Surface and finite-size effects on $N-\mathrm{Sm}\text{−}A-\mathrm{Sm}\text{−}C$ phase transitions in free-standing films

The present study is devoted to the investigation of surface anchoring and finite-size effects on nematic–smectic-$A$–smectic-$C$ $(N-\mathrm{Sm}\text{−}A-\mathrm{Sm}\text{−}C)$ phase transitions in free-standing films. Using an extended version of the molecular theory for smectic-$C$ liquid crystals, we analyze how surface anchoring and film thickness affect the thermal behavior of the order parameters in free-standing smectic films. In particular, we determine how the transition temperature depends on the surface ordering and film thickness. We show that the additional orientational order imposed by the surface anchoring may lead to a stabilization of order parameters in central layers, thus modifying the nature of the phase transitions. We compare our results with experimental findings for typical thermotropic compounds presenting a $N-\mathrm{Sm}\text{−}A-\mathrm{Sm}\text{−}C$ phase sequence.

Figure 1

(a) Schematic representation of the long molecular axis $\vec{a}$ in the Sm-$C$ phase, for a Cartesian coordinate system with the $z$ axis being normal to the smectic layer plane. For convenience, we assume that the director $\vec{n}$ is restricted to the $z\text{−}x$ plane, defined as the tilt plane. Here, the vector $\vec{c}$ corresponds to the projection of the director in the $x$ axis, while $\omega$ represents the tilt angle of the director $\vec{n}$ in relation to the smectic wave vector $\vec{q}$. The orientation of the long molecular axis is defined by polar and azimuthal angles $\theta$ and $\varphi$, respectively. $\psi$ is the relative angle between the long molecular axis $\vec{a}$ and director $\vec{n}$. (b) Representation of the layer contraction induced by the molecular tilt, with $d_0$ being the layer spacing in the Sm-$A$ phase.

Figure 2

(a) Helmholtz free energy as a function of the tilt angle $\omega$, at the vicinity of the Sm-$C-\mathrm{Sm}\text{−}A$ transition temperature, $T_{CA}$. We use representative values of the model parameters: ${\alpha }_0=0.85$ and $\beta =0.33$. Notice that a nonnull tilt angle corresponds to a minimum of the Helmholtz free energy for $T<T_{CA}$. (b) Temperature dependence of tilt, $\eta$, translational, $\sigma$, and orientational, $s$, order parameters for ${\alpha }_0=0.85$ and $\beta =0.33$. We observe that $\eta$ decays continuously as the system temperature is increased, while $\sigma$ and $s$ stay finite, signaling a second-order Sm-$C-\mathrm{Sm}\text{−}A$ phase transition.

Figure 3

The tilt angle as a function of $T/T_C$ for $\beta =0.33$ and three representative values of the geometric parameter ${\alpha }_0$: ${\alpha }_0=0.75$ (black solid line), ${\alpha }_0=0.85$ (red dashed line), and ${\alpha }_0=0.98$ (blue dotted line), where $T_C$ represents the temperature where $\eta$ order parameter vanishes. Notice that the continuous decay of the tilt angle is replaced by a discontinuous behavior as the parameter ${\alpha }_0$ is increased.

Figure 4

The phase diagram in the reduced temperature vs ${\alpha }_0$ plane for different values of the parameter $\beta$. (a) $\beta =0.30$, (b) $\beta =0.33$, (c) $\beta =0.40,$ and (d) $\beta =0.70$. Solid (dotted) lines corresponds to first-order (second-order) transitions. As the parameter $\beta$ increases, the Sm-$A$ phase disappears.

Figure 5

Tilt angle profile for a $N=15$ layer film, for distinct temperatures: $T=0.880T_{NI}$ (black circles) and $T=0.906T_{NI}$ (red squares). We consider ${\omega }_0=0.26$, ${\alpha }_0=0.85,$ and $\beta =0.33$. For these parameters, bulk Sm-$C-\mathrm{Sm}\text{−}A$ transition temperature is estimated as $T_{CA}=0.902T_{NI}$. Notice that a pronounced reduction takes place in the tilt angle of central layers when $T>T_{CA}$, while a nonnull tilt persists in outermost layers.

Figure 6

Profiles of (a) orientational, (b) translational, and (c) tilt-order parameters for a free-standing film with $N=15$, at different temperatures: $T=0.880T_{NI}$ (black circles) and $T=0.906T_{NI}$ (red squares). The model parameters are the same used in Fig. 5, where the bulk Sm-$C-\mathrm{Sm}\text{−}A$ transition temperature takes place at $T_{CA}=0.902T_{NI}$. We use $W_0/V_0=0.25$, corresponding to the regime of weak anchoring condition. Notice that the tilt-order parameter stays finite in the surface layers above the bulk Sm-$C-\mathrm{Sm}\text{−}A$ transition temperature, even in the regime of weak surface anchoring.

Figure 7

Profiles of (a) orientational, (b) translational, and (c) tilt-order parameters for a free-standing smectic film, at the vicinity of bulk Sm-$C-\mathrm{Sm}\text{−}A$ transition temperature ($T_{CA}=0.902T_{NI}$). We consider different film temperatures: $T=0.880T_{NI}$ (black circles) and $T=0.906T_{NI}$ (red squares). The model parameters are ${\alpha }_0=0.85$, $\beta =0.33$, ${\omega }_0=0.26$, $N=15,$ and $W_0/V_0=2.5$. We notice an enhancement in the tilt-order parameter due to the strong anchoring condition.

Figure 8

Thickness dependence of order parameters in central layer of free-standing Sm-$C$ films: nematic ($s_{cl}$, black circles), smectic (${\sigma }_{cl}$, red squares), and tilt (${\eta }_{cl}$, blue diamonds) order parameters. The tilt angle of central layer is also shown (${\omega }_{cl}$, green triangles). Different regimes of surface anchoring are considered: (a) $W_0=0.25$ and (b) $W_0=2.50$. The model parameters are ${\alpha }_0=0.85$, $\beta =0.33,$ and ${\omega }_0=0.26$. Notice that the surface anchoring plays an important role in central order parameters of thin films.

Figure 9

Temperature dependence of order parameters at the central layer in smectic films with different thicknesses and anchoring conditions: (a) $N=7$ and $W_0=0.25$, (b) $N=7$ and $W_0=2.50$, (c) $N=25$ and $W_0=0.25,$ and (d) $N=25$ and $W_0=2.50$. Notice that the interplay of finite size effects and surface anchoring may change the phase diagram of free-standing films. The gray areas represent the temperature regions where a dramatic reduction in the smectic order takes place, corresponding to an unstable film.