Free energy calculations for rings and chains formed by dipolar hard spheres.

We employ a method based on Monte Carlo grand-canonical simulations to precisely calculate partition functions of non-interacting chains and rings formed by dipolar hard spheres (DHS) at low temperature. The extended low temperature region offered by such cluster calculations, compared to what had been previously achieved with standard simulations, opens up the possibility of exploring a part of the DHS phase diagram which was inaccessible before. The reported results offer the unique opportunity of verifying well-established theoretical models based on the ideal gas of cluster approximation in order to clarify their range of validity. They also provide the basis for future studies in which cluster-cluster interactions will be included.

Revealing the signature of dipolar interactions in dynamic spectra of polydisperse magnetic nanoparticles.

We investigate, via a modified mean field approach, the dynamic magnetic response of a polydisperse dipolar suspension to a weak, linearly polarised, AC field. We introduce an additional term into the Fokker-Planck equation, which takes into account dipole-dipole interaction in the form of the first order perturbation, and allows for particle polydispersity. The analytical expressions, obtained for the real and imaginary dynamic susceptibilities, predict three measurable effects: the increase of the real part low-frequency plateaux; the enhanced growth of the imaginary part in the low-frequency range; and the shift of the imaginary part maximum. Our theoretical predictions find an experimental confirmation and explain the changes in the spectrum.

Sedimentation equilibria in polydisperse ferrofluids: critical comparisons between experiment, theory, and computer simulation.

The sedimentation equilibrium of dipolar particles in a ferrofluid is studied using experiment, theory, and computer simulation. A theory of the particle-concentration profile in a dipolar hard-sphere fluid is developed, based on the local-density approximation and accurate expressions from a recently introduced logarithmic free energy approach. The theory is tested critically against Monte Carlo simulation results for monodisperse and bidisperse dipolar hard-sphere fluids in homogeneous gravitational fields. In the monodisperse case, the theory is very accurate over broad ranges of gravitational field strength, volume fraction, and dipolar coupling constant. In the bidisperse case, with realistic dipolar coupling constants and compositions, the theory is excellent at low volume fraction, but is slightly inaccurate at high volume fraction in that it does not capture a maximum in the small-particle concentration profile seen in simulations. Possible reasons for this are put forward. Experimental measurements of the magnetic-susceptibility profile in a real ferrofluid are then analysed using the theory. The concentration profile is linked to the susceptibility profile using the second-order modified mean-field theory. It is shown that the experimental results are not consistent with the sample being monodisperse. By introducing polydispersity in the simplest possible way, namely by assuming the system is a binary mixture, almost perfect agreement between theory and experiment is achieved.

Temperature-dependent dynamic correlations in suspensions of magnetic nanoparticles in a broad range of concentrations: a combined experimental and theoretical study.

The interweave of competing individual relaxations influenced by the presence of temperature and concentration dependent correlations is an intrinsic feature of superparamagnetic nanoparticle suspensions. This unique combination gives rise to multiple applications of such suspensions in medicine, nanotechnology and microfluidics. Here, using theory and experiment, we investigate dynamic magnetic susceptibility in a broad range of temperatures and frequencies. Our approach allows, for the first time to our knowledge, to separate clearly the effects of superparamagnetic particle polydispersity and interparticle magnetic interactions on the dynamic spectra of these systems. In this way, we not only provide a theoretical model that can predict well the dynamic response of magnetic nanoparticles systems, but also deepen the understanding of the dynamic nanoparticle self-assembly, opening new perspectives in tuning and controlling the magnetic behaviour of such systems in AC fields.

The initial magnetic susceptibility of polydisperse ferrofluids: A comparison between experiment and theory over a wide range of concentration.

