Test of Halperin-Lubensky-Ma crossover function at the $N–\mathrm{Sm}-A$ transition in liquid crystal binary mixtures via high-resolution birefringence measurements

We report optical birefringence data for a series of mixtures of the liquid crystals octylcyanobiphenyl (8CB) and decylcyanobiphenyl (10CB). Nematic order parameter $S$ data in the nematic and smectic $A$ phases have been derived from phase angle changes obtained in temperature scans with a rotating analyzer method. These $S$ values have been used to arrive at values for possible entropy discontinuities at the smectic $A$ to nematic phase transition temperature $T_{NA}$. The 10CB mole fraction dependence of the obtained entropy discontinuities could be well fitted with a crossover function consistent with the mean-field free-energy expression with a nonzero cubic term arising from the coupling between the smectic-$A$ order parameter and the orientational order parameter director fluctuations in the Halperin-Lubensky-Ma theory. The obtained results are in good agreement with existing results from adiabatic scanning calorimetry. By exploiting the fact that the temperature derivative of the order parameter $S\left(T\right)$ near $T_{NA}$ exhibits the same power law divergence as the specific heat capacity, we have extracted the effective critical exponent $\alpha$ values for the compositions under study. The critical exponent $\alpha$ has been observed to reach the tricritical value ${\alpha }_{\mathrm{TCP}}=0.5$ for the 10CB mole fraction of $x=0.330$.

Figure 1

Partial phase diagram of 8CB-10CB binary mixtures as a function of mole fraction $x$ of 10CB. I: isotropic phase, N: nematic phase, $SmA$: smectic-$A$ phase. Notice that the solid symbols are the optical birefringence data, and the open symbols refer to the calorimetric data from Ref. [14]. For pure 8CB the N-I and $N-\mathrm{Sm}A$ transition temperatures, extracted from $\mathrm{\Delta }n\left(T\right)$ data, are from Ref. [35].

Figure 2

Temperature dependence of the measured birefringence $\mathrm{\Delta }n$ for the binary mixtures $8{\text{CB}}_{1-x}10{\text{CB}}_x$, with $x$ mole fraction of 10CB. (a) $x=0$, pure 8CB, from Ref. [35], $x=0.099, 0.179$, and $0.199$, (b) $x=0.300, 0.330, 0.400, 0.430, 0.499, 0.569$, and $0.629$. Different mole fractions $x$ of 10CB are given.

Figure 3

The derived reduced entropy discontinuity $\overline{s}=\frac{\mathrm{\Delta }\overline{S}}{R}$($R$ being the gas constant) vs mole fraction $x$ of 10CB fitted to the universal scaling function of Eq. (8) to extract $x^{*}$ and ${\overline{s}}^{*}=\frac{\mathrm{\Delta }{\overline{S}}^{*}}{R}$ of the LTP.

Figure 4

The reduced entropy discontinuity $\overline{s}$ of Fig. 3 vs the mole fraction $x$ of 10CB on a semilogarithmic scale. Different lines represent fits with different data sets as indicated.

Figure 5

Comparison in a semilogarithmic plot of the different data sets with the universal curve resulting from Eq. (8).

Figure 6

Temperature behavior of the quotient $\mathrm{\Delta }Q\left(T\right)$ in the vicinity of the $N-\mathrm{Sm}A$ transition for the binary mixtures $0.099\le x\le 0.199$ together with pure 8CB. Solid lines are the fits to Eq. (15). The different mole fractions of 10CB are indicated in the figure.

Figure 7

The $x$ dependence of the effective critical exponent $\alpha$ (asterisks), and the reduced entropy discontinuity $\overline{s}$ (solid circles) for the binary mixtures $8{\text{CB}}_{1-x}10{\text{CB}}_x$. Here $x^{*}$ refers to the mole fraction of the Landau tricritical point (see text).