Analysis of the optical transmission of a ferrofluid by an electromagnetic mixture law

Abstract

Evolution of the optical transmission of a ferrofluid after magnetic field commutation is analyzed by means of an approach based on the so-called mixture laws: expressions which predict the effective permittivity of heterogeneous media as a function of their constituents' permittivities, their proportions and the way they are arranged. In particular, this work is based on a law proposed by Sihvola and Kong for the effective permittivity of a host substance with ellipsoidal inclusions. Ferrofluids are peculiar examples of this kind of media: with the solvent as host, the inclusions are nanoparticle agglomerates whose shapes become modified by magnetic field exposure. In this work, experimental optical transmission of a ferrofluid is compared with predictions based on Sihvola&Kong's law. A remarkable coincidence is obtained both in absence of magnetic field, without using any fitting parameter, and in presence of magnetic field, employing the inclusions' average ellipticity as fitting parameter. The results obtained for time dependent optical transmission of a ferrofluid after magnetic field switch on or off allow one to estimate how the average shape of the agglomerates evolves along time. On the other hand, mixture laws are proven to be an interesting alternative to scattering concepts to model the optical transmission changes experienced by ferrofluids once they are exposed to magnetic fields.

Numerical method for analysis of the correlation between ferrofluid optical transmission and its intrinsic properties

“…The trends obtained in Fig. 3 are in agreement with experimental results [23]: parallel and perpendicular field applications produce an increase and decrease on the attenuation constant respectively, and therefore, a loss in transmission in parallel configuration and an increase in the perpendicular one. Furthermore, it is logical to expect that, the higher the particle concentration, the bigger the change in transmission ( Fig.…”

“…The above mentioned calculation method is highly time-consuming, so that a detailed calculation of how the attenuation constant evolves during the process of chain formation becomes unapproachable. For this reason, we propose an alternative method based on a mixture law proposed in [22] and successfully applied to ferrofluids [23]. This law provides an expression for the complex effective permittivity of a heterogeneous sample composed of a host material with spheroidal inclusions: (4) where (5) In the former expressions, the asterisk denotes a complex magnitude, is the mixture effective permittivity in the U direction (U = or ⊥ , parallel or perpendicular to the polar axis of the spheroids), ε e and are the host and inclusion permittivities, respectively, p is the volume fraction occupied by the inclusions and N U is the spheroid depolarization factor in the U direction.…”

“…This law provides an expression for the complex effective permittivity of a heterogeneous sample composed of a host material with spheroidal inclusions: (4) where (5) In the former expressions, the asterisk denotes a complex magnitude, is the mixture effective permittivity in the U direction (U = or ⊥ , parallel or perpendicular to the polar axis of the spheroids), ε e and are the host and inclusion permittivities, respectively, p is the volume fraction occupied by the inclusions and N U is the spheroid depolarization factor in the U direction. This factor is related with the polar and equatorial semiaxes and a ⊥ by means of: (6) As shown in [23], the latter expression is determined by the spheroid aspect ratio . As it can be seen in Eq.…”

“…In ferrofluids, inclusions are single particles in absence of magnetic field and a certain combination of single particles and chains in presence of a magnetic field. In [23] it is demonstrated that, as chains remain basically oriented parallel to the applied field, the anisotropy in the heterogeneous mixture can be correctly modeled by a medium with equivalent spheroidal inclusions characterized by their aspect ratio.…”

“…All the dimensions are set in μm. culates the sample optical transmission by means of two methods that prove to be complementary: on the one hand, by numerical propagation of an electromagnetic wave over the particles' distributions and, on the other hand, by means of a mixture law [22,23]. This way, a general method is proposed to obtain the magneto-optical response of any ferrofluid from its intrinsic parameters.…”