| DNA and certain species of cationic and neutral lipids self-assemble into lamellar complexes when mixed in water. X-ray scattering experiments have determined that these complexes are highly organized with DNA molecules located in the galleries between lipid bilayers. Within each gallery, DNA molecules form a two-dimensional (2D) smectic lattice. We model the lipid and DNA complexes as three-dimensional stacks of weakly-coupled 2D smectic lattices and study the phase diagram of this model as a function of temperature. The various phases are distinguished by the strength of the translational and orientational correlations between neighboring smectic lattices. For example, it is possible for a columnar phase to form at low temperature with long-range translational correlations between smectic lattices. In the columnar phase, the DNA molecules form a 2D crystal lattice in the plane perpendicular to their column axes, and there is a nonzero shear modulus for sliding neighboring lattices relative to each other. As temperature is increased, thermal fluctuations may induce a second-order phase transition to the sliding columnar phase. This phase is characterized by strong in-plane smectic correlations, a vanishing shear modulus for sliding lattices across each other, and a nonvanishing orientational rigidity for rotating lattices relative to each other. Thus, neighboring lattices are able to slide but not rotate relative to each other without energy cost. We calculate several important structural properties of the sliding columnar phase, for instance, the sliding columnar density-density correlation and structure functions. The sliding columnar correlation function is unique in that in-plane correlations are weaker than any power-law and decay with separation r as exp[-- ln2 r]. We also calculate the energy cost for edge dislocations in the sliding columnar phase and show that, at the longest lengthscales, a dislocation unbinding transition from the columnar to the nematic lamellar phase precludes the columnar to sliding columnar phase transition. However, if further-neighbor orientational interactions between smectic lattices are permitted, there is a temperature range where the sliding columnar phase is thermodynamically stable. |
