Magnetic properties in monolayers of a model polydisperse ferrofluid

The influence of polydispersity on the equilibrium properties of monolayers of three-dimensional dipolar spheres with short-range repulsive interactions is studied by means of Monte Carlo simulations and a high field approximation perturbation theory. The particle distribution in the simulations is realized in the semigrand ensemble by tuning appropriately the underlying particle distribution density. The magnetization curves are calculated as functions of density and temperature, and the obtained results are compared with the data determined in a monodisperse equivalent of the system. In-plane and out-of-plane initial magnetic susceptibilities are determined using external fields applied parallel or normal to the monolayer plane. Susceptibility data for the true two- and three-dimensional counterparts of the system are also calculated for comparison. Our findings for the magnetic properties can partly be explained by the structural characteristics obtained from the simulations.

Figure 1

The applied particle distribution density for the polydisperse systems: The continuous line indicates the target distribution. Data points show the average distribution obtained from a semigrand ensemble simulation in a q2D system at $T^{*}=1.0$, ${\rho }^{*}=0.60$, and $H_z^{*}=3.0$ (the field direction is perpendicular to the plane), and the dashed line is the underlying (sampling) distribution used here in the final simulation cycle.

Figure 2

(Color online) (a) Simulated dimensionless magnetization as a function of the external magnetic field in q2D systems obtained at $T^{*}=1.0$ (dark) and $T^{*}=1.1$ (light-colored), where ‖ and ⊥ stand for the direction of the external field (in-plane and out-of-plane, respectively). The triangles and the dotted lines correspond to ferrofluids at ${\rho }^{*}=0.40$, the circles and the dashed lines to ferrofluids at ${\rho }^{*}=0.50$, the squares and the solid lines to ferrofluids at ${\rho }^{*}=0.60$. Open and half-closed symbols denote the monodisperse and the polydisperse systems, respectively. The statistical uncertainties of the simulation results do not exceed the symbol size. (b) Comparison of the theoretical predictions (lines) with the simulation results (symbols) at ${\rho }^{*}=0.40$. The continuous lines represent the magnetization of the ideal ferrocolloid gas $[M_L^{*}=m^{*}{\rho }^{*}L\left(\alpha \right)]$.

Figure 3

(Color online) Dimensionless in-plane (a) and out-of-plane (b) magnetizations as a function of the external magnetic field in q2D systems obtained at $T^{*}=1.0$ (dark) and $T^{*}=1.1$ (light-colored). The meaning of the lines and symbols is the same as in Fig. 2. The straight lines are the linear fitting results, the slope of which gives the initial susceptibility.

Figure 4

Initial susceptibilities for 2D (triangles and dotted lines), q2D (circles and solid lines), and 3D (squares and dashed lines) systems as a function of the dimensionless parameter $y$. Symbols represent the simulation results and lines are the theoretical predictions. Open and half-closed symbols denote the monodisperse and the polydisperse systems, respectively. Note that these results are obtained from calculations at two different temperatures.

Figure 5

(Color online) Pair correlation functions for 2D systems at ${\rho }^{*}=0.40$ (dotted lines), q2D systems at ${\rho }^{*}=0.50$ (solid lines), and 3D systems at ${\rho }^{*}=0.60$ (dashed lines) in the absence of the external field at $T^{*}=1.0$. Light-colored and dark curves denote the monodisperse and the polydisperse systems, respectively.

Figure 6

(Color online) Projections of the pair correlation functions. The meaning of the lines is the same as in Fig. 5.

Figure 7

(Color online) Projections of the pair correlation functions in q2D systems at $T^{*}=1.0$ and ${\rho }^{*}=0.50$. The solid lines represent the zero field results, the dashed and the dotted lines stand for the results obtained at $H_z^{*}=3.0$ (the field direction is perpendicular to the plane) and at $H_x^{*}=2.0$ (the field direction is parallel to the plane), respectively. Light-colored and dark curves denote the monodisperse and the polydisperse systems, respectively.