Open Access Article

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Alexey O.
Ivanov
*^{a},
Vladimir S.
Zverev
^{a} and
Sofia S.
Kantorovich
^{ab}
^{a}Ural Federal University, Lenin av. 51, 620000, Ekaterinburg, Russia
^{b}University of Vienna, Sensengasse 8, 1090, Vienna, Austria. E-mail: sofia.kantorovich@univie.ac.at

Received
29th October 2015
, Accepted 3rd February 2016

First published on 4th February 2016

We investigate, via a modified mean field approach, the dynamic magnetic response of a polydisperse dipolar suspension to a weak, linearly polarised, AC field. We introduce an additional term into the Fokker–Planck equation, which takes into account dipole–dipole interaction in the form of the first order perturbation, and allows for particle polydispersity. The analytical expressions, obtained for the real and imaginary dynamic susceptibilities, predict three measurable effects: the increase of the real part low-frequency plateaux; the enhanced growth of the imaginary part in the low-frequency range; and the shift of the imaginary part maximum. Our theoretical predictions find an experimental confirmation and explain the changes in the spectrum.

In this manuscript, we present a new theoretical approach based on the solution of the Fokker–Planck equation with an additional term in the form of the modified mean field. This term is the first order perturbation that allows for the magnetic dipole–dipole interaction in the system. In this framework, the particle polydispersity enters the equation in a very natural and straight-forward way. The resulting expressions for the initial dynamic susceptibility, i.e. dynamic response of the system to a weak, linearly polarised, harmonic, external field, have a closed analytical form and represent the first-order density corrections to the Debye spectrum. We apply our formalism to describe three measurable effects stemming from the interparticle correlations: the low-frequency regime of the real part of the dynamic susceptibility; its imaginary part maximum shift; and the low-frequency growth of the imaginary part. Analysing the effect of the dilution on the experimentally obtained spectra, we not only find a very good agreement of our theoretical results and the experimental data, but also conclude that the measured spectra do not reflect the superposition of the individual particle relaxations, but are the consequences of the complex interplay between these individual particles' dynamics, their polydispersity, and the interparticle interactions, both sterically and magnetically.

Although, here, we will focus on the simplest case of magnetic dipoles and magnetic fields, the approach put forward in this manuscript is rather generic and can be applied to the dipoles of different nature. Here, it is worth saying that for electric dipoles the approach has to be extended as both higher order corrections to the dipolar interactions and the currents might become relevant, see for example ref. 7 and references therein.

(1) |

(2) |

(3) |

(4) |

(5) |

_{0}(1) = (α_{1}/sinhα_{1})exp(α_{1}cosθ_{1}), |

(6) |

(7) |

g_{2}(1,2) = (1)_{0}(2)Θ(1,2), | (8) |

(9) |

(10) |

(11) |

Returning to a non-equilibrium case, eqn (1)^{2} transforms into eqn (12)

(12) |

U_{e}(1) = U_{H}(1) + n〈W_{0}(2)U_{dd}(1,2)Θ(1,2)〉_{2}. | (13) |

(14) |

(15) |

W(1) = 1 + α_{1}A_{1}(ω)cosθ_{1}e^{iωt}, | (16) |

(17) |

(18) |

The novelty of the described above approach is to introduce the second term for the effective field produced by all dipole moments (n〈W_{0}(2)U_{dd}(1,2)Θ(1,2)〉_{2}) into the expression for U_{e}(1) to construct eqn (12).

Fig. 1 Dynamic susceptibility 4πχ(ω) of F1: comparison of the Debye ideal gas model to the expressions in eqn (18). The frequency range is given on the log-scale. Here, φ_{m} = 0.06; T = 293 K; K = 20 kJ m^{−3}; effective viscosity η = 2 × 10^{−3} Pa s; M_{0} = 480 kA m^{−1}. |

This term has a meaning of the effective field acting on each particle due to the presence of all others. In the case of the static applied field (ω → 0), the perturbation theory of the first order in n correctly leads to a well-tested modified mean field approach^{47} expression for the initial susceptibility:

χ(0) = χ_{L}(1 + 4πχ_{L}/3). | (19) |

Below, as one of the measurable characteristics of the spectrum, we analyse the dependence of the frequency ω*, at which the maximum of the imaginary part is reached, which, as mentioned above, does not depend on n in the framework of standard eqn (1). We can define the value of ω* in our formalism by solving

(20) |

(21) |

(22) |

(23) |

In the following we will test expressions from eqn (18) by comparing the latter to the experimental results for moderately interacting magnetic fluids.

In order to illustrate the model proposed above for the dynamic response, we use two ferrofluids with well-defined particle-size distributions that can be accurately described using gamma-distribution:

(24) |

Fig. 2 Reduced dynamic susceptibility χ(ω)/χ(0) of F1: the effect of dilution. Three spectra are plotted for φ_{m} = 0.12, 0.06, 0.01 in blue, orange and bordeaux, respectively. The frequency range is given on the log-scale. Inset: The dependence ω*(φ_{m})/ω*(0). The solid line is the solution of eqn (21); the dotted line is a corresponding monodisperse sample eqn (23). |

Fig. 3 Dynamic susceptibility 4πχ(ω) of F2: comparison of the experimental data to the expressions in eqn (18). The frequency range is given on the log-scale. Here, the mass fraction of iron for the basic sample is 0.061; T = 293 K; K = 100 kJ m^{−3}; effective viscosity η = 5 × 10^{−3} Pa s; M_{0} = 1490 kA m^{−1}. Reference sample: orange, filled symbols; 10 time diluted sample: blue, empty symbols (rescaled to be visible). |

The other measurable effect can be observed when analysing the initial slope κ of χ′′:

(25) |

In Fig. 4 we plot κ(φ_{m}) for F1 and F2 as predicted by Debye ideal gas model eqn (4) and by the modified mean field approach eqn (18). The results of the two models differ substantially. The absolute value of these deviations is different for F1 and F2. In the case of F1, the quadratic correction from χ_{L} is not very large, because the particles in F1 are rather magnetically weak. On the other hand, particles in F2 are predominantly magnetically hard (very high crystallographic magnetic anisotropy), and in this case, the quadratic term in χ_{L} is much larger. The actual values of the slope are determined by the value of τ_{char}. Even though p(x) is broader in the case of F1, the value of τ_{char} for F2 is approximately an order of magnitude larger, due to the dominance of the Brownian slower relaxations in it. Notice that, had we used the Debye model to extract τ_{char} from the experimental measurements, at a finite magnetic phase concentration of approximately 5 percent, the error due to the neglected interparticle correlations would have been as high as the factor of two.

Fig. 4 Initial slope (κ) as a function of φ_{m}: comparison of the Debye ideal gas model to the expressions in eqn (25). The results for F1 (black) are associated with the left ordinate axis; the results for F2 (orange) with the right one. Inset: Particle size distributions versus magnetic core diameter (x). |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5sm02679b |

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