## Abstract

In this chapter, we review the experimental and theoretical modeling of structural and dynamical properties of colloidal magnetic fluids at equilibrium. Presently, several prototype experimental systems are very well characterized. We survey the different models, which help to reach a comprehensive knowledge of these complex magnetic fluids. One prime example is the ongoing investigation of the realistic interparticle potentials that drive the formation of the different phase states observed experimentally. Further, a stochastic equation approach for the description of tracer diffusion, viscoelasticity, and dielectric relaxation at equilibrium in colloidal ferrofluids is discussed.

### Keywords

- ferrofluids
- viscoelasticity
- colloidal magnetic fluids
- magnetohydrodynamics
- dielectric relaxation
- diffusion stochastic dynamics

## 1. Introduction

Colloidal magnetic fluids are made up of nanometer and micron-size ferromagnetic particles dispersed either in electrolytes or in organic solvents [1]. Due to their easy manipulation with magnetic or electric fields, they are having a significant impact on diverse technological applications ranging from biomedicine [2] and photonic devices [3, 4] up to fundamental studies that motivate the development of modern theories of nonequilibrium condensed matter [5, 6, 7, 8, 9, 10]. In this chapter, we review the studies of static structural and dynamical properties in colloidal magnetic fluids, which provide a comprehensive description of their bulk phase behavior at thermal equilibrium. Presently, model magnetic colloidal systems can be prepared with tunable interaction among particles of the hard sphere and long-range dipolar types [11, 12, 13, 14], especially the colloidal system constituted of micron-size polystyrene spheres that are electrically charged and dispersed in organic solvents [11]. In such a system, particles acquire an induced electric dipole moment under the external static electric field. Confocal microscopy has allowed the determination of its three-dimensional bulk phase diagram as a function of applied electric field and by taking into account the presence of a 1:1 salt. The electric and magnetic dipole pair interaction potential is symmetric. Consequently, a corresponding change of physical units leads to an equivalent description of the magnetic fluid phase diagrams. Novel Monte Carlo simulation methods have confirmed the different experimental phases [15, 16]. Integral equations and density functional theories have assisted in the understanding of the phases formed by molecular liquids [17], and they have been used to gain qualitative insight into the expected phases as a function of volume fraction and dipolar strength of colloidal magnetic fluids [18, 19, 20, 21, 22, 23]. Likewise, the structure factor and diffusion coefficient of magnetic suspensions made of maghemite nanoparticles dispersed in water and equilibrium with electrolyte solutions have also been reported [24, 25, 26, 27, 28, 29, 30, 31]. The measured structure factor provides the microscopic arrangements of particles accurately. Here, we provide a Langevin stochastic approach that allows the determination of the translational and rotational diffusion of the particles in ferrofluids [32, 33]. These dynamic properties, in turn, allow quantifying the viscoelastic and dielectric moduli of the structural evolution of the fluid toward their equilibrium states [34]. This approach may help to interpret recent experiments of passive microrheology [35, 36] and Altern Current spectroscopic techniques that measure the viscoelastic dynamics and dielectric relaxation of colloidal magnetic fluids. It is expected these theoretical methods be extended to help to understand the corresponding experimental observations of similar dynamics in two-dimensional paramagnetic colloids [37] and under external fields.

## 2. Experimental observations of structural properties of magnetic fluids

The importance of knowing the phase diagram of a ferrofluid resides on the information it provides about the undetermined underlying effective interaction of the ferromagnetic particles [13]. For maghemite nanoparticles in aqueous solutions, there have been attempts to determine such electrostatic potentials by using the measured parameters such as dipole strength and fluid density as inputs in Brownian dynamics simulations to reproduce the observed bulk structure factor [30, 31].

