Condensation, demixing, and orientational ordering of magnetic colloidal suspensions

In this work we study the phase behavior of magnetic particles suspended in a simple nonmagnetic solvent. Magnetic particles are modelled as spherical particles carrying a three-dimensional, classical Heisenberg spin, whereas solvent molecules are treated as spherically symmetric Lennard-Jones particles. The binary mixture of magnetic particles and solvent is studied within the framework of classical density functional theory (DFT). Within DFT pair correlations are treated at the modified mean-field level at which they are approximated by orientation dependent Mayer $f$ functions. In the absence of an external magnetic field four generic types of phase diagrams are observed depending on the concentration of magnetic particles. In this case we observe liquid-liquid phase coexistence between an orientationally ordered (polarized) and a disordered phase characterized by slightly different concentrations of magnetic particles. Liquid-liquid phase coexistence is suppressed by an external field and vanishes completely if the strength of the field is sufficiently large.

Figure 1

Phase diagrams in temperature $T$ versus total number density $\rho$ representation. Data have been obtained for ${\varepsilon }_H=0.12$ and ${\varepsilon }_{12}=1.40$. (a) $x_2^{\prime \prime }=0.25$, GI phase coexistence (, ) and (b) $x_2^{\prime \prime }=0.55$, GP (, ) and GI phase coexistence (, ). Also shown is the critical line (, ) starting at a critical end point (see text).

Figure 2

As Fig. 1 but for (a) $x_2^{\prime \prime }=0.85$ where one has GP (, ), GI (, ), and IP (, ) phase coexistence; (b) $x_2^{\prime \prime }=0.95$ where a nonpolar fluid phase $\left(\rho \lesssim 0.50\right)$ coexists with a P phase (, ). Also shown is the beginning of a critical line (, ) starting at a tricritical point.

Figure 3

Variation of the gas- (isotropic) liquid critical temperature $T_c$ with the concentration of magnetic particles in the liquid phase $x_2^{\prime \prime }$. Data have been obtained for ${\varepsilon }_H=0.12$ and ${\varepsilon }_{12}=1.40$. The point () does not correspond to a true critical point but rather represents the inflection point visible at about $\rho \simeq 0.25$ in Fig. 2.

Figure 4

The order parameter ${\mathcal{P}}_1$ as a function of temperature $T$ along the phase boundaries of P and I phases. Both curves have been obtained for ${\varepsilon }_H=0.12$ and ${\varepsilon }_{12}=1.40$ but for $x_2^{\prime \prime }=0.55$ (, ) [see Fig. 1] and $x_2^{\prime \prime }=0.85$ (, ) [see Fig. 2].

Figure 5

As Fig. 1 but for ${\varepsilon }_{12}=0.80$.

Figure 6

“Reduced” concentration of magnetic particles $x_2^{\prime }/x_2^{\prime \prime }$ as functions of temperature $T$ along the coexistence lines. (a) (, ) $x_2^{\prime \prime }=0.25$ [see Fig. 1], (, ) $x_2^{\prime \prime }=0.55$ [see Fig. 1], and (, ) $x_2^{\prime \prime }=0.95$ [see Fig. 2]; (b) as in part (a) but for $x_2^{\prime \prime }=0.85$ and GP (, ), GI (, ), and IP phase coexistence (, ) [see Fig. 2]. Solid lines in both parts of the figure are intended to guide the eye. In all cases ${\varepsilon }_H=0.12$ and ${\varepsilon }_{12}=1.40$.

Figure 7

(a) As in Fig. 6 but for $x_2^{\prime \prime }=0.05$ (, ) (right ordinate) and $x_2^{\prime \prime }=0.55$ (, ) (left ordinate). (b) As in Fig. 4 but for the same “reduced” concentrations as in part (a) of the figure. In both parts data have been generated for ${\varepsilon }_{12}=0.80$ and ${\varepsilon }_H=0.12$.

Figure 8

Overview of types of phase diagrams encountered depending on the strength of spin-spin coupling ${\varepsilon }_H$, the strength of interaction between unlike molecules ${\varepsilon }_{12}$, and the mole fraction of magnetic particles in the denser phase $x_2^{\prime \prime }$. Squares: ${\varepsilon }_H=0.06$; circles: ${\varepsilon }_H=0.09$; diamonds: ${\varepsilon }_H=0.12$. Color is used to identify specific types of phase diagrams. Black symbols: type 0; red symbols: type I; green symbols: type II; blue symbols: type III. In regions in which symbols are missing no stable solutions of Eqs. (4.1) and (4.2) could be obtained.

Figure 9

As in Fig. 1 but for ${\varepsilon }_H=0.12,{\varepsilon }_{12}=1.20$, and $H=0.00$ (, ), $H=0.10$ (, ), $H=1.00$ (, ), and $H=10.00$ (, ); (a) $x_2^{\prime \prime }=0.95$, (b) $x_2^{\prime \prime }=0.75$. For data sets corresponding to $H=0.00$ critical lines have been omitted for the sake of clarity of the plots in both parts of the figure.

Figure 10

As in Fig. 6 but for $H=0.000$ (, ), $H=0.001$ (, ), $H=0.005$ (, ), and $H=0.010$ (, ). For all curves ${\varepsilon }_H=0.12,{\varepsilon }_{12}=1.20$, and $x_2^{\prime \prime }=0.75$ [cf. Fig. 9]. Inset: An enhancement showing only the short branch corresponding to IP phase coexistence (see text).

Figure 11

Variation of the order parameter ${\mathcal{P}}_1$ with temperature $T$ along the entire coexistence curve and for ${\varepsilon }_H=0.12,{\varepsilon }_{12}=1.20$, and $H=0.00$ (, ), $H=0.10$ (, ), $H=1.00$ (, ), and $H=10.00$ (, ); (a) $x_2^{\prime \prime }=0.95$, (b) $x_2^{\prime \prime }=0.75$.