How chains and rings affect the dynamic magnetic susceptibility of a highly clustered ferrofluid

The dynamic magnetic susceptibility, $\chi \left(\omega \right)$, of a model ferrofluid at a very low concentration (volume fraction, approximately $0.05\%$), and with a range of dipolar coupling constants ($1\le \lambda \le 8$), is examined using Brownian dynamics simulations. With increasing $\lambda$, the structural motifs in the system change from unclustered particles, through chains, to rings. This gives rise to a nonmonotonic dependence of the static susceptibility $\chi \left(0\right)$ on $\lambda$ and qualitative changes to the frequency spectrum. The behavior of $\chi \left(0\right)$ is already understood, and the simulation results are compared to an existing theory. The single-particle rotational dynamics are characterized by the Brownian time, ${\tau }_B$, which depends on the particle size, carrier-liquid viscosity, and temperature. With $\lambda \le 5.5$, the imaginary part of the spectrum, ${\chi }^{\prime \prime }\left(\omega \right)$, shows a single peak near $\omega \sim {\tau }_B^{-1}$, characteristic of single particles. With $\lambda \ge 5.75$, the spectrum is dominated by the low-frequency response of chains. With $\lambda \ge 7$, new features appear at high frequency, which correspond to intracluster motions of dipoles within chains and rings. The peak frequency corresponding to these intracluster motions can be computed accurately using a simple theory.

Figure 1

(a) The static susceptibility $\chi \left(0\right)$ divided by the Langevin susceptibility ${\chi }_{\mathrm{L}}$ as a function of the dipolar coupling constant $\lambda$, for fluids with the reduced concentration $\rho {\sigma }^3=0.001$. The dashed black line represents noninteracting particles, the solid red line is from Eq. (12), and the symbols are from BD simulations with the nominal and effective DHS values of the dipolar coupling constant. (b) The fractions of particles with zero, one, two, three, and four nearest neighbors as a function of $\lambda$. In (a) and (b), the dotted black lines show the approximate boundaries between states where the structure is primarily unclustered particles, chains, and rings.

Figure 2

Simulation snapshots for systems with the reduced concentration $\rho {\sigma }^3=0.001$ and dipolar coupling constants (a) $\lambda =5$, (b) $\lambda =6$, (c) $\lambda =7$, and (d) $\lambda =8$.

Figure 3

The magnetization autocorrelation function $C\left(t\right)$ for fluids with the reduced concentration $\rho {\sigma }^3=0.001$: (a) $\lambda =5$; (b) $\lambda =6$; (c) $\lambda =7$; (d) $\lambda =8$. The solid red line shows the raw BD simulation data, and the dashed black line is a fit using Eq. (5).

Figure 4

The dynamic susceptibility $\chi \left(\omega \right)$ divided by the static susceptibility $\chi \left(0\right)$ for fluids with the reduced concentration $\rho {\sigma }^3=0.001$: (a) $\lambda =5$; (b) $\lambda =6$; (c) $\lambda =7$; (d) $\lambda =8$. The solid black line represents the real part ${\chi }^{\prime }\left(\omega \right)$, and the dashed red line the imaginary part ${\chi }^{\prime \prime }\left(\omega \right)$. In (d), the dotted green line and the dot-dashed blue line show, respectively, the real and imaginary parts of the Debye function with effective relaxation time ${\tau }_{\mathrm{eff}}=0.0510{\tau }_B$, scaled to give the correct peak height in ${\chi }^{\prime \prime }$.

Figure 5

Heat map showing ${\chi }^{\prime \prime }\left(\omega \right)/{\chi }_{\max }^{\prime \prime }$ from BD simulations, as a function of $\lambda$ and ${log}_{10}\omega {\tau }_B$ for fluids with the reduced concentration $\rho {\sigma }^3=0.001$. ${\chi }_{\max }^{\prime \prime }$ is the maximum value of ${\chi }^{\prime \prime }$ for a given value of $\lambda$.

Figure 6

Peak positions (a) and peak heights (b) in ${\chi }^{\prime \prime }\left(\omega \right)/\chi \left(0\right)$ for fluids with the reduced concentration $\rho {\sigma }^3=0.001$. The peaks are separated into three arbitrarily labeled groups (1, 2, and 3), with peak 1 representing motions of unclustered particles, peak 2 representing motions of chains, and peak 3 representing motions within chains and rings. In (a), the solid blue and dashed magenta lines represent, respectively, the theory/BD and theory/chain predictions given by Eq. (26), as detailed in Sec. 3c. In (b), the peak heights ${\chi }_{\max }^{\prime \prime }$ are plotted relative to the static susceptibility $\chi \left(0\right)$.