Second-order-like cluster-monomer transition within magnetic fluids and its impact upon the magnetic susceptibility
© Zhong et al; licensee Springer. 2012
Received: 18 November 2011
Accepted: 5 March 2012
Published: 5 March 2012
The low-field (below 5 Oe) ac and dc magnetic response of a magnetic fluid [MF] sample in the range of 305 to 360 K and 410 to 455 K was experimentally and theoretically investigated. We found a systematic deviation of Curie's law, which predicts a linear temperature dependence of inverse initial susceptibility in the range of our investigation. This finding, as we hypothesized, is due to the onset of a second-order-like cluster-to-monomer transition with a critical exponent which is equal to 0.50. The susceptibility data were well fitted by a modified Langevin function, in which cluster dissociation into monomers, at the critical temperature [T*], was included. In the ac experiments, we found that T* was reducing from 381.8 to 380.4 K as the frequency of the applied field increases from 123 to 173 Hz. In addition, our ac experiments confirm that only monomers respond for the magnetic behavior of the MF sample above T*. Furthermore, our Monte Carlo simulation and analytical results support the hypothesis of a thermal-assisted dissociation of chain-like structures.
PACS: 75.75.-C; 75.30.Kz; 75.30.Cr.
The interest in magnetic fluids [MFs] has increased enormously in the last decade, particularly due to the opportunities they provide for applications in the medical field [1–7]. Among others, MFs have been used as an excellent material platform for the development of magnetic immunoassay [1, 2], contrast agents for magnetic resonance imaging [3, 4], and material devices for magnetohyperthermia [5–7]. Deep understanding of magnetic susceptibility, however, is a key issue not only from the fundamental point of view, but also while tailoring nanosized magnetic materials for medical applications [1–7]. The design of nanosized magnetic particles, taking into account the maximization of the materials' response in terms of their use for diagnosis, imaging, and therapy, requires the knowledge of the temperature dependence of the magnetic susceptibility under the action of applied dc and ac fields.
In this context, the widely accepted concept is a linear relationship between the inverse initial magnetic susceptibility (1/χ) and the temperature (T), which is accounted for by the first-order Langevin function. Nevertheless, unusual deviations of linearity at temperatures within the range of interest for the medical applications, with no conclusive explanation yet, have been reported . Since an interaction among particles in a MF sample, either modulated or not by external fields, cannot be ignored and leads to expontaneous agglomeration in clusters or chain-like structures (dimers, trimers, etc.) [9–11], the superlinear deviation of the 1/χ versus T curve has been attributed to magnetic dipolar interaction among the nanosized particles . The Langevin function, however, includes no interaction among the suspended particles, and therefore, clusters are ruled out from the classical description. Nevertheless, at high enough temperatures, the phenomenon of thermally assisted cluster disruption within MF samples has been reported . Therefore, to understand the underlying physics of this nonlinearity in magnetization, a more complete physical model is highly demanded. The new model should take into account both monomers and clusters in MFs.
In this study, we report the unusual superlinear deviation in the temperature dependence of the inverse initial magnetic susceptibility in a magnetite-based (Fe3O4) MF sample in the temperature range of 305 to 360 K. The observed breakdown of Curie's law, which scales linearly the inverse susceptibility with temperature, indicates that besides the usual tendency of alignment of magnetic moments with the applied field, there exists an additional thermally assisted physical process connected to cluster disruption within MF samples. Therefore, we propose a model in which a chain-like disruption at a typical transition temperature (T*) is incorporated. We found that the extended model reproduces quite well the observed superlinear deviation. Additionally, the above-mentioned superlinear deviation of the (1/χ) × T data for chains of particles is supported by Monte Carlo [MC] simulation and herein incorporated.
In order to verify the universality of the experiment regarding the relationship of inverse initial susceptibility and temperature, we explored the experiments under dc and ac magnetic fields, the latter at different frequencies. The MF sample used in our experiment was a commercial magnetic colloid (EFH1, Ferrotec Corporation, Santa Clara, CA, USA), consisting of Fe3O4 magnetite nanoparticles (mean particle diameter of 10 nm) suspended in light mineral oil. The applied magnetic fields were 5 and 2 Oe (amplitude) for dc and ac experiments, respectively. The ac experiments were performed at 123 and 173 Hz.
Model and discussion
The susceptibility model for pure monomers
Equation 2 describes the (1/χ) × T data based on the first-order Langevin function. Note that the dashed line in Figure 2 is the curve fitting of the data (open symbols) using Equation 2, showing the expected linearity in the temperature range of interest. However, our experimental data (open symbols in both Figures 2 and 3) revealed a superlinear trend at the high temperature end, making the fitting procedure using Equation 2 visibly poor (see inset (a) of Figure 2). We then hypothesized that the assumption of only monomers in the MF sample in the temperature range of 305 to 360 K no longer holds.
