Positional ordering and close packing of hard spheres in nanochannels

The phase behavior of a quasi-one-dimensional fluid of hard spheres is studied in hard cylindrical and rectangular straight channels using the transfer operator method. The radius of the channel (R) is chosen such that only first-neighbor interactions are present, which allows only fluidlike and zigzag solidlike structures to form. It is managed to show that the fluid-solid structural change occurs approximately at ρ=1/〈σ_{∥}〉, where ρ is the one-dimensional density and 〈σ_{||}〉 is the average of the longitudinal contact distance in the case of uniform particle distribution. At this density, the pressure of positionally disordered structure diverges, but the pressure ratio (P/P_{T}), where P is the pressure of the system and P_{T} is the pressure of an effective one-dimensional hard rod fluid, exhibits a peak. At the close packing, while the positional fluctuation (〈Δr^{2}〉) vanishes with P^{β[over ̃]}, where β[over ̃]=-2 for any R and cross section, P/P_{T} (we call it α[over ̃]) goes to 2 in the rectangular and 2.5 in the cylindrical channel. The most striking difference can be observed in the transversal position correlation length (ξ). It exhibits exp(ΔP) dependence in the rectangular channel with Δ being the extra length necessary to create a defect. However, ξ goes with P^{γ[over ̃]} in the cylindrical channel, where γ[over ̃]=1. Interestingly, the cylindrically confined hard spheres obey the sum rules α[over ̃]+β[over ̃]=1/2 and β[over ̃]+γ[over ̃]=-1 for any R.