Annotation of the results obtained in 2017
The project is devoted to the investigation of dynamic magnetic properties of ferrofluids. A complex approach is used to handle the problem: theoretical analysis is accompanied by both experimental measurements and Brownian dynamics computer simulations. During the year 2017 the following investigations were performed and the following scientific results were obtained. We performed a theoretical study of the dynamic susceptibility spectrum of a ferrofluid under the influence of a weak probing alternating magnetic field. Dynamic properties were calculated using an analytical solution of the Fokker-Planck equation, which was modified to take into account interparticle correlations and the polydispersity of a magnetic fluid. Interparticle correlations were calculated within the framework of the second-order thermodynamic perturbation theory and within the classical approach of the Weiss mean field. The consistency of the results is established in the limiting cases with the one-particle Debye theory, with the static modified mean-field theory of the first order and its extension to the dynamic case developed in the previous years of the project. An increase in the concentration of magnetic particles leads to the growth of the value of the real part in the region of low frequencies and the value of the maximum of the imaginary part with a small shift of its position to the region of low frequencies. The results of the computer simulation of the systems under study demonstrate a good agreement with theoretical predictions of the main characteristics of the dynamic magnetic response. The model of the dynamic response, based on the Weiss mean field, demonstrates the widest range of agreement between theoretical predictions and the results of a computer simulations. We developed a theory of the dynamic magnetic response of a moderately concentrated ferrofluid, exposed simultaneously to the action of a crossed tatic and alternating linearly polarized magnetic fields. The theory takes into account dipole-dipole interparticle interactions in the framework of the modified mean-field theory of the first and second orders and assumes that the relaxation of the magnetic moment of particles occurs according to the Brownian mechanism. The model is based on the Fokker-Planck-Brown-Yvon equation for the orientation probability density of the magnetic moment, which was solved analytically (for small amplitudes of the alternating field) and numerically elsewise. The solutions obtained were used to determine the dynamic magnetic susceptibility of the system. It is established that an increase in the static field intensity leads to a decrease in the relaxation time of the system, and it also contributes to the weakening of the magnetic response of the liquid to the alternating magnetic field. It is established that the relaxation times of magnetic moments in a strongly magnetized system are insensitive to changes in concentration. A computer simulation of the dynamic magnetic properties of a monodisperse system of dipole hard spheres located in crossed and coaxial static and alternating magnetic fields is carried out. The results of computer simulation demonstrate a good agreement with theoretical results for all main characteristics of the dynamic magnetic response for moderately concentrated ferrofluids with weak and moderate interparticle dipole-dipole interactions at any intensity of the static magnetic field. The effects introduced by polydispersity into the spectrum of the dynamic magnetic susceptibility of a ferrofluid placed in crossed and coaxial static and alternating magnetic fields are theoretically described. The initial state (in an absence of the external magnetic field) of the ferrogel sample in relation to the particle concentration and strength of the dipolar interaction is modelled. The nonmonotonic dependence of the degree of particle aggregation on the concentration is found for the sample with high values of particle anisotropy energy. It means that in the most loaded system particle mobility is hamper visibly by the elastic reaction of the polymer lattice on the particle rotation. The simulation of the cycles of quasi-static magnetization does not show any notable hysteresis of the field-dependence of the particle aggregation parameter, but some lag of the response of orientation of particle clusters on the field decrease is observed. The magnetic relaxation in the ensemble of small amount (up to 64) mobile single-domain particles with uniaxial magnetic anisotropy, dispersed in the Stokesian fluid, is considered based on the solution of the inertialess variant of motion equations. In the systems where strength of particle dipolar interaction is higher, then some threshold value, the orientational magnetic relaxation proceeds similar to the non-interacted case. With the overcoming of this threshold, particles form chain-like aggregates rapidly, that prevents to employ of the linear response theory for the results analysis. The threshold practically does not depend on the particle anisotropy and fluid viscosity. The raspberry model of the colloid particle is used in the simulation of the rotational magnetic relaxation of clusters, consisted of both magnetic and “passive” particles. There are considered a several types of the clusters differ from each other by spatial particle distribution, shape and size