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ABSTRACT
By use of a standard theoretical method, we arrive at a coarse-grained elastic energy density for twist-bend nematic (Ntb) phases which features two fields, a director t designating the outer macroscopic optic axis and a variable cone angle reflecting the inner heliconical microscopic structure. The theory describes the Ntb phase through a nematic-like energy with an extra scalar order parameter. The coupling with an external field is similarly described at the same coarse-grained level and a field-induced transition to the ordinary nematic phase is also predicted to take place in a confined Frederiks cell, in the simplified setting where natural boundary conditions are prescribed on the cell’s plates.
Graphical Abstract
Acknowledgments
We are grateful to both anonymous referees who offered critical and insightful readings of our paper. We are equally grateful to the anonymous Arbiter of our paper who helped us reconciling the two (contrasting) Referees’ reports. In particular, we found it both wise and inspiring to remind us of Boltzmann’s theoretical pluralism. In the Arbiter’s own words (which we take the liberty of reproducing here): ‘Perhaps it is convenient to invoke the testimony of L. Boltzmann who, in several parts of his writings, asserts that no scientific theory could exclude other theories because this eliminates the possibility of progress in science’ (See, L. Boltzmann, Theoretical and philosophical problems: selected writings, edited by B. McGuiness, Reidel, Dordrecht, 1974).
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1. Often in the literature, cholesterics are also called chiral nematics; we shall refrain from using this appellation to avoid confusion with the phase chirality (with no molecular origin) encountered in this paper.
2. In an earlier contribution [Citation1], Meyer had envisaged the same phase without proposing any elastic theory for it.
3. The ideal cone angle is an equilibrium parameter, as it designates the preferred value of the cone angle
in the absence of any elastic distortion. In the presence of distortions, we may allow
to vary in space.
4. It is a bit unfortunate, but overall inessential that twist tensors in represent director fields with negative helicity, whereas twist tensors in
represent director fields with positive helicity.
5. The microscopic elastic theory of [Citation14] could also be termed a ‘two-director theory’.
6. Though perhaps already convincing, this argument is further expounded in Appendix 1.
7. Arriving at (18) is a bit more laborious than arriving at (10). The interested reader is referred to Appendix 3 for an illustration of the method followed here. See also [35, p. 263] and [26, p. 146] for other applications of this method.
8. Some useful indications for the critical reader are given in Appendix 2.
9. This also agrees with the view expressed in [Citation15,Citation31] that should be used as a scalar order parameter to describe the nematic-to-Ntb phase transition; x vanishes in the classical nematic phase and grows to
, either continuously or through a jump as the Ntb-phase establishes itself.
10. Our way of writing the electric energy calls for a word of caution. The simple formula (35) applies when the local electric field is taken to be as if produced in a vacuum, ignoring that the dielectric displacement must solve the Maxwell equations. Coupling the molecular field with Maxwell equations, as done classically in [Citation32], here would require coarse-graining them too, which is perhaps a project worth pursuing.
11. In general, the choice of appropriate boundary conditions for the coarse-grained free energy functional is not an easy task, as it is not clear how different anchoring devices for the molecular director m translate into conditions for the coarse-grained fields t and
. Equation (38) has the merit of being simple enough.
12. Not to mention the coupling between tilt and polarisation that is the basis for the electroclinic effect in Ntb phases, also predicted in [Citation33] by a different approach, in analogy to the classical chiral smectic-A phases [Citation34].