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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 (B2 and B3) are evaluated as functions of the dipolar coupling constant λ, and the Langevin parameter a. The formula for S3 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 A. and a 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 ≲ 2 and φ ≲ 0.1. This is demonstrated explicitly by fitting osmotic equations of state for real ferrofluids measured recently by analytical centrifugation. [ABSTRACT FROM AUTHOR]

The ferrofluid emulsion, made of kerosene-based ferrofluid droplets suspended in nonmiscible aviation oil, demonstrates experimentally the nonmonotonic dependence of the effective magnetic permeability as a function of the uniform static magnetic field. In weak fields the emulsion permeability rapidly grows; it reaches its maximum at fields on the order of 1 kA/m; after that, it decays to zero. The theoretical explanation of the effect, as we show here, could be based on the following idea: In a weak magnetic field the growth of the induced droplet magnetic moment is faster than the linear one due to the droplet elongation accompanied by the reduction of the demagnetizing field. Further increase of the external magnetic field strength cannot lead to a significant decrease of the demagnetizing field, as the droplets are already highly elongated. On the other hand, the magnetic susceptibility of the ferrofluid reduces with the field strength. Thus, the effective magnetic permeability of the ferrofluid suspension starts decreasing. The developed theoretical model describes well the experimental observations. [ABSTRACT FROM AUTHOR]

The dynamic magnetic response of a ferrofluid to a weak ac magnetic field is studied using statistical mechanical theory and Brownian dynamics simulations, taking account of dipole-dipole interactions between the constituent ferromagnetic colloidal particles, and the presence of a range of particle sizes. The effects of interactions and polydispersity on the frequency dispersion are shown to be substantial: the amplitude of the response can be about twice that of a noninteracting system; the frequency for peak power loss can be reduced by about one half; and polydispersity effects can even change the qualitative appearance of the susceptibility spectrum. [ABSTRACT FROM AUTHOR]

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)] 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. [ABSTRACT FROM AUTHOR]

The dynamic susceptibility of concentrated ferrofluids of magnetite-kerosene type is studied experimentally to clarify the effect of interparticle interactions on the magnetization reversal dynamics and the ferrofluid relaxation time spectrum. We synthesize six ferrofluid samples, four of which have the same wide particle size distribution with a high (more than 2kT) average energy of magnetic dipole interactions. These samples differ in particle concentration and dynamic viscosity. The two remaining samples have a lower content of large particles and a moderate energy of magnetic dipole interactions. For all samples, we measure the dynamic susceptibility in the weak probing field at frequencies up to 160 kHz and the field amplitude dependence of the susceptibility at a frequency of 27 kHz. The results show that the susceptibility dispersion at frequencies up to 10 kHz is due to the rotational diffusion of colloidal particles and aggregates. Steric and hydrodynamic interparticle interactions are the main reason for the strong concentration dependence of the viscosity and so they also strongly influence the frequency dependence of the susceptibility. The influence of van der Waals and magnetic dipole interactions on the susceptibility is manifested indirectly, through the formation of multiparticle clusters. The contribution of clusters to the low-frequency susceptibility reaches 80%. Their large sizes (about 100 nm) shift the dispersion region to frequencies of 1-100 Hz, depending on the temperature and particle concentration. Experiments at 27 kHz demonstrate the increase in the dynamic susceptibility with increasing field amplitude. This growth is unexpected since all spectral amplitudes in the Debye function expansion of the dynamic susceptibility decrease monotonically with increasing field. To clarify the situation, the auxiliary problem of the magnetodynamics of a uniaxial particle in the alternating field is solved numerically. The Fokker-Planck-Brown rotational diffusion equation is used. It is shown that an increase in the field amplitude reduces the anisotropy barrier and the Néel relaxation time of particles and increases the dynamic susceptibility by one to two orders of magnitude compared to the weak-field limit. The calculation results are in qualitative agreement with the experimental data and allow us to propose a consistent interpretation of these data. We find that the increase in dynamic susceptibility with increasing amplitude is observed when two necessary conditions are met: (i) The suspension viscosity and the field frequency are high enough to cause the blocking of the rotational degrees of freedom of particles and aggregates and (ii) particles with a large magnetic anisotropy are present in the ferrofluid. [ABSTRACT FROM AUTHOR]