ratio of the particles. In all cases the account of the hydrodynamic interactions leads to the increase of relaxation rate. This tendency is heightened with increase of fluid viscosity and/or particle concentration. Theory of orientation motion of a Brownian magnetic nanoparticle embedded in a viscoelastic medium and subjected to a time-dependent uniform magnetic field is developed. The rheology of the viscoelastic environment of the particle is modelled by the Jeffreys scheme which under variation of a minimal number of parameters is able to resemble a wide range of soft materials: from weakly structured (nearly Newtonian) polymer solutions to gels. It is shown that in the Jeffreys model the diffusional orientation motion of a particle is a combination of two modes which could be associated with a fast motion within the polymer mesh cell and a slow displacement that involves deformation of the mesh, respectively. The dependencies of the reference times of both relaxation modes on the Jeffreys viscous and elastic parameters and temperature are found. It turns out that in substantially viscoelastic media the rate of the slow mode (it dominates in relaxation) quadratically depends on the matrix temperature. This effect does not have analogues in linearly viscous systems. For an ensemble of magnetic nanoparticles in viscoelastic and gel Jeffreys matrices: (1) the dynamic magnetic susceptibility is derived and evaluated both in the framework of exact approach and in a simple approximation; (2) the problem of magnetic relaxometry, i.e., evolution of magnetization after a step-wise turning on / off the field is solved; (3) the specific power loss caused by viscous dissipation generated by the particles under an AC field is analyzed as a function of the rheological parameters. Agreement of the theoretical results with experimental data of the German team (universities of Köln and Braunschweig) is confirmed. Results (1) and (2) provide simple models for magnetic nanorheology; considerations (3) advance the physics of magnetic hyperthermia in viscoelastic and gel-like media. A theory of ferromagnetic resonance in superparamagnetic core-shell nanoparticles is proposed. The model is applied to bimagnetic nanoparticles, where the core is a uniaxial ferromagnet in a single-domain state, and the shell is a set of subnanograins with antiferromagnetic, ferromagnetic or spin-glass structure. It is found that core-shell interaction via a domain-wall-like transition layer leads to onset of exchange anisotropy. The latter includes three terms – fixed unidirectional one, rotatable unidirectional and rotatable uniaxial ones. It is shown that all contributions to exchange anisotropy are explicitly identified with ferromagnetic resonance spectra. Furthermore, it is sufficiently to measure only resonance lines for an ensemble of randomly oriented particles; exactly this case is the most typical in experiments. The key point in the identification of parameters of exchange anisotropy is temperature “scanning” of resonance spectra. In the experimental part, during the reporting year, the main focus was on the centrifugation of the magnetic fluid as a method/tool to vary the dispersed composition of particles in a ferrofluid and to obtain a sample with the widest particle size distribution and the maximum intensity of magnetic dipole interactions. As a result of the work done, several samples of magnetic liquids of the "magnetite + liquid hydrocarbon + oleic acid" type with different dispersed composition and particle concentration and initial static susceptibility from one and a half to eighteen SI units were synthesized. The most valuable information was obtained from four samples with an anomalously wide particle size distribution (relative distribution width 0.56) and a record high intensity of magnetic dipole interactions - three times higher than the energy of thermal motion. It was found that samples with high energy of magnetic dipole interactions demonstrate a number of physical effects that are not characteristic to previously studied ferrofluids: 1) a steep maximum in the relaxation time spectrum in the 10-1-10-2c region at low temperatures and significant spread of the susceptibility at low frequencies of the order of 10 Hz; 2) the increase in the "high-frequency" susceptibility with the amplitude of the probe field (in ordinary ferrofluids, it decreases); 3) the temperature dependence of the equilibrium susceptibility is weaker, in comparison to the modified effective field model. All these observations might be attributed to the existence of a large number of clusters consisting mainly of superparamagnetic particles and formed due to van der Waals interactions in the magnetic fluid. The role of magnetodipole interactions is reduced to the correlation of the magnetic moments of the particles within the cluster and to the blocking of the Neel reversal magnetization mechanism. The existence of such clusters consisting of several dozen particles was predicted earlier in a number of experimental studies on the rheology of magnetic fluids, the diffusion of colloidal particles, the dynamics of magnetization, and magnetophoresis. An estimate for the amplitude of the probe field that removes the blocking of Neel particles (H 5 - 12 kA / m) is obtained. This estimate agrees perfectly with the experimental data on the amplitude dependence of the linear susceptibility.

