Abstract
Electron microscopy observations of replicas of freeze-fractured samples of two columnar hexagonal phases of different nature (a lyotropic one, the inverse AOT in water; a thermotropic one; C8HET) yield very different results: most defects at microscopic scales are screw dislocations in the lyotropic phase, longitudinal edge dislocations in the thermotropic phase. A possible way to interpret these differences is as follows: in the lyotropic the Lame coefficients lambda and mu and the bend modulus K-3 would not display any anomaly compared to expected values; in the thermotropic the shear modulus I-L would be ten times smaller than the compressibility modulus lambda, while K-3 would Still be comparable to (but larger than) the bend modulus of a small molecules liquid crystal. We present an elementary theoretical model of the latter case which could explain the anomalous measurements of Kg and of the longitudinal compressibility B-parallel to (Ref. [10]) without contradicting more recent measurements of (Refs. [17,22]). Essentially, the C8HET hexagonal phase would be a phase with defects (longitudinal dislocations) akin to an hexatic phase but with some differences.