Temperature dependencies of the static initial magnetic susceptibility for ferrofluids at various concentrations are studied using experiment and statistical-mechanical theories. Magnetic susceptibility measurements are carried out for twelve samples of magnetite-based fluids stabilized with oleic acid over a wide range of temperatures (210 K ≲T ≲ 390 K); all samples have the same granulometric composition but different volume ferroparticle concentrations (0.2 ≲ φ ≲ 0.5). Experimental results are analyzed using three theories: the second-order modified mean-field theory (MMF2) [A. O. Ivanov and O. B. Kuznetsova, Phys. Rev. E 64, 41405 (2001)]; its correction for polydisperse ferrofluids arising from Mayer-type cluster expansion and taking into account the first terms of the polydisperse second virial coefficient [A. O. Ivanov and E. A. Elfimova, J. Magn. Magn. Mater 374, 327 (2015)]; and a new theory based on MMF2 combined with the first terms of the polydisperse second and third virial contributions to susceptibility. It turns out that the applicability of each theory depends on the experimental sample density. If twelve ferrofluid samples are split into three groups of strong, moderate, and low concentrated fluids, the temperature dependences of the initial magnetic susceptibility in each group are very precisely described by one of the three theories mentioned above. The determination of a universal formula predicting a ferrofluid susceptibility over a broad range of concentrations and temperatures remains as a challenge.

The effect of links on the interparticle dipolar correlations in supramolecular magnetic filaments.

We present a combined computational and analytical study of supramolecular magnetic filaments, i.e., permanently linked chains of ferromagnetic nanocolloids. We put forward two different models for the interparticle connectivity within the chain. In the first model, the magnetic dipoles of the particles are free to rotate independently from the permanent links. The second model penalises the misalignment of the dipoles by coupling their orientations to the chain backbone. We show that the effect of the long-range magnetic dipolar interactions on the zero field net magnetic moment of the chain becomes less significant in the second case. However, the overall magnetic response in the model of freely rotating dipoles is much weaker.

Temperature-induced structural transitions in self-assembling magnetic nanocolloids.

With the help of a unique combination of density functional theory and computer simulations, we discover two possible scenarios, depending on concentration, for the hierarchical self-assembly of magnetic nanoparticles on cooling. We show that typically considered low temperature clusters, i.e. defect-free chains and rings, merge into more complex branched structures through only three types of defects: four-way X junctions, three-way Y junctions and two-way Z junctions. Our accurate calculations reveal the predominance of weakly magnetically responsive rings cross-linked by X defects at the lowest temperatures. We thus provide a strategy to fine-tune magnetic and thermodynamic responses of magnetic nanocolloids to be used in medical and microfluidics applications.

Pair correlations of the easy magnetisation axes of superparamagnetic nanoparticles in a ferrofluid/ferrocomposite.

The widespread use of magnetic nanoparticles in modern technologies and medical applications highlights the need for reliable theoretical models that can predict their physical properties. The pair correlation function of two randomly selected superparamagnetic nanoparticles in a ferrofluid/ferrocomposite is studied to depict the joint probability density of the easy magnetisation axes across the planes of parameters of major importance; these are the interaction of ferroparticles with an external magnetic field, the energy of magnetic anisotropy inside the superparamagnetic nanoparticle, and the interparticle magnetic dipole-dipole interaction. Assuming the rotational symmetry of the system, we come to the conclusion that the pair correlations of interest are dependent only on the polar angles, determining the inclinations of the ferroparticle easy axes from the direction of an external magnetic field. The dimer configuration, where two ferroparticles are in close contact along a magnetic field with their easy magnetisation axes aligned, is the most probable. This configuration becomes more pronounced with increasing anisotropy energy, dipolar coupling constant, and external magnetic field.

Structure and dynamics in suspensions of magnetic platelets.