 

Publications

---

Annotation of the results obtained in 2015
To describe the dynamic magnetic response of a system of interacting magnetic nanoparticles in a ferrofluid, we proposed a novel approach, based on the introduction of an additional term in the dynamic equation for the probability density of the orientation of a randomly selected particles (in the Fokker-Planck-Brown). This additional term describes a collective magnetic field, produced by the magnetic moments of all the other particles in the system, that acts on the chosen particle in addition to the main applied external ac magnetic field. Analytical expressions describing the frequency spectrum of the real and imaginary part of the dynamic initial susceptibility we obtained contain the extension to the Debye spectrum. This extension is quadratic in ferroparticles' concentration and in the intensity of the interparticle magnetic-dipole interaction. Theoretical study of the spectra carried out for polydisperse ferrofluids, for which both both the Brownian and Néel relaxation mechanisms of the magnetic moments of nanoparticles should be taken into consideration, revealed that the most sensitive to the influence of interparticle interactions are the following areas: a low-frequency behavior of the real part of the susceptibility, low-frequency increase in the imaginary part of the susceptibility, the position of the maximum of the imaginary part. Due to the interparticle interaction, the low-frequency plateaux of the susceptibility real part shifts strongly upwards in the region of larger values. The presence of interaction leads to a significant increase in the initial slope of the susceptibility imaginary part at low frequencies. The maximum of the imaginary part shifts up and to the left to lower the frequency of the field. This means that the effective relaxation time in a system of interacting ferroparticles exceeds that of a system of non-interacting magnetic moments. The concentration and temperature dependence of the relaxation time for collective polydisperse system ferroparticles was determined numerically. The results indicate that the dynamic susceptibility spectra are not a simple superposition of the individual magnetic moments’ relaxations, but to a great extent they are determined by a complex interplay of factors, size distribution, individual relaxation mechanisms of particles, the intensity of the interparticle interaction and concentration. The problem of static magnetic response of ferrogel was considered. Modeled sample possesses quasi-cubic structure of polymer matrix involves limited number (~100) of interacted single-domain particles, each with magnetic easy axis. The method of coarse-grained molecular dynamics coupled with Langevin dynamics was used to obtain the equilibrium magnetization curves for a wide range of particle anisotropy energy. Moreover, the reorganization of the filler structure in the field was examined. It was observed, that magnetic anisotropy embarrasses the magnetization, leads to more noticeable field-induced deformation of the sample and, at the same time, to shortening of the particle chains, formed in the field. Another branch of work on the project is devoted to magnetodynamics of interacted single-domain particles in liquid media. The two approaches for the simulation of such a processes are considered. First of them uses the simple approximation of the fluid’s impact by the introducing of Stokes friction with known dependencies of a mobility coefficient on particle shape and fluid viscosity. The ESPResSo package (used for simulation in the framework of coarse-grained molecular dynamics) was modified in order to introduce to it an ability to integrate the inertialess variant of motion, more suitable for the simulation of long-time behavior of nanoparticles (the real prototypes of the considered magnetic objects). Test calculation of magnetic relaxation of ensemble of non-interacted particles was performed. Model was verified based on the comparison of simulation results with the data, obtained by other method (integrating of the stochastic magnetodynamical equation). The second approach to examination of particle magnetodynamics is oriented to more rigorous simulation of hydrodynamics interactions in system. For this purpose the combination of two methods is used: 1) lattice Boltzmann method is charged with description of fluid media and 2) coarse-grained molecular dynamics is employed for simulation of magnetic particles. This quite complex computational scheme was tuned (a necessary calculation parameters were determined) relying on the simulation of magnetic relaxation of single particle. Ferromagnetic resonance in ferrite nanoparticles with core-shell architecture is studied. It is demonstrated that in ferromagnet nanoparticles the superparamagnetic effect “dresses” with temperature dependencies only those anisotropy contributions, whose tensor rank is 2 or higher. In other words, in nanoparticles, the intrinsic field caused by a vector (unidirectional) anisotropy does not “thaw” upon heating as it happens, for example, with the fields of uniaxial or cubic bulk anisotropy, whose contributions are characterized by second and fourth rank tensors, respectively. The results obtained are used for interpreting the data on high-frequency magnetic susceptibility of MnFe2O4-Fe2O3 nanoparticles. Withib the experimental part of the project the following resuts were obtained. The setup, which uses the sweep method for the measurement of magnetization curves, has been upgraded. The usage of a cooled solenoid as a source of static magnetic field in combination with a new highly sensitive ADC has improved measurement accuracy. Mutual inductance bridge has been upgraded with ADC (LAI-24) and digital phase meter. Seven samples of magnetic fluid have been synthesized. Samples have different particle concentrations and the same disperse composition. The initial magnetic susceptibility of samples at room temperature increases exponentially from two to sixteen SI units. Granulometric analysis has been implemented to process experimentally obtained magnetization curves of fluid samples. Saturation magnetization, initial magnetic susceptibility, average magnetic moment of particles, dispersion of magnetic moments, average diameter of particle magnetic core and relative width of particle size distribution have been determined. Despite the high concentration of colloidal particles, all investigated samples demonstrate classical Newtonian rheology, i.e. linear relation between the shear stress and the shear rate. We consider the Newtonian behavior of magnetic fluid as a sign of its high quality. It indicates the negligible concentration of free oleic acid in solution and the low content of large aggregates, which are sensible to the shear flow. Temperature dependence of viscosity has been investigated for the five most concentrated samples, starting from the sample №3. Dependencies of the dynamic susceptibility on the AC field frequency have been measured for all synthesized samples at five different temperatures: -41C; -22C; -0.5C; +27C and +64C. The results of measurements are presented in table form.Concentration dependence of the dynamic susceptibility has been studied in the frequency range from 1 Hz to 100 kHz at different temperatures. It has been found that at low temperatures the phase shift between magnetization and applied field decreases with concentration (contrary to the initial expectations). We currently believe that the reason for the decrease of energy losses in concentrated samples is the mutual rotation of particles in opposite directions due to their hydrodynamic interaction.

