Polydisperse Ferrofluids Research Articles

Magnetization relaxation dynamics in polydisperse ferrofluids.

When a ferrofluid is magnetized in a strong magnetic field, and then the field is switched off, the magnetization decays from its saturation value to zero. The dynamics of this process are controlled by the rotations of the constituent magnetic nanoparticles, and for the Brownian mechanism, the respective rotation times are strongly influenced by the particle size and the magnetic dipole-dipole interactions between the particles. In this work, the effects of polydispersity and interactions on the magnetic relaxation are studied using a combination of analytical theory and Brownian dynamics simulations. The theory is based on the Fokker-Planck-Brown equationfor 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.

Effects of interactions, structure formation, and polydispersity on the dynamic magnetic susceptibility and magnetic relaxation of ferrofluids

Linear response theory relates the decay of equilibrium magnetisation fluctuations in a ferrofluid to the frequency-dependent response of the magnetisation to a weak ac external magnetic field. The characteristic relaxation times are strongly affected by interactions between the constituent particles. Similarly, the relaxation of an initially magnetised system towards equilibrium in zero field occurs on a range of timescales depending on the structure of the initial state, and the interactions between the particles. In this work, ferrofluids are modelled as colloidal suspensions of spherical particles carrying point dipole moments, and undergoing Brownian motion. Recent theoretical and simulation work on the relaxation and linear response of these model ferrofluids is reviewed, and the effects of interactions, structure formation, and polydispersity on the characteristic time scales are outlined. It is shown that: (i) in monodisperse ferrofluids, the timescale characterising the collective response to weak fields increases with increasing interaction strength and/or concentration; (ii) in monodisperse ferrofluids, the initial, short-time decay is independent of interaction strength, but the asymptotic relaxation time is the same as that characterising the collective response to weak fields; (iii) in the strong-interaction regime, the formation of self-assembled chains and rings introduces additional timescales that vary by orders of magnitude; and (iv) in polydisperse ferrofluids, the instantaneous magnetic relaxation time of each fraction varies in a complex way due to the role of interactions.

Dynamical effective field model for interacting ferrofluids: I. Derivations for homogeneous, inhomogeneous, and polydisperse cases

Quite recently I have proposed a nonperturbative dynamical effective field model (DEFM) to quantitatively describe the dynamics of interacting ferrofluids. Its predictions compare very well with the results from Brownian dynamics simulations. In this paper I put the DEFM on firm theoretical ground by deriving it within the framework of dynamical density functional theory, taking into account nonadiabatic effects. The DEFM is generalized to inhomogeneous finite-size samples for which the macroscopic and mesoscopic scale separation is nontrivial due to the presence of long-range dipole–dipole interactions. The demagnetizing field naturally emerges from microscopic considerations and is consistently accounted for. The resulting mesoscopic dynamics only involves macroscopically local quantities such as local magnetization and Maxwell field. Nevertheless, the local demagnetizing field essentially couples to magnetization at distant macroscopic locations. Thus, a two-scale parallel algorithm, involving information transfer between different macroscopic locations, can be applied to fully solve the dynamics in an inhomogeneous sample. I also derive the DEFM for polydisperse ferrofluids, in which different species can be strongly coupled to each other dynamically. I discuss the underlying assumptions in obtaining a thermodynamically consistent polydisperse magnetization relaxation equation, which is of the same generic form as that for monodisperse ferrofluids. The theoretical advances presented in this paper are important for both qualitative understanding and quantitative modeling of the dynamics of ferrofluids and other dipolar systems.

Generic theory of the dynamic magnetic response of ferrofluids.

Ferrofluids belong to an important class of highly functional soft matter, benefiting from their magnetically controllable physical properties. Therefore, it is of central importance to quantitatively predict the dynamic magnetic response of ferrofluids. Traditional dynamic theories, however, are often restricted to the near-equilibrium regime and/or only apply to nearly ideal ferrofluids that are monodisperse, dilute enough, and weakly interacting. In this paper I develop a self-consistent and nonperturbative dynamical mean field theory for typical ferrofluids which are often polydisperse, concentrated, and strongly interacting, possibly driven far from equilibrium. I obtain a general nonperturbative expression for the dynamic magnetic susceptibility, quantitatively agreeing with the spectra obtained from Brownian Dynamics simulations on both mono- and bidisperse samples. Furthermore, I derive a generic magnetization relaxation equation (MRE) for both mono- and polydisperse ferrofluids by employing the projection operator technique in nonequlibrium statistical mechanics. This MRE is in simple closed form and independent of which model is employed to approximate the equilibrium magnetization curve. Existing models can be recovered as low-order approximations of my generic and nonperturbative MRE. My theory can play a key role in studying the dynamics of ferrofluids and other polar fluids. It may also have substantial and immediate consequences to various ferrofluid applications.

