Phase behavior of the thermotropic melt of asymmetric V-shaped molecules

The phase behavior of the monodisperse melt of V-shaped molecules composed of two rigid segments of different lengths joined at their ends at an external angle $\alpha$ has been examined within the Landau–de Gennes approach. Each rigid segment consists of a sequence of monomer units; the anisotropic interactions in the system are assumed to be of the Maier-Saupe form. The coefficients of the Landau–de Gennes free-energy expansion have been found from a microscopic model of V-shaped molecule. A single Landau point at which the system undergoes direct continuous transition from isotropic to biaxial nematic phase is found for asymmetry parameter $\varphi =1/3$ or $\varphi =2/3$, where $\varphi$ is the number fraction of monomer units in one of the segments. Two Landau points are found in a range $1/3<\varphi <2/3$. Only isotropic and nematic states are found to be stable for $0\le \varphi <1/3$ and $2/3<\varphi \le 1$. Regions of stability of isotropic, prolate uniaxial, and biaxial nematic phases are found for $\varphi =1/3$ and $\varphi =2/3$. In addition, a stable oblate uniaxial phase is revealed if asymmetry parameter falls in the range $1/3<\varphi <2/3$. The region of stability of the biaxial nematic phase becomes smaller as parameter $\varphi$ increases from $1/3$ to $1/2$ (or decreases from $2/3$ to $1/2$). Coefficients of the gradient terms have been found; for certain values of asymmetry parameter these coefficients can become negative in some ranges of angles.

Figure 1

Sketch of V-shaped molecule composed of two rigid rods joined at an external angle $\alpha$. Rods consist of $N_1$ and $N_2=N-N_1$ numbers of monomer units.

Figure 2

Plots of values of the solutions ${\alpha }_1^{*}$ and ${\alpha }_2^{*}$ of Eq. (17) as function of asymmetry parameter.

Figure 3

Order parameter jump at I-N transition. The positions of points ${\alpha }_1^{*}$ and ${\alpha }_2^{*}$ are indicated by filled circles for each curve that contains such points. There is only one point ${\alpha }_1^{*}={\alpha }_2^{*}$ for curve with $\varphi =1/3$.

Figure 4

(a) Lines of the first order ${\text{I-N}}_{\mathrm{U}}^{+}$ phase transitions for $\varphi <1/3$. The isotropic ($\mathrm{I}$) and prolate uniaxial nematic (${\mathrm{N}}_{\mathrm{U}}^{+}$) phases are stable below and above of each curve, respectively. (b) Plots of eigenvalues Eqs. (11) for $\varphi =0.3$ and $\tilde{\omega }=10$ with ${\lambda }_1>{\lambda }_2={\lambda }_3$. The vertical dash-dot lines correspond to the boundaries of isotropic phase where all eigenvalues vanish.

Figure 5

Phase diagram of melt of V-shaped molecules for different values of the asymmetry parameter: $\varphi =1/3$ (a), $\varphi =0.335$ (b), $\varphi =0.34$ (c), and $\varphi =0.4$ (d). The solid line represents a first-order transition, while the dashed lines correspond to a second-order transition. The Landau points are indicated by the filled circles.

Figure 6

(a) Plots of eigenvalues Eqs. (11) for $\varphi =1/3$ and $\tilde{\omega }=18$ with ${\lambda }_1>{\lambda }_2>{\lambda }_3$. (b) Plots of eigenvalues Eqs. (11) for $\varphi =0.335$ and $\tilde{\omega }=16$ in oblate uniaxial and biaxial nematic phases. For both figures the vertical dash-dot lines correspond to the boundaries of the biaxial phase where two of three eigenvalues become identical.

Figure 7

Plots of rescaled elastic constants $\left(L_1\right)/{\left(Nb\right)}^2$ (a) and $\left(L_2\right)/{\left(Nb\right)}^2$ (b) as function of an external angle $\alpha$ of V-shaped molecule ($0\le \alpha <\pi$) for several values of asymmetry parameter.