Influence of a bias dc field and an ac field amplitude on the dynamic susceptibility of a moderately concentrated ferrofluid.

In this paper, we study the effect of a bias dc field on the dynamic response of a moderately concentrated ferrofluid to an ac magnetic field of arbitrary amplitude. The ferrofluid is modeled by an ensemble of interacting moving magnetic particles; the reaction of particle magnetic moments to ac and dc magnetic fields occurs according to the Brownian mechanism; and the ac and dc magnetic fields are parallel. Based on a numerical solution of the Fokker-Planck equation for the probability density of the orientation of the magnetic moment of a random magnetic particle, dynamic magnetization and susceptibility are determined and analyzed for various values of the ac field amplitude, the dc field strength, and the intensity of dipole-dipole interactions. It is shown that the system's magnetic response is formed under the influence of competing interactions, such as dipole-dipole, dipole-ac field, and dipole-dc field interactions. When the energies of these interactions are comparable, unexpected effects are observed: the system's susceptibility can either increase or decrease with increasing ac field amplitude. This behavior is associated with the formation of nose-to-tail dipolar structures under the action of the dc field, which can hinder or promote the system's dynamic response to the ac field. The obtained results provide a theoretical basis for predicting the dynamic properties of ferrofluids to improve their use in biomedical applications, such as, in magnetic induction hyperthermia.

Effect of Polydispersity on the Structural and Magnetic Properties of a Magnetopolymer Composite.

When using magnetopolymer composites in high-precision industrial and biomedical technologies, the problem of predicting their properties in an external magnetic field arises. In this work, we study theoretically the influence of the polydispersity of a magnetic filler on a composite's equilibrium magnetization and on the orientational texturing of magnetic particles formed during polymerization. The results are obtained using rigorous methods of statistical mechanics and Monte Carlo computer simulations in the framework the bidisperse approximation. It is shown that by adjusting the dispersione composition of the magnetic filler and the intensity of the magnetic field at which the sample's polymerization occurs, it is possible to control the composite's structure and magnetization. The derived analytical expressions determine these regularities. The developed theory takes into account dipole-dipole interparticle interactions and therefore can be applied to predict the properties of concentrated composites. The obtained results are a theoretical basis for the synthesis of magnetopolymer composites with a predetermined structure and magnetic properties.

Influence of field amplitude and dipolar interactions on the dynamic response of immobilized magnetic nanoparticles: Perpendicular mutual alignment of an alternating magnetic field and the easy axes.

In this paper, the dynamic magnetic properties of an ensemble of interacting immobilized magnetic nanoparticles with aligned easy axes in an applied ac magnetic field directed perpendicular to the easy axes are considered. The system models soft, magnetically sensitive composites synthesized from liquid dispersions of the magnetic nanoparticles in a strong static magnetic field, followed by the carrier liquid's polymerization. After polymerization, the nanoparticles lose translational degrees of freedom; they react to an ac magnetic field via Néel rotation, when the particle's magnetic moment deviates from the easy axis inside the particle body. Based on a numerical solution of the Fokker-Planck equation for the probability density of the magnetic moment orientation, the dynamic magnetization, frequency-dependent susceptibility, and relaxation times of the particle's magnetic moments are determined. It is shown that the system's magnetic response is formed under the influence of competing interactions, such as dipole-dipole, field-dipole, and dipole-easy-axis interactions. The contribution of each interaction to the magnetic nanoparticle's dynamic response is analyzed. The obtained results provide a theoretical basis for predicting the properties of soft, magnetically sensitive composites, which are increasingly used in high-tech industrial and biomedical technologies.

Structure and magnetization of a magnetoactive ferrocomposite.

This work is devoted to the theoretical study of the structural and magnetic properties of an ensemble of single-domain interacting magnetic nanoparticles immobilized in a non-magnetic medium. This model is typical for describing magnetically active soft materials, "smart" polymer ferrocomposites, which have been applied in science-intensive industrial and biomedical technologies. It is assumed that the ferrocomposite is obtained by solidification of the carrier medium in a ferrofluid under an external magnetic field, the intensity of which is determined by the Langevin parameter αp; after the solidification of the carrier liquid, the nanoparticles retain the spatial distribution and orientation of their easy magnetization axes. The features of the orientational texture formed in the sample are analyzed depending on the intensity of the magnetic field αp and interparticle dipole-dipole interactions. The magnetization of a textured ferrocomposite in the magnetic field α is also investigated. Our results show that in the case of a co-directional arrangement of the considered fields and if α < αp, the ferrocomposites are magnetized much more efficiently than ferrofluids due to their texture. In the fields α > αp, the ferrocomposite is magnetized less efficiently than the ferrofluid due to the internal magnetic anisotropy of the nanoparticles. The analytical expressions presented here make it possible to predict the magnetization of a ferrocomposite depending on its internal structure and synthesis conditions, which is the theoretical basis for the synthesis of ferrocomposites with a predetermined magnetic response in a given magnetic field.

