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.

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.

Dynamic susceptibility of a concentrated ferrofluid: The role of interparticle interactions.

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.

Magnetic properties of polydisperse ferrofluids: a critical comparison between experiment, theory, and computer simulation.

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.

Sedimentation equilibrium of magnetic nanoparticles with strong dipole-dipole interactions.

Langevin dynamics simulation is used to study the suspension of interacting magnetic nanoparticles (dipolar spheres) in a zero applied magnetic field and in the presence of a gravitational (centrifugal) field. A particular emphasis is placed on the equilibrium vertical distribution of particles in the infinite horizontal slab. An increase in the dipolar coupling constant λ (the ratio of dipole-dipole interaction energy to thermal energy) from zero to seven units causes an increase in the particle segregation coefficient by several orders of magnitude. The effect of anisotropic dipole-dipole interactions on the concentration profile of particles is the same as that of the isotropic van der Waals attraction modeled by the Lennard-Jones potential. In both cases, the area with a high-density gradient separating the area with high and low particle concentration is formed on the profiles. Qualitative difference between two potentials manifests itself only in the fact that in the absence of a gravitational field the dipole-dipole interactions do not lead to the "gas-liquid" phase transition: no separation of the system into weakly and highly concentrated phases is observed. At high particle concentration and at large values of λ, the orientational ordering of magnetic dipoles takes place in the system. Magnetic structure of the system strongly depends on the imposed boundary conditions. Spontaneous magnetization occurs in the infinite horizontal slab (i.e., in the rectangular cell with two-dimensional periodic boundary conditions). Replacement of the infinite slab by the finite-size hard-wall vertical cylinder leads to the formation of azimuthal (vortex-like) order. The critical values of the coupling constant corresponding to the transition into an ordered state are very close for two geometries.

Self-organization of magnetic moments in dipolar chains with restricted degrees of freedom.

Equilibrium behavior of a single chain of dipolar spheres is investigated by the method of molecular dynamics in a wide range of the dipolar coupling constant λ. Two cases are considered: rodlike and flexible chains. In the first case, particle centers are immovably fixed on one axis, but their magnetic moments retain absolute orientational freedom. It has been found that at λ≳1.5 particle moments are chiefly aligned parallel to the chain axis, but the total moment of the chain continuously changes its sign with some mean frequency, which exponentially decreases with the growth of λ. Such behavior of the rodlike chain is analogous to the Néel relaxation of a superparamagnetic particle with a finite energy of magnetic anisotropy. In the flexible chain particles are able to move in the three-dimensional space, but the distance between centers of the first-nearest neighbors never exceeds a given limiting value r(max). If r(max)≃d (d is the particle diameter) then the most probable shape of the chain of five or more particles at λ≳6 is that of a ring. The behavior of chains with r(max)≥2d is qualitatively different: At λ≃4 long chains collapse into dense quasispherical globules and at λ≳8 these globules take toroidal configuration with a spontaneous azimuthal ordering of magnetic dipoles. With the increase of r(max) to larger values (r(max)>10d) globules expand and break down to form separate rings.

Low-temperature susceptibility of concentrated magnetic fluids.

The initial susceptibility of concentrated magnetic fluids (ferrocolloids) has been experimentally investigated at low temperatures. The results obtained indicate that the interparticle dipole-dipole interactions can increase the susceptibility by several times as compared to the Langevin value. It is shown that good agreement between recent theoretical models and experimental observations can be achieved by introducing a correction for coefficients in the series expansion of susceptibility in powers of density and aggregation parameter. A modified equation for equilibrium susceptibility is offered to sum over corrections made by Kalikmanov (Statistical Physics of Fluids, Springer-Verlag, Berlin, 2001) and by B. Huke and M. Lucke (Phys. Rev. E 67, 051403, 2003). The equation gives good quantitative agreement with the experimental data in the wide range of temperature and magnetic particles concentration. It has been found that in some cases the magnetic fluid solidification occurs at temperature several tens of kelvins higher than the crystallization temperature of the carrier liquid. The solidification temperature of magnetic fluids is independent of particle concentration (i.e., magneto-dipole interparticle interactions) and dependent on the surfactant type and carrier liquid. This finding allows us to suggest that molecular interactions and generation of some large-scale structure from colloidal particles in magnetic fluids are responsible for magnetic fluid solidification. If the magnetic fluid contains the particles with the Brownian relaxation mechanism of the magnetic moment, the solidification manifests itself as the peak on the "susceptibility-temperature" curve. This fact proves the dynamic nature of the observed peak: it arises from blocking the Brownian mechanism of the magnetization relaxation.