$XY$-spin fluids in an external magnetic field: An integral equation approach

We develop an integral equation approach to study anisotropic fluids with planar spins in the presence of an external field. As a result, the integral equation calculations for these systems appear to be no more difficult than those for ordinary isotropic liquids. The method presented is applied to the investigation of phase coexistence properties of ferromagnetic $XY$-spin fluids in a magnetic field. The soft mean spherical approximation is used for the closure relation connecting the orientationally dependent two-particle direct and total correlation functions. The Lovett-Mou-Buff-Wertheim and Born-Green-Yvon equations are employed to describe the one-particle orientational distribution. The phase diagrams are obtained in the whole range of varying the external field for a wide class of $XY$-spin fluid models with various ratios of the strengths of magnetic to nonmagnetic Yukawa-like interactions. The influence of changing the screening radii of the interaction potentials is also considered. Different types of the phase diagram topology are identified. They are characterized by the existence of critical, tricritical, critical end, and triple points related to transitions between gas, liquid, and para- and ferromagnetic states, accompanied by different external field dependencies of critical temperatures and densities corresponding to the gas-liquid and liquid-liquid transitions. As is demonstrated, the integral equation approach leads to accurate predictions of the complicated phase diagram behavior which coincide well with those evaluated by the cumbersome Gibbs ensemble simulation and multiple-histogram reweighting techniques.

Figure 1

The gas-liquid coexistence densities ${\rho }^{*}$ as a function of temperature $T^{*}$ obtained within the AOZ-SMSA-BGY integral equation approach (bold curves) for the ideal $(R=\infty )$ soft-core $XY$-spin fluid with $z_1^{*}=1$ and different values of the external field, $H^{*}=0$, 0.5, 2, 5, 10, 20, 40, and $\infty$. The AOZ-SMSA-LMBW and SCMF results are plotted correspondingly by bold short- and thin long-dashed curves. The GEMC and MHR simulation data are shown as open and full circles, respectively. The paramagnetic-ferromagnetic phase transition [at $H^{*}=0$, subset (a)] is presented by the dotted curve.

Figure 2

The liquid-gas coexistence densities as a function of temperature obtained within the AOZ-SMSA-BGY integral equation approach for the soft-core ideal $XY$-spin fluid at different values, $z_1^{*}=0.5$, 1, 2, and 3, of the inverse screening length [subsets (a), (b), (c), and (d), respectively] as well as different values of the external field, namely, top to bottom (the set of dashed curves), $H^{*}=0.01$, 0.1, 0.3, and 2, as well as, bottom to top (the set of solid curves), $H^{*}=5$, 10, 20, 40, 100, 1000, and $\infty$ (within each subset). The bold curves correspond to the case $H=0$. The paramagnetic-ferromagnetic phase transition (at $H^{*}=0$) is plotted by the dotted lines.

Figure 3

The critical temperature $T_{\mathrm{c}}^{*}$ [subset (a)] and critical density ${\rho }_{\mathrm{c}}^{*}$ [subset (b)] as functions of the external field $H^{*}$, evaluated within the AOZ-SMSA-BGY approach for the soft-core ideal $XY$-spin fluid at various values of the inverse screening length $z^{*}$. The AOZ-SMSA-LMBW function is plotted for $z^{*}=1$ by the dotted curves. At $z^{*}\to 0$, the result corresponds to the SCMF theory (dashed curves).

Figure 4

(a) The paramagnetic-ferromagnetic transition temperature $T_{\lambda }^{*}$ obtained at $H^{*}=0$ within the AOZ-SMSA-BGY integral equation approach for the soft-core ideal $XY$-spin fluid for different values of inverse screening length, from right to left, $z^{*}=0.5$, 1, 2, 3, and 5. The result of the standard HSMF theory is plotted as the dashed straight line and relates to the case $z^{*}\to 0$. (b) The results of the SCMF and AOZ-SMSA-BGY theories for the case $z^{*}=1$ are shown as dashed and solid curves, respectively, in comparison with canonical MC simulation data (circles).

Figure 5

The ratio of the first-harmonic coefficient in the expansion of the one-particle distribution function evaluated within the AOZ-SMSA-BGY approach for the ideal $XY$-spin fluid model $(z_1^{*}=1)$ to the effective magnetic field obtained by the MF theory [subsets (a) and (b)]. The influence of the second- and third-harmonic coefficients is shown in subsets (c) and (d), respectively. The results are plotted versus density at several temperatures and external field values by the bold solid, short-dashed, thin solid, and long-dashed curves for the cases $H^{*}=0$, 2, 5, and 10, respectively.