In this research, we employ Brownian dynamics simulations, density functional theory, and mean-field theory to explore the profound influence of shape anisotropy of magnetic nanoplatelets on suspension magnetic response. Each platelet is modelled as an oblate cylinder with a longitudinal point dipole, with an emphasis on strong dipolar interactions conducive to self-assembly. We investigate static structural and magnetic properties, characterising the system through pair distribution function, static structure factor, and cluster-size distribution. The findings demonstrate that shape-specific interactions and clustering lead to significant changes in reorientational relaxation times. Under zero field, distinctive modes in the dynamic magnetic susceptibility identify individual particles and particle clusters. In the presence of an applied field, the characteristic relaxation time of clusters increases, while that of single particles decreases. This research provides insights into the intricate interplay between shape anisotropy, clustering, and magnetic response in platelet suspensions, offering valuable perspectives for recent experimental observations.

Nonmonotonic magnetic susceptibility of dipolar hard-spheres at low temperature and density.

We investigate, via numerical simulations, mean field, and density functional theories, the magnetic response of a dipolar hard sphere fluid at low temperatures and densities, in the region of strong association. The proposed parameter-free theory is able to capture both the density and temperature dependence of the ring-chain equilibrium and the contribution to the susceptibility of a chain of generic length. The theory predicts a nonmonotonic temperature dependence of the initial (zero field) magnetic susceptibility, arising from the competition between magnetically inert particle rings and magnetically active chains. Monte Carlo simulation results closely agree with the theoretical findings.

Branching points in the low-temperature dipolar hard sphere fluid.

In this contribution, we investigate the low-temperature, low-density behaviour of dipolar hard-sphere (DHS) particles, i.e., hard spheres with dipoles embedded in their centre. We aim at describing the DHS fluid in terms of a network of chains and rings (the fundamental clusters) held together by branching points (defects) of different nature. We first introduce a systematic way of classifying inter-cluster connections according to their topology, and then employ this classification to analyse the geometric and thermodynamic properties of each class of defects, as extracted from state-of-the-art equilibrium Monte Carlo simulations. By computing the average density and energetic cost of each defect class, we find that the relevant contribution to inter-cluster interactions is indeed provided by (rare) three-way junctions and by four-way junctions arising from parallel or anti-parallel locally linear aggregates. All other (numerous) defects are either intra-cluster or associated to low cluster-cluster interaction energies, suggesting that these defects do not play a significant part in the thermodynamic description of the self-assembly processes of dipolar hard spheres.

Magnetization relaxation dynamics in polydisperse ferrofluids.

When a ferrofluid is magnetized in a strong magnetic field, and then the field is switched off, the magnetization decays from its saturation value to zero. The dynamics of this process are controlled by the rotations of the constituent magnetic nanoparticles, and for the Brownian mechanism, the respective rotation times are strongly influenced by the particle size and the magnetic dipole-dipole interactions between the particles. In this work, the effects of polydispersity and interactions on the magnetic relaxation are studied using a combination of analytical theory and Brownian dynamics simulations. The theory is based on the Fokker-Planck-Brown equation for Brownian rotation and includes a self-consistent, mean-field treatment of the dipole-dipole interactions. The most interesting predictions from the theory are that, at short times, the relaxation of each particle type is equal to its intrinsic Brownian rotation time, while at long times, each particle type has the same effective relaxation time, which is longer than any of the individual Brownian rotation times. Noninteracting particles, though, always relax at a rate controlled only by the Brownian rotation times. This illustrates the importance of including the effects of polydispersity and interactions when analyzing the results from magnetic relaxometry experiments on real ferrofluids, which are rarely monodisperse.

Theory and simulation of anisotropic pair correlations in ferrofluids in magnetic fields.

