Grain boundaries and stacking faults in a Pm3n cubic mesophase

A monodendron that over a broad temperature range self-assembles in quasi-spherical supramolecular aggregates organized on a thermotropic cubic Pm3n lattice was examined by TEM. Grain boundaries were found parallel to the (1 0 0) and (3 2 0) planes. In well annealed specimens, in which grain boundaries are rare, two types of stacking fault were numerous, suggesting their relatively low energy. In the original Pm 3n lattice and in the two stacking fault arrangements, the supramolecular aggregates are distorted to an oblate shape, which is likely to be favoured by a crowded microscopic interface between core and tail moieties. Symmetry and phase selection for the ordered arrangement of amphiphilic molecules is discussed.     

Further Accurate Representations of Partial Differential Equations by Finite-difference Methods

A fourth-order finite-difference scheme recently introducedfor the solution of second-order partial differential equationsis developed. The resulting scheme retains all the advantagesof the original, but is more satisfactory in that the simultaneousalgebraic equations to be solved are more amenable to solutionby numerical techniques in many cases. Some numerical solutionsof the Navier-Stokes equations are considered.     

Accurate Representations of Partial Differential Equations by Finite-difference Schemes

A Specialized method introduced some years ago for obtainingfinite-difference approximations to both ordinary and partialdifferential equations is developed to give representationsof higher accuracy. A numerical example is considered whichis based on a model solution of the two-dimensional Navier-Stokesequations for the flow of an incompressible fluid.