Nonlinear magnetic susceptibility of ferrofluids

In ferrofluids, external magnetic fields can produce a nonlinearity in the dependence of magnetic susceptibilities on the field strength, due to both normal saturation and abnormal saturation. We derive the nonlinear magnetic susceptibility by using a Langevin model. Then, we investigate the nonlinear ac responses of the magnetic susceptibility by means of a perturbation expansion method and an orthogonal numerical method. The two methods are numerically shown in excellent agreement. We find that the responses are sensitive to suspension structures and field frequencies. Thus, by detecting the ac responses, it seems possible to real-time-monitor the structure of ferrofluids. 2006 Elsevier B.V. All rights reserved. Ferrofluids (magnetic fluids) are colloidal suspensions containing single domain nanosized ferromagnetic particles dispersed in a carrier liquid [1]. These particles are usually stabilized against agglomeration by coating particles with long-chain molecules (sterically) or decorating them with charged groups (electrostatically). Since these particles can interact easily in the presence of applied magnetic fields, which in turn can affect the viscosity and structural properties tremendously, ferrofluids possess a wide variety of potential applications in various fields, ranging from mechanical engineering [2] to biomedical applications [3]. Thus, ferrofluids have received much attention [4–6]. The static magnetic properties of magnetic fluids have been studied extensively (e.g., see Ref. [7]). The dynamic (ac) magnetic properties have also been studied (e.g., Ref. [5]). For instance, the experimental results of the frequency-dependent susceptibility enables one to understand the dynamics of the magnetic particles in a partly frozen state of magnetic moments [8]. The measurement of an ac complex magnetic susceptibility of magnetic fluids is a suitable method to study the relaxation process of the magnetic dipoles of colloidal particles in magnetic fluids [9]. For 0009-2614/$ see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.02.010 * Corresponding author. Fax: +86 21 65118743. E-mail address: [email protected] (J.P. Huang). magnetic fluids, Fannin and his coworkers measured ac complex magnetic susceptibility v = Re[v] iIm[v] at room temperature [10]. Here Re[v] and Im[v] denote real and imaginary parts of v, respectively. The experimental results are possible to be explained by the Debye model [11], in which one may introduce the mutual interaction of the particles [12]. The magnetization-induced second-order harmonic generation is the phenomenon that the magnetization along the specific direction activates the originally silent tensor components for the second-order nonlinear optical susceptibility. So far, this phenomena have been observed for surfaces or interfaces of ferromagnetic materials (thin films) [13], or for polar antiferromagnets [14] and polar ferromagnets [15]. For the system under our consideration, the nonlinearity can be caused to appear by two effects, namely, normal saturation and anomalous saturation. In detail, the normal saturation arises from the higher terms of the Langevin function at large field intensities [16]. In contrast, the anomalous saturation results from the equilibrium between entities (e.g., particle chains) with higher and lower dipole moments which is shifted under the influence of the field [16]. Our formalism will hold for the coupling between the two effects. When a suspension having nonlinear characteristics is subjected to a sinusoidal (ac) electric/magnetic G. Wang, J.P. Huang / Chemical Physics Letters 421 (2006) 544–548 545 field, the electric/magnetical response will in general consist of ac fields at frequencies of higher-order harmonics [17,18]. That is, the harmonics of the magnetic susceptibility can be induced to appear. For treating magnetic responses, the formalism used for electro-magnetorheological fluids [19] can be directly extended to ferrofluids. We first briefly review the formulae for nonlinear characteristics of ferrofluids. Under the application of a magnetic field along z axis, a nonlinear characteristic appears in a ferrofluid, due to the suspended particles with a permanent magnetic dipole moment p0. Inside the model ferrofluid, the dependence of the magnetic induction B on the external field H0 will be nonlinear [16] B 1⁄4 leH0 þ 4pnH 20H0; ð1Þ where n and le stand for the nonlinearity coefficient and effective permeability for the longitudinal field case, respectively. In this case, the effective permeability le may be determined by the anisotropic Clausius–Mossotti equation [20,21] gLðle l2Þ l2 þ gLðle l2Þ 1⁄4 4p 3 N 1 b þ p0 3kBT ð1þ i2pf s1Þ ; ð2Þ where l2 is the permeability of the host fluid,N1 the number density of the particles, kB the Boltzmann constant, T the absolute temperature, f the frequency of the applied magnetic field, i 1⁄4 ffiffiffiffiffiffi 1 p , s1 = sBsN/(sB + sn) the effective relaxation time [22] of the particles due to the existence of the distribution of particle sizes, where sB and sN are the Brownian relaxation time and the Néel relaxation time, respectively. Since the ferromagnetic particles of interest generally have a radius of 10 nm or so, the magnetizability b in Eq. (2) is negligible and thus it is set to be zero throughout the work. Our model describes the aggregation behavior in an external field, by introducing the longitudinal demagnetization factor gL which is located in the Clausius–Mossotti equation [Eq. (2)]. The degree of anisotropy of the system is just measured by how gL is deviated from 1/3, gL 6 1/3 in the longitudinal field case. There is a sum rule [23] gL + 2gT = 1, where gT denotes the transverse demagnetizing factor. The substitution of gL(=gT) = 1/3 into Eq. (2) yields the well-known (isotropic) Clausius–Mossotti equation [16], as expected. In Eq. (2), we have used a linear relation for the Langevin function L = p0H0/(3kBT). This is because it is well established that the effective third-order nonlinear susceptibility (of our interest) can be calculated from the linear field [24], while the effective higher-order nonlinearity must depend on the nonlinear field [25]. Due to the same reason, in what follows we shall omit the contribution from the nonlinear field, too. So far, the desired field-dependent incremental permeability lH is lH 1⁄4 oB oH0 1⁄4 le þ 12pnH 2 0 [16]. Then, the nonlinear magnetic effect is characterized by Dl=H 20 [16] Dl H 20 1⁄4 lH le H 20 1⁄4 12pn. ð3Þ Next, we consider the orientation magnetization Mor of a sphere of volume V, containing n1 particles with a permanent magnetic dipole moment pd (see below) embedded in a continuum with permeability l1 (see below). The sphere is surrounded by an infinite medium with the same macroscopic properties as the sphere. The average component in the direction of the field of the magnetic dipole moment due to the dipoles in the sphere ÆMd Æ eæ = VMor is given by hMd ei 1⁄4 VMor 1⁄4 R dX1Md ee u=kT R dX1e u=kT ; ð4Þ where e denotes the unit vector in the direction of the external field, andX stands for the set of position and orientation variables of all particles. Here u is the energy related to the dipoles in the sphere, and it consists of three parts: the energy of the dipoles in the external field, the magnetostatic interaction energy of the dipoles, and the non-magnetostatic interaction energy between the dipole which is responsible for the short-range correlation between orientations and positions of the dipoles, e.g., the London–Van der Waals interaction energy. In Eq. (4), Md is given by