Director anchoring on a simple edge dislocation at the surface of induced smectic-C_{A} films

We present a detailed analysis of c-director anchoring measurements on simple edge dislocations at the surface of smectic-C_{A} films (steps). Indications show that the c-director anchoring on the dislocations originates from a local and partial melting of the dislocation core that depends on the anchoring angle. The SmC_{A} films are induced on isotropic puddles of 1-(methyl)-heptyl-terephthalylidene-bis-amino cinnamate molecules by the surface field, while the dislocations are located at the isotropic-smectic interface. The experimental setup is based on the connection of a three-dimensional smectic film sandwiched between a one-dimensional edge dislocation on its lower surface, and a two-dimensional surface polarization spread over the upper surface. Applying an electric field produces a torque that balances the anchoring torque of the dislocation. The film distortion that results is measured under a polarizing microscope. Exact calculations on these data, anchoring torque versus director angle, yield the anchoring properties of the dislocation. A specificity of our sandwich configuration is to improve the measurement quality by a factor of N^{3/2}∼600, where N=72 is the number of smectic layers in the film. We fit a second-order Fourier series on the torque-anchoring angle data, which has the advantage of converging uniformly over the entire anchoring angle range, i.e., over more than 70^{∘}. The two corresponding Fourier coefficients, k_{a1}^{F2} and k_{a2}^{F2}, are anchoring parameters that generalize the usual anchoring coefficient. When changing the electric field E, the anchoring state evolves along paths in a torque-anchoring angle diagram. Two cases occur depending on the angle α_{∞} of E relative to the unit vector S, perpendicular to the dislocation and parallel to the film. When α_{∞}<130^{∘}, the operating point Q that represents the anchoring state in the diagram follows reversible and "at-equilibrium" paths. Its free displacement velocity is infinitely slow, so that we have to push it with electric torque steps smaller than the experimental error bar δΓ∼10^{-14}N. On the other hand, for α_{∞}>130^{∘}, Q describes a hysteresis loop similar to the usually encountered ones in solids. This loop connects two states that exhibit broken and nonbroken anchorings, respectively. The paths that join them in an out-of-equilibrium process are irreversible and dissipative. When coming back to a nonbroken anchoring state, both the dislocation and smectic film spontaneously heal back in the very same state they were before the anchoring broke. The process does not produce any erosion thanks to their liquid nature, including at the microscopic scale. The energy that is dissipated on these paths is roughly estimated in terms of the c-director rotational viscosity. Similarly, we can evaluate the maximum time of flight along the dissipative paths to be of the order of a few seconds, which is consistent with qualitative observations. In contrast, the paths located inside each domain of these anchoring states are reversible and can be followed in an "at equilibrium" manner all along. This analysis should provide a basis for understanding the structure of multiple edge dislocations in terms of parallel simple edge dislocations interacting with each other through pseudo-Casimir forces arising from c-director thermodynamic fluctuations between them.