Entropic chiral symmetry breaking in self-organized two-dimensional colloidal crystals
Long-range chiral symmetry breaking (CSB) has been recently observed in 2D self-organized rhombic crystals of hard, achiral, 72 degree rhombic microparticles. However, purely entropic selection of a CSB crystal in an idealized system of hard achiral shapes, in which attractions are entirely absent and the shape does not dictate a chiral tiling, has not yet been quantitatively predicted. Overcoming limitations of a purely rotational cage model, we investigate a translational–rotational cage model (TRCM) of dense systems of hard achiral rhombs and quantitatively demonstrate that entropy can spontaneously drive the preferential self-organization of a chiral crystal composed of achiral shapes that also tile into an achiral crystal. At different particle area fractions, ϕA, we calculate the number of accessible translational–rotational microstates, Ω, of a mobile central rhomb in a static cage of neighboring rhombs, which can have different orientation angles, γ, relative to the bisector of the crystalline axes. As we raise ϕA, two maxima emerge in Ω(γ) at non-zero cage orientation angles, ±γmax. These maxima correspond to additional translational microstates that become accessible in the CSB crystalline polymorph through reduced translational tip–tip interference. Thus, entropy, often associated with structural disorder, can drive CSB in condensed phase systems of non-attractive achiral objects that do not tile into chiral structures. The success of the TRCM in explaining the entropic origin of CSB in systems of hard rhombs indicates that the TRCM will have significant utility in predicting the self-organized behavior of dense systems of other hard shapes in 2D.