Measurement of the Continuous Lehmann Rotation of Cholesteric Droplets Subjected to a Temperature Gradient

Abstract

In 1900, Otto Lehmann observed the continuous rotation of cholesteric drops when subjected to a temperature gradient. This thermomechanical phenomenon was predicted 68 years later by Leslie from symmetry arguments but was never reobserved to our knowledge. In this Letter, we present an experiment allowing quantitative analysis of the Lehmann effect at the cholesteric-isotropic transition temperature. More precisely, we measure the angular velocity of cholesteric drops as a function of their size and the temperature gradient and we show that applying an electric field can stop the drop rotation. From these observations and a theoretical model we estimate the Lehmann coefficient nu.

Lehmann rotation of the cholesteric helix in droplets oriented by an electric field

“…By writing that this torque equilibrates with the viscous torque and by assuming that there is no flow of matter (we will come back later to this assumption), it was shown that the droplets rotate with an angular velocity ω given by [2,3] − νG γ 1 ω = 1 + drop e z · ∂ n ∂θ × n + ∂ n ∂θ 2 dV drop e z · ∂ n ∂θ × n + 1 − ( e z · n) 2 dV . (2) In this expression, e z is the unit vector parallel to the temperature gradient, θ the polar angle, and γ 1 the rotational viscosity. This equation shows that the texture angular velocity is proportional to G and has the form ω = −(1 + A)νG/γ 1 , where A is a dimensionless coefficient which depends on the director field and on the drop geometry.…”

“…The experiment was first reproduced in 2008 by using a compensated cholesteric (mixture of octyloxycyanobiphenyl and cholesteryl chloride in equal proportions) [2,3] and 1 year later by using diluted cholesteric mixtures (i.e., nematic phase doped with a small amount of chiral molecules) [4,5]. In all of these experiments, the droplets coexist with their isotropic liquid and have a banded texture, indicating that the helical axis is rather perpendicular to the temperature gradient.…”

“…[2,3] by changing the orientation of the helix in the droplets. For this purpose, we use a cholesteric liquid crystal of negative dielectric anisotropy and apply an ac electric field parallel to the temperature gradient to change the orientation of the helix.…”

“…This equation shows that the texture angular velocity is proportional to G and has the form ω = −(1 + A)νG/γ 1 , where A is a dimensionless coefficient which depends on the director field and on the drop geometry. This coefficient was calculated by making different assumptions on the shape of the droplet (flattened cylindrical box or spherical cap [7,8]) and the structure of the director field (perfect helix [7,8] or helix strongly deformed by the the anchoring conditions at the surface of the droplet [2]). In all cases, it was shown that A * patrick.oswald@ens-lyon.fr was an even and rapidly increasing function of the product kR, where k is the wave vector of the banded texture (in practice, always close to q, the equilibrium twist of the phase) and R the radius of the droplet.…”

Minimal Energy Dissipation Rate and Director Orientation Relative to External Dissipative Fields such as Temperature and Velocity Gradients in Nematic and Cholestric Liquid Crystals

“…On the other hand, molecular dynamics simulation of nematic phases of calamitic and discotic soft ellipsoids [17][18][19] clearly show that they orient perpendicularly and parallel, respectively, to the temperature gradient, so that the heat flow and thereby the energy dissipation rate are minimized. However, one system, where the director definitely orients perpendicularly to the temperature gradient, is the cholesteric liquid crystal, where the cholesteric axis orients parallel to the temperature gradient, so that the director becomes perpendicular to this gradient and the heat flow is minimized [1,2,[20][21][22]. This is also in agreement with the above-mentioned dissipation theorem even though the torque orienting the director is proportional to the square of the temperature gradient, whereas it is directly proportional to the velocity gradient in shear flow and elongational flow.…”