### 2.1 Importance of an experimental phase diagram of maghemite colloid

There is a set of experiments in a well-characterized system of electrically stabilized maghemite nanometer-size particles, which are dispersed in water [24, 25, 26, 27, 28, 29]. Its stabilization is reached with the citrate electrolyte. By fine-tuning the salt concentration, the particle’s interaction was shifted from repulsive, where they form a solid glass phase, into long-range attractive interactions that yield a fluid or gas state behavior. These experiments have prompted the determination of the corresponding pairwise interaction among particles [13]. The proposed interaction consists of a Yukawa repulsive part, a Van der Waals short-range attraction and includes an angular averaged attractive long-range pair dipole potential. There have not been attempts to predict the phase diagram with these potentials. However, the modeled interactions were used to determine with Brownian dynamics the observed structure factor [30, 31]. By using a generic model of pair dipole interactions together with an effective attraction associated with density gradients, Lacoste, et al. [38] considered a quasi-two-dimensional layer of a dilute ferrofluid subjected to a perpendicular magnetic field, and with the help of a mean field energy approach, they found modulated phases given by stripes and hexagonal formations. The Gibbs free energy they considered for inhomogeneous systems has the general shape

It has been recognized that ferromagnetic particles of 10 -nm size remain dispersed [39]; however, when particles have larger sizes up to the micron scale, experiments find they do assemble into chains, rings, and nets were several chains bound together into an amorphous structure [40]. A mean field theory that takes into account such topological structures was developed to explain the liquid-gas phase transition by the formation of Y-like bounds of chains in the network [41]. There remains a quantitative verification of this theory with its experimental counterpart and with computer simulations. Another method to validate the proposed model of particle’s interaction is based on the parameters that appear in the Lennard-Jones potential that represents the Van der Waals attractions together with the pair dipole-dipole potential, and by fitting the predicted magnetization curve as a function of the applied magnetic field to the experimental one [42]. Even though there is not yet a realistic model interaction potential that represents the interaction of ferromagnetic particles, the use of fitting parameters as mentioned above has led to the prediction of the structure factor and the birefringence as a function of the applied field, which shows qualitative agreement with the experimental values [22, 29, 30, 31].

### 2.2 Experimental phase diagram of a charged and sterically stabilized electrorheological colloidal suspension under an electric field

Yethiraj et al. [11] made a colloidal system of charged and sterically stabilized polymethyl methacrylate spheres of micron size in an organic solvent to have dipole moments on particles induced by an electric field. This monodisperse suspension shows both long-range repulsion, and attractive anisotropic interaction potential that is fixed with the addition of salt, whereas the dipolar interaction is controlled by the external electric field. For low Reynolds numbers and in an infinite fluid, a solid spherical particle that sinks in the fluid reaches a terminal velocity due to a balance of the friction force exerted by the fluid that opposes the gravitational force on the particle. If the fluid has a dynamic viscosity *Q*^{∗} is the effective electric charge of a particle that can be measured with electrophoresis experiments.

### 2.3 Theoretical models of phase diagrams of dipolar colloidal fluids: computer simulation and optimization techniques

Finite-temperature calculations of ground state free energies with Yukawa repulsion and dipolar-dipolar interactions of varying strength were performed by Hynninen and Dijkstra [15]. In their calculations, they used canonical Monte Carlos simulations to obtain the free energy minimum as a function of volume fraction and dipolar strength in three-dimensional model systems. They found a phase diagram that contains the same phases as was observed by Yethiraj et al. [11]. However, additionally, they predicted a new hexagonal close-packed phase at the hard sphere repulsion limit and body-centered orthorhombic when the repulsion becomes long range by lowering the content of electrolyte in solution. Their method allows quicker and reliable determinations of the three-dimensional phase diagram than the evolutionary algorithm method that searches for the ground state at zero temperature for two-dimensional systems [45]. The bulk phase diagram has not been searched yet with genetic algorithm (GA) at finite temperature. A particular detail of GA is that to look for the minimum of the free energy, the derivative of the energy function at each evolution step is required for finding the best-adapted crystal structures (individuals) in the population, which is made up of several crystal structures produced by the algorithm. The search of the best-adapted individuals (crystal phase) is done in the energy landscape. For fixed pressure and finite temperature, the Gibbs free energy as a function of pairwise interaction strength and particles concentration in the fluid yields a three-dimensional plot of the free energy known as energy landscape, which has local minima and maxima. The global minimum of an equilibrium crystal structure is reached with the assistance of a local gradient algorithm that requires the derivative of the energy. Such a procedure makes an indirect search of the crystal structure by sampling the energy landscape. Thus, it becomes technically more involved to implement than the Hynninen et al. method of MC ground state energy searches. In Figure 1, we provide two nonequilibrium states of local minima that result from the application of a genetic algorithm approach at zero temperature and fixed pressure.

