Introduction

Magnetic nanoparticles exposed to external magnetic fields can assemble into ordered structures including one-dimensional (1D) nanochains, 2D sheets, and 3D photonic crystals (He et al. 2012; Wang et al. 2013a, b; Ohno et al. 2018; He et al. 2011; Zhang et al. 2010; Helseth 2005). 1D chain-like structures represent the simplest ordered distribution when magnetic nanoparticles transform from a random state. The shape anisotropy and periodic structures give 1D nanostructures obvious advantages in potential applications such as microelectronics, magnetic recording, drug delivery, sensing, interference, photovoltaic, spintronics, etc. (Helseth 2005; Yuan et al. 2011; Laughlin et al. 2000; Kamble and Mathe 2008; Richardson and Jones 2009; Chen et al. 2018; Ravi and Prabvin 2013; Ravi and Shashikanth 2017).

Driven by the excellent performance and broad application prospects, considerable efforts have been devoted to magnetic assembly of chain-like structures, including experimental observations and numerical simulations (Kralj and Makovec 2015; Wang et al. 2010; Ye et al. 2010; Hu et al. 2011; Wang et al. 2013a, b; Satoh et al. 1996, 2000; Enomoto et al. 2003; Peng et al. 2009; Xue et al. 2017). Prior work has been carried out to reveal the assembly mechanism in colloidal suspensions (Tran et al. 2015; Furst and Gast 2000; Bertoni et al. 2011; Lalatonne et al. 2004). The key factors that determine the chain-like structures of magnetic nanoparticles are two major types of interactions: van der Waals and magnetic dipolar interactions. Under weak dipolar interactions, such as magnetic nanocrystals, van der Waals interactions are expected to be the dominant contribution to the total interparticle potential, so isotropic van der Waals interactions induce the formation of disordered aggregates. Under weak van der Waals interactions, dominant dipolar interactions can control the formation of chain-like structures of individual nanoparticles. Furthermore, the interparticle interactions depend on magnetic controlling parameters, such as the magnetic field intensity, magnetization, particle radius and concentration (Xue and Furlani 2014, 2015). However, research on the magnetic assembly mechanism is mainly from the perspective of interparticle interactions, and the quantitative relationship between essential magnetic controlling parameters and assembled chain-like structures still needs to be further studied.

In this work, we use a Monte Carlo method to simulate the chain-like structure formation of magnetic nanoparticles. The aim is to investigate the influences of essential magnetic controlling parameters on chain-like structures in theory. The results presented here are in agreement with the experimental observations of assembled chain-like structures, suggesting the possibility of guiding the construction of other assembled structures consisting of building blocks.

Methods

Experiment

MnFe2O4 nanospheres were obtained by the solvothermal method. The preparation procedure and characterization results are described in detail in the Supplementary Material. In addition, the synthesis process of the chain-like structures was presented as follows. A total of 1.5 mL of MnFe2O4 aqueous solution was dispersed in ethanol (10 mL) and tetraethoxysilane (TEOS, 50 μL) under sonication. After that, the mixture was transferred into a flask under mechanical stirring. After 12 min, aqueous ammonia (0.5 mL) was quickly injected, and then the mixture was transferred to a beaker (25 mL) and placed inside a Helmholtz coil creating a uniform magnetic field for 1 h. The magnetically assembled structure was stabilized and fixated by the deposition of an additional layer of silica using the hydrolyzation and condensation of TEOS. After being removed from the magnetic field, the sample was allowed to sit undisturbed for another 30 min, after which it was washed with ethanol three times and redispersed in ethanol.

