Effects of interactions on magnetization relaxation dynamics in ferrofluids

The dynamics of magnetization relaxation in ferrofluids are studied with statistical-mechanical theory and Brownian dynamics simulations. The particle dipole moments are initially perfectly aligned, and the magnetization is equal to its saturation value. The magnetization is then allowed to decay under zero-field conditions toward its equilibrium value of zero. The time dependence is predicted by solving the Fokker-Planck equation for the one-particle orientational distribution function. Interactions between particles are included by introducing an effective magnetic field acting on a given particle and arising from all of the other particles. Two different approximations are proposed and tested against simulations: a first-order modified mean-field theory and a modified Weiss model. The theory predicts that the short-time decay is characterized by the Brownian rotation time ${\tau }_B$, independent of the interaction strength. At times much longer than ${\tau }_B$, the asymptotic decay time is predicted to grow with increasing interaction strength. These predictions are borne out by the simulations. The modified Weiss model gives the best agreement with simulation, and its range of validity is limited to moderate, but realistic, values of the dipolar coupling constant.

Figure 1

The fractional magnetization as a function of time in systems with $\phi =0.125$: (a) ${\chi }_L=1$ and $\lambda =1$; (b) ${\chi }_L=2$ and $\lambda =2$; (c) ${\chi }_L=3$ and $\lambda =3$; and (d) ${\chi }_L=4$ and $\lambda =4$. The points are from simulations: black circles are for noninteracting particles; red squares are from method A; green diamonds are from method B; and blue up-triangles are from method C. The lines are from theory: the dashed lines are for noninteracting particles [Eq. (6)], and they lie perfectly on top of the simulation points; the dotted lines are from MMF theory for interacting particles [Eq. (15)]; and the solid lines are from MW theory for interacting particles [Eq. (24)].

Figure 2

The fractional magnetization as a function of time in systems with $\lambda =1$: (a) ${\chi }_L=1$ and $\phi =0.125$; (b) ${\chi }_L=2$ and $\phi =0.250$; (c) ${\chi }_L=3$ and $\phi =0.375$; and (d) ${\chi }_L=4$ and $\phi =0.500$. The points are from simulations: red squares are from method A and green diamonds are from method B. The lines are from theory: the dashed lines are for noninteracting particles [Eq. (6)]; the dotted lines are from MMF theory for interacting particles [Eq. (15)]; and the solid lines are from MW theory for interacting particles [Eq. (24)].

Figure 3

Snapshots of the initial configurations of systems with $\phi =0.010$ from method A [(a), (c), and (e)] and method C [(b), (d), and (f)]: (a) $\lambda =2$, method A; (b) $\lambda =2$, method C; (c) $\lambda =4$, method A; (d) $\lambda =4$, method C; (e) $\lambda =8$, method A; and (f) $\lambda =8$, method C. In each case, the particle dipole moments relax from perfect alignment in the vertical direction.

Figure 4

Equilibrium cluster-size distributions before relaxation in the systems with $\phi =0.010$ from method A (a) and method C (b). The mean values $\langle n\rangle$ are shown for $\lambda =1$, 2, 3, and 4. Results for $\lambda =8$ are omitted, because the clusters become too large, but the mean cluster sizes are $\langle n\rangle \simeq 90$ (method A) and $\langle n\rangle \simeq 110$ (method C).

Figure 5

The fractional magnetization as a function of time in systems with $\phi =0.010$: (a) ${\chi }_L=0.08$ and $\lambda =1$; (b) ${\chi }_L=0.16$ and $\lambda =2$; (c) ${\chi }_L=0.32$ and $\lambda =4$; and (d) ${\chi }_L=0.64$ and $\lambda =8$. The points are from simulations: red squares are from method A and blue up-triangles are from method C. The lines are from theory: the dashed lines are for noninteracting particles [Eq. (6)]; the dotted lines are from MMF theory for interacting particles [Eq. (15)]; and the solid lines are from MW theory for interacting particles [Eq. (24)]. The dotted and solid lines (MMF and MW theories, respectively) are indistinguishable on the scale of the plot.

Figure 6

Instantaneous decays times, as defined in Eq. (27), for systems with $\lambda =1$ and $\phi =0.125$, 0.250, 0.375, and 0.500. The points are from simulations with method A. The dashed lines are obtained from fits to the simulation data for $t\ge {\tau }_B$. The solid lines are from Eqs. (24) and (27).

Figure 7

Instantaneous decays times, as defined in Eq. (27): [(a) and (b)] systems with $\phi =0.125$ and $\lambda =1$, 2, 3, and 4; [(c) and (d)] systems with $\phi =0.010$ and $\lambda =1$, 2, 3, 4, and 8. In (a) and (c) the points are from simulations with method A. In (b) and (d) the points are from simulations method C. The dashed lines are obtained from fits to the simulation data for $t\ge {\tau }_B$. The solid lines are from Eqs. (24) and (27).