Dynamic magnetic response of a ferrofluid in a static uniform magnetic field

A theory for the frequency-dependent magnetic susceptibility of a ferrofluid in a static uniform magnetic field is developed, including the dipolar interactions between the constituent particles. Interactions are included within the framework of modified mean-field theory. Predictions are given for the linear responses of the magnetization to a probing ac field both parallel and perpendicular to the static field and are tested against results from Brownian dynamics simulations. The effects of the particle concentration and dipolar coupling constant on the field-dependent static susceptibilities and the frequency dispersions are shown to be substantial, which justifies taking proper account of the interactions between particles. The theory is reliable provided that the volume concentration and dipolar coupling constant are not too large and within the range of values for real ferrofluids.

Figure 1

Field-dependent properties of the one-particle model describing an ideal superparamagnetic gas with Langevin parameter ${\alpha }_z$. (a) $M_z/M_z\left(\infty \right)$ is the fractional magnetization in the $z$ direction. The solid line is the exact result [Eq. (6)] and the symbols are from BD simulations. (b) ${\chi }_{xy}^{\mathrm{id}}\left(0\right)$ and (c) ${\chi }_z^{\mathrm{id}}\left(0\right)$ are the field-dependent static susceptibilities in the $xy$ and $z$ directions, respectively. Solid lines are the exact results [Eqs. (8) and (9)], dashed lines are the approximate results [Eqs. (20) and (42)], and symbols are from BD simulations. In (b), the exact and approximate results are indistinguishable on the scale of the graph. (d) ${\mathrm{\Omega }}_{xy}^{\mathrm{id}}$ and ${\mathrm{\Omega }}_z^{\mathrm{id}}$ are the peak frequencies in the imaginary parts of the susceptibility spectra in the $xy$ and $z$ directions, respectively. ${\tau }_{\mathrm{B}}$ is the Brownian rotation time, (12). Solid lines are ${\mathrm{\Omega }}_{\parallel }{\tau }_{\mathrm{B}}={\tau }_{\mathrm{B}}/{\tau }_{\parallel }$ and ${\mathrm{\Omega }}_{\perp }{\tau }_{\mathrm{B}}={\tau }_{\mathrm{B}}/{\tau }_{\perp }$ from Eqs. (13) and (35), respectively; dashed lines are the approximate results [by numerical differentiation of Eqs. (19) and (41)]; and symbols are from BD simulations.

Figure 2

Magnetization curves for ferrofluids with (a) $\lambda =1.0$ and (b) $\lambda =2.5$ and with concentrations $0.0\le {\rho }^{*}\le 0.5$. Lines are from MMF1 theory, (31), and symbols from BD simulations.

Figure 3

Static field-dependent susceptibilities of ferrofluids with (a, b) $\lambda =1.0$ and (c, d) $\lambda =2.5$ and with concentrations $0.0\le {\rho }^{*}\le 0.5$: (a, c) ${\chi }_{xy}\left(0\right)$; (b, d) ${\chi }_z\left(0\right)$. Results are shown divided by the respective ideal-gas susceptibilities ${\chi }_{xy}^{\mathrm{id}}\left(0\right)$ and ${\chi }_z^{\mathrm{id}}\left(0\right)$. Lines are from theory [(32), (51)], and symbols from BD simulations.

Figure 4

Static concentration-dependent susceptibilities of ferrofluids with (a, b) $\lambda =1.0$ and (c, d) $\lambda =2.5$ and with Langevin parameters $0\le {\alpha }_z\le 5$: (a, c) ${\chi }_{xy}\left(0\right)$; (b, d) ${\chi }_z\left(0\right)$. Results are shown divided by the respective ideal-gas susceptibilities ${\chi }_{xy}^{\mathrm{id}}\left(0\right)$ and ${\chi }_z^{\mathrm{id}}\left(0\right)$. Lines are from theory [(32), (51)], and symbols from BD simulations.

Figure 5

Frequency-dependent susceptibilities of a ferrofluid with $\lambda =1.0$ and ${\rho }^{*}=0.1$. Real parts of (a) ${\chi }_{xy}\left(\omega \right)$ and (b) ${\chi }_z\left(\omega \right)$. Imaginary parts of (c) ${\chi }_{xy}\left(\omega \right)$ and (d) ${\chi }_z\left(\omega \right)$. Lines are from theory [(30), (50)], and symbols from BD simulations.

Figure 6

Frequency-dependent susceptibilities of a ferrofluid with $\lambda =1.0$ and ${\rho }^{*}=0.5$. Real parts of (a) ${\chi }_{xy}\left(\omega \right)$ and (b) ${\chi }_z\left(\omega \right)$. Imaginary parts of (c) ${\chi }_{xy}\left(\omega \right)$ and (d) ${\chi }_z\left(\omega \right)$. Lines are from theory [(30), (50)], and symbols from BD simulations.

Figure 7

Frequency-dependent susceptibilities of a ferrofluid with $\lambda =2.5$ and ${\rho }^{*}=0.1$. Real parts of (a) ${\chi }_{xy}\left(\omega \right)$ and (b) ${\chi }_z\left(\omega \right)$. Imaginary parts of (c) ${\chi }_{xy}\left(\omega \right)$ and (d) ${\chi }_z\left(\omega \right)$. Lines are from theory [(30), (50)], and symbols from BD simulations.

Figure 8

Frequency-dependent susceptibilities for a ferrofluid with $\lambda =2.5$ and ${\rho }^{*}=0.5$. Real parts of (a) ${\chi }_{xy}\left(\omega \right)$ and (b) ${\chi }_z\left(\omega \right)$. Imaginary parts of (c) ${\chi }_{xy}\left(\omega \right)$ and (d) ${\chi }_z\left(\omega \right)$. Lines are from theory [(30), (50)], and symbols from BD simulations.

Figure 9

Field-dependent peak frequencies of ferrofluids with (a, b) $\lambda =1.0$ and (c, d) $\lambda =2.5$ and with concentrations $0.0\le {\rho }^{*}\le 0.5$: (a, c) ${\mathrm{\Omega }}_{xy}$; (b, d) ${\mathrm{\Omega }}_z$. Results are shown divided by the respective ideal-gas peak frequencies ${\mathrm{\Omega }}_{xy}^{\mathrm{id}}$ and ${\mathrm{\Omega }}_z^{\mathrm{id}}$. Lines are from theory, and symbols from BD simulations.

Figure 10

Concentration-dependent peak frequencies of ferrofluids with (a, b) $\lambda =1.0$ and (c, d) $\lambda =2.5$ and with Langevin parameters $0\le {\alpha }_z\le 5$: (a, c) ${\mathrm{\Omega }}_{xy}$; (b, d) ${\mathrm{\Omega }}_z$. Results are shown divided by the respective ideal-gas susceptibilities ${\mathrm{\Omega }}_{xy}^{\mathrm{id}}$ and ${\mathrm{\Omega }}_z^{\mathrm{id}}$. Lines are from theory, and symbols from BD simulations.