Demixing in simple dipolar mixtures: Integral equation versus density functional results

Using reference hypernetted chain (RHNC) integral equations and density functional theory in the modified mean-field (MMF) approximation we investigate the phase behavior of binary mixtures of dipolar hard spheres. The two species ($A$ and $B$) differ only in their dipole moments $m_A$ and $m_B$, and the central question investigated is under which conditions these asymmetric mixtures can exhibit demixing phase transitions in the fluid phase regime. Results from our two theoretical approaches turn out to strongly differ. Within the RHNC (which we apply to the isotropic high-temperature phase) demixing does indeed occur for dense systems with small interaction parameters $\Gamma =m_B^2∕m_A^2$. This result generalizes previously reported observations on demixing in mixtures of dipolar and neutral hard spheres $\left(\Gamma =0\right)$ to the case of true dipolar hard sphere mixtures. The RHNC approach also indicates that these demixed fluid phases are isotropic at temperatures accessible by the theory, whereas isotropic-to-ferroelectric transitions occur only at larger $\Gamma$. The MMF theory, on the other hand, yields a different picture in which demixing occurs in combination with spontaneous ferroelectricity at all $\Gamma$ considered. This discrepancy underlines the relevance of correlational effects for the existence of demixing transitions in dipolar systems without dispersive interactions. Indeed, supplementing the dipolar interactions by small, asymmetric amounts of van der Waals–like interactions (and thereby supporting the systems tendency to demix) one finally reaches good agreement between MMF and RHNC results.

Figure 1

Stability limits of binary DHS mixtures with interaction ratios $\Gamma ⩽0.4$ and $\Gamma ⩾0.4$ (inset) in the concentration-temperature plane $\left(T^{*}=k_{B}T{\sigma }^3∕m_A^2,c_A={\rho }_A∕\rho \right)$. The total density is fixed at ${\rho }^{*}=0.7$.

Figure 2

Dielectric constant as function of the inverse temperature for the pure $A$ fluid and for mixtures at $c_A=0.2$ and various values of $\Gamma$.

Figure 3

The eigenvalues ${\lambda }_{\mathrm{F}}$ (solid lines) and ${\lambda }_{\mathrm{D}}$ (dashed lines) as functions of the inverse temperature at $\Gamma =0.8$ and $c_A=0.2$ (a) and $c_A=0.8$ (b).

Figure 4

The eigenvalues ${\lambda }_{\mathrm{F}}$ (solid lines) and ${\lambda }_{\mathrm{D}}$ (dashed lines) as functions of the inverse temperature at $\Gamma =0.2$ and $c_A=0.2$ (a) and $c_A=0.8$ (b). The inset of (a) shows results for $c_A=0.2$ and $\Gamma =0$.

Figure 5

Eigenvalues ${\lambda }_{\mathrm{F}}$ (solid lines) and ${\lambda }_{\mathrm{D}}$ (dashed lines) at the stability limits for $\Gamma =0$ (a), $\Gamma =0.1$ (b), $\Gamma =0.2$ (c), and $\Gamma =0.4$ (d).

Figure 6

MMF density-temperature phase diagrams for DHS-HS mixtures $\left(\Gamma =0\right)$ at $\Delta {\mu }^{*}\to -\infty$ (solid and dotted lines), $\Delta {\mu }^{*}=2.0$ (dashed lines), and $\Delta {\mu }^{*}=4.0$ (dot-dashed lines). The abbreviations are explained in the main text.

Figure 7

MMF results for the temperatures related to the boundary of the isotropic phase at ${\rho }^{*}=0.7$ and various values of $\Gamma$. Dashed (solid) lines denote second-order (first-order) phase transitions.

Figure 8

MMF density-temperature phase diagrams of pure $\left(A\right)$ Stockmayer fluids characterized by ${ϵ}_{AA}^{*}=0.3$ (a) and ${ϵ}_{AA}^{*}=0.4$ (b).

Figure 9

MMF phase diagrams of Stockmayer/HS mixtures ($\Gamma =0$, ${ϵ}_{AA}^{*}=0.3$). (a) Density-temperature diagrams at $\Delta {\mu }^{*}=1.8$ (solid lines) and $\Delta {\mu }^{*}=3.0$ (dashed lines). (b) Concentration-temperature diagram at $\Delta {\mu }^{*}=1.8$.

Figure 10

Polarization and concentrations of $A$ particles as functions of the chemical potential at $\Delta {\mu }^{*}=1.8$ and $T^{*}=0.85$ ($\Gamma =0$, ${ϵ}_{AA}^{*}=0.3$).

Figure 11

MMF results for the temperatures related to the boundary of the isotropic phase for Stockmayer mixtures at ${\rho }^{*}=0.7$ and various values of $\Gamma$. The reduced LJ parameters (${ϵ}_{ab}^{*}:={ϵ}_{ab}{\sigma }^3∕m_A^2$) are chosen as ${ϵ}_{AA}^{*}=0.3$, ${ϵ}_{BB}^{*}=0.3\Gamma$, and ${ϵ}_{AB}^{*}={ϵ}_{BA}^{*}=0.3\sqrt{\Gamma }$. Dashed (solid) lines denote second-order (first-order) phase transitions.

Figure 12

RHNC results (solid lines) for the stability limits of Stockmayer/HS mixtures ($\Gamma =0$) at ${\rho }^{*}=0.7$ and various values of ${ϵ}_{AA}^{*}$. Also shown (dashed line) is the MMF result for the boundary of the isotropic phase at ${\rho }^{*}=0.7$ and ${ϵ}_{AA}^{*}=0.3$.