de Vries behavior in smectic liquid crystals near a biaxiality-induced smectic-$\mathit{A}$–smectic-$\mathit{C}$ tricritical point

We show that a generalized Landau theory for the smectic-$\mathit{A}$–smectic-$\mathit{C}$ (Sm-$\mathit{A}$–Sm-$\mathit{C}$) phases exhibits a biaxiality induced Sm-$\mathit{A}$–Sm-$\mathit{C}$ tricritical point. Proximity to this tricritical point depends on the degree of orientational order in the system; for sufficiently large orientational order the Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition is three-dimensional $XY$-like, while for sufficiently small orientational order, it is either tricritical or first order. We investigate each of the three types of Sm-$\mathit{A}$–Sm-$\mathit{C}$ transitions near tricriticality and show that for each type of transition, small orientational order implies de Vries behavior in the layer spacing, an unusually small layer contraction. This result is consistent with, and can be understood in terms of, the “diffuse cone” model of de Vries. Additionally, we show that birefringence grows upon entry to the Sm-$\mathit{C}$ phase. For a continuous transition, this growth is more rapid the closer the transition is to tricriticality. Our model also predicts the possibility of a nonmontonic temperature dependence of birefringence.

Figure 1

Phase diagram in temperature- $\left(T\text{−}\right)$ concentration $\left(c\right)$ space. For materials with excluded volume interactions, increasing the concentration would lead to an increase in the orientational order. The solid line represents the continuous Sm-$\mathit{A}$–Sm-$\mathit{C}$ boundary while the dashed line represents the first-order Sm-$\mathit{A}$–Sm-$\mathit{C}$ boundary. These two boundaries meet at the tricrtical point: $\left(T_{\text{TC}},c_{\text{TC}}\right)$. The dotted line indicates the region in which the behavior in the Sm-$\mathit{C}$ phase crosses over from $XY$-like to tricritical. The region in which the behavior is $XY$-like shrinks to zero as the tricritical point is approached. Also shown as double ended arrows, are the three distinct classes of transitions (at fixed concentration): $XY$-like, tricritical, and first order.

Figure 2

A schematic showing the layer normal and optical axis. The layers are shown as dashed lines. The transition from the Sm-$\mathit{A}$ to -$\mathit{C}$ phase occurs via a tilting, by angle $\theta$, of the optical axis away from the layer normal.

Figure 3

The tilt angle $\theta$ as a function of reduced temperature $t\equiv \left(1-\frac{T}{T_C}\right)$ near the Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition temperature $T_C$, i.e., for $t⪡1$. Upon entry to the Sm-$\mathit{C}$ phase the growth of the tilt angle scales as ${\left|t\right|}^{1/2}$ for a mean-field $XY$-like transition. For a tricritical transition it scales as ${\left|t\right|}^{1/4}$ and is thus more rapid. For a first-order transition there is a jump in the tilt angle upon entry to the Sm-$\mathit{C}$ phase.

Figure 4

The specific heat $c_V$ as a function of reduced temperature $t\equiv \left(1-\frac{T}{T_C}\right)$ near the continuous Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition temperature $T_C$, i.e., for $t⪡1$. As the transition is approached from the Sm-$\mathit{C}$ phase, the specific heat grows as $c_V\propto {\left(1-\frac{T}{T_m}\right)}^{-1/2}$, where $T_m>T_C$. This growth is cut off at $T=T_C$, where it reaches a maximum value $\Delta c_V$. If the transition becomes tricritical $T_m\to T_C$ and $c_V$ diverges at the transition. Note that the specific heat shown here only includes the contribution from the piece of the free energy density associated with the ordering as the system moves into the Sm-$\mathit{C}$ phase. For a first-order transition there will be a latent heat absorbed in going from the Sm-$\mathit{C}$ phase to the Sm-$\mathit{A}$ phase.

Figure 5

The layer contraction ${\Delta }_d\equiv \left(d_{AC}-d_C\right)/d_{AC}$ as a function of ${\theta }^2$ near the Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition. For any type of transition the contraction will scale as $M_0{\theta }^2$. Thus, the slope of ${\Delta }_d$ versus ${\theta }^2$ is proportional to the orientational order $M_0$ in the system. Near tricriticality, the orientational order is small and $M_0⪡1$ and so the contraction is also small. Also shown is the layer contraction for a system with strong orientational order $M_0\approx 1$, for which the contraction will be sizable. For a first-order transition there will be a jump in the tilt angle $\theta$ at the transition and, thus, the ${\Delta }_d$ vs ${\theta }^2$ line does not extend all the way to zero.