Figure 6

The gas-liquid coexistence densities as a function of temperature obtained within the AOZ-SMSA-BGY approach for the soft-core nonideal $XY$-spin fluid with $z_1^{*}=z_2^{*}=1$ at $H^{*}=0$ for large values of $R$. The MHR simulation data for $R=1$ are shown as circles.

Figure 7

The gas-liquid and liquid-liquid coexistence densities as a function of temperature obtained within the AOZ-SMSA-BGY approach for the soft-core nonideal $XY$-spin fluid with $z_1^{*}=z_2^{*}=1$ at $H=0$ for moderate [subset (a)] and small [subset (b)] values of $R$. The MHR simulation data for $R=0.415$, 0.33, and 0.2 are shown as circles.

Figure 8

The coexistence densities as a function of temperature $T^{*}$ obtained within the AOZ-SMSA-BGY approach for the nonideal $XY$ spin fluid at $R=0.415$ [subset (a)] and $R=0.26$ [subset (b)] corresponding to different values $z_1^{*}=z_2^{*}=0.5$, 1, 2, and 3, of the inverse screening radius.

Figure 9

The same as in Fig. 8 but for $z_1\ne z_2$ at $R=0.415$ [subsets (a) and (b)] and $R=0.26$ [subsets (c) and (d)].

Figure 10

The paramagnetic-ferromagnetic Curie curves evaluated at $H=0$ using the AOZ-SMSA-BGY theory for the soft-core nonideal $XY$-spin fluid with $z_1^{*}=z_2^{*}=1$ at different values of the system parameter, top to bottom, $R=0.2$, 0.23, 0.28, 0.33, 0.415, 0.6, 1, and $\infty$. The results of the HSMF and SCMF approaches are plotted as the dotted and dashed curves, respectively. The canonical MC simulation points are shown by circles $(R=0.33)$ and squares $(R=0.2)$.

Figure 11

The complete thermodynamic phase diagrams obtained for the soft-core nonideal $XY$-spin fluid with $z_1^{*}=z_2^{*}=1$ using the AOZ-SMSA-BGY approach at some typical values of $R=5$, 1, 0.5, 0.415, 0.37, 0.3, 0.26, and 0.2 [subsets (a), (b), (c), (d), (e), (f), (g), and (h), respectively]. The families of the coexistence curves in each of the subsets correspond to different values of the external field, namely, top to bottom (the set of dashed curves), $H^{*}=0.1$ and 2, as well as, bottom to top (the set of solid curves), $H^{*}=5$, 10, 20, 40, 100, and $\infty$. The bold curves relate to the case $H=0$. The MHR simulation data for $R=0.37$ and $H^{*}=0$, 2, 10, 40, and $\infty$ are shown in subset (e) as circles. The paramagnetic-ferromagnetic phase transition (at $H^{*}=0$) is plotted by the dotted curves.

Figure 12

The thermodynamic phase diagrams obtained within the AOZ-SMSA-BGY approach for the soft-core nonideal $XY$-spin fluid with $z_1^{*}=z_2^{*}=1$ at $R=R_{\mathrm{vL}}=0.376$ and $H^{*}=0$, 0.1, 1, 2, 3, 5, 10, 20, 40, 100, and $\infty$ [subset (a)]. A more detailed phase behavior near the asymmetric tricritical point is presented in subset (b) for (bottom to top) $H^{*}=0.5$, 1, 1.3, 1.5, 1.7, 1.9, 2.1, and 2.4. Samples of the MHR simulation data are presented in subset (b) for $H^{*}=0.5$ and 1.9 by circles. The gas-liquid (on the left) and liquid-liquid (on the right) critical points are shown as open and full squares, respectively, connected by dashed curves. The curves meet in the tricritical point (star). The triple points are represented by the horizontal dashed lines. The critical temperature and critical density of the gas-liquid and liquid-liquid phase transitions are plotted as functions of $H^{*}$ in subsets (c) and (d), respectively.

Figure 13

The critical temperature of the wing lines $T_{\mathrm{c}\mathrm{w}}^{*}$, subsets (a) and (b), and the gas-liquid critical lines $T_{\mathrm{cgl}}^{*}$, subsets (c) and (d), as a function of the external magnetic field $H^{*}$ for the nonideal $XY$-spin fluid at different values of parameter $R$.

Figure 14

The critical density of the wing lines ${\rho }_{\mathrm{c}\mathrm{w}}^{*}$, subsets (a) and (b), and the gas-liquid critical lines ${\rho }_{\mathrm{cgl}}^{*}$, subsets (c) and (d), as a function of the external magnetic field $H^{*}$ for the nonideal $XY$-spin fluid at different values of $R$.