By use of a director representation, it is proved that the disordered phase in cholesterics becomes thermodynamically unstable with respect to a localized mode, It is argued that this instability provides a basis for understanding the experimentally observed "fog" or "gray" phase.PACS numbers: 64.70.Eve The anomalous phases which appear in many cholesteric liquid crystals below their clearing point, and which are known collectively as the "blue phase" (BP), are being intensively investigated. ' ' AIthough the BP region is relatively narrow (1-2 K), it consists of at least three subregions. Two of these (BP I, BP II) have been identified as having cubic structures' ' while the third (BP III, also known as the "gray" or "fog" phase) seems to be amorphous. "' By use of Landau theory based upon a biaxial order parameter, it has been shown" that thermodynamically stable cubic phases can indeed exist in a narrow temperature region between the disordered and ordinary cholesteric phases. Recently, however, an alternate approach has been proposed, "'" in which the order parameter is constrained to be locally uniazial and a cubic lattice is obtained by packing space with an array of interlaced right circular cylinders, in each of which the uniaxial director rotates in a 'curling mode" configuration" (defined below). Here, we (1) derive the connection between Landau theory and the uniaxial approach, (2) show rigorously that the disordered phase becomes thermodynamically unstable, under suitable conditions, with respect to a localized mode possessing cylindrical symmetry while remaining stable with respect to the usual cholesteric phase, and (3) argue that this localized mode provides a basis for understanding BP III.The Landau free energy for chol. esterics is" Z= jd'x(-. 'Iae, , '+c, e. . . '+ ec. . . e", -2de. .. e,"e, ", ] Pe, , e-, , e".+y(e, , ')'].(1) As usual, a is proportional to a reduced temperature; c» c» d, P, and y are temperature-independent parameters; the local biaxial order parameter e, , (x) is the anisotropic part of the dielectric tensor, i.e. , e, , (x) = e, , '(x) --', 5, , Tr(e" ); e. . . =Be, , /Bx» and summation on repeated indices is understood.
