Dipolar interaction and demagnetizing effects in magnetic nanoparticle dispersions: Introducing the mean-field interacting superparamagnet model

Aiming to analyze relevant aspects of interacting magnetic nanoparticle systems (frequently called interacting superparamagnets), a model is built from magnetic dipolar interaction and demagnetizing mean-field concepts. By making reasonable simplifying approximations, a simple and useful expression for effective demagnetizing factors is achieved, which allows the analysis of uniform and nonuniform spatial distributions of nanoparticles, in particular the occurrence of clustering. This expression is a function of demagnetizing factors associated with specimen shape and clusters shape, and of the mean distances between near neighbor nanoparticles and between clusters, relative to the characteristic sizes of each of these two types of objects, respectively. The model explains effects of magnetic dipolar interactions, such as the observation of apparent nanoparticle magnetic moments smaller than real ones and approaching to zero as temperature decreases. It is shown that by performing a minimum set of experimental determinations along principal directions of geometrically well-defined specimens, model application allows retrieval of nanoparticle intrinsic properties, like mean volume, magnetic moment, and susceptibility in the absence of interactions. It also permits the estimation of mean interparticle and intercluster relative distances, as well as mean values of demagnetizing factors associated with clusters shape. An expression for average magnetic dipolar energy per nanoparticle is also derived, which is a function of specimen effective demagnetizing factor and magnetization. Experimental test of the model was performed by analysis of results reported in the literature and of original results reported here. The first case corresponds to oleic-acid-coated 8-nm magnetite particles dispersed in PEGDA-600 polymer, and the second one to polyacrilic-acid-coated 13-nm magnetite particles dispersed in PVA solutions from which ferrogels were later produced by a physical cross-linking route. In both cases, several specimens were studied covering a range of nanoparticle volume fractions between 0.002 and 0.046. Magnetic response is clearly different when prism-shaped specimens are measured along different principal directions. These results remark the importance of reporting complete information on measurement geometry when communicating magnetic measurement results of interacting magnetic nanoparticles. Intrinsic nanoparticle properties as well as structural information on particles spatial distribution were retrieved from our analysis in addition to, and in excellent agreement with, analysis previously performed by other authors and/or information obtained from FESEM images. In the studied samples, nanoparticles were found to be in close contact to each other within almost randomly oriented clusters. Intercluster mean distance, relative to cluster size, was found to vary between 2.2 and 7.5, depending on particles volume fraction.

Figure 1

Ratio ${\zeta }_u(\nu ,\theta )={\chi }_{u0}/\left(3{\chi }_L\right)$ for different orientation of easy axes relative to applied field direction. Shadowed regions correspond to $\nu \ll 1$, where the distribution of easy axes orientations is irrelevant, and to $\nu \gg 1$, where this distribution plays a crucial role.

Figure 2

Sketch of nanoparticles and clusters. (a) Left-top inset: NP shape (ellipse), sphere of diameter $D$ with same volume $V$ as NP (continuous contour), and sphere with diameter equal to mean near-neighbor distance $d=\gamma D$ (dotted contour, see also right-bottom inset). Main figure: random distribution of NPs in a nonmagnetic matrix. Packing fraction of dotted spheres is $\phi$. (b) Nonrandom distribution of NPs. Dotted spheres of diameter $D_c$ have same volume $V_c$ as clusters. Mean distance between near-neighbor clusters is $d_c={\gamma }_c{D}_c$. Dashed spheres diameter is $d_c$. Packing fraction of dashed spheres is ${\phi }_c$.

Figure 3

$\delta$ as a function of ${\gamma }_c=d_c/D_c$ for the case of spheres (a) and wires (b) and (c) with the orientation sketched in the inset.

Figure 4

(a) $N_x^E$, $N_z^E$ for the case of clusters preferentially oriented. Specimen $l_x$ and $l_y$ dimensions are identical. (b) $N_x^E$, $N_y^E$, and $N_z^E$ for a high aspect ratio specimen with clusters randomly oriented. In both examples the NPs relative distance parameter was set at $\gamma =1.5$.

