Critical behavior at nematic–to–smectic-Aphase transitions for smectic-A1and reentrant smectic-Adphases

We have carried out high-resolution x-ray scattering and heat-capacity studies of the nematic (N)→smectic-A (SmA) transitions in 4’-(4’ ’-n-alkoxybenzyloxy)-4-cyanostilbene with the alkyl chain of length 7 (T7) and 8 (T8). The material T7 exhibits only a single nematic–to–smectic-$A_1$ transition where the Sm$A_1$ period is commensurate with the molecular length l. T8 exhibits with decreasing temperature the doubly reentrant sequence N-Sm$A_d$-N-Sm$A_1$ where the Sm$A_d$ period is incommensurate with d∼1.2l. For both T7 and T8 we find second-order N-Sm$A_1$ transitions with identical reduced amplitudes and exponents; the critical exponents so obtained are γ=1.22±0.06, ${\nu }_{?}$=0.69±0.03, ${\nu }_{\perp }$=0.63±0.03, and α=0.06±0.06. These are close, but not identical, to those expected for an ideal XY transition and they give ${\nu }_{?}$+2${\nu }_{\perp }$+α-2=0.01±0.15, in good agreement with anisotropic hyperscaling. The lower N-Sm$A_d$ transition in T8 gives γ=1.53±0.1, ${\nu }_{?}$=0.88±0.05, and ${\nu }_{\perp }$=0.76±0.05, in disagreement with all models. The heat capacity associated with this transition is too small to yield a reliable value for the critical exponent α. In the reentrant nematic phase of T8, the Sm$A_d$ and Sm$A_1$ fluctuations are essentially independent; the Sm$A_d$ fluctuations change over from being SmA-like to SmC-like with decreasing temperature in the reentrant nematic phase.