Calculation

Simulations were carried out using a three-dimensional Monte Carlo method based on the Metropolis algorithm (Peng et al. 2009). The Monte-Carlo method is a statistical tool that models the probability of different outcomes and has been used to analyze magnetic properties and the Curie temperature (Miller and Drillon 2005; Ravi 2020). The simulation system consisted of N = 200 particles with diameter d dispersed in a 3D cube. The particles were covered by a steric layer of thickness δ = 0.01d. The initial configuration of particles was taken as random. The maximum displacement of a particle in one trial move was 0.025d, and a cluster was defined as a group of particles in which adjacent particles existed within 1.025d. The moments of magnetic particles were fixed along the field direction. Periodic boundary conditions were applied in the X, Y and Z directions. Each simulation was carried out up to 20,000 steps to obtain a stable assembled structure.

Our computational model took into account several dominant assembly mechanisms, including the induced magnetostatic energy \( {u}_i^{(H)} \), magnetic dipole-dipole interactions \( {u}_{ij}^{(m)} \), van der Waals potential energy \( {u}_{ij}^{(v)} \) and surfactant interaction \( {u}_{ij}^{(r)} \) caused by surfactant-surfactant contact. The details of the model are briefly summarized as follows (Xue et al. 2017):

$$ {u}_i^{(H)}=-{\mu}_o{\mathbf{m}}_{\mathbf{i}}\cdotp \mathbf{H} $$
(1)
$$ {u}_{ij}^{(m)}=\frac{\mu_o}{4\pi {r}_{ij}^3}\cdotp \left[{\mathbf{m}}_{\mathbf{i}}\cdotp {\mathbf{m}}_{\mathbf{j}}-\frac{3}{r_{ij}^2}\left({\mathbf{m}}_{\mathbf{i}}\cdotp {\mathbf{r}}_{\mathbf{i}\mathbf{j}}\right)\left({\mathbf{m}}_{\mathbf{j}}\cdotp {\mathbf{r}}_{\mathbf{i}\mathbf{j}}\right)\right] $$
(2)
$$ {\mathrm{u}}_{\mathrm{ij}}^{\left(\mathrm{v}\right)}=-\frac{\mathrm{A}}{6}\left[\frac{2{\mathrm{R}}_{\mathrm{p},\mathrm{i}}\cdotp {\mathrm{R}}_{\mathrm{p},\mathrm{j}}}{{\mathrm{r}}_{\mathrm{ij}}^2-{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}+{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}^2}+\frac{2{\mathrm{R}}_{\mathrm{p},\mathrm{i}}\cdotp {\mathrm{R}}_{\mathrm{p},\mathrm{j}}}{{\mathrm{r}}_{\mathrm{ij}}^2-{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}-{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}^2}+\ln \frac{{\mathrm{r}}_{\mathrm{ij}}^2-{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}+{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}^2}{{\mathrm{r}}_{\mathrm{ij}}^2-{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}-{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}^2}\right] $$
(3)
$$ {\mathrm{u}}_{\mathrm{ij}}^{\left(\mathrm{r}\right)}=2\uppi \frac{{\mathrm{R}}_{\mathrm{p},\mathrm{i}}^2\cdotp {\mathrm{R}}_{\mathrm{p},\mathrm{j}}^2}{{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}+{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}^2}{\mathrm{N}}_{\mathrm{s}}{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\left[2-\frac{{\mathrm{r}}_{\mathrm{ij}}-\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}+{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)}{\updelta}-\frac{{\mathrm{r}}_{\mathrm{ij}}}{\updelta}\ln \left(\frac{\left({\mathrm{R}}_{\mathrm{p},\mathrm{i}}+{\mathrm{R}}_{\mathrm{p},\mathrm{j}}\right)+2\updelta}{{\mathrm{r}}_{\mathrm{ij}}}\right)\right] $$
(4)

where μo is the magnetic permeability of free space, H is the applied magnetic field, mi and mj are the magnetic moments of the i’th and j’th particles, Rp, i and Rp, j are the radii of the i’th and j’th particles, respectively (in our study, Rp, i=Rp, j), rij is the magnitude of the vector rij drawn from particle i to j, A is the Hamaker constant, kB is Boltzmann’s constant, T is the absolute temperature of the suspension, δ is the thickness of the surfactant layer and Ns is the surface density of surfactant molecules.

