Mean-field model for a mixture of biaxial nematogens and dipolar nanoparticles

We analyze a mean-field model for mixtures involving biaxial nematogens and dipolar nanoparticles, taking into account not only orientational and isotropic pair interactions between nematogens but also orientational nematogen-nanoparticle interactions. We determine bulk equilibrium phase diagrams for a wide range of interaction strengths, identifying in each case the effect of the nanoparticles on the stability of nematic phases and on the appearance of multicritical points. Special attention is given to the limit of a low concentration of nanoparticles, in which their effect on the temperatures of both the first-order uniaxial-isotropic and the continuous biaxial-uniaxial transitions is investigated in detail.

Figure 1

Temperature versus nematogen shape biaxiality $\mathrm{\Delta }$ for $U=0$. (a) Pure system ($\varphi =1$). (b) Dipole-doped system, with $\varphi =0.975$ and $B=A/2$. (c) Dipole-doped system, with $\varphi =0.975$ and $B=3A$. (d) Dipole-doped system, with $\varphi =0.975$ and $B=-3A$. (e) Dipole-doped system, with $\varphi =0.975$ and $B=-5A$. $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{U}}^{-}$: Discotic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. ${\mathrm{C}}_1$: Coexistence between $\mathrm{ISO}$ and ${\mathrm{N}}_{\mathrm{B}}$. ${\mathrm{C}}_2$: Coexistence between $\mathrm{ISO}$ and ${\mathrm{N}}_{\mathrm{U}}^{-}$. $\mathrm{L}$ is a Landau multicritical point. $\mathrm{CE}$ in (b) is a critical end point (see Sec. 3) and the contiguous dashed red (dark gray) line is a line of CE points separating ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$.

Figure 2

Energy levels as a function of the degree of biaxiality $\mathrm{\Delta }$. Level crossings account for variations in the entropic contributions to the free energy. Solid lines correspond to energy levels associated with interactions between two nematogens, while dashed lines indicate energy levels arising from interactions involving NPs and nematogens. Line thickness is proportional to the degeneracy of the corresponding level.

Figure 3

Phase diagram in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for an intrinsically uniaxial system ($\mathrm{\Delta }=0$), different values of anisotropic interaction between nematogens, and dipolar NPs and in absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. Short-dashed lines are the boundaries of the biphasic region (gray).

Figure 4

Phase diagram in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for an intrinsically uniaxial system ($\mathrm{\Delta }=0$), different values of anisotropic interaction between nematogens, and dipolar NPs and in absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. Short-dashed lines are the boundaries of biphasic region (gray).

Figure 5

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for biaxiality degree $\mathrm{\Delta }=4/5$ and in the absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{U}}^{-}$: Discotic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray). Red dash-dotted line: critical end point ($\mathrm{CE}$). Short-dotted red line in the insets mark triple points. $\mathrm{TC}$ is a tricritical point.

Figure 6

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for biaxiality degree $\mathrm{\Delta }=4/5$ and in the absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}, {\mathrm{N}}_{{\mathrm{U}}_{\mathrm{I}}}^{+}$, and ${\mathrm{N}}_{{\mathrm{U}}_{\mathrm{II}}}^{+}$: Calamitic uniaxial nematic phases. ${\mathrm{N}}_{\mathrm{U}}^{-}$: Discotic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray). Red dash-dotted line: Critical end point ($\mathrm{CE}$). Short-dotted red line in the inset mark triple points. $\mathrm{L}$ is a Landau multicritical point.

Figure 7

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for biaxiality degree $\mathrm{\Delta }=1$ and in the absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. Short-dashed lines are the boundaries of biphasic region (gray). Red dash-dotted line: critical end point ($\mathrm{CE}$). $\mathrm{L}$ is a Landau multicritical point.

Figure 8

(a) Coexistence regions in the plane of normalized isotropic-uniaxial transition temperature versus concentration ${\varphi }_{\mathrm{NP}}$ of dipolar NPs for intrinsically uniaxial nematogens ($\mathrm{\Delta }=0$) and different values of $\kappa =B/A$. (b) Normalized isotropic-uniaxial transition temperature as a function of $\kappa =B/A$ for intrinsically uniaxial nematogens ($\mathrm{\Delta }=0$) and different values of ${\varphi }_{\mathrm{NP}}$. Black dotted line indicates the clearing temperature ($T=T_{\mathrm{I}-\mathrm{U}}^{\left(0\right)}$) of the LC host.