Anisotropic pair correlations in ferrofluids exposed to magnetic fields are studied using a combination of statistical-mechanical theory and computer simulations. A simple dipolar hard-sphere model of the magnetic colloidal particles is studied in detail. A virial-expansion theory is constructed for the pair distribution function (PDF) which depends not only on the length of the pair separation vector, but also on its orientation with respect to the field. A detailed comparison is made between the theoretical predictions and accurate simulation data, and it is found that the theory works well for realistic values of the dipolar coupling constant (λ = 1), volume fraction (φ ≤ 0.1), and magnetic field strength. The structure factor is computed for wavevectors either parallel or perpendicular to the field. The comparison between theory and simulation is generally very good with realistic ferrofluid parameters. For both the PDF and the structure factor, there are some deviations between theory and simulation at uncommonly high dipolar coupling constants, and with very strong magnetic fields. In particular, the theory is less successful at predicting the behavior of the structure factors at very low wavevectors, and perpendicular Gaussian density fluctuations arising from strongly correlated pairs of magnetic particles. Overall, though, the theory provides reliable predictions for the nature and degree of pair correlations in ferrofluids in magnetic fields, and hence should be of use in the design of functional magnetic materials.

Superdiffusion in self-reinforcing run-and-tumble model with rests.

This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equation for the probability density of random-walk position, from which we obtain the second moment in the long-time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least 2/3 of the mean resting time for superdiffusion to be possible. Monte Carlo simulations validate this theoretical result. This work demonstrates the possibility of extending the telegrapher's (or Cattaneo) equation by adding self-reinforcing directionality so that superdiffusion occurs even when rests are introduced.

Magnetophoresis, sedimentation, and diffusion of particles in concentrated magnetic fluids.

A dynamic mass transfer equation for describing magnetophoresis, sedimentation, and gradient diffusion of colloidal particles in concentrated magnetic fluids has been derived. This equation takes into account steric, magnetodipole, and hydrodynamic interparticle interactions. Steric interactions have been investigated using the Carnahan-Starling approximation for a hard-sphere system. In order to study the effective interparticle attraction, the free energy of the dipolar hard-sphere system is represented as a virial expansion with accuracy to the terms quadratic in particle concentration. The virial expansion gives an interpolation formula that fits well the results of computer simulation in a wide range of particle concentrations and interparticle interaction energies. The diffusion coefficient of colloidal particles is written with regard to steric, magnetodipole and hydrodynamic interactions. We thereby laid the foundation for the formulation of boundary-value problems and for calculation of concentration and magnetic fields in the devices (for example, magnetic fluid seals and acceleration sensors), which use a concentrated magnetic fluid as a working fluid. The Monte-Carlo methods and the analytical approach are employed to study the magnetic fluid stratification generated by the gravitational field in a cylinder of finite height. The coefficient of concentration stratification of the magnetic fluid is calculated in relation to the average concentration of particles and the dipolar coupling constant. It is shown that the effective particle attraction causes a many-fold increase in the concentration inhomogeneity of the fluid if the average volume fraction of particles does not exceed 30%. At high volume concentrations steric interactions play a crucial role.

How chains and rings affect the dynamic magnetic susceptibility of a highly clustered ferrofluid.

The dynamic magnetic susceptibility, χ(ω), of a model ferrofluid at a very low concentration (volume fraction, approximately 0.05%), and with a range of dipolar coupling constants (1≤λ≤8), is examined using Brownian dynamics simulations. With increasing λ, the structural motifs in the system change from unclustered particles, through chains, to rings. This gives rise to a nonmonotonic dependence of the static susceptibility χ(0) on λ and qualitative changes to the frequency spectrum. The behavior of χ(0) is already understood, and the simulation results are compared to an existing theory. The single-particle rotational dynamics are characterized by the Brownian time, τ_{B}, which depends on the particle size, carrier-liquid viscosity, and temperature. With λ≤5.5, the imaginary part of the spectrum, χ^{''}(ω), shows a single peak near ω∼τ_{B}^{-1}, characteristic of single particles. With λ≥5.75, the spectrum is dominated by the low-frequency response of chains. With λ≥7, new features appear at high frequency, which correspond to intracluster motions of dipoles within chains and rings. The peak frequency corresponding to these intracluster motions can be computed accurately using a simple theory.