Results and discussion

From the mathematical expression of the interaction model, the factors that influence the interactions determining the assembled structures are the radius of particles R, magnetic moment of particles m, center-center distance between particles r determined by the volume fraction of particles φ, and magnetic field intensity H.

An applied magnetic field induces a magnetic moment in a magnetic particle m=χHV=MV (Wang et al. 2013a, b), where χ is the volume susceptibility of the particle, H is the local magnetic field intensity, M is the magnetization intensity of the particle, and V is the volume of the particle. For magnetically assembled systems, the radius of particles R and magnetic field intensity H control assembled structures mainly by influencing the magnetic moment m. Hence, the major magnetic controlling parameters are the magnetic moment m and volume fraction φ of particles. The effects of the magnetic moment and volume fraction on the assembled chain-like structures are discussed in the following section.

Aggregates of magnetic nanoparticles

According to the magnetic moment m = MV, we can adjust the magnetic moment by changing the magnetization intensity. The assembled distributions of magnetic nanoparticles at low magnetic moments were studied, using snapshots of the assembled structures in equilibrium for φ= 0.02%, d = 360 nm, and M = 9 kA/m, 10 kA/m, 12 kA/m, and 15 kA/m, namely the magnetic moment was m=\( \frac{1}{6}\uppi M{d}^3 \)=2.2×10−16 A∙m2, 2.4×10−16 A∙m2, 2.9×10−16 A∙m2, and 3.7×10−16 A∙m2, as shown in Fig. 1. At m = 2.2×10−16 A∙m2 and 2.4×10−16 A∙m2, which are low magnetic moments, magnetic nanoparticles did not arrange along the direction of the magnetic field but formed disordered aggregates as shown in the illustration of Fig. 1a and 1b. Figure 2 shows the TEM image of assembled disordered aggregates of magnetic nanoparticles for the stirring speed ω= 280 r/min, the volume fraction φ= 0.02%, the magnetic field intensity H = 3978.9 A/m, and the TEM image qualitatively supports the simulation results in Fig. 1a and 1b.

Fig. 1
figure 1

Assembled structure of magnetic nanoparticles for φ= 0.02% and d = 360 nm: (a) m = 2.2×10−16 A∙m2; (b) m = 2.4×10−16 A·m2; (c) m = 2.9×10−16 A∙m2; (d) m = 3.7×10−16 A∙m2

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Fig. 2
figure 2

TEM image of assembled disordered aggregates of magnetic nanoparticles for ω = 280 r/min, φ= 0.02%, H = 3978.9 A/m

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Upon further increasing the magnetic moment, magnetic nanoparticles gradually assembled into chain-like aggregates rather than nanochains of individual particles, as shown in Fig. 1c–1d. For a low magnetic moment, the value of the magnetic moment was too small to produce significant magnetic dipolar interactions. Therefore, van der Waals interactions were expected to be the dominant contribution to the total interparticle potential. Consequently, the individual magnetic nanoparticles formed aggregates due to the van der Waals interactions, and the strength of the dipole moment of aggregates can facilitate the formation of chain-like clusters.

Nanochain formation of individual magnetic nanoparticles

As the magnetic moment of the magnetic nanoparticles further increased, sufficient magnetic dipolar interactions versus van der Waals interactions were produced. As shown in Fig. 3a, the aggregates in the nanochains disappeared, and the clusters displayed the formation of nanochains of individual particles for m = 1.5×10−15 A∙m2, indicating that dipolar interactions overwhelmed van der Waals interactions to drive the assembly of nanochains.

