# Chain Formation and Phase Separation in Ferrofluids: The Influence on Viscous Properties

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Main Structures and Approaches

**m**and

_{1}**m**is equal to the energy ${U}_{dd}$ of the interaction between two point dipoles situated at the centers of the particles:

_{2}**r**stands for the radius vector that links the centers of the particles. It is convenient to introduce a dimensionless parameter λ for the interaction energy between two identical closely situated particles, each with magnetic moment m and diameter d:

**r**. This conclusion is very important for the development of theoretical approaches used to study the physics of ferrofluids. Analysis has shown that the theory presented in [30] considers the homogeneous fluctuations (“clouds”) of a local density of the particles in ferrofluids. Therefore, the validity of its application for the description of heterogeneous chains and other aggregates is, at best, disputable.

## 3. Chain-Like Structures and Their Effect on Macroscopic Properties of Ferrofluids

#### 3.1. Magnetoviscous Effect

#### 3.2. Viscoelastic Effects

## 4. Condensation Phase Transitions in Ferrofluids

^{−6}N/m [39,40,41,43,52,54]), the elongation degree might be very large, even under the influence of a weak-to-moderate strength external field; therefore, the experimental observations will very often demonstrate the columnar-like structures, as shown, for example, in Figure 1.

#### 4.1. Effect of the Drop Aggregates on the Viscous Properties of Ferrofluids

#### 4.2. Quasi-Elastic and Yield Stress Effects

## 5. Ferrofluids with Multicore Particles

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Typical photo [34] of the drop-like aggregates aligned along an applied magnetic field.

**Figure 2.**Illustration of the “head-to-tail” (left) and “side-by-side” (right) relative alignments of the particles.

**Figure 3.**Photo [59] of the dense cylindrical domains in a gap that is filled with ferrofluid. The field is perpendicular to the figure plane. Published with permission of the American Physical Society (license RNP/20/AUG/029405).

**Figure 4.**(

**a**–

**f**) Patterns obtained using small-angle neutron scattering for a ferrofluid based on cobalt nanoparticles for different magnetic field strengths and shear rates. The image was sourced from [77], copyright Elsevier.

**Figure 5.**Scattering patterns calculated using Monte Carlo simulations for an idealized monodisperse ferrofluid at rest and experiencing a shear rate of 1 s

^{−1}with a magnetic field strength of 200 mT applied for two different directions between a neutron beam and a magnetic field. The images were kindly provided by Patrick Ilg (University of Reading) and were obtained using the methods given in [66].

**Figure 7.**Illustration of a linear chain. The arrows indicate the magnetic moments of the particles.

**Figure 8.**The dependence of the reduced magnetic effective viscosity on magnetic field H inside the ferrofluid [68]. The diameter of the magnetic core of the ‘‘large’’ particles was ${d}_{l}=16.5$ nm, while their hydrodynamic volume concentration was ${\phi}_{l}=0.017$. The dots are experimental data, while the lines correspond to calculations: (

**a**) $\dot{\gamma}=0.1{\mathrm{s}}^{-1}$ (1), $0.5{\mathrm{s}}^{-1}$ (2), and $0.9{\mathrm{s}}^{-1}$ (3); (

**b**) $\dot{\gamma}=1.05{\mathrm{s}}^{-1}$ (1) and $5.23{\mathrm{s}}^{-1}$ (2).

**Figure 9.**The time dependence [92] of the effective viscosity $\eta $ after a stepwise decrease of the shear rate $\dot{\gamma}$ from 16 s

^{−1}to 1.6 s

^{−1}at t = 0. Curve 1—the dimensionless magnetic field $\kappa =3$ Curve 2—$\kappa =1$. Parameters of the system: the hydrodynamic diameter (with the surface layers) of the particles was 16 nm, the volume concentration of the particles was $\phi =$ 0.015, the dipolar coupling was $\lambda $ = 2.75, the viscosity of the carrier liquid ${\eta}_{0}=0.13$ Pa∙s. Published with the permission of the American Physical Society (license RNP/20/AUG/029407).