A theory for the frequency-dependent magnetic susceptibility of a ferrofluid in a static uniform magnetic field is developed, including the dipolar interactions between the constituent particles. Interactions are included within the framework of modified mean-field theory. Predictions are given for the linear responses of the magnetization to a probing ac field both parallel and perpendicular to the static field and are tested against results from Brownian dynamics simulations. The effects of the particle concentration and dipolar coupling constant on the field-dependent static susceptibilities and the frequency dispersions are shown to be substantial, which justifies taking proper account of the interactions between particles. The theory is reliable provided that the volume concentration and dipolar coupling constant are not too large and within the range of values for real ferrofluids. [ABSTRACT FROM AUTHOR]

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)] 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 χL. The theory is in quantitative agreement with the simulation results as long as χL≲1. [ABSTRACT FROM AUTHOR]

The dynamics of the growth and relaxation of the magnetization in ferrofluids are determined using theory based on the Fokker-Planck-Brown equation, and Brownian-dynamics simulations. Magnetization growth starting from an equilibrium nonmagnetized state in zero field, and following an instantaneous application of a uniform field of arbitrary strength, is studied with and without interparticle interactions. Similarly, magnetization relaxation is studied starting from an equilibrium magnetized state in a field of arbitrary strength, and following instantaneous removal of the field. In all cases, the dynamics are studied in terms of the time-dependent magnetization m(t). The field strength is described by the Langevin parameter α, the strength of the interparticle interactions is described by the Langevin susceptibility χ_{L}, and the individual particles undergo Brownian rotation with time τ_{B}. For noninteracting particles, the average growth time decreases with increasing α due to the torque exerted by the field, while the average relaxation time stays constant at τ_{B}; with vanishingly weak fields, the timescales coincide. The same basic picture emerges for interacting particles, but the weak-field timescales are larger due to collective particle motions, and the average relaxation time exhibits a weak, nonmonotonic field dependence. A comparison between theoretical and simulation results is excellent for noninteracting particles. For interacting particles with χ_{L}=1 and 2, theory and simulations are in qualitative agreement, but there are quantitative deviations, particularly in the weak-field regime, for reasons that are connected with the description of interactions using effective fields.

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.

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.

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.

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.

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.

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.

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle's position increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

A theoretical formalism to predict the structure factors observed in dipolar soft-sphere fluids based on a virial expansion of the radial distribution function is presented. The theory is able to account for cases with and without externally applied magnetic fields. A thorough comparison of the theoretical results to molecular-dynamics simulations shows a good agreement between theory and numerical simulations when the fraction of particles involved in clustering is low; i.e., the dipolar coupling parameter is lambda less, similar 2, and the volume fraction is phi less, similar 0.25. When magnetic fields are applied to the system, special attention is paid to the study of the anisotropy of the structure factor. The theory reasonably accounts for the structure factors when the Langevin parameter is smaller than 5.

Experimental magnetization curves for a polydisperse ferrofluid at various concentrations are examined using analytical theories and computer simulations with the aim of establishing a robust method for obtaining the magnetic-core diameter distribution function p(x). Theoretical expressions are fitted to the experimental data to yield the parameters of p(x). It is shown that the majority of available theories yield results that depend strongly on the ferrofluid concentration, even though the magnetic composition should be fixed. The sole exception is the second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)] which yields consistent results over the full experimental range of ferrofluid concentration. To check for consistency, extensive molecular dynamics and Monte Carlo simulations are performed on systems with discretized versions of p(x) corresponding as closely as possible to that of the real ferrofluid. Essentially perfect agreement between experiment, theory, and computer simulation is demonstrated. In addition, the MMF2 theory provides excellent predictions for the initial susceptibility measured in simulations.

