Quadruple reentrance (nematic-smectic-${\mathit{A}}_{\mathit{d}}$-nematic-smectic-${\mathit{A}}_{\mathit{d}}$-nematic-smectic-${\mathit{A}}_1$) from the frustrated spin-gas model of liquid crystals

The microscopic ‘‘spin-gas’’ model reproduces and physically explains the sequence of liquid-crystal phases, nematic–smectic-$A_d$–nematic–smectic-$A_d$ –nematic–smectic-$A_1$, which was observed with decreasing temperature in recent experiments. The microscopic frustration inherent to smectic structures of dipolar molecules is relieved by permeation fluctuations at the atomic and librational length scales. The molecular tail length theoretically required for this quadruple reentrance agrees with experiment. The calculated ratios of layer thicknesses in the partial bilayer ($A_d$) and monolayer ($A_1$) smectic phases are also in good agreement with experiment.