Magnetic properties of textured ferrocomposite consisting of immobilized superparamagnetic nanoparticles.

Wide use of magnetic nanoparticles in modern technologies and biomedical applications requires reliable theoretical models capable of predicting physical properties. Solidification of a ferroparticle suspension under the action of permanent magnetic field allows us to obtain a ferrocomposite, characterized by some orientational texture of the nanoparticle easy magnetization axes. The static magnetic response of this ferrocomposite differs from that of the parent magnetic suspension due to "freezing" of nanoparticle translational and rotational degrees of freedom. Here the superparamagnetic fluctuations of the nanoparticle magnetic moments play a key role in the formation of the ferrocomposite magnetic response depending on the degree of orientational ordering, obtained during synthesis of a ferrocomposite. With the help of statistical mechanics we calculate the magnetization and the initial magnetic susceptibility of the textured ferrocomposite for various temperatures and magnetic field strengths. The easy axis texturing leads to a significant increase of the magnetic properties, and the effect intensifies with the growth of nanoparticle magnetocrystalline anisotropy. Theoretical predictions are supported by Monte Carlo simulations. The obtained results evidence that the texturing of a ferroparticle suspension and transforming it into a textured ferrocomposite are a real way to enhance the magnetic response of these magnetic soft materials.

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

Ferrofluids have attracted considerable interest from researchers and engineers due to their rich set of unique physical properties that are valuable for many industrial and biomedical applications. Many phenomena and features of ferrofluids' behavior are determined by internal structural transformations in the ensembles of particles, which occur due to the magnetic interaction between the particles. An applied magnetic field induces formations, such as linear chains and bulk columns, that become elongated along the field. In turn, these structures dramatically change the rheological and other physical properties of these fluids. A deep and clear understanding of the main features and laws of the transformations is necessary for the understanding and explanation of the macroscopic properties and behavior of ferrofluids. In this paper, we present an overview of experimental and theoretical works on the internal transformations in these systems, as well as on the effect of the internal structures on the rheological effects in the fluids.

Static magnetic response of multicore particles.

We present theoretical calculations of the characteristics of the static magnetic response of multicore magnetic nanoparticles. These particles contain a considerable number (∼10^{2}) of single-domain magnetic nanocrystallites, which are modeled as uniformly magnetized balls with uniaxial magnetocrystalline anisotropy, the energetic barrier of which is comparable with the thermal energy. Thus, we model a multicore magnetic nanoparticle as an ensemble of superparamagnetic nanoparticles, the position and the easy magnetization axis of which are fixed but randomly distributed. Summing up the magnetic moments of the nanocrystallites inside a multicore particle, we thus obtain the magnetic moment of the multicore particle under the assumption that magnetic interactions between the nanocrystallites can be neglected. It is found that the weak-field magnetic response of these multicore particles is independent of anisotropy constant. The model is compared with recent experimental data and good agreement is found.

Effects of interactions on magnetization relaxation dynamics in ferrofluids.

The dynamics of magnetization relaxation in ferrofluids are studied with statistical-mechanical theory and Brownian dynamics simulations. The particle dipole moments are initially perfectly aligned, and the magnetization is equal to its saturation value. The magnetization is then allowed to decay under zero-field conditions toward its equilibrium value of zero. The time dependence is predicted by solving the Fokker-Planck equation for the one-particle orientational distribution function. Interactions between particles are included by introducing an effective magnetic field acting on a given particle and arising from all of the other particles. Two different approximations are proposed and tested against simulations: a first-order modified mean-field theory and a modified Weiss model. The theory predicts that the short-time decay is characterized by the Brownian rotation time τ_{B}, independent of the interaction strength. At times much longer than τ_{B}, the asymptotic decay time is predicted to grow with increasing interaction strength. These predictions are borne out by the simulations. The modified Weiss model gives the best agreement with simulation, and its range of validity is limited to moderate, but realistic, values of the dipolar coupling constant.