Figure 6

(Color online) (a) An oversimplified schematic showing the arrangement of molecules in the Sm-$\mathit{A}$ phase, in which the orientational order is perfect. Such a model predicts that, as the system moves into the Sm-$\mathit{C}$ phase, the layer spacing should contract according to ${\Delta }_d\equiv \left[1-\text{cos}\left(\theta \right)\right]$, where ${\Delta }_d=\left(d_{AC}-d_C\right)/d_{AC}$. (b) A more realistic arrangement of the molecules in which the molecular axes are tilted away from the optical axis, but in azimuthally random directions. The more that the molecules are tilted, the smaller the orientational order. As the system moves into the Sm-$\mathit{C}$ phase, the “pretilted” molecules do not need to tilt but rather need only to order azimuthally, thus leading to an unusually small layer contraction. Thus, the smaller the orientational order in the Sm-$\mathit{A}$ phase, the more pretilted the molecules will be and the smaller the layer contraction will be, an interpretation consistent with our result, Eq. (3). The figure also shows that, as a result of the azimuthal ordering as the system moves into the Sm-$\mathit{C}$ phase, it should become more orientationally ordered.

Figure 7

The fractional change in birefringence ${\Delta }_{\Delta n}\equiv \frac{\Delta n-\Delta n_{AC}}{\Delta n_{AC}}$ as a function of reduced temperature $t\equiv \left(1-\frac{T}{T_C}\right)$ near the Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition temperature $T_C$, i.e., for $t⪡1$. For materials with excluded volume interactions, we expect the birefringence $\Delta n$, and thus ${\Delta }_{\Delta n}$, to decrease as the Sm-$\mathit{A}$–Sm-$\mathit{C}$ transition is approached from within the Sm-$\mathit{A}$ phase. For all three types of transitions ($XY$-like, tricritical, first-order) this decrease will scale linearly $\propto t$ with reduced temperature. Upon entry to the Sm-$\mathit{C}$ phase the birefringence $\Delta n$, and thus ${\Delta }_{\Delta n}$, will grow. The growth is linear $\propto \left|t\right|$ for a mean-field $XY$-like transition. For a tricritical transition the growth scales as $\propto {\left|t\right|}^{1/2}$ and is thus more rapid. For a first-order transition there will be a jump in birefringence as the system enters the Sm-$\mathit{C}$ phase.

Figure 8

(Color online) The unit eigenvectors ${\hat{\mathbf{e}}}_1$, ${\hat{\mathbf{e}}}_2$, ${\hat{\mathbf{e}}}_3$ of the orientational order tensor $\mathcal{Q}$. These are shown as solid arrows, with ${\hat{\mathbf{e}}}_1$ pointing into the page. Also shown, as a dotted arrow, is the layering direction $\hat{\mathbf{z}}$, which is normal to the plane of the layers. The eigenvector ${\hat{\mathbf{e}}}_3$ corresponds to the average direction of the molecules’ long axes. The order parameter $\mathbf{c}$ for the Sm-$\mathit{C}$ phase is the projection of ${\hat{\mathbf{e}}}_3$ onto the plane of the layers, and is shown as a dashed arrow. The angle $\theta$, by which the optical axis tilts, is also shown.

Figure 9

The phase diagram in $t_s\text{−}{M}_0$ space near the tricritical point $\left(t_{s_{\text{TC}}},M_{0_{\text{TC}}}\right)$. The quantity $M_0$ is a measure of how much bare orientational order the system possesses and for de Vries materials is effectively athermal. Increasing concentration should increase $M_0$. The quantity $t_s$ is a monotonic function of temperature so that for a given material, decreasing the temperature corresponds to moving horizontally from right to left. The topology of the corresponding phase diagram in temperature-concentration space should essentially be the same. The solid line represents the continuous Sm-$\mathit{A}$–Sm-$\mathit{C}$ boundary while the dashed line represents the first-order Sm-$\mathit{A}$–Sm-$\mathit{C}$ boundary. These two boundaries meet at the tricritical point $\left(t_{s_{\text{TC}}},M_{0_{\text{TC}}}\right)$. The dotted line indicates the region in which the behavior crosses over from $XY$-like to tricritical. The region in which the behavior is $XY$-like shrinks to zero as the tricritical point is approached. The slopes of the first-order and continuous Sm-$\mathit{A}$–Sm-$\mathit{C}$ boundaries are equal at the tricritical point. Also shown as double ended arrows are the three distinct classes of transitions: $XY$-like, tricritical, and first order.

Figure 10

The size of the specific heat jump $\Delta c_V$ as a function of the system’s orientational order $M_0$. As $M_0\to M_{\text{TC}}$ the transition becomes tricritical and the specific heat jump diverges. For systems with athermal $M_0$ it should be experimentally possible to drive the system to tricriticality by varying the concentration.