Theory of static magnetization of magnetopolymer composites: The second virial approximation.

This paper deals with a theoretical study of the static magnetization of a composite, consisting of nanodisperse single-domain ferromagnetic particles immobilized in a nonmagnetic medium. This situation is typical for magnetopolymer systems-smart materials, which found applications in high industrial and biomedical applications. It is supposed that the composite was polymerized without an external magnetic field; after curing the particles retain the spatial distribution and orientation of their axes of easy magnetization that they had before the host medium polymerization. Our results demonstrate that, in contrast to ferrofluids with a liquid host medium, a magnetic interparticle interaction provokes a decrease of the macroscopic magnetization of the system.

Dynamic magnetic susceptibility of a ferrofluid: The influence of interparticle interactions and ac field amplitude.

Based on numerical results of dynamic susceptibility, a simple theory of the dynamic response of a ferrofluid to an ac magnetic field is obtained that includes both the effects of interparticle dipole-dipole interactions and the dependence on field amplitude. Interparticle interactions are incorporated in the theory using the so-called modified mean-field approach. The new theory has the following important characteristics: in the noninteracting regime at a weak ac field, it gives the correct single-particle Debye theory results; it expands the applicability of known theories valid for high concentrations [Ivanov, Zverev, and Kantorovich, Soft Matter 12, 3507 (2016)10.1039/C5SM02679B] or large values of ac field amplitudes [Yoshida and Enpuku, Jpn. J. Appl. Phys. 48, 127002 (2009)10.1143/JJAP.48.127002], in accordance with their applicability. The susceptibility spectra are analyzed in detail. It is demonstrated that interparticle dipole-dipole interactions and an increase in field amplitude have an opposite effect on the dynamic response of ferrofluids, so that at certain field amplitudes, relaxation processes in the system of interacting particles are determined by the characteristic relaxation times for an ideal paramagnetic gas. The new theory correctly predicts the dynamic susceptibility and characteristic relaxation times of a ferrofluid at high ac field amplitudes as long as the Langevin susceptibility χ_{L}≲1, which is a complex characteristic of ferrofluid density and the intensity of interparticle dipole-dipole interactions.

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.

Static magnetization of immobilized, weakly interacting, superparamagnetic nanoparticles.

The magnetization curve and initial susceptibility of immobilized superparamagnetic nanoparticles are studied using statistical-mechanical theory and Monte Carlo computer simulations. The nanoparticles are considered to be distributed randomly within an implicit solid matrix, but with the easy axes distributed according to particular textures: these are aligned parallel or perpendicular to an external magnetic field, or randomly distributed. The magnetic properties are calculated as functions of the magnetic crystallographic anisotropy barrier (measured with respect to the thermal energy by a parameter σ), and the Langevin susceptibility (related to the dipolar coupling constant and the volume fraction). It is shown that the initial susceptibility χ is independent of σ in the random case, an increasing function of σ in the parallel case, and a decreasing function of σ in the perpendicular case. Including particle-particle interactions enhances χ, and especially so in the parallel case. A first-order modified mean-field (MMF1) theory is accurate as compared to the simulation results, except in the parallel case with a large value of σ. These observations can be explained in terms of the range and strength of the (effective) interactions and correlations between particles, and the effects of the orientational degrees of freedom. The full magnetization curves show that a parallel texture enhances the magnetization, while a perpendicular texture suppresses it, with the effects growing with increasing σ. In the random case, while the initial response is independent of σ, the high-field magnetization decreases with increasing σ. These trends can be explained by the energy required to rotate the magnetic moments with respect to the easy axes.

Modified mean-field theory of the magnetic properties of concentrated, high-susceptibility, polydisperse ferrofluids.

The effects of particle-size polydispersity on the magnetostatic properties of concentrated ferrofluids are studied using theory and computer simulation. The second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)1063-651X10.1103/PhysRevE.64.041405] has been extended by calculating additional terms of higher order in the dipolar coupling constant in the expansions of the initial magnetic susceptibility and the magnetization curve. The theoretical predictions have been tested rigorously against results from Monte Carlo simulations of model monodisperse, bidisperse, and highly polydisperse ferrofluids. Comparisons have been made between systems with the same Langevin susceptibility and the same saturation magnetization. In all cases, the new theoretical magnetization curve shows better agreement with simulation data than does the MMF2 theory. As for the initial susceptibility, MMF2 theory is most accurate for the monodisperse model, while the new theory works best for polydisperse systems with a significant proportion of large particles. These results are important for the analysis and characterization of recently synthesized polydisperse ferrofluids with record-breaking values of the initial magnetic susceptibility.

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.

Influence of dipolar interactions on the magnetic susceptibility spectra of ferrofluids.

