Modified mean-field theory of the magnetic properties of concentrated, high-susceptibility, polydisperse ferrofluids

The effects of particle-size polydispersity on the magnetostatic properties of concentrated ferrofluids are studied using theory and computer simulation. The second-order modified mean-field (MMF2) theory of Ivanov and Kuznetsova [Phys. Rev. E 64, 041405 (2001)] has been extended by calculating additional terms of higher order in the dipolar coupling constant in the expansions of the initial magnetic susceptibility and the magnetization curve. The theoretical predictions have been tested rigorously against results from Monte Carlo simulations of model monodisperse, bidisperse, and highly polydisperse ferrofluids. Comparisons have been made between systems with the same Langevin susceptibility and the same saturation magnetization. In all cases, the new theoretical magnetization curve shows better agreement with simulation data than does the MMF2 theory. As for the initial susceptibility, MMF2 theory is most accurate for the monodisperse model, while the new theory works best for polydisperse systems with a significant proportion of large particles. These results are important for the analysis and characterization of recently synthesized polydisperse ferrofluids with record-breaking values of the initial magnetic susceptibility.

Figure 1

Full magnetization curves of monodisperse [(a), (c), (e), and (g)] and polydisperse [(b), (d), (f), and (h)] ferrofluids at $T=295\text{K}$ and various concentrations: (a) and (b) $M\left(\infty \right)=25\text{kA}{\text{m}}^{-1}$; (c) and (d) $M\left(\infty \right)=50\text{kA}{\text{m}}^{-1}$; (e) and (f) $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$; and (g) and (h) $M\left(\infty \right)=100\text{kA}{\text{m}}^{-1}$. The points are from MC simulations, the solid lines are from MMF2 theory, and the dashed lines are from MMF2+ theory.

Figure 2

Low-field magnetization curves of monodisperse [(a), (c), (e), and (g)] and polydisperse [(b), (d), (f), and (h)] ferrofluids at $T=295\text{K}$ and various concentrations: (a) and (b) $M\left(\infty \right)=25\text{kA}{\text{m}}^{-1}$; (c) and (d) $M\left(\infty \right)=50\text{kA}{\text{m}}^{-1}$; (e) and (f) $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$; and (g) and (h) $M\left(\infty \right)=100\text{kA}{\text{m}}^{-1}$. The points are from MC simulations, the solid lines are from MMF2 theory, and the dashed lines are from MMF2+ theory.

Figure 3

Initial susceptibility as a function of temperature for monodisperse [(a), (c), (e), and (g)] and polydisperse [(b), (d), (f), and (h)] ferrofluids at various concentrations: (a) and (b) $M\left(\infty \right)=25\text{kA}{\text{m}}^{-1}$; (c) and (d) $M\left(\infty \right)=50\text{kA}{\text{m}}^{-1}$; (e) and (f) $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$; and (g) and (h) $M\left(\infty \right)=100\text{kA}{\text{m}}^{-1}$. The points are from MC simulations, the solid lines are from MMF2 theory, and the dashed lines are from MMF2+ theory.

Figure 4

The initial susceptibilities for (a) polydisperse and (b) bidisperse ferrofluids divided by those for monodisperse ferrofluids at various concentrations. The points are from MC simulations, the solid lines in (a) and (b) are from MMF2+ theory for the polydisperse/bidisperse ferrofluids and MMF2 theory for the monodisperse ferrofluids, and the dashed lines in (b) are from MMF2+ theory for all ferrofluids.

Figure 5

Simulation snapshots for the polydisperse ferrofluid in zero field and at $T=221.25\text{K}$: (first row) $M\left(\infty \right)=25\text{kA}{\text{m}}^{-1}$; (second row) $M\left(\infty \right)=50\text{kA}{\text{m}}^{-1}$; (third row) $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$; (fourth row) $M\left(\infty \right)=100\text{kA}{\text{m}}^{-1}$. The first column shows all of the particles, the second column shows the small particles, the third column shows the medium particles, and the fourth column shows the large particles.

Figure 6

Radial distribution functions of polydisperse ferrofluids at (a),(c),(e) $T=221.25\text{K;}$ and (b),(d),(f) $T=\infty$, and various concentrations: (a) and (b) small ($\mathrm{S}$) particles; (c) and (d) medium ($\mathrm{M}$) particles; and (e) and (f) large ($\mathrm{L}$) particles. Results are shown for $M\left(\infty \right)=25$, 50, 75, and $100\text{kA}{\text{m}}^{-1}$ as indicated, and they are separated from one another by one unit for clarity. Simulation results are shown as solid lines, and the MMF2+ predictions are shown as dashed lines.

Figure 7

Static structure factors of polydisperse ferrofluids at (a), (c), (e) $T=221.25\text{K;}$ and (b), (d), (f) $T=\infty$, and various concentrations: (a) and (b) small ($\mathrm{S}$) particles; (c) and (d) medium ($\mathrm{M}$) particles; (e) and (f) large ($\mathrm{L}$) particles. Results are shown for $M\left(\infty \right)=25$, 50, 75, and $100\text{kA}{\text{m}}^{-1}$ as indicated, and they are separated from one another by one unit for clarity. Simulation results are shown as solid lines, and the MMF2+ predictions are shown as dashed lines.

Figure 8

Full magnetization curves of monodisperse and bidisperse ferrofluids at $T=295\text{K}$ and with $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$: (a) all simulation results; (b) monodisperse configuration; (c) bidisperse configuration 1; and (d) bidisperse configuration 2. The points are from MC simulations, the solid lines are from MMF2 theory, and the dashed lines are from MMF2+ theory.

Figure 9

Low-field magnetization curves of monodisperse and bidisperse ferrofluids at $T=295\text{K}$ and with $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$: (a) all simulation results; (b) monodisperse configuration; (c) bidisperse configuration 1; and (d) bidisperse configuration 2. The points are from MC simulations; in (a) the lines are the linear parts of the magnetization curves with the initial susceptibility calculated using Eq. (35), and in (b)–(d) the solid lines are from MMF2 theory and the dashed lines are from MMF2+ theory.

Figure 10

Initial susceptibility as a function of temperature for monodisperse and polydisperse ferrofluids with $M\left(\infty \right)=75\text{kA}{\text{m}}^{-1}$: (a) all simulation results; (b) monodisperse configuration; (c) bidisperse configuration 1; and (d) bidisperse configuration 2. The points are from MC simulations, the solid lines are from MMF2 theory, and the dashed lines are from MMF2+ theory.