fibres were placed as spacers between slides. This gap thickness was chosen in order that the effect of hydrodynamic interactions between the nanorods and the substrates could be neglected. In our earlier publication [28], we confirmed that the chosen gap thickness is optimal; measuring viscosity of different standard liquids with same set-up and using magnetic rotational spectroscopy, we reproduced viscosity of these standards [11,46].

3.2 Optical cell

The schematic of the optical cell is shown in figure 1a. Two cylindrical magnets were used in this experiment. The rear face of the smaller magnet was attached to the front face of the larger one so that both magnets had a common axis as shown in figure 1b. The smaller magnet was 1.6 mm in diameter and 1.6 mm in length, and the larger magnet was 12.7 mm in diameter and 12.7 mm in length (Grade N52, K&J Magnetics). This construction allowed for a sufficiently strong field of the order of 1 T. The gradient changes by as much as two orders of magnitude within a distance of approximately 5 mm from the front face of the smaller magnet as illustrated in figure 1c. The field distribution was simulated using COMSOL, taking magnetization as 1.48×10⁶ A m⁻¹ for both magnets and placing the origin of coordinates at the free surface of the small magnet 2.

This construct was positioned under the microscope (Olympus BX 51) with the common axis parallel to the optical stage. The position of the construct was controlled by a linear stage (VT-21, MICOS). The Olympus BX 51 microscope was equipped with a digital camera (SPOT Imaging Solutions, Inc.) allowing for the application of dark field imaging. The sample was positioned under the microscope in front of the smaller cylindrical magnet as shown in figure 1a, and the behaviour of nanorods was studied by focusing the camera on four different spots along the common axis of the magnets at the following positions: y=5, 3, 2 and 1.5 mm.

3.3 Experimental protocol

The main challenge in studying the interactions between nanorods subject to a non-uniform field is that the nanorods always move within the fluid. The nanorods keep moving towards the region of a stronger field until they reach a boundary, e.g. the liquid–air interface. When a nanorod hits a boundary, it does not move anymore and stays pinned to the boundary. It is therefore convenient to focus the camera to a pinned nanorod and watch the behaviour of incoming nanorods (figure 2b).

Initially, the magnets were placed far away from the sample to eliminate any translational motion of the nanorods. The recording started when the magnets were brought closer to the sample. Figure 3 shows two sequences of images illustrating different scenarios of the nanorods landing: (a) landing on top of the pinned nanord and (b) landing next to each other.

4. Magnetostatic interactions between nanorods and external field

In the focal plane of observation, it is convenient to introduce the local Cartesian system of coordinates (X, Y) with the Y-axis aligned along the common axis of the magnets. The origin of the local system of coordinates is taken at the centre of mass of the pinned nanorod as illustrated in figure 2b. The position of the incoming nanorod is measured by its centre of mass (X, Y). The length of the pinned nanorod and the length of the incoming nanorod are denoted as L and l, respectively.

As illustrated in figure 2c, the x-component of the magnetic field B is almost zero at these points. Therefore, the magnetization M of all nanorods is expected to point in the y-direction. If the nanorods were non-interacting, they would move only in the y-direction. Within the small field of view (approx. 20×20 μm²) shown in figure 2b, the dB_y/dy gradient is almost constant, dB_y/dy=α. For a strong gradient (α∼100 T m⁻¹), the variation of the field over a 20 μm distance is approximately B_y∼2 mT. On the other hand, the magnitude of the field is greater than |B|>0.1 T. Therefore, we assume that the magnetization of nanorods is constant and that the two neighbour nanorods have the same magnetization M. In experiments, the applied magnetic field was sufficiently strong to ensure that the nanorods were not able to change their orientations even in close proximity to each other.

As the distance between the nanorods is comparable with their lengths, the nanorods cannot be treated as point dipoles [7]. We therefore use the model of ‘magnetic charges’ [7,47,48]. To determine the charge Q, we divide the magnetic nanorod into a chain of infinitesimally small magnets as shown in figure 2a. Each elementary magnet has the moment m_i=QΔl with an elementary ‘magnetic charge’ Q. If we sum up all elementary magnets within the nanorod, all internal poles of the opposite sign will be cancelled out and ‘magnetic charges’ of the opposite sign will remain only at the ends of the nanorod. For a nanorod with diameter d and magnetization M, the ‘magnetic charge’ Q can be therefore calculated as: Q=πd²M/4.

Following the chosen system of coordinates, figure 2b, the energy of the nanorod subject to an external field can be calculated by introducing magnetostatic potential φ(X, Y). It has to satisfy the Laplace equation written in cylindrical coordinates as: ∂²φ/∂X²+(1/X)(∂φ/∂X)+∂²φ/∂Y ²=0. In the vicinity of the central axis, the potential is represented as: φ=−B₀Y −α(X²−2Y ²)/4, where B₀ is the constant component of a non-uniform external magnetic field taken at the centre of mass of the pinned nanorod (0, 0). In our experiments, the gradient α=−dB_y/dy is always positive. This implies that the nanorods tend to move to the boundary (X, −L/2) as illustrated in figure 2b.

With the given potential, the magnetic field is obtained as B=−∇φ=(αX/2,B₀−αY). The magnetostatic energy of the incoming nanorod in the external magnetic field is calculated as: φ(X,Y +l/2)Q−φ(X,Y −l/2)Q=QlY α−QlB₀. The second term is independent of the position of the incoming nanorod, and hence it does not contribute to the force balance. The total magnetostatic energy of two interacting magnetic nanorods under the field gradient is therefore written as: