Behavior of bulky ferrofluids in the diluted low-coupling regime: Theory and simulation

A theoretical formalism to predict the structure factors observed in dipolar soft-sphere fluids based on a virial expansion of the radial distribution function is presented. The theory is able to account for cases with and without externally applied magnetic fields. A thorough comparison of the theoretical results to molecular-dynamics simulations shows a good agreement between theory and numerical simulations when the fraction of particles involved in clustering is low; i.e., the dipolar coupling parameter is $\lambda \lesssim 2$, and the volume fraction is $\phi \lesssim 0.25$. When magnetic fields are applied to the system, special attention is paid to the study of the anisotropy of the structure factor. The theory reasonably accounts for the structure factors when the Langevin parameter is smaller than 5.

Figure 1

Anisotropy of the pair distribution function in a magnetic field. The contour plots are presented for function (11) in the plane $O{r}_{\parallel }{r}_{\perp }$, where $r_{\parallel }$ and $r_{\perp }$ are the radius-vector components, parallel and perpendicular to the magnetic-field direction. The dipolar coupling constant is chosen as $\lambda =1$, and the volume fraction is $\phi =0.1$ for plots (a)–(c), and $\phi =0.2$ for plots (d)–(f). The values of the dimensionless magnetic field are $\alpha =0$ for plots (a) and (d), $\alpha =1$ for plots (b) and (e), and $\alpha =5$ for plots (c) and (f).

Figure 2

Simulation results for the fraction of particles in the system that is not forming clusters. We use the energy criteria at 70%; i.e., if the dipolar energy between two particles is larger than 70% of the maximum energy two particles can have at a distance equal to the diameter of one particle, then we say that the particles are linked and form a cluster. Plots (a), (b), and (c) show results for $\lambda =1$, 1.5, and 2, respectively. Different intensities of external magnetic field are shown ranging from zero $\alpha =0$ up to a value of the Langevin parameter (see text) of $\alpha =5$. As one would expect, by increasing either the dipolar coupling $\lambda$ or the magnetic field $\alpha$ the number of “free” particles decreases. In the most unfavorable case, around 22% of particles participate in the formation of clusters.

Figure 3

A comparison of the structure factor data obtained at $\phi =0.15$ with no applied field, one from the simulations (circles) and one from the analytical theory (solid line). The value of $\lambda$ changes from top to bottom from $\lambda =1$, to $\lambda =1.5$, and $\lambda =2$.

Figure 4

Same as in Fig. 3 but for volume fraction $\phi =0.25$.

Figure 5

A comparison between theory and simulations for the value of $q\sigma$ at which the main peak in the structure factor occurs as a function of the volume fraction $\phi$. The comparison is done for $\lambda =1$, 1.5, and 2 at zero field $\alpha =0$.

Figure 6

A comparison of the structure factors parallel to the field predicted by the theory and those obtained in numerical simulations is depicted for several volume fractions $\phi =0.05$, 0.15, and 0.25 at different strengths of the magnetic field $\alpha$ and dipolar coupling parameter $\lambda$. Plot (a) is for $\lambda =1$ and Langevin parameter $\alpha =1$; (b) is for $\lambda =1$ and $\alpha =5$; (c) is for $\lambda =2$ and $\alpha =5$.

Figure 7

Same as in Fig. 6 but for the component of the structure factor perpendicular to the magnetic field.

Figure 8

The perpendicular structure factor in the presence of a magnetic field for different values of the dipolar coupling parameter $\lambda =1$, 1.5, and 2 at several volume fractions and magnetic-field values: in plot (a) $\phi =0.05$ and $\alpha =1$; in plot (b) $\phi =0.05$ and $\alpha =5$; in plot (c) $\phi =0.250$ and $\alpha =5$.