Computer simulations of the structure of colloidal ferrofluids

The structure of a ferrofluid under the influence of an external magnetic field is expected to become anisotropic due to the alignment of the dipoles into the direction of the external field, and subsequently to the formation of particle chains due to the attractive head to tail orientations of the ferrofluid particles. Knowledge about the structure of a colloidal ferrofluid can be inferred from scattering data via the measurement of structure factors. We have used molecular-dynamics simulations to investigate the structure of both monodispersed and polydispersed ferrofluids. The results for the isotropic structure factor for monodispersed samples are similar to previous data by Camp and Patey that were obtained using an alternative Monte Carlo simulation technique, but in a different parameter region. Here we look in addition at bidispersed samples and compute the anisotropic structure factor by projecting the $q$ vector onto the $XY$ and $XZ$ planes separately, when the magnetic field was applied along the $z$ axis. We observe that the $XY$-plane structure factor as well as the pair distribution functions are quite different from those obtained for the $XZ$ plane. Further, the two-dimensional structure factor patterns are investigated for both monodispersed and bidispersed samples under different conditions. In addition, we look at the scaling exponents of structure factors. Our results should be of value to interpret scattering data on ferrofluids obtained under the influence of an external field.

Figure 1

(Color online) Monodispersed case: Structure factor $S\left(q_{xyz}\right)$ for (a) ${\chi }_L=0.4\pi$ and (b) ${\chi }_L=1.6\pi$ as a function of $\lambda$ at zero field $(\alpha =0)$. It is worth noting that all the quantities plotted in Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are dimensionless naturally [e.g., $S\left(q\right)$ and $g\left(r\right)$], or that they were already normalized to be dimensionless (e.g., $r$, $q$, $x$, and $y$) in the text.

Figure 2

(Color online) Monodispersed case: Ratio of the nonzero-field structure factor to the zero-field (a,b) ${\phantom{\mid }S\left(q_{xy}\right)\mid }_{\alpha =9}∕{\phantom{\mid }S\left(q_{xy}\right)\mid }_{\alpha =0}$ and (c,d) ${\phantom{\mid }S\left(q_{xz}\right)\mid }_{\alpha =9}∕{\phantom{\mid }S\left(q_{xz}\right)\mid }_{\alpha =0}$, for (a,c) ${\chi }_L=0.4\pi$ and (b,d) ${\chi }_L=1.6\pi$ as a function of $\lambda$. In (c,d), the solid (or long-dashed) line is a guide for the eye for $\lambda =4$ (or $\lambda =8$).

Figure 3

(Color online) Monodispersed case: Pair distribution function (a,b) $g\left(r_{xy}\right)$ and (c,d) $g\left(r_{xz}\right)$ for (a,c) ${\chi }_L=0.4\pi$ and (b,d) ${\chi }_L=1.6\pi$ as a function of $\lambda$ at nonzero field $(\alpha =9)$. For clarity, the interesting parts of the figures are enlarged in the insets, respectively.

Figure 4

Projection of particle locations onto the $XY$ plane, to account for the depletion observed for $\lambda =8$ in Fig. 3a.

Figure 5

(Color online) Monodispersed case: Log-log plots of the structure factors, (a) $S\left(q_{xyz}\right)$, (b) $S\left(q_{xy}\right)$, and (c) $S\left(q_{xz}\right)$, for (a) zero field $(\alpha =0)$ and (b,c) nonzero field $(\alpha =9)$ as a function of $\lambda$ and/or ${\chi }_L$. The curves denote the fitting: (a) dotted line for $\lambda =8$ and $\rho {\sigma }^3=0.0375$ $({\chi }_L=0.4\pi )$; dashed line for $\lambda =8$ and $\rho {\sigma }^3=0.15$ $({\chi }_L=1.6\pi )$; (b) solid line for $\lambda =4$ and $\rho {\sigma }^3=0.075$ $({\chi }_L=0.4\pi )$; dotted line for $\lambda =8$ and $\rho {\sigma }^3=0.0375$ $({\chi }_L=0.4\pi )$; dashed line for $\lambda =8$ and $\rho {\sigma }^3=0.15$ $({\chi }_L=1.6\pi )$. The corresponding scaling exponents $\nu$ are indicated in the figure.

Figure 6

Monodispersed case: 2D structure factor $S(q_{\perp },q_{\Vert })$ of monodispersed cases, with $-\pi ⩽q_{\perp }⩽\pi$ and $-\pi ⩽q_{\Vert }⩽\pi$, with an applied magnetic field perpendicular to the horizontal axis. From left to right panels: $\alpha =0,9$. From upper to lower panels: $\lambda =4,8$ (or $\rho =0.075,0.0375$). Bright (dark) regions indicate high (low) intensities.

Figure 7

(Color online) Bidispersed case: Structure factor (a,b) $S\left(q_{xy}\right)$ and (c,d) $S\left(q_{xz}\right)$ as a function of the volume fraction of large particles ${\varphi }_L$, at (a,c) zero field and (b,d) nonzero field. The total volume fraction of the large and small particles is fixed to $\varphi =0.07$. The number density $\rho =0.1237,0.1051,0.0615,0.0326$ corresponds to the system of the volume fraction of the large particles ${\varphi }_L=0.007,0.02,0.05,0.07$, respectively.

Figure 8

(Color online) Bidispersed case: Same as Fig. 7, but for the pair distribution function (a,b) $g\left(r_{xy}\right)$ and (c,d) $g\left(r_{xz}\right)$.

Figure 9

Projection of particle locations onto the $XY$ plane, to account for the depletion observed for $\rho =0.0326$ (or ${\varphi }_L=0.07$) in Fig. 8b.

Figure 10

Bidispersed case: 2D structure factor $S(q_{\perp },q_{\Vert })$ of bidispersed cases, with $-\pi ⩽q_{\perp }⩽\pi$ and $-\pi ⩽q_{\Vert }⩽\pi$, with an applied magnetic field perpendicular to the horizontal axis, at a fixed total volume fraction of both small and large particles $\varphi =0.07$. From left to right panels: $\alpha =0,2.9,6.1,8.9$. From top to bottom panels: volume fraction of large particles ${\varphi }_L=0.007,0.02,0.05,0.07$.