**Figure 10.**Relaxation time $\tau $ versus the applied magnetic field H. Curves 1 and 2—theoretical calculations [92] for a ferrofluid with the same parameters as in Figure 9. Theory: the shear rate $\dot{\gamma}$ changed in a stepwise manner from 16 s

^{−1}to 1.6 s

^{−1}(line 1) and back (line 2). Dots 3—experiments [70] with an oscillating shear flow. Published with the permission of the American Physical Society (license RNP/20/AUG/029408).

**Figure 11.**Three-dimensional visualization of the (

**a**) short chains (obtained at H = 10 kA/m) and (

**b**) bulk structures (obtained at H = 450 kA/m) of magnetic particles. The images were kindly provided by Dmitry Borin (TU Dresden) and were obtained using methods equivalent to those given in [103].

**Figure 12.**Microscopic images of a single thin chain (

**a**) and bulk structure/thick chain (

**b**) of magnetic particles similar to the structures shown in Figure 11. The images were kindly provided by Dmitry Borin (TU Dresden) and were obtained using methods equivalent to those given in [103] and shown in Figure 11. Adopted from [103].

**Figure 13.**Total stress σ versus the shear rate $\dot{\gamma}$. Lines represent theoretical predictions and the dots represent experimental data: line 1 and open squares correspond to a field of 8.6 kA/m, while line 2 and filled squares correspond to a field of 5.7 kA/m. Adopted from [122]. Further details are given in [122].

**Figure 14.**Lamellar structures detected in computer simulations [88] (left; Figure 17 of [88]) and in laboratory experiments [127] (right, Figure 3 of [127]). Published with the permission of the Royal Society of Chemistry (license 1055535-1) and American Physical Society (license RNP/20/AUG/029406).

**Figure 15.**TEM images [137] of the clustered particles for two different samples. The mean diameter of the magnetic nanoparticles is 13 nm (

**a**) and 17 nm (

**b**).

**Figure 16.**Microscopic images of the magnetic fluid based on multicore magnetite nanoparticle structures at various magnetic field strengths; the concentration of the magnetic phase was φ ≈ 0.013 vol.% [125].

**Figure 17.**The magnetoviscous effect in a multicore-based ferrofluid was measured and calculated using various shear rates as a function of the applied magnetic field. The solid lines are the results of the calculations performed according to the chain model (hydrodynamic diameter of the particle cluster was assumed to be 90 nm, λ = 2.3) [143].

**Figure 18.**Comparison of the theoretical prediction and experimental results for the residual stress τ

_{r}versus the magnetic field strength. The proportion of magnetic material that was part of the bulk drops was 0.12. Details of the experiment and calculations are given in [144]. The plots were redrawn by the authors using our own raw data.

**Figure 19.**Comparison of the theoretical prediction and experimental results of the stress relaxation for various magnetic field strengths after a stepwise change of the shear rate from 0.02 to 0 s

^{−1}. Details of the experiment and calculations are given in [144]. The plots were redrawn by the authors using our own raw data.

**Figure 20.**Structures formed by a multicore-based ferrofluid under the influence of a magnetic field of H = 20 kA/m (

**a**) and H = 80 kA/m (

**b**) for several time steps after the application of the respective magnetic field strength [126].

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ivanov, A.O.; Zubarev, A. Chain Formation and Phase Separation in Ferrofluids: The Influence on Viscous Properties. *Materials* **2020**, *13*, 3956.
https://doi.org/10.3390/ma13183956

**AMA Style**

Ivanov AO, Zubarev A. Chain Formation and Phase Separation in Ferrofluids: The Influence on Viscous Properties. *Materials*. 2020; 13(18):3956.
https://doi.org/10.3390/ma13183956

**Chicago/Turabian Style**

Ivanov, Alexey O., and Andrey Zubarev. 2020. "Chain Formation and Phase Separation in Ferrofluids: The Influence on Viscous Properties" *Materials* 13, no. 18: 3956.
https://doi.org/10.3390/ma13183956