Emergence of biaxial nematic phases in solutions of semiflexible dimers

We investigate the isotropic, uniaxial nematic and biaxial nematic phases, and the transitions between them, for a model lyotropic mixture of flexible molecules consisting of two rigid rods connected by a spacer with variable bending stiffness. We apply density-functional theory within the Onsager approximation to describe strictly excluded-volume interactions in this athermal model and to self-consistently find the orientational order parameters dictated by its complex symmetry, as functions of the density. Earlier work on lyotropic ordering of rigid bent-rod molecules is reproduced and extended to show explicitly the continuous phase transition at the Landau point, at a critical bend angle of ${36}^{\circ }$. For flexible dimers with no intrinsic biaxiality, we find that a biaxial nematic phase can nevertheless form at a sufficiently high density and low bending stiffness. For bending stiffness $\kappa >0.86k_{B}T$, this biaxial phase manifests as dimer bending fluctuations occurring preferentially in one plane. When the dimers are more flexible, $\kappa <0.86k_{B}T$, the modal shape of the fluctuating dimer is a $\mathsf{V}$ with an acute opening angle, and one of the biaxial order parameters changes sign, indicating a rotation of the directors. These two regions are separated by a narrow strip of uniaxial nematic in the phase diagram, which we generate in terms of the spacer stiffness and particle density.

Figure 1

(a) Flexible dimers consisting of two hard rods connected by a flexible spacer. The angle $\psi$, with equilibrium value $\psi =0$, measures the degree of bending. (b) Biaxial nematic phase of rigid bent rods studied in [8].

Figure 2

Principal axes and standard Euler angles defining the orientation of a biaxial bent-rod molecule in the laboratory-fixed frame $(x,y,z)$. The shape of the molecule can be embedded in a parallelepiped of thickness $D$, with long principal axis $\mathit{w}$ and short principal axis $\mathit{u}\perp \mathit{w}$. The long axis is inclined at the angle $\theta$ from $z$, the short axis lies in the plane that crosses the $x-y$ plane at angle $\varphi$, and $\mathit{u}$ is rotated counterclockwise by angle $\chi$ in this plane.

Figure 3

(a) Order parameter and excess free energy (in units of $k_{B}T$) plotted for $\psi =0^{\circ }$. There is a strong first-order phase transition, with only $S_1$ present. (b) The order parameter and excess free energy (in units of $k_{B}T$) for $\psi ={36}^{\circ }$ reflect phase transitions just below the Landau point. The I-${\mathrm{N}}_{\mathrm{U}}^{+}$ transition is very nearly continuous, reflecting the proximity to the Landau point. Arrows highlight the singularities observed at the ${\mathrm{N}}_{\mathrm{U}}-{\mathrm{N}}_{\mathrm{B}}^{+}$ transition.

Figure 4

(a) Order parameter and excess free energy (in units of $k_{B}T$) reflect phase transitions for $\psi ={37}^{\circ }$, just above the Landau point. The first transition here is to an oblate uniaxial ${\mathrm{N}}_{\mathrm{U}}^{-}$ phase; arrows highlight the singularities at the ${\mathrm{N}}_{\mathrm{U}}-{\mathrm{N}}_{\mathrm{B}}$ transition. (b) For a large bending angle, $\psi ={45}^{\circ }$, the molecule is essentially oblate and the corresponding ${\mathrm{N}}_{\mathrm{U}}^{-}$ phase is the only outcome in the range of densities investigated.

Figure 5

$\mathrm{I}-{\mathrm{N}}_{\mathrm{B}}$ phase transitions for $\psi =36.{32}^{\circ }$, at the Landau point. (a) Obtained with all initial $\left\{S_i\right\}$ positive; (b) obtained with negative initial $S_1$ and $S_2$. The transition is manifestly continuous, directly from the isotropic to the biaxial nematic phase, and the free energy of both resulting phases is nearly the same (to $\sim {10}^{-3}{k}_{B}T$ per molecule). This is consistent with the change in behavior from prolate- to oblatelike order across the Landau point but implies that there are two degenerate biaxial phases at the point itself, ${\mathrm{N}}_{\mathrm{B}}^{+}$ and ${\mathrm{N}}_{\mathrm{B}}^{-}$.

Figure 6

Order parameters and excess free energy (in units of $k_{B}T$) plotted versus reduced density for three dimers: stiff, with $\kappa =100k_{B}T$ (plain curves); medium, with $\kappa =1k_{B}T$ (curves with a circle); and very flexible, with $\kappa =0.01k_{B}T$ (curves with a square). In all cases we find a strong first-order transition to the prolate ${\mathrm{N}}_{\mathrm{U}}^{+}$ phase, which, in the case of nonstiff dimers, is followed by an ${\mathrm{N}}_{\mathrm{U}}^{+}-{\mathrm{N}}_{\mathrm{B}}$ transition (marked by arrows).

Figure 7

Order parameters and excess free energy (in units of $k_{B}T$) plotted versus reduced density for dimers with bending stiffness $\kappa$ crossing over from the ${\mathrm{N}}_{\mathrm{B}}^{+}$ to the ${\mathrm{N}}_{\mathrm{B}}^{-}$ phase. For $\kappa =0.855k_{B}T$ (curves with a square) the second transition is into the phase with a negative $S_4$; for $\kappa =0.856k_{B}T$ (curves with a circle) the emerging phase has a positive $S_4$. The transitions at $\kappa =1k_{B}T$ (plain curves) are also shown for reference; the biaxial transition for this bending stiffness is indicated by the arrow.

Figure 8

Reduced density–bending stiffness phase diagram of flexible dimers. The $\mathrm{I}-{\mathrm{N}}_{\mathrm{U}}^{+}$ phase transition boundary (lower curve) reaches the asymptote of $c\approx 1.35$ at very large $\kappa$. For small $\kappa$ the flexible dimers also form the biaxial phase at higher densities. Regions corresponding to the I, ${\mathrm{N}}_{\mathrm{U}}^{+}, {\mathrm{N}}_{\mathrm{B}}^{+}$, and ${\mathrm{N}}_{\mathrm{B}}^{-}$ phases are labeled. Vertical arrows indicate the three density scans presented in Fig. 6. Inset: The horizontal arrow indicates the stiffness scan explored in Fig. 9.

Figure 9

‘Stiffness scan’ of the phase diagram across the crossover region between the ${\mathrm{N}}_{\mathrm{B}}^{-}$ and ${\mathrm{N}}_{\mathrm{B}}^{+}$ phases, at $c=4.1$. (a) The four order parameters, with the notable feature that the value of $S_4$ in ${\mathrm{N}}_{\mathrm{B}}^{+}$ is just minus its continuation from ${\mathrm{N}}_{\mathrm{B}}^{-}$ (as illustrated by the dashed line). (b) Similarly, we see that the two biaxial phases, ${\mathrm{N}}_{\mathrm{B}}^{+}$ and ${\mathrm{N}}_{\mathrm{B}}^{-}$, have essentially the same free energy (by extrapolation). Likewise, the free energy difference of the uniaxial phase ${\mathrm{N}}_{\mathrm{U}}$ sandwiched between the two biaxial phases is the continuation of that of the uniaxial phase at larger $\kappa$ (as illustrated by dashed lines).