The evolution of a colloidal fractal-like aggregate system is studied theoretically with account for the reduction of the single-particle concentration in the space between these growing aggregates. Two different mechanisms of the initial aggregate formation are considered. They are heterogeneous aggregation, when the initial aggregates appear on foreign nuclei (the biggest particles, pollution, etc.), and homogeneous nucleation, when the initial clusters appear due to a random meeting and combination of single Brownian particles. The principles of colloid supersaturation decrease and the aggregate distribution in size are obtained in both cases as well as the internal structure of the clusters. Unlike the usual condensation-type phase transition, the system of discrete clusters is formed at the final stage of the aggregation process. The internal particle distribution in these clusters does not obey the universal scaling power law, which differs from the model case of separate cluster growth in an infinite colloidal medium with permanent concentration of free particles.

The theory of particle association in flexible chains in dilute ferrofluids is generalized to the case of an arbitrarily strengthened magnetic field. The chain distribution in dynamic equilibrium is obtained on the basis of free energy minimization method under the neglect of interchain interaction. The chain partition function is calculated analytically with the help of the rotation matrix technique under the condition when the interparticle dipole-dipole interaction between the nearest neighboring ferroparticles in each chain is taken into account. At weak fields, the chain distribution and the initial susceptibility are shown to be dependent on the value of the correlation coefficient describing the zero field mutual orientational correlations between the magnetic moments of two neighboring ferroparticles in a chain. The internal chain orientational correlations and the field dependent chain lengthening result in higher magnetization of the aggregated ferrofluid in comparison with the Langevin magnetization.

The theory of particle association in chains in dilute ferrofluids and dipole fluids is generalized to the case of polydisperse systems. The chains could be formed by ferroparticles of different sizes, so various types of chain aggregates are considered. The probabilities of chain structure appearance are calculated, and the phase diagram, allowing to find the most probable structure with only the continuous particle size distribution known, is built. Our results demonstrate that in spite of a very weak dipole-dipole interaction between the small size fraction particles, their presence exerts a decisive influence on the ferrofluid microstructure. The chain shortening caused by the small particles sticking to the edges of chains formed by large particles is discovered theoretically. The latter effect is proved to exist by the recent computer simulations on bidisperse ferrofluid makeup. The application of the developed model to the description of magnetic birefringence phenomena in weak external magnetic fields shows a very good agreement with experimental data.

The magnetization properties of aggregated ferrofluids are calculated by combining the chain formation model developed by Zubarev with the modified mean-field theory. Using moderate assumptions for the inter- and intrachain interactions we obtain expressions for the magnetization and initial susceptibility. When comparing the results of our theory to molecular dynamics simulations of the same model we find that at large dipolar couplings (lambda>3) the chain formation model appears to give better predictions than other analytical approaches. This supports the idea that chain formation is an important structural ingredient of strongly interacting dipolar particles.

This paper is devoted to the theoretical justification of spontaneous orientational order in magnetic fluids. We study the self-consistent solutions of the Bogoliubov-Born-Green-Kirkwood-Yvon equation connecting the one-particle distribution function with the pair correlation function. This self-consistent approach is used in the specific density functional method and proves to be equivalent to the mean field theory. On the basis of the second-order perturbation method over the intensity of dipole-dipole interparticle interaction the following effect is discovered: the self-consistent density functional approach leads to the spontaneous "ferrimagnetic" state of the magnetic fluid induced by the dipole-dipole interaction. This strange result seems to be physically meaningless and prejudices the validity of the density functional methods and mean field theories applied to orientational microstructure in ferrofluids.

We are concerned with large-scale magnetic-field dynamo generation and propagation of magnetic fronts in turbulent electrically conducting fluids. An effective equation for the large-scale magnetic field is developed here that takes into account the finite correlation times of the turbulent flow. This equation involves the memory integrals corresponding to the dynamo source term describing the alpha effect and turbulent transport of magnetic field. We find that the memory effects can drastically change the dynamo growth rate, in particular, nonlocal turbulent transport might increase the growth rate several times compared to the conventional gradient transport expression. Moreover, the integral turbulent transport term leads to a large decrease of the speed of magnetic front propagation.