Magnetization relaxation dynamics in polydisperse ferrofluids

When a ferrofluid is magnetized in a strong magnetic field, and then the field is switched off, the magnetization decays from its saturation value to zero. The dynamics of this process are controlled by the rotations of the constituent magnetic nanoparticles, and for the Brownian mechanism, the respective rotation times are strongly influenced by the particle size and the magnetic dipole-dipole interactions between the particles. In this work, the effects of polydispersity and interactions on the magnetic relaxation are studied using a combination of analytical theory and Brownian dynamics simulations. The theory is based on the Fokker-Planck-Brown equation for Brownian rotation and includes a self-consistent, mean-field treatment of the dipole-dipole interactions. The most interesting predictions from the theory are that, at short times, the relaxation of each particle type is equal to its intrinsic Brownian rotation time, while at long times, each particle type has the same effective relaxation time, which is longer than any of the individual Brownian rotation times. Noninteracting particles, though, always relax at a rate controlled only by the Brownian rotation times. This illustrates the importance of including the effects of polydispersity and interactions when analyzing the results from magnetic relaxometry experiments on real ferrofluids, which are rarely monodisperse.

Figure 1

Magnetic relaxation curves for bidisperse configuration A. The points are from BD simulations, the dashed lines are for noninteracting particles, and the solid lines are from the Weiss theory: red circles and lines, small particles ($k=1$); blue diamonds and lines, large particles ($k=2$); black triangles and lines, all particles.

Figure 2

Magnetic relaxation curves for bidisperse configuration B. The points are from BD simulations, the dashed lines are for noninteracting particles, and the solid lines are from the Weiss theory: red circles and lines, small particles ($k=1$); blue diamonds and lines, large particles ($k=2$); black triangles and lines, all particles. The green dot-dashed lines are fits to the data in the range $t\ge 5\langle \tau \rangle$.

Figure 3

Magnetic relaxation curves for tridisperse configuration C. The points are from BD simulations, the dashed lines are for noninteracting particles, and the solid lines are from the Weiss theory: red circles and lines, small particles ($k=1$); green squares and lines, medium particles ($k=2$); blue diamonds and lines, large particles ($k=3$); black triangles and lines, all particles.

Figure 4

Magnetic relaxation curves for bidisperse configuration D. The points are from BD simulations, the dashed lines are for noninteracting particles, and the solid lines are from the Weiss theory: red circles and lines, small particles ($k=1$); blue diamonds and lines, large particles ($k=2$); black triangles and lines, all particles.

Figure 5

Effective relaxation times from the Weiss theory for (a) bidisperse configuration A, (b) bidisperse configuration B, (c) tridisperse configuration C, and (d) bidisperse configuration D: red dotted lines, small particles; green dot-dashed lines, medium particles; blue dashed lines, large particles; black solid lines, all particles.

Figure 6

Magnetic relaxation curves (a) and effective relaxation times (b) from the Weiss theory for configuration E: red dotted lines, small particles ($k=1$); blue dashed lines, large particles ($k=2$); black solid lines, all particles.

Figure 7

Ratios of the magnetization of different fractions in (a) bidisperse configuration A, (b) bidisperse configuration B, (c) tridisperse configuration C, and (d) bidisperse configuration D. The points are from BD simulations, the dashed lines are for noninteracting particles, and the solid lines are from the Weiss theory. The small/large and small/medium ratios are shown as red circles and red lines, the medium/large ratio in (c) is shown as green squares and green lines, and the total/large ratios are shown as blue diamonds and blue lines. For each configuration, the theoretical and simulation results for the total/large ratio are in good agreement within the estimated errors, and so the points and lines overlap.