Birefringence in the vicinity of the smectic-$A$ to smectic-$C$ phase transition: Crossover from $XY$ critical to tricritical behavior

High-resolution birefringence ($\mathrm{\Delta }n$) measurements are carried out to probe the critical behavior at the smectic-$A$–smectic-$C$ (Sm-$A$–Sm-$C$) phase transition in a binary system. The critical behavior of this transition is explored with the aid of a differential quotient extracted from the $\mathrm{\Delta }n$ values. The results obtained reveal that the Sm-$A$–Sm-$C$ and nematic–smectic-$A$ ($N$–Sm-$A$) transitions exhibit nonuniversal behaviors with effective exponents lying between the tricritical and three-dimensional $XY$ values and follow two distinctly different curves with decreasing width of the Sm-$A$ and $N$ phases, respectively. The origin of such critical behavior is a unique feature for the respective phase transitions.

Figure 1

Partial phase diagram of the binary system comprised of PhP2 and PhP1. Here $x_{\mathrm{PhP}1}$ denotes the mole fraction of PhP1; $I$: isotropic phase; $N$: nematic phase; Sm-$A$: smectic-$A$ phase; Sm-$C$: smectic-$C$ phase; $○$: isotropic to nematic transition temperature; $•$: nematic to smectic-$A$ transition temperature; $\blacksquare$: smectic-$A$ to smectic-$C$ transition temperature; The structure of the PhP compound is shown. For PhP1, $n=2$; Crystal (359.7 K) $N$ (363.5 K) $I$; For PhP2, $n=7$; Crystal (330 K) Sm-$C$ (352.45 K) Sm-$A$ (363.5 K) $N$ (367.55 K) $I$. The Inset shows the concentration dependence of the Sm-$A$ and nematic ranges for the present system. Solid lines are drawn to guide the eye.

Figure 2

Temperature dependence of birefringence data for the mixture $x_{\mathrm{PhP}1}=0.2$. Vertical arrows show the $I-N, N$–Sm-$A$, and Sm-$A$–Sm-$C$ phase transitions for the mixtures. In the insets variation of birefringence in the vicinity of the Sm-$A$–Sm-$C$ transition [inset (a)] and $N$–Sm-$A$ transition [inset (b)] has been presented.

Figure 3

Temperature dependence of $Q\left(T\right)$ in the vicinity of the (a) Sm-$A$–Sm-$C$ transition and (b) $N$–Sm-$A$ transition in mixtures of $\mathrm{PhP}1+\mathrm{PhP}2$. 1: $x_{\mathrm{PhP}1}=0.0$; 2: $x_{\mathrm{PhP}1}=0.05$; 3: $x_{\mathrm{PhP}1}=0.1$; 4: $x_{\mathrm{PhP}1}=0.15$; 5: $x_{\mathrm{PhP}1}=0.2$; 6: $x_{\mathrm{PhP}1}=0.3$; 7: $x_{\mathrm{PhP}1}=0.35$; 8: $x_{\mathrm{PhP}1}=0.4$; 9: $x_{\mathrm{PhP}1}=0.45$.

Figure 4

Variation of effective critical exponent ${\alpha }^{\prime }$ with $T_{NA}/T_{IN}$ for the $N$–Sm-$A$ transition and $T_{AC}/T_{NA}$ for the Sm-$A$–Sm-$C$ transition. The vertical dash-dotted lines and solid line correspond to the TCP and 3D $XY$ limit, respectively. The solid (red) lines are linear fit to the data.