 

Publications

---

Annotation of the results obtained in 2016
The project is dedicated to the investigations of dynamic magnetic response of magnetic composites. We use a combined approach: theoretical investigations are accompanied by experiments and molecular dynamics computer simulations (using Brownian dynamics). During 2016 the following tasks were accomplished and the following results were obtained. Theoretical investigations of the dynamic susceptibility spectra was carried out in the case of two collinear fields, one of which was constant and another oscillating. Dynamic properties were calculated with the help of Fokker-Planck equation. The latter was modified to allow for dipolar correlations in polydisperse magnetic fluids. In the limiting cases (when the interaction goes to zero), the results of a new theoretical approach coincide with that of Debye theory, whereas in the case of zero frequency the results transform into modified mean field theory for the static case. It was shown that the increase of a static field drives to an overall decrease of the relaxation times in the system and also weakens the response to the oscilating field. The growth of magnetic particle concentration leads to the increase in the real part of the dynamic susceptibility, especially in the region of low frequences. The concentration growth also shifts the imiganary part maximum towards lower frequences. Theoretical results are verified by a good agreement with the simulation data. We also investigated the question of the magnetically induced deformation of a small fragment (bellow 100 particles) of a magnetic gel. The system response (magnetisation change, the variation of the size and the volume of the sample and also its microstructure evolution) as a function of an applied magnetic field, anisotropy energy of magnetic particles and their concentration in the sample was analysed. Our modelling revealed that for a noticeable magneto-deformation, the concentration of magnetic particles should exceed a certain critical value. In the samples with lower concentration either a negligible deformation is observed or the aggregate formation of particles does not influence geometrical and mechanical properties of the gel. Modelling of the ferrofluid small volume using Stocks friction was performed via molecular dynamics in the inertia-free case. This allowed a more efficient calculation when describing system linear response. Small ensembles of particles (N ≤ 64) were studied in the case of particles being magnetised to the saturation and particle centres fixed. This way a complex influence of magnetic interactions on the ensemble dynamic properties was discovered. The model of a magnetic particle in the liquid carrier was developed considering hydrodynamics via combination of molecular dynamics and Lattice-Boltzmann method. To construct this simulation, a raspberry model of a particle was implemented. This helped us to investigate the dynamic orientational relaxation of two particles in the fluid in case of particle centres fixed. We studied the influence of hydrodynamic interactions and carrier liquid viscosity on the orientational relaxation for two cases: zero magnetic interaction and week magnetic interaction. Additionally, a model is proposed that explains the origin of magnetization “memory” experimentally discovered in magnetic nanoparticle suspensions based on mesomorphic substances (thermotropic nematics). The main cause of the effect is the interference of anisotropic surface tension (it turns up below the phase transition point) to the energy balance that defines configurations of the nanoparticle aggregates. Those small aggregates emerge after magnetization of the sample in the isotropic state (a full analog of a ferrofluid). Minimizing their magnetic energy, they assume anisometric (elongated) shape. Below the transition point, the mesomorphic matrix undergoes an orientational transformation and turns into a nematic liquid crystal. This transition is accompanied by the increase of energy at the matrix-particle interface, and this energy becomes comparable with the magnetic interparticle energy. In that case, under those conditions, the net energy minimum requires the aggregates to be drop-shaped. In such a “drop” the magnetic moments of the nanoparticles are not aligned, their coupling is reduced, and the integrity of the aggregate is maintained mostly by the surface energy. Evidently, the magnetic susceptibility of those aggregates is weakened. Under heating of the