# Magnetic Properties of Nanocomposites
## 3 Polymer Nanocomposites
### Polymer Nanocomposites Containing Weakly Interacting Magnetic Nanoparticles
Then, tetraethoxysilane (TEOS) was added and the reaction was stirred at room temperature for 48 h. Subsequently, the reaction mixture was washed with ethanol and particles underwent different cycles of centrifugation and redispersion to remove the surfactant. Finally, silica particles containing magnetite NPs (Fe\({}_{3}\)O\({}_{4}\)@silica) were dried under vacuum. The average size of silica particles was 25.6 nm; TEM image analysis revealed that each silica shell hosts 3-4 magnetite NPs [11]. The appearance of the NC containing Fe\({}_{3}\)O\({}_{4}\)@silica NPs is shown in Figure 1 (top right). Although the hysteretic properties of this NC are generally similar to the ones measured in the NC hosting bare Fe\({}_{3}\)O\({}_{4}\) NPs, both the coercive field at low temperatures and the fictive dipolar interaction temperature T* are much higher (Figure 1, bottom right and 2, bottom panel). This behavior is consistent with an increase in the interparticle interactions in the NC. It should be remarked that the T*/T curve of Fe\({}_{3}\)O\({}_{4}\)@silica nanopowder and of the associated NC is identical (Figure 2, bottom panel), indicating that the strength of interaction among NPs is basically the same; in fact, the dominant interparticle interaction is the one among the magnetite NPs hosted by the same silica shell, while interaction among different silica particles is much smaller. Both NCs containing almost unimodal, pure-Fe\({}_{3}\)O\({}_{4}\) NPs (either bare or surrounded by the silica shell) are thus apt to be studied through a model, which applies to independent (non-interacting) magnetic units. In one case, these units are coincident with the bare NPs themselves; in the other case, the magnetic units are silica particles containing a small number of interacting magnetite NPs. Figure 2: Top panels: Scaling laws in the polymer NC containing bare Fe\({}_{3}\)O\({}_{4}\) NPs. **Left**: standard superparamagnetic (SP) scaling; **right**: interacting superparamagnetic (ISP) scaling; **bottom**: temperature behavior of the ratio T*/T in the same system. The model, based upon a rate-equation approach to ordering/disordering of double-well systems [27], allows one to evaluate the size distribution \(p(D)\) of a system of non-interacting NPs starting from the ZFC curve:  
\[p(D)=\frac{\pi}{k_{B}}\left(\frac{K_{eff}}{25}\right)^{2}D^{2}\frac{1}{M_{s}^{2 }H}\left\{M^{\prime}_{ZFC}+\frac{1}{T}\left[M_{ZFC}-\frac{M_{s}^{2}H}{4K_{eff}} \right]\right\} \tag{1}\]  
where \(K_{eff}\) is an effective magnetic anisotropy constant (considered to be temperature-independent over narrow temperature intervals); \(M_{ZFC}\), \(M^{\prime}_{ZFC}\) are the ZFC curve and its derivative with respect to temperature; \(M_{s}\) is the NP saturation magnetization; and \(H\) is the measurement field. An example of \(p(D)\) curves resulting from the application of Equation (1) to the experimental ZFC curves of both NCs (shown in Figure 3, top panel) is given in the middle panel of Figure 3.  
Figure 3: (**Top panel**) experimental ZFC magnetization curves of the Fe\({}_{3}\)O\({}_{4}\) and Fe\({}_{3}\)O\({}_{4}\)@silica NCs; (**center panel**) corresponding normalized \(p(D)\) curves obtained from Equation (1); (**bottom panel**) corresponding volume distribution functions \(p(V)\). \(<\)\(V\)\(>\) is the resulting mean value of magnetic-unit volume. The mean size of bare magnetic NPs is in good agreement with the TEM data; instead, the mean volume of magnetic units in the Fe\({}_{3}\)O\({}_{4}\)@silica NC appears to be 3.6 times larger than the mean volume of bare Fe\({}_{3}\)O\({}_{4}\) NPs. This is in very good agreement with the number of magnetite NPs per silica particle estimated from TEM image analysis (3-4 magnetite NPs). The \(p(V)\) curve shown in Figure 3 (bottom panel) is directly obtained from the \(p(D)\) function through the relation \(p(V)=\frac{1}{3}\big{(}\frac{6}{\pi}\big{)}^{3}V^{-\frac{2}{3}}p(D\to V)\), which applies to spheroidal particles of diameter \(D\) and volume \(V=\frac{\pi}{6}D^{3}\).