Fig. 3
figure 3

Assembled structure of magnetic nanoparticles for φ= 0.02% and d = 360 nm: (a) m = 1.5×10−15 A∙m2; (b) m = 7.3×10−15 A∙m2; (c) m = 9.8×10−15 A·m2; (d) m = 1.2×10−14 A∙m2

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For the case of m = 7.3×10−15 A∙m2, the short nanochains gradually merged into long nanochains, and the arrangement of the nanochains became straighter, in agreement with the TEM image (ω= 280 r/min, φ= 0.02%, H = 19098.6 A/m) in Fig. 4. When the magnetization intensity further increased to m = 9.8×10−15 A∙m2 and even m = 1.2×10−14 A∙m2, the distribution of nanochains was almost the same as that in the case of m = 7.3×10−15 A∙m2 based on observing Fig. 3b–3d and comparing the curves in Fig. 5. This occurred because at a high magnetic moment and a low volume fraction, the nanochains were too rigid and the dipolar interactions between nanochains were repulsive in the far field, making it difficult for these nanochains to interact to merge into longer nanochains (Furst and Gast 2000).

Fig. 4
figure 4

TEM images of nanochains of individual magnetic nanoparticles for ω = 280 r/min, φ= 0.02%, H = 19098.6 A/m

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Fig. 5
figure 5

Number distribution of particles in a chain

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Transition from nanochains to nanobundles

At a high magnetic moment and a low volume fraction, the rigid nanochains with interchain repulsion interactions were dilute and well separated, and the distribution of chain-like assembled structures was mainly determined by the volume fraction. Hence, the assembly process of magnetic nanoparticles at high magnetic moment and volume fraction was investigated with m = 7.3×10−15 A∙m2, d = 360 nm, and φ= 1, 2, 6, and 8%. For reasons of computation time, the volume fraction of magnetic nanoparticles was adjusted by changing the side length of the computational domain.

Fig. 6 shows snapshots of the assembled structures in equilibrium. As the volume fraction of magnetic particles increased, the nanochains gradually attracted each other and coarsened into nanobundles, and these bundle-like structures were qualitatively in agreement with the experimental observation (ω= 280 r/min, φ= 8%, H = 19098.6 A/m) in Fig. 7. These results indicated that for dilute dipolar nanochains, the interchain interactions, mainly magnetic dipole-dipole repulsion interactions, kept the chains away from each other. However, at higher volume fractions, the separation between nanochains decreased, and the interchain repulsive interactions significantly increased. When the volume fraction of the nanochains, as well as the dipolar repulsive interactions increased to a certain extent, the interchain repulsive interactions were sufficiently strong such that the assembled system became thermodynamically unstable. Furthermore, the near-field behavior of fluctuating nanochains exhibited a strong and sudden change between attractive and repulsive interactions, resulting in a relative displacement between the nanochains, and the nanochains started to aggregate into nanobundles at this moment (Malik et al. 2012; Islam et al. 2003).

Fig. 6
figure 6

Assembled structure of magnetic nanoparticles for m = 7.3×10−15 A∙m2 and d = 360 nm: (a) φ= 1%; (b) φ= 2%; (c) φ= 6%; (d) φ= 8%

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Fig. 7
figure 7

SEM images of 3D nanobundles for ω = 280 r/min, φ= 8%, H = 19098.6 A/m

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Conclusions

The influences of essential magnetic controlling parameters on the chain-like structures of magnetic nanoparticles were studied by making use of a Monte Carlo 3D algorithm. The magnetic moment and volume fraction of magnetic nanoparticles were found to play crucial roles in the arrangement of the nanoparticles. The simulation results were in agreement with experimental evidence, summarized as follows: In the limit of low magnetic moment m < 3.7×10−16 A∙m2, magnetic nanoparticles did not linearly arrange along the direction of the magnetic field but formed aggregates. At magnetic moment m > 1.5×10−15 A∙m2 and volume fraction φ< 0.02%, the formation effect of nanochains, which consisted of individual particles, first enhanced and then remained unchanged with increasing magnetic moment. For magnetic moment m > 7.3×10−15 A∙m2 and volume fraction φ > 1%, nanochains gradually coarsened into nanobundles with increasing volume fraction. The results may lay the foundation for researching other assembled structures consisting of building blocks.