Magnetic properties of polydisperse ferrofluids: A critical comparison between experiment, theory, and computer simulation

Experimental magnetization curves for a polydisperse ferrofluid at various concentrations are examined using analytical theories and computer simulations with the aim of establishing a robust method for obtaining the magnetic-core diameter distribution function $p\left(x\right)$. Theoretical expressions are fitted to the experimental data to yield the parameters of $p\left(x\right)$. It is shown that the majority of available theories yield results that depend strongly on the ferrofluid concentration, even though the magnetic composition should be fixed. The sole exception is the second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)] which yields consistent results over the full experimental range of ferrofluid concentration. To check for consistency, extensive molecular dynamics and Monte Carlo simulations are performed on systems with discretized versions of $p\left(x\right)$ corresponding as closely as possible to that of the real ferrofluid. Essentially perfect agreement between experiment, theory, and computer simulation is demonstrated. In addition, the MMF2 theory provides excellent predictions for the initial susceptibility measured in simulations.

Figure 1

Magnetic-core diameter distribution functions: $p\left(x\right)$ (3) with $a=4.9518$ and $x_0=1.2266\mathrm{nm}$ (line); $p_1\left(x\right)$ for configuration 1 ($N=1000$ particles) with $1.5⩽x⩽16.4\mathrm{nm}$ and $\Delta x=0.1\mathrm{nm}$ (impulses).

Figure 2

Magnetic-core diameter distribution functions: $p\left(x\right)$ (3) with $a=4.9518$ and $x_0=1.2266\mathrm{nm}$ (line); $p_2\left(x\right)$ for configuration 2 ($N=500$ particles) with $\Delta x=2x_0$ (bars). Also shown are the numbers of particles in each fraction (with magnetic core diameter $x∕x_0=1,3,5,\dots ,17$).

Figure 3

(Color online) Magnetic-core diameter distributions obtained from the apparent third and sixth moments extracted from experimental data at high concentration $(M_{\infty }=57.0\mathrm{kA}{\mathrm{m}}^{-1})$ using various theories.

Figure 4

(Color online) Magnetic-core diameter distributions obtained from the apparent third and sixth moments extracted from experimental data at low concentration $(M_{\infty }=5.0\mathrm{kA}{\mathrm{m}}^{-1})$ using various theories. The results from Weiss, MSA, HTA, MMF1, and MMF2 theories are identical.

Figure 5

(Color online) Low-field portions of the magnetization curves, $M\left(H\right)$, for ferrofluids at different concentrations with saturation magnetizations (from top to bottom) $M_{\infty }=57.0$, 37.8, 25.3, 16.9, 11.2, 7.8, and $5.0\mathrm{kA}{\mathrm{m}}^{-1}$. Filled circles are from experiment [8], open circles are from MD simulations with $p_1\left(x\right)$, solid lines are from MMF2 theory with $p\left(x\right)$, and dashed lines are from MMF2 theory with $p_1\left(x\right)$.

Figure 6

(Color online) Low-field portions of the magnetization curves, $M\left(H\right)$, for ferrofluids at different concentrations with saturation magnetizations (from top to bottom) $M_{\infty }=57.0$, 37.8, 25.3, 16.9, 11.2, 7.8, and $5.0\mathrm{kA}{\mathrm{m}}^{-1}$. Filled circles are from experiment [8], open circles are from MC simulations with $p_2\left(x\right)$, open triangles are from MD simulation with $p_2\left(x\right)$, solid lines are from MMF2 theory with $p\left(x\right)$, and dashed lines (almost coincident with solid lines) are from MMF2 theory with $p_2\left(x\right)$. The statistical uncertainties in the simulation points are smaller than the symbol size.

Figure 7

(Color online) Complete magnetization curves, $M\left(H\right)$, for ferrofluids at different concentrations with saturation magnetizations (from top to bottom) $M_{\infty }=57.0$, 37.8, 25.3, $16.9\mathrm{kA}{\mathrm{m}}^{-1}$, 11.2, 7.8, and $5.0\mathrm{kA}{\mathrm{m}}^{-1}$. Filled circles are from experiment [8], open circles are from MC simulations with $p_2\left(x\right)$, open triangles are from MD simulation with $p_2\left(x\right)$, solid lines are from MMF2 theory with $p\left(x\right)$, and dashed lines (almost coincident with solid lines) are from MMF2 theory with $p_2\left(x\right)$. The statistical uncertainties in the simulation points are smaller than the symbol size.

Figure 8

Initial susceptibility $\chi$ vs Langevin susceptibility ${\chi }_L$ from the Langevin theory (dot double-dashed line), Weiss theory (dot-dashed line), MSA (dashed line), HTA/MMF1 theories (dotted line), MMF2/CE theories (solid line), and MC simulations with $p_2\left(x\right)$ (open circles). The simulation points correspond to systems with saturation magnetizations of, from left to right, $M_{\infty }=5.0$, 7.8, 11.2, 16.9, 25.3, 37.8, and $57.0\mathrm{kA}{\mathrm{m}}^{-1}$. The statistical uncertainties in the simulation points are smaller than the symbol size.