  The nematic twist-bend (TB) phase, exhibited by certain achiral thermotropic
liquid crystalline (LC) dimers, features a nanometer-scale, heliconical
rotation of the average molecular long axis (director) with equally probable
left- and right-handed domains. On meso to macroscopic scales, the TB phase may
be considered as a stack of equivalent slabs or "pseudo-layers", each one
helical pitch in thickness. The long wavelength fluctuation modes should then
be analogous to those of a smectic-A phase, and in particular the hydrodynamic
mode combining "layer" compression and bending ought to be characterized by an
effective layer compression elastic constant $B_{eff}$ and average director
splay constant $K_1^{eff}$. The magnitude of $K_1^{eff}$ is expected to be
similar to the splay constant of an ordinary nematic LC, but due to the absence
of a true mass density wave, $B_{eff}$ could differ substantially from the
typical value of $\sim 10^6$ Pa in a conventional smectic-A. Here we report the
results of a dynamic light scattering study, which confirms the "pseudo-layer"
structure of the TB phase with $B_{eff}$ in the range $\sim 10^3-10^4$ Pa. We
show additionally that the temperature dependence of $B_{eff}$ at the TB to
nematic transition is accurately described by a coarse-grained free energy
density, which is based on a Landau-deGennes expansion in terms of a heli-polar
order parameter that characterizes the TB state and is linearly coupled to bend
distortion of the director.