The frequency-dependent magnetic susceptibility of a ferrofluid is calculated under the assumption that the constituent particles undergo Brownian relaxation only. Brownian-dynamics simulations are carried out in order to test the predictions of a recent theory [A. O. Ivanov, V. S. Zverev, and S. S. Kantorovich, Soft Matter 12, 3507 (2016)1744-683X10.1039/C5SM02679B] that includes the effects of interparticle dipole-dipole interactions. The theory is based on the so-called modified mean-field approach and possesses the following important characteristics: in the low-concentration, noninteracting regime, it gives the correct single-particle Debye-theory results; it yields the exact leading-order results in the zero-frequency limit; it includes particle polydispersity correctly from the outset; and it is based on firm theoretical foundations allowing, in principle, systematic extensions to treat stronger interactions and/or higher concentrations. The theory and simulations are compared in the case of a model monodisperse ferrofluid, where the effects of interactions are predicted to be more pronounced than in a polydisperse ferrofluid. The susceptibility spectra are analyzed in detail in terms of the low-frequency behavior, the position of the peak in the imaginary (out-of-phase) part, and the characteristic decay time of the magnetization autocorrelation function. It is demonstrated that the theory correctly predicts the trends in all of these properties with increasing concentration and dipolar coupling constant, the product of which is proportional to the Langevin susceptibility χ_{L}. The theory is in quantitative agreement with the simulation results as long as χ_{L}≲1.

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.

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.

The effects of polydispersity on the initial susceptibilities of ferrofluids.

The effects of particle-size polydispersity on the initial susceptibilities of concentrated ferrofluids are analyzed using a combination of theory and computer simulation. The study is focused on a model ferrofluid with a prescribed magnetic-core diameter distribution, a fixed non-magnetic surface layer (corresponding to a demagnetized layer and adsorbed surfactant) and a combination of dipolar and hard-core interactions. The non-trivial effects of polydispersity are identified by comparing the initial susceptibilities of monodisperse and polydisperse ferrofluids with the same Langevin susceptibility. The theory is based on a correction to the second-order modified mean-field theory arising from a formal Mayer-type cluster expansion; this correction is dependent on a parameter similar to the normal dipolar coupling constant, except that it contains a complicated double average over the particle-size distribution, which means that the initial susceptibility should depend significantly on polydispersity. Specifically, the theory predicts that the initial susceptibility is enhanced significantly by polydispersity. This prediction is tested rigorously against results from Monte Carlo simulations and is found to be robust. The qualitative agreement between theory and simulation is already satisfactory, but the quantitative agreement could be improved by a systematic extension of the cluster expansion. The overall conclusion is that polydispersity should be accounted for carefully in magnetogranulometric analyses of real ferrofluids.

Thermodynamics of ferrofluids in applied magnetic fields.

The thermodynamic properties of ferrofluids in applied magnetic fields are examined using theory and computer simulation. The dipolar hard sphere model is used. The second and third virial coefficients (B(2) and B(3)) are evaluated as functions of the dipolar coupling constant λ, and the Langevin parameter α. The formula for B(3) for a system in an applied field is different from that in the zero-field case, and a derivation is presented. The formulas are compared to results from Mayer-sampling calculations, and the trends with increasing λ and α are examined. Very good agreement between theory and computation is demonstrated for the realistic values λ≤2. The analytical formulas for the virial coefficients are incorporated in to various forms of virial expansion, designed to minimize the effects of truncation. The theoretical results for the equation of state are compared against results from Monte Carlo simulations. In all cases, the so-called logarithmic free energy theory is seen to be superior. In this theory, the virial expansion of the Helmholtz free energy is re-summed in to a logarithmic function. Its success is due to the approximate representation of high-order terms in the virial expansion, while retaining the exact low-concentration behavior. The theory also yields the magnetization, and a comparison with simulation results and a competing modified mean-field theory shows excellent agreement. Finally, the putative field-dependent critical parameters for the condensation transition are obtained and compared against existing simulation results for the Stockmayer fluid. Dipolar hard spheres do not undergo the transition, but the presence of isotropic attractions, as in the Stockmayer fluid, gives rise to condensation even in zero field. A comparison of the relative changes in critical parameters with increasing field strength shows excellent agreement between theory and simulation, showing that the theoretical treatment of the dipolar interactions is robust.

Thermodynamics of dipolar hard spheres with low-to-intermediate coupling constants.

The thermodynamic properties of the dipolar hard-sphere fluid are studied using theory and simulation. A new theory is derived using a convenient mathematical approximation for the Helmholtz free energy relative to that for the hard-sphere fluid. The approximation is designed to give the correct low-density virial expansion. New theoretical and numerical results for the fourth virial coefficient are given. Predictions of thermodynamic functions for dipolar coupling constants λ=1 and 2 show excellent agreement with simulation results, even at the highest value of the particle volume fraction φ. For higher values of λ, there are deviations at high volume fractions, but the correct low-density behavior is retained. The theory is compared critically against the established thermodynamic perturbation theory; it gives significant improvements at low densities and is more convenient in terms of the required numerics. Dipolar hard spheres provide a basic model for ferrofluids, and the theory is accurate for typical experimental parameters λ

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.

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.