Chiral symmetry breaking in confined simple fluids.

It has long been recognized that the packing of even simple objects can lead to complicated, large-scale structures. The hard-core, excluded volume fluid has extremely simple interactions [1].  No two particles can come within a particle diameter  of each other.  This interaction leads to non-ideal corrections to the equation of state at high density, the evolution of an oscillatory structure in the sphere pair-correlation function, and an entropy-driven freezing transition to a solid state.  While much is known about the behavior of this model, both analytically, experimentally, and through direct simulation, even this simply stated statistical mechanical system is too
complicated to admit a first-principles, analytic solution.  When the constituent objects in the fluid have a rich internal structure, the situation is even more interesting and complicated. For example, the  excluded volume  polymer solution is composed of long, flexible chains which can cross neither themselves nor their neighbors [2].  A single such chain in three dimensions swells far beyond the ideal random walk chain dimension in response to forbidding any two monomers to occupy the same point in space.  The screening of this excluded volume interaction, and the consequently surprising return of single-chain dimensions to ideal scaling is also governed solely
by this trivial-to-state interaction.  Liquid crystals also exhibit ordering as a result of a subtle competition between reducing the contact of long, rigid objects and maximizing their orientational (for the nematic phase) and translational (in the more ordered smectic phases) entropy [3].

These microscopic interactions are more properly thought of as constraints.  To  solve  the hard-sphere fluid, one merely needs to index all possible positions of all the spheres in the system such that no two spheres overlap.  It is quite possible to add  external constraints to the  internal  non-overlapping constraints in all of these types of systems.  A simple fluid exhibits a featureless ensemble-averaged density field in the bulk, but displays density ripples when brought into contact with a hard, smooth surface [4].  The oscillating structure of the pair correlation indicates that solvation layering occurs in the bulk fluid, and in the presence of a wall this layering results in regions of high and low sphere density.  Thus, even a simple fluid exhibits non-trivial structure in the presence of external constraints.  The presence of an impenetrable wall also effects microscopic ordering in thin films of block copolymers [5], the location of the glass transition temperature [6], and provides boundary conditions determining the degree and type of local liquid crystal order for rod-solutions [3].

Another type of external constraint, confinement, inside a cylindrical tube, is the focus of this project.  Cylindrical confinement is a generic physical condition, applicable to studies of filtration of colloidal suspensions, catalysis inside zeolyte pores [7], and adsorption and drug delivery by nano-tubules [8].  As shown in Fig. 1, packing alone results in rich structure.  Here, I show the close-packed situation for unit diameter spheres packed into a cylinder of diameter.  Radial layering is clearly evident.  Equally evident is another type of ordering not found in bulk liquids: helical.  This ultra-confined example (the confinement dimension is larger than the spheres themselves) exhibits a three-armed helix winding around the cylinder axis to the left.  With equal probability, this configuration could have occurred with the order winding around to the right.  This system exhibits a spontaneously broken chiral symmetry.  This situation is similar to that encountered in minimizing repulsive contacts between point-sites held on the surface of a cylinder: cylindrical phyllotaxis [9].  There, the degree and morphology of the helical order have been determined theoretically under the assumption that all the sites exist on a single helix.  That is, some of the packed states have been characterized for this problem.  What of the less-than-maximally-packed solid, and, even more interestingly, the liquid phases?  Does this broken symmetry characterize these phases?  If so, then I will have demonstrated the existence of a confined chiral liquid that could represent a new class of liquid crystal-like materials: a cholesteric phase induced by the confinement of a simple fluid.  The characterization of such a new phase could be of considerable importance in filtration and enantioselective separation applications [8], and rational drug design.

But, there is one other tantalizing possibility.  Virtually all of the biomolecules active in the processes of life possess a single, distinct, handedness [10].  As these biomolecules can be synthesized from achiral smaller molecules (water, simple hydrocarbons, etc.) a spontaneous breaking of chiral symmetry seems to be necessary in the environment of the prebiotic Earth.  It does not seem likely that cylindrical packing of simple molecules is the dominant mechanism through which this global stereoselection was effected.  It is possible, however, that a combination of pairwise anisotropic, yet achiral, interactions between the spheres can induce a  self-enforced  axial confinement.  Thus, with the correct microscopic interactions, achiral elements can self-assemble into collective structures with strong chirality.  Under these conditions, even a small influence (such as that mediated by the chiral electroweak interaction) can be magnified.  Therefore, the sensitivity to external (chiral) influences of the self-assembled chiral structures is an important quantity to determine.  This study points toward a  steric origin of biochirality  hypothesis.

We have produced an animation showing how the helical ordering changes as the cylinder barrel increases in diameter.  The animation makes a brief pause at each structural transition.  The generic situation is that the structure is chiral, but that the right-handed and left-handed helices appear with equal probability.

We have a preprint on this topic:  chiral.pdf .

References

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last updated: May 16, 2000