Recently, Spiteri and Messina [16] have proposed an efficient nonlinear optimization technique that allows predicting the crystal phases of dipolar colloids at zero temperature without the need to use derivatives of the free energies. It was found that for a monodisperse repulsive hard sphere plus dipole-dipole interaction, the predicted phase diagram has the same phases as was previously found by Hynninen et al. [15] and that a knew so-called clinohexagonal prism span all known ground state structures at any density. There remains to be verified if this prediction fulfills at a finite temperature also.

### 2.4 Experimental structure factors and diffusion coefficients

The microstructural order inside of the ferrofluid is determined by the potential interaction among particles. The measured structure factor yields the details of the particles’ spatial geometric arrangement. The water-based maghemite ferrofluid is electrostatically stabilized. The interparticle interaction was modeled in Ref. [13] with a Yukawa repulsion part together with the anisotropic dipolar potential. The highest peak of the structure factor regarding wave number

The contact values at k = 0 yield the compressibility modulus. The material parameters of maghemite colloid were used [40]. At low densities and magnetic moment, there is liquid order. For higher dipole moments, formation of chain appears.

Figure 3 is a plot of the collective diffusion coefficient

The researchers in Ref. [46] also measured the wave vector-dependent collective translational diffusion coefficient in the absence of an external field using X-ray correlation spectroscopy. Such a dynamical property has a known linear dependence on the particles’ hydrodynamic interactions (HIs). The proposed theoretical hydrodynamic function that captures the HI among particles overestimates the experimental data. On the other hand, Meriguet et al. [29] used small-angle neutron scattering to measure the incoherent scattering function at the fixed magnetic field. The incoherent part gives in the long-time overdamped regime, the single translational diffusion coefficient for long wave vectors k, and the coherent part provides the collective diffusion as a function of k. Using liquid theory definitions of these two dynamical functions, their comparison with the experimental values yields good agreement at all wave scattering vectors k. The same collective diffusion coefficient can be measured by forced Rayleigh scattering also [26]. In this technique, the fluctuations in concentrations of the particles are considered a dynamic variable. We note that similar studies have been undertaken in the case of rodlike macromolecules in suspension such as Tobacco mosaic virus [48]. Explicit statistical mechanics derivations for the dynamical collective correlation functions (intermediate scattering function) have been given that fit well Brownian dynamics simulations just at long wave numbers but fail at intermediate and short k values. In Refs. [32, 33, 34], we proposed a Langevin equation theory for a tracer ferroparticle whose motion is coupled to the cloud of the other colloidal magnetic particles with which it interacts.

with

Here,

where the Onsager coefficient

The propagator

These correlation function components are obtained from the Fourier transform of their definitions

In Figure 4, Ref. [34] is provided a plot of the translational and rotational self-diffusion coefficients of such a theory. In this picture, we compare the diffusion coefficients with Langevin dynamics simulation results for the same diffusion coefficients by using typical parameters of

### 2.5 Diffusion microrheology of colloidal magnetic fluids under no external electric or magnetic fields

Ferrofluids are complex fluids whose viscoelastic response to weekly applied strain rates

where

We notice that the effective viscosity at the overdamped regime

### 2.6 Dynamic susceptibility response of a ferrofluid

The time-dependent magnetic susceptibility

where

This equation is valid for colloidal suspensions of particles with axially symmetric potentials, where the memory function

Alternative kinetics methods for low-density ferrofluids are given in [40].

## 3. Conclusion

This review highlights several active research lines on the structure and dynamics in ferrofluids. These investigations are motivated by many experimental observations that colloidal ferromagnetic particles both in bulk or in thin film aggregate, building several macroscopic structures of practical and scientific interest. As for all applications where colloidal materials are being processed, it is necessary to understand the interparticle interactions to have better control of the resulting structures and their properties. The interactions are known experimentally under specific conditions through the measured phase diagrams [11]; it has been found [29] that a useful potential that represents well the structure factor of ferrofluids is a Lennard-Jones plus dipolar interaction between pairs of particles. Therefore, our use of this potential in an evolutionary genetic algorithm leads to Figure 1, which shows transient metastable crystalline structures of a monodisperse ferrofluid at

## Acknowledgments

The authors acknowledge the General Coordination of Information and Communications Technologies (CGSTIC) at CINVESTAV for providing HPC resources on the Hybrid Supercomputer ″Xiuhcoatl″ that has contributed to the research results reported within this chapter.

## Conflict of interest

There are no conflicts to declare.