Anisotropic self-diffusion in ferrofluids studied via Brownian dynamics simulations

The anisotropic self-diffusion in ferrofluids in the presence of a magnetic field is studied by Brownian dynamics simulations. It is found that the diffusion parallel to the magnetic field is hindered when compared to the case without field. Depending on the strength of dipolar interactions, the diffusion perpendicular to the applied field is either enhanced or hindered due to the field. The average diffusion coefficient is found to decrease with increasing concentration and dipolar interaction strength, but to be independent of the magnetic field strength for moderate dipolar interactions. Comparison to mean-field models show agreement for moderate dipolar interaction strengths but significant deviations in the regime of strong dipolar interactions.

Figure 1

(Color online) Snapshots of the configurations after equilibration. The volume fraction was chosen as $\varphi =0.02$, dipolar interaction parameter $\lambda =1$. A magnetic field was applied in the vertical direction. The dimensionless strength of the magnetic field was chosen $h=10$.

Figure 2

The Cartesian components of the mean-square displacement $⟨{\left(\Delta r_{\alpha }\right)}^2⟩∕d_{\mathrm{c}}^2$ as a function of reduced time. The upper two sets of data correspond to perpendicular displacements, while the lower one corresponds to parallel displacements with respect to the magnetic field. The volume fraction $\varphi =0.05$, dipolar interaction parameter $\lambda =4$, and Langevin parameter $h=20$ have been chosen.

Figure 3

The normalized average diffusion coefficient $\overline{D}∕D_{\mathrm{sp}}$ as a function of the volume fraction $\varphi$. Circles and squares correspond to Langevin parameters of $h=0$ and $h=20$, respectively. Solid, open, and shaded symbols correspond to $\lambda =1.0$, 2.0, and 4.0, respectively. The dashed line is the prediction of Ref. 8 for dipolar hard spheres.

Figure 4

The normalized average diffusion coefficient $\overline{D}∕D_{\mathrm{sp}}$ as a function of the dipolar interaction strength $\lambda$ defined in Eq. (3). Circles and squares correspond to $\varphi =0.02$ and $\varphi =0.05$, solid and open symbols to Langevin parameters of $h=0$ and $h=20$, respectively. The dashed line is the prediction of Ref. 8 for dipolar hard spheres.

Figure 5

The normalized average diffusion coefficient $\overline{D}∕D_{\mathrm{sp}}=(D_{\Vert }+2D_{\perp })∕3D_{\mathrm{sp}}$ is shown as a function of the Langevin parameter $h$. Circles, diamonds, and squares correspond to volume fractions of $\varphi =0.02$, 0.03, and 0.05, respectively. Solid, open, and shaded symbols correspond to $\lambda =1.0$, 2.0, and 4.0, respectively.

Figure 6

The anisotropy ratio $R=(D_{\Vert }-D_{\perp })∕(D_{\Vert }+2D_{\perp })$ of the diffusion coefficients as a function of the Langevin parameter $h$. The same symbols as in Fig. 5 are used. Solid lines are the predictions of the mean-field model, Eqs. (12, 13).

Figure 7

The normalized diffusion coefficients $[D_{\Vert ,\perp }\left(h\right)-D\left(0\right)]∕D\left(0\right)$ as a function of the orientational order parameter $L_2$ defined in the text. The dipolar interaction parameter was chosen as $\lambda =2$. Circles, diamonds, and squares correspond to volume fractions of $\varphi =0.02$, 0.03, and 0.05, respectively. Full symbols correspond to parallel diffusion coefficients; open symbols to perpendicular diffusion coefficients. Solid and dashed lines are the result of a linear fit.

Figure 8

The normalized diffusion coefficients $[D_{\Vert ,\perp }\left(h\right)-D\left(0\right)]∕D\left(0\right)$ for $h=20$ as a function of the dipolar interaction parameter $\lambda$ defined in Eq. (3). Circles and squares correspond to volume fractions of $\varphi =0.02$ and 0.05, respectively. Full symbols correspond to parallel diffusion coefficients; open symbols to perpendicular diffusion coefficients.

Figure 9

The average cluster size $S_{\mathrm{av}}$, defined in Eq. (15), as a function of the Langevin parameter $h$ for $\varphi =0.05$. Solid, open, and shaded circles correspond to $\lambda =1$, 2, and 4, respectively, while solid squares correspond to $\lambda =8$.