system above the mesomorphic transition point, the molecular order in the matrix is destroyed, the interface energy goes down, and thermal motion readily “evaporates” the aggregates. This process does not have much of an effect on the magnetic susceptibility, and the latter remains unchanged. The state with the enhanced susceptibility could be restored, however, if one again exposes the sample for a short time (a few seconds) to a static field. According to the experiment, the above-described cycle could be multiply repeated. A theory of ferromagnetic resonance in superparamagnetic core-shell nanoparticles is proposed. It is shown that exchange interaction between separate magnetic phases results in the so called rotatable anisotropy. It is found that the symmetry of that anisotropy could be identified by analysing the temperature dependence of the resonance spectra. If the rotatable anisotropy contribution to the resonance field is temperature-independent, than the anisotropy is unidirectional. Alternatively, the decrease of the specified contribution with temperature indicates the uniaxial rotatable anisotropy. Next, we investigated experimentally the linear susceptibility of magnetic fluid, i.e. its response in the alternating magnetic field on the triple frequency divided by the magnitude of the probing field. In order to do so, we designed, constructed and adjusted the new version of the mutual inductance bridge. The key features of the new bridge are the possibility to work with alternating magnetic fields of relatively large magnitudes (up to 10 kA/m), the usage of the audio frequency signal generator with a low level of harmonics and the two-channel lock-in amplifier Anfatec eLockIn 203. Nonlinear susceptibility was shown to depend stronger on the interparticle interactions than the linear one, which gave us a hope to obtain new information about the effect of magnetic dipole interactions on the dynamics of ferrofluid magnetization. In general, these hopes were justified. We obtained data on the linear and nonlinear susceptibility in the temperature range from -10 to +17 C. All measurements were carried out on the fixed series of frequencies of the probing field: 140 Hz, 250 Hz, 500 Hz, 1 kHz, 2.5 kHz, 5 kHz, 10 kHz and 100 kHz. At relatively low frequency (250 Hz), the system is magnetized quasi-statically, and all the results qualitatively match our expectations. Nonlinear susceptibility monotonically grows with the field strength, reaches maximum and then gradually falls back. Qualitatively different picture can be seen at higher frequencies, starting approximately from1 kHz. In this case, for fields up to 200-300 A/m the triple-frequency signal from the measuring coil is at the level of the experimental error, and the absolute value of the dynamic susceptibility remains constant. As the field magnitude farther increases (up to 400 A/m), the nonlinear susceptibility grows abruptly, forming a sharp peak, while the linear susceptibility decreases in such a way that a kink is clearly seen on its field dependence. Since sharp peaks of the nonlinear susceptibility are absent in the quasi-static magnetization mode, we interpret the observed effect as a purely dynamic phenomenon, which cannot be explained in the framework of simple (linear) models of the relaxation. As a working hypothesis we propose the assumption that there exist some characteristic time scale about 10-3 – 10-5s, which is responsible for the formation and destruction of some structural elements. The most probable type of these elements is chains or spherical clusters. The dispersion composition, dynamical susceptibility and the spectrum of magnetization relaxation time shave been experimentally determined for six samples of magnetic fluid, obtained by the centrifugation of two base solutions of magnetite in kerosene. Base solutions were different in concentration of the magnetic phase and the width of the particle size distribution. We also perform the cluster analysis, which allowed to estimate characteristic sizes of aggregates with non-compensated magnetic moment. The results of magneto-granulometric and cluster analysis were provided. The strong effect of the centrifugation on physical properties of obtained fractions, which is due the spatial redistribution of particles and many-particle aggregates, was demonstrated. The presence of aggregates in magnetic fluids is considered to be the main reason of the low frequency (0.1 – 10 kHz) dispersion of the dynamic susceptibility.

 

Publications

---