Drops of nematic liquid crystals
The problem of finding the equilibrium shapes of a drop of nematic liquid crystal in contact with an isotropic fluid is fascinating, but hard to solve. I attempt here to crack this problem. Following the pioneers, in Sect. 1 I write the energy functional. Sect. 2 is devoted to the equations which govern the equilibrium configurations of the drop in a class of admissible shapes whose boundaries may have sharp edges. In Sect. 2 a heuristic argument leads to the statement of two limiting problems which apply, respectively, when the drop is small or large, in a sense which does not pertain to the geometric size only. In Sect. 4 a standard surface potential is adopted which depends on a dimensionless parameter \(\omega\). The equilibrium shapes of a small drop are then determined with the aid of a geometric construction due to Wulff. A criterion is worked out which predicts some qualitative properties of the set delivered by Wulff's construction. Use of such a criterion shows that the shapes that have been observed so far are like those the theory predicts when \(\omega >1.\) In Sect. 5 I consider a large drop floating on a dense isotropic liquid and in contact with an isotropic fluid all around the emerging boundary. The surface potential in Sect 4 is employed on the submerged boundary. Under this and further assumptions, the equilibrium shapes of the floating drop are determined for all values of \(\omega\). Once again, the shapes the theory predicts when \(\omega >1\) agree with those that have been observed. I believe that there is enough evidence in this paper to foresee that phase transitions involving the shape are likely to be observed in drops of nematic liquid crystals.
scientific article; zbMATH DE number 4128614