Discussion of the susceptibility model including dimers
The explanation of the superlinear behavior observed on the (1/χ) × T experimental data displayed in Figures 2 and 3 (open symbols), as we claim, is due to the presence of a fraction of chain-like structures within the MF sample, in addition to monomers, more likely dimers [13, 14], and the temperature dependence of the dimer fraction in a critical way around T*, as discussed below.
The probability of agglomeration and disruption of suspended nanoparticles in MFs is assumed to be dependent upon the relative strength of magnetic, van de Waals, electrostatic, and steric interactions and thermal energy. The literature [15–17] describes that magnetic dipolar interaction held particles together, whereas electrostatic interaction and thermal energy work together taking nearby particles apart. It can be inferred from the experimental observations that there is a dynamic balance between the relative content of monomers and dimers within the simplest model picture of a magnetically textured MF sample. This balance will be broken when there is a change on the parameters governing the energy terms involved, and the relative content of monomers and dimers will change accordingly. As the thermal energy increases, the dimers tend to disrupt into monomers, leading to a decrease of the dimers' relative content. Inversely, within this simplest model picture, when lowering the temperature of a MF sample, the monomers tend to agglomerate into dimers, leading to a decrease of the monomers' relative content.
The low-field dc susceptibility data are shown in Figure 2. The solid line represents (see Figure 2) the best curve fitting of the (1/χ) × T experimental data (open squares) using Equation 5. Included here for comparison, the dashed line in Figure 2 represents the fitting of the data using Equation 2. Note that ferromagnetic resonance data (resonance line splitting) have been used to describe dimer disruption in a nickel ferrite-based ionic MF sample, providing values of T* (340 K) and β (0.42), the latter in reasonable agreement with the value we found in the present study . Our data indicate that above T*, the suspended particles within the MF sample investigated are essentially isolated (monomers). The inset (a) of Figure 2 shows a detail of the (1/χ) × T data, emphasizing the superlinear behavior at temperatures close to the typical dimer disruption temperature (T*). The inset (b) of Figure 2 shows the temperature dependence of P 2 , here, describing the order parameter associated to a second-order-like phase transition.
Experiments on low-field ac susceptibility at different frequencies (123 and 173 Hz) are shown (symbols) in Figure 3. The saturation magnetization of monomers (MS 1) and dimers (MS 2) in the same MF sample was considered to be constant. MS 1and MS 2were firstly acquired by the fitting of any applied field (dc or ac) and frequency, and then both MS 1and MS 2were used as known parameters to fit different data sets. We found different susceptibility responses while changing the frequency of the applied field. The fittings of the experimental data (solid lines) using Equation 5 show that the critical temperature decreases from T* = 381.8 down to T* = 380.4 as the frequency of the ac excitation field increases from 123 to 173 Hz, in agreement with recently reported results .
Monte Carlo simulation of second-order-like dimer-monomer transition
where the sum is extended to the nearest neighbors only, J is the nearest neighbor exchange integral, and h= M s H. The equilibrium magnetic moment configuration is obtained by a MC simulation using the standard Metropolis algorithm. It is interesting to find out that for systems such as MFs in which suspended nanoparticles are found as monomers and dimers, the magnetization and susceptibility can be calculated analytically through the partition function of the system, , where E n is the energy of the system in the n th configuration. For instance, we can use the following four configurations in order to describe dimers |σ1σ2>=|++>,|+->,|-+>,|-->.
The pure monomer scenario above T*
Note that A in Equation 8 means just a scale factor for the experimental data. The solid line in Figure 5 has an excellent agreement with the experimental data and presents the best curve fitting achieved using Equation 8. Furthermore, the monomer's saturation magnetization (M S ) obtained from the data recorded in the temperature range above T* (from 410 to 455 K) is very much close to the value found from the data recorded in the temperature range below T* (from 305 to 360 K). Although fitting of the data presented in Figure 5 (open symbols) can be performed using the dimer-monomer model, represented by Equation 5, there is no agreement between the MS 1obtained from below and above T*. This indicates that dimers can be assumed to be totally disrupted into monomers when the temperature is increased above T*, and the critical temperature T* can be actually used to describe a MF sample.
In conclusion, the usual linear temperature dependence of the inverse initial susceptibility (dc or ac) of MFs at lower temperatures is found whereas at higher temperatures, an upward deviation of linearity is observed. This superlinear behavior is attributed to a thermal-assisted disruption of dimers into monomers which is described by the classical Landau's approach for second-order phase transitions. The experimental observations in the temperature ranges below and above the critical temperature T* are well fitted by the model picture proposed here, in which an extended Langevin function including the thermal criticality of the MF system is adopted below T*. Our findings are strongly supported by both MC simulation and analytic analysis.
The work was supported by NSFC 61174008 and 11104089, MOST 0S2012GR0121, R&D HB 2010BFA013, and GBIE HF09062011184.
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