Lehmann effect in nematic and cholesteric liquid crystals: a review

ABSTRACT

The Lehmann effect is the continuous rotation of cholesteric droplets subjected to a temperature gradient. Discovered by Otto Lehmann in 1900, this effect was re-observed recently by several authors not only in cholesterics but also in nematics when the director field is twisted inside the droplets. In most experiments, the droplets coexist with their isotropic liquid, but the Lehmann effect can also be observed when the droplets are dispersed in an isotropic liquid in which the LC is partly miscible. After a brief history on the Lehmann effect and its first explanation by Leslie in 1968, we will review the main experimental results obtained on this subject from 2008. In particular, the role of the temperature gradient, of the size of the droplets, of the textures and their orientation with respect to the temperature gradient, of the confinement effects, of the impurities and of the concentration of chiral molecules will be described. A special emphasis will also be placed on the research of hydrodynamic effects to answer the fundamental question of whether it is just the texture or the droplet itself that rotates. We will then review the different models proposed in the literature to explain the Lehmann effect. Among them are two thermomechanical models directly based on the Leslie explanation (named TM1 and TM2 models), a thermomechanical model of rotating texture ‘surfing’ on a heat wave (TM3 model), a model of melting-growth (MG model) that only applies when the droplets coexist with their own isotropic liquid, and a pure hydrodynamic model (H model) based on the existence of Marangoni flows – currently, only evidenced in emulsified cholesterics. The strengths and weaknesses of each model will be discussed in relation with the experimental results.

KEYWORDS:

• Nematic and cholesteric liquid crystals
• thermomechanical effects
• out-of-equilibrium thermodynamics

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that because of the errors on the measurement of $C^{\star }$, a very slow rotation due to the Leslie effect as the one described in Ref. [81] cannot be excluded at this concentration.

2 Note that this mechanism does not depend on the polar or azimuthal nature of the anchoring: if we assume an easy axis $\mathit{p}$ (which can be planar, homeotropic or tilted with respect to the surface) for the interaction energy between the cholesteric droplet and the particle, the particle will simply minimize this interaction energy by shifting its position to a point where $\mathit{n}=\pm \mathit{p}$ (the two signs are due to the symmetry $\mathit{n}\to -\mathit{n}$).

3 Note that the thermomechanical effects of Leslie, Akopyan and Zel'dovich are very much relevant in experiments performed below the transition temperature with fully cholesteric samples [28,33,34,76–78,81,96] or nematic samples with a deformed texture [83]. Such systems are outside the scope of this paper, which only focus on the Lehmann rotation of cholesteric droplets.

4 In the original paper of Akopyan and Zel'dovich, the phenomenological equations do not respect Onsager reciprocity relations. This problem is addressed in the rigorous framework used by Pleiner and Brand to derive the same type of terms.

5 In principle, this torque can also contain viscous and thermomechanical contributions coming from the surfacic production of entropy, but these terms are generally neglected in cholesterics and nematics.

6 Indeed, in the Ericksen–Leslie theory the thermomechanical stress and torque of Leslie, Akopyan and Zel'dovich just appear as additional terms in the general expression of the internal viscous stress and torque. These terms proportional to the temperature gradient are allowed by symmetries only in pure cholesterics (Leslie terms) and in deformed nematics and cholesterics (Akopyan and Zel'dovich terms).

7 Note that this equation could also been obtained by applying the angular momentum theorem to a ball of radius $R_b$ larger than R (the radius of the droplet) and by then making $R_b$ tend to R.

8 Note that in this calculation we consider that f (or W) is an explicit function of T and the spatial coordinates: $f(T,r,\theta ,z)$. For this reason $\left(\mathrm{\partial }f/\mathrm{\partial }T\right)\left(\mathrm{\partial }T/\mathrm{\partial }\theta \right)$ is in general different from $\mathrm{\partial }f/\mathrm{\partial }\theta$, the integral of which is equal to 0 as underlined by one of the reviewers. Indeed, let us suppose that $f(T,r,\theta ,z)=Th(r,\theta ,z)$. Then, $\left(\mathrm{\partial }f/\mathrm{\partial }T\right)\left(\mathrm{\partial }T/\mathrm{\partial }\theta \right)=h\left(\mathrm{\partial }T/\mathrm{\partial }\theta \right)\ne \mathrm{\partial }f/\mathrm{\partial }\theta =T\left(\mathrm{\partial }h/\mathrm{\partial }\theta \right)+h\left(\mathrm{\partial }T/\mathrm{\partial }\theta \right)$ (by assuming, this time, that f is a function of $(r,\theta ,z)$ only).

9 This can be checked by noting that the rigorous (covariant) expression of $\mathrm{\partial }T/\mathrm{\partial }\theta$ is $(\overrightarrow{u}\times \overrightarrow{r})\cdot \overrightarrow{\mathrm{\nabla }}T$, where $\overrightarrow{u}$ is the polar axis and $\overrightarrow{r}$ is the spatial position. Since $(\overrightarrow{u}\times \overrightarrow{r})$ is a pseudo-vector (it changes sign under the mirror transformation) and $\overrightarrow{\mathrm{\nabla }}T$ is a true vector (it stays invariant under the mirror transformation), $\mathrm{\partial }T/\mathrm{\partial }\theta$ changes its sign under the transformation $q\to -q$ (equivalent to a mirror transformation).

10 These results show that the droplets do not work as mechanical turbines at all, as might be intuitively thought.