Observation of soft glassy behavior in a magnetic colloid exposed to an external magnetic field.

We provide the first experimental evidence for soft glassy behavior in a sterically stabilized magnetic colloid (ferrofluid) of relatively low volume fraction (φ = 0.037) when a uniform magnetic field is applied at a sufficiently high rate (fast quench). Fast magnetic-field quenches favor structural arrest of field-induced aggregates, owing to insufficient time to settle into lower energy states, thereby pushing the system to a frustrated metastable configuration like a repulsive glass. Brownian dynamics simulations are used to show that the polydisperse ferrofluid (as in experiments) forms thick ropes aligned along the field direction, while a monodisperse ferrofluid does not. The simulations show that there is practically no ordering of the thin, monodisperse chains, while the thick, polydisperse ropes show positional ordering with a typical center-center separation between the particles in different ropes of about 0.39μm. As a consequence of structural arrest, the ferrofluid exhibits aging with broken time-translational invariance, a hallmark of glassy dynamics. The superposition of strain and creep compliance curves obtained from rheological measurements at different waiting times in the effective time domain corroborates the soft glassy behavior when exposed to a magnetic field applied at a fast ramp rate.

Shear rate dependence of viscosity and normal stress differences in ferrofluids

In this work we investigate numerically the viscosity and normal stress differences of mono and polydisperse ferrofluids undergoing a simple shear flow in the presence of a transverse applied magnetic field. The simulations are carried out by means of a Brownian Dynamics integration algorithm. For all simulations we assume that the dipole moments are fixed to the magnetic particles. In the case of polydisperse suspensions the particles present a typical log-normal distribution of magnetic diameters which are covered with a layer of surfactant of constant width. The long-range dipolar interaction between the magnetic moments are computed by Ewald summation technique and the surfactant influence is modeled as a steric repulsive force between pairs of particles. The attractive van der Waals force is also incorporated in the numerical simulations. The numerical procedure is validated by comparing the simulation results of the non-dimensional relative viscosity increase as a function of the magnetic field intensity with experimental data. The present numerical simulations show that the non-dimensional relative viscosity increment as a function of the non-dimensional shear rate is strongly influenced by the particle volume fraction and the strength of the dipolar interactions parameter. By examining the suspension microstructure into the numerical box, we can see anisotropic structures like chains of particles for the low shear rate regimes, specially in the polydisperse cases. The results also indicate the presence of a suspension shear-thinning behavior with two different non-dimensional shear rate dependence. In addition, non-dimensional normal stress differences are observed even in the absence of dipolar interactions, as a consequence of stress anisotropy produced by the repulsive forces during the particle collisions induced by the imposed steady-shear flow. The dipolar interactions increase these non-dimensional normal stress differences and make their dependence on the non-dimensional shear rate non-monotonic. For low non-dimensional shear rates, the dipolar interactions dominate and the formation of large-structures like chains oriented in the direction of the magnetic fields leads to an increase in the non-dimensional first normal stress difference. For very high non-dimensional shear rates, the flow breaks the anisotropic structures and the effects of polydispersity and dipolar interactions become unimportant. The numerical simulations provide a useful tool for understanding the complex microphysical and rheology of real ferrofuids under shear flows.

The initial magnetic susceptibility of high-concentrated, polydisperse ferrofluids: Universal theoretical expression

In this article, we propose the universal theoretical expression of the ferrofluid initial magnetic susceptibility, which is valid in a wide range of concentrations. The new formula is tested against experimental susceptibility measurements and compared with the well-known second-order modified mean-field theory (MMF2) [Phys. Rev. E 64, 041405 (2001)] and the extended version of MMF2 theory (MMF2+) [Phys. Rev. E 96, 052609 (2017)]. It is shown that the new formula is more reliable than MMF2 and MMF2 + theories and predicts initial magnetic susceptibility correctly for the polydisperse ferrofluids within a wide range of both temperature and concentration.

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.

Characterization of aggregate state of polydisperse ferrofluids: Some aspects of anisotropy analysis of 2D SAXS in magnetic field

2D small-angle X-ray scattering (SAXS) patterns for dilute ferrofluids with different degree of particle interaction under external magnetic field are analyzed to obtain structural characteristics of aggregates formed by polydisperse particles. Classical ferrofluids based on organic low-polarity solvents (toluene, decalin) with dispersed nanoparticles of magnetite and cobalt coated with oleic acid and water-based ferrofluid with dispersed magnetite nanoparticles coated with double layer of sodium oleate, are studied. All systems show the Langevin type behavior regarding the orientation of the anisotropic aggregates in them with the saturation at a relatively weak magnetic field strength. The field-induced anisotropy in the SAXS patterns is considered in the frame of a simple approximation.

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.

