Lamellar Phases in Nonuniform Electric Fields: Breaking the In-Plane Rotation Symmetry and the Role of Dielectric Constant Mismatch

We consider orientational transitions of lamellar phases under the influence of a spatially nonuniform electric field. The transition between parallel and perpendicular lamellar stackings with respect to the substrate is investigated as a function of the system parameters. The dielectrophoretic energy and the energy penalty for having dielectric interfaces perpendicular to the field’s direction are identified as linear and quadratic terms in a free energy expansion in the dielectric constant mismatch. We find that if the dielectric constant mismatch $\mathrm{\Delta }\text{eps}$ is smaller than some critical value $\mathrm{\Delta }{\text{eps}}_c$, parallel lamellar stacking will be realized, no matter how large the voltage difference between electrodes is. At $\mathrm{\Delta }\text{eps}>\mathrm{\Delta }{\text{eps}}_c$, perpendicular stacking will appear if the voltage is high enough. Nonuniform fields remove the in-plane degeneracy present in the more common uniform fields. We therefore calculate the energy of grains of different orientations. The torque acting on the grains leads to the preference of only one orientation. The results have direct implications to block copolymer orientation and to surface patterning on the nanometer scale.