PHYSICAL REVIEW E 90, 062303 (2014) Multifractality in dilute magnetorheological fluids under an oscillating magnetic field R. E. Moctezuma * and J. L. Arauz-Lara Instituto de F´ ısica “Manuel Sandoval Vallarta,” Universidad Aut´ onoma de San Luis Potos´ ı, Alvaro Obreg´ on 64, 78000 San Luis Potos´ ı, San Luis Potos´ ı, Mexico F. Donado Instituto de Ciencias B ´ asicas e Ingenier´ ıa de la Universidad Aut´ onoma del Estado de Hidalgo-AAMF, Pachuca 42184, Pachuca, Mexico (Received 11 June 2014; published 4 December 2014) A study of the multifractal characteristics of the structure formed by magnetic particles in a dilute magnetorheo- logical fluid is presented. A quasi-two-dimensional magnetorheological fluid sample is simultaneously subjected to a static magnetic field and a sinusoidal magnetic field transverse to each other. We analyzed the singularity spectrum f (α) and the generalized dimension D(q ) of the whole structure to characterize the distribution of the aggregates under several conditions of particle concentration, magnetic field intensities, and liquid viscosity. We also obtained the fractal dimension D g , calculated from the radius of gyration of the chains, to describe the internal distribution of the particles. We present a thermodynamic interpretation of the multifractal analysis, and based on this, we discussed the characteristics of the structure formed by the particles and its relation with previous studies of the average chain length. We have found that this method is useful to quantitatively describe the structure of magnetorheological fluids, especially in systems with high particle concentration where the aggregates are more complex than simple chains or columns. DOI: 10.1103/PhysRevE.90.062303 PACS number(s): 83.80.Gv, 05.45.Df , 61.43.Hv I. INTRODUCTION Objects having irregular forms or a complex mass dis- tribution are commonly observed in nature, in man made materials, and in mathematical sets and functions [1]. These objects, that sometimes appear structureless, exhibit scaling properties having fractal or multifractal characteristics, which provide a good mathematical description of the physical processes in such materials. A multifractal system is an object that needs more than a single fractal dimension to describe its distribution, i.e., a discrete or continuous spectrum of exponents is needed [2]. Since the last century, fractals have been of mathematical, scientific, engineering, and purely artistic interest, and their mathematical language has been used as a powerful tool to characterize diverse systems in almost all science disciplines [3,4]. In physics, fractals have been used to study the kinetics and structure of disordered materials, such as polymers, colloids, aerosols, and gels [5–7]. They also have applications in numerous other areas, including transport phe- nomena, dynamics of random materials, the growth and form of complex patterns, hydrodynamic instabilities, etc. [8–10]. Moreover, fractal and multifractal concepts have been used to study systems as the bifurcating structure of trees, blood ves- sels, geochemical patterns, fractured surfaces of materials, mu- sic analysis, and galaxy distributions, among others [11–14]. It has been shown that fractals are of great utility as they may reflect the underlying physical process driving physical phenomena and they may act as diagnostics of anomalous behavior. In this paper, we present a quantitative analysis of the geometry of the structures formed in a magnetorheologi- cal (MR) fluid in the presence of two different magnetic fields. In magnetorheological fluids, which are dispersions of * rosario@ifisica.uaslp.mx micrometric magnetic particles in nonmagnetic liquids, when a static magnetic field is applied, aggregates are formed due to the magnetic moment induced in the particles [15–18]. These aggregates cause noticeable changes in the physical properties of the systems, and their characteristics depend mainly on the magnetic field intensity and particle con- centration [19]. The study of the formed structure enables us to see a description of the physical properties of MR fluids, such as yield stress, viscosity, magnetization, and elastic modulus [20–22]. At low particle concentrations, the formation of chains is followed by the formation of thicker aggregates due to the coalescence of the chains. This system can be described as an ensemble of chains having an exponential distribution [23]. As we consider higher particle concentrations, the aggregates become more complex forming interconnected structures whose description in terms of chains and columns is insufficient. Some of the fractal and multifractal properties of this kind of structure have been previously stud- ied [20,24–26]. However, a comprehensive study of the mul- tifractal characteristics in a magnetic dispersion is carried out here. When a sinusoidal magnetic field is applied in addition to the static field, the effective oscillating magnetic field drives the system to different configurations [19,27]. In Ref. [19], the average chain length for different values of frequency, particle concentration, viscosity, and magnetic field intensities was studied. In this paper, we use the same large collection of digital photographs obtained in that study to analyze the complexity of the different distributions of the aggregates within the sample and the distribution of the particles in a single chain or aggregate. For the former analysis, we use multifractal measurements, whereas for the latter we calculate the radius of gyration of the aggregates. In Sec. II we briefly revise the scheme of the multifractal formalism. In Sec. III the multifractal characteristics of the distribution of the chains under several conditions in terms of 1539-3755/2014/90(6)/062303(9) 062303-1 ©2014 American Physical Society