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 dependence of initial magnetic susceptibility of polydisperse ferrofluids: A critical comparison between experiment and theory

Temperature dependence of initial magnetic susceptibility of polydisperse ferrofluids: A critical comparison between experiment and theory

Temperature dependence of initial magnetic susceptibility of polydisperse ferrofluids: A critical comparison between experiment and theory

Temperature dependence of initial magnetic susceptibility of polydisperse ferrofluids: A critical comparison between experiment and theory

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.

Low-temperature magnetic susceptibility of concentrated ferrofluids: The influence of polydispersity

In this paper we address the question of theoretical explanation of extremely high low-temperature initial magnetic susceptibility of concentrated ferrofluids . These laboratory synthesized samples [A.F. Pshenichnikov, A.V. Lebedev, J. Chem. Phys. 121(11) (2004) 5455; Colloid J. 67(2) (2005) 189] demonstrated the record-breaking values χ ~ 120 – 150 at temperatures ~ 230–240 K. The existing models predict such high susceptibility only under the assumption of unreasonably large dipolar coupling constant, which is out of the range of applicability. Here we calculate the second virial contribution to susceptibility for polydisperse ferrofluid, modeled by the dipolar hard sphere fluid. In the resulting expression there exists the parameter, which plays a part of dipolar coupling constant and which is defined in a form of double averaging of high powers of particle sizes over the granulometric distribution. For real particle size distribution this effective parameter at least twice exceeds the commonly defined polydisperse dipolar coupling constant. We show that the low-temperature magnetic susceptibility of the record-breaking ferrofluids could be explained theoretically on the basis of the first terms of the polydisperse second virial contribution in combination with the second-order modified mean field model. • Record-breaking magnetic susceptibility of ferrofluids at low temperatures. • Second virial contribution to magnetic susceptibility of polydisperse ferrofluids. • Interparticle dipole correlations are more pronounced in dense ferrofluids. • Presented model describes the ferrofluid susceptibility χ ~ 120 at temperature 240 K.

Comparison between theory and simulations for the magnetization and the susceptibility of polydisperse ferrofluids

The influence of polydispersity on the magnetization of ferrofluids is studied based on a previously published magnetization equation of state (Szalai and Dietrich, 2011 J. Phys.: Condens. Matter 23 326004) and computer simulations. The polydispersity of the particle diameter is described by the gamma distribution function. Canonical ensemble Monte Carlo simulations have been performed in order to test these theoretical results for the initial susceptibility and the magnetization. The results for the magnetic properties of the polydisperse systems turn out to be in quantitative agreement with our present simulation data. In addition, we find good agreement between our theory and experimental data for magnetite-based ferrofluids.

The quasi-magnetic-hysteresis behavior of polydisperse ferrofluids with small coupling constant

The magnetization behaviors of ferrofluids based on γ-Fe 2 O 3 /Ni 2 O 3 composite nanoparticles of size about 11 nm have been investigated. The dipole coupling constant λ of these particles is so small (0.43) that they cannot form aggregates through magnetic interaction alone. Experimental results have shown that for a polydisperse ferrofluid with a particle volume fraction of ϕ V =2.4%, the magnetization curve exhibits quasi-magnetic-hysteresis behavior, i.e., the demagnetization curve lies above the magnetization curve in a high field. However, for a more dilute γ-Fe 2 O 3 /Ni 2 O 3 ferrofluid with ϕ V =0.94%, the magnetization curve does not show such behavior. According to the bidisperse model for polydisperse ferrofluids, these magnetization behaviors may be attributed to field-induced effects of self-assembled pre-existing chain-like aggregates. For such pre-existing chain-like aggregates, the orientation of the moments inside the particles is not co-linear, so that during the magnetization and demagnetization processes, their apparent magnetizations at the high-field limit are different. As a consequence, the magnetization curve of the ferrofluid with ϕ V =2.4% displays quasi-magnetic-hysteresis.

The structural force arising from magnetic interactions in polydisperse ferrofluids

Ionic ZnFe2O4 is a weak magnetic ferrofluid with different particle sizes (a polydisperse ferrofluid), in which the dipolar coupling constant λ is less than 2. By comparing the reduced magnetization curves and initial magnetic susceptibilities of ferrofluids with different particle volume fractions ϕ, it is found that there are field-induced attractive interactions between the colloidal particles, which make them aggregate; this interaction decreases with ϕ. The dipolar coupling constant is so small that particle aggregation cannot be induced by magnetic interaction alone. Thus, it can be concluded from the experimental results that a nonmagnetic attractive interaction is stimulated during the magnetization process for ferrofluids. Using a model based on “large” and “small” particles (the bidisperse model), this nonmagnetic interaction can be explained as a field-induced structural force. This force is enhanced following magnetic interactions between the large particles as the applied magnetic field increases. The field-induced structural force is similar to a depletion force and can also arise in strong magnetic ferrofluids (λ>2) with polydisperse particle sizes under an applied external magnetic field.

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.