Figure 9

(a) Normalized isotropic-uniaxial transition temperature as a function of the concentration of dipolar NPs ${\varphi }_{\mathrm{NP}}$ for intrinsically biaxial nematogens $\mathrm{\Delta }=4/5$ and different values of $\kappa =B/A$. (b) Normalized uniaxial-biaxial transition temperature as a function of ${\varphi }_{\mathrm{NP}}$ for intrinsically biaxial nematogens $\mathrm{\Delta }=4/5$ and different values of $\kappa =B/A$.

Figure 10

Phase diagram in terms of temperature $T$ (in units of $A$) and $\kappa =B/A$, for biaxiality degree $\mathrm{\Delta }=4/5$, concentration of nematogens $\varphi =0.975$, and in the absence of isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{U}}^{-}$: Discotic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. $\mathrm{L}$ is a Landau multicritical point. The red (light gray) point has coordinates $({\kappa }_o,T_o)$.

Figure 11

Temperature dependence of the equilibrium values of (a) $S^{★}=S/\varphi$ (black solid line) and $R^{★}=R/(1-\varphi )$ (red or light gray solid line) and (b) ${\eta }^{★}=\eta /\varphi$ (black solid line) and ${\zeta }^{★}=\zeta /(1-\varphi )$ (red or light gray solid line), for $\kappa =-3.25$ and $\mathrm{\Delta }=4/5$. Temperature scale is given in units of $A$. Red thin-dashed straight lines (three leftmost lines in each plot) signal second-order transitions and blue thin-dashed straight lines (rightmost in each plot) signal first-order transitions.

Figure 12

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for intrinsically uniaxial nematogens ($\mathrm{\Delta }=0$) and different values of the isotropic interaction. $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray).

Figure 13

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for intrinsically uniaxial nematogens ($\mathrm{\Delta }=0$) and for different values of the isotropic interaction. ${\mathrm{I}}_{\mathrm{DR}}$: Dipole-rich isotropic phase. ${\mathrm{I}}_{\mathrm{DP}}$: Dipole-poor isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray). Red dash-dotted line is an ${\mathrm{I}}_{\mathrm{DR}}-{\mathrm{I}}_{\mathrm{DP}}-{\mathrm{N}}_{\mathrm{U}}^{+}$ triple point. $\mathrm{C}$ is a simple critical point.

Figure 14

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for biaxiality degree $\mathrm{\Delta }=19/20$ and different values of the isotropic interaction ($U=0$). $\mathrm{ISO}$: Isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{U}}^{-}$: Discotic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray). Red dash-dotted line: Critical end point ($\mathrm{CE}$). $\mathrm{L}$ is a Landau point.

Figure 15

Phase diagrams in terms of temperature $T$ (in units of $A$) and concentration $\varphi$ of nematogens for biaxiality degree $\mathrm{\Delta }=1$ and different values of the isotropic interaction. $\mathrm{ISO}$: Isotropic phase. ${\mathrm{I}}_{\mathrm{DR}}$: Dipole-rich isotropic phase. ${\mathrm{I}}_{\mathrm{DP}}$: Dipole-poor isotropic phase. ${\mathrm{N}}_{\mathrm{U}}^{+}$: Calamitic uniaxial nematic phase. ${\mathrm{N}}_{\mathrm{B}}$: Biaxial nematic phase. Short-dashed lines are the boundaries of biphasic regions (gray). Red dash-dotted line: Critical end point ($\mathrm{CE}$). Dotted red line marks a triple point. $\mathrm{L}$ is a Landau point. $\mathrm{C}$ is a simple critical point.

Figure 16

Candidate multicritical Landau points in diagrams of temperature or nematogen concentration $\varphi$ as functions of the molecular degree of biaxiality $\mathrm{\Delta }$, for different values of $\kappa =B/A$. Gray region is unphysical.