Figure 5

Comparison of distribution functions $f$ and $g$ appearing in Eq. (16) for the case ${\chi }_u/{\kappa }_u=10/3$. Lognormal distribution has been used.

Figure 6

Dipolar energy per NP for specimens whose demagnetizing factors are illustrated in Figs. 4 and 4 (identified by the scripts a and b respectively). $x$ and $z$ identify the magnetization direction. A value of $M={10}^5$ A/m has been used for the calculation.

Figure 7

Relative interparticle distances, $\gamma$, and intercluster distances, ${\gamma }_c$, as a function of volume fraction $x_V$ for all PEG specimens listed in Table 1. The value corresponding to DP specimen is indicated by an arrow for completeness.

Figure 8

FESEM image of FG6 specimen. Clusters of NPs are indicated.

Figure 9

ZFC-FC-TRM curves from FG9aP1 specimen. Cooling part of ZFC and TRM measurements were performed under zero field. Warming part of ZFC and FC measurements were performed under a 8 kA/m field. Field was applied parallel to the longest ($l_x$) prism dimension. Inset shows a close look of ZFC-FC around irreversible temperature ($T_{\text{irr}}$) of dry specimen

Figure 10

$M\left(H^A\right)$ cycles for FG9P1 specimen at several temperatures. Field was applied parallel to the longest ($l_x$) prism dimension. Inset shows coercive field ($H_c$) vs. temperature.

Figure 11

Saturation magnetization $M^S$ versus temperature $T$ for FG9aP1 specimen. Dots were obtained from the analysis of Fig. 10 data. Line corresponds to fitting with $M^S\left(T\right)=A{(1-T/B)}^C$. The values obtained were $A=3.97\left(1\right)\times {10}^2$ kA/m, $B=350\left(8\right)$ K, and $C=0.100\left(7\right)$

Figure 12

$\rho =\langle {\mu }^2\rangle /{\langle \mu \rangle }^2$ versus temperature $T$ were $\mu$ is the NP magnetic moment. Dots were obtained from the analysis of Fig. 10 data. Line corresponds to fitting with a linear function $\rho =AT+B$. The values obtained were $A=3.1\left(1\right)\times {10}^{-3}{\text{K}}^{-1}$ and $B=0.97\left(1\right)$.

Figure 13

Inverse of apparent susceptibility $1/{\kappa }_x$, obtained from ZFC and FC measurements, as a function of $T/\rho {M^S}^2$. Straight line is the fit of the linear region (specimen magnetization in thermal equilibrium).

Figure 14

Apparent susceptibility ${\kappa }_x$ and corrected (true) susceptibility ${\chi }_x$. Vertical dash line corresponds to $T=T_{\mathrm{irr}}$, therefore correction is only reliable at $T\ge T_{\mathrm{irr}}$ (black symbols for ${\chi }_x$).

Figure 15

NP magnetic moment $\mu$: ${\mu }_a$ is the apparent NP moment, ${\mu }_{\chi }$ is the corrected NP moment obtained with the present model, and ${\mu }_{Ms}$ is the NP moments obtained through saturation magnetization data. Vertical dash line corresponds to $T=T_{\mathrm{irr}}$, therefore correction ${\mu }_{\chi }$ is only reliable at $T\ge T_{\mathrm{irr}}$ (black-star symbols).

Figure 16

$\sigma \left(H\right)$ cycles for specimen FG6P6 (raw data).

Figure 17

Linear region of $M\left(H\right)$ curves, measured in FG9aP1 specimen with the applied field along the three prism directions. Curves were normalized at high fields as described in the text.

Figure 18

Measured (apparent) susceptibilities ${\kappa }_u$ versus demagnetizing factors $N_{su}$ corresponding to specimen shape.

Figure 19

Relative distance parameters $\gamma$ and ${\gamma }_c$ versus NPs volume fraction $x_V$.

Figure 20

Cluster demagnetizing factors $N_{cu}$, as a function of NPs volume fraction $x_V$. Dashed line stands for the expected value for random cluster orientations, $N_{cu}=1/3$.

Figure 21

Linear part of room temperature $M$ vs. $H^E$ cycles for all of the specimens, after correcting by demagnetizing effects.