Chiral segregation in three microscopic statistical-mechanical models.

We consider three microscopic model molecular systems, each containing an equimolar mixture of a chiral molecule and its nonsuperimposable mirror image. The molecules in each model are assumed to lie on a thin film in such a way that they occupy the sites of a honeycomb lattice. Although neither enantiomorph is externally favored at low temperatures, we prove that for one range of interactions, chiral segregation into ordered phases containing a single enantiomorph occurs for two of the models and, in a second range of interactions, ordered racemic phases (containing equal numbers of each enantiomorph) occur for the two models. For a third range of interactions, each of the two models has an infinite number of ground-state configurations and, moreover, an associated residual entropy. In all three ranges of interactions considered, the third model has an infinite number of ground-state configurations and a residual entropy.

The microscopic structure of an exactly solvable model binary solution that exhibits two closed loops in the phase diagram.

An exactly solvable lattice model describing a binary solution is considered where rodlike molecules of types AA and BB cover the links of a honeycomb lattice, the neighboring molecular ends having three-body and orientation-dependent bonding interactions. At phase coexistence of AA-rich and BB-rich phases, the average fraction of each type of triangle of neighboring molecular ends is calculated exactly. The fractions of the different types of triangles are then used to deduce the local microscopic structure of the coexisting phases for a case of the model that contains two closed loops in the phase diagram.

Low temperature phases of the Andelman-de Gennes model of chiral discrimination: rigorous results.

We determine the ordered low temperature phases of the Andelman-de Gennes model of chiral discrimination, using rigorous statistical mechanical methods. The system is considered in the close-packed regime, equivalent to placing the molecules at every site of a honeycomb lattice. If the system contains an equimolar mixture of each of a pair of enantiomers, we prove in general that a heterochiral phase (disfavoring enantiomeric segregation) as well as a homochiral phase (favoring the segregation) is possible, depending on the types of intermolecular interactions. We apply our general results to the specific examples of the interactions considered by Andelman and de Gennes and provide a comparison with their conjectures that were based on two-molecule partition functions.

Exact phase boundary of an amphiphile-rich phase in a ternary solution on a Bethe lattice.

A model ternary solution is considered in which the bonds of a three-coordinate Bethe lattice are covered by rodlike molecules of types AA, BB, and AB, the AB-type molecule representing an amphiphile. The phase boundary in temperature-composition space of an ordered amphiphile-rich phase is calculated exactly.

Phase separation in an exactly solvable model binary solution with three-body interactions and intermolecular bonding.

A model is presented in which the bonds of a honeycomb lattice are covered by rodlike molecules of types AA and BB, molecular ends near a common site having both three-body interactions and orientation-dependent bonding between two A molecular ends and between an A and a B molecular end. Phase diagrams corresponding to the separation into AA-rich and BB-rich phases are calculated exactly. Depending on the relative strengths of the interactions, one of several qualitatively different types of phase diagrams can result, including diagrams containing phenomena such as a double critical point or two separate asymmetric closed loops. The model is essentially a limiting case of a previously considered ternary solution model, and it is equivalent to a two-component system of interacting A and B molecules on the sites of a kagomé lattice.

An exactly solvable model for a ternary solution with three-body interactions and orientationally dependent bonding.

A model is considered in which the bonds of a honeycomb lattice are covered by rodlike molecules of types AA, BB, and AB. Neighboring molecular ends have three-body and orientation-dependent interactions. The model is shown to be equivalent to a spin-1/2 Ising model on the same lattice with a field, but with only pairwise interactions. Symmetric and asymmetric coexistence surfaces for the separation into an AA-rich and a BB-rich phase are calculated exactly.