Theory of rotational columnar structures of soft spheres

Abstract

There is a growing interest in cylindrical structures of hard and soft particles. A promising new method to assemble such structures has recently been introduced by Lee et al. [T. Lee, K. Gizynski, and B. Grzybowski, Adv. Mater. 29, 1704274 (2017)]. They used rapid rotation around a central axis to drive spheres of lower density than the surrounding fluid towards this axis. This resulted in different structures as the number of spheres is varied. Here we present comprehensive analytic energy calculations for such self-assembled structures, based on a generic soft sphere model, from which we obtain a phase diagram. It displays interesting features, including peritectoid points. These analytic calculations are complemented by preliminary numerical simulations for finite sample sizes with soft spheres. A similar analytic approach could be used to study packings of spheres inside cylinders of fixed dimensions, but with a variation in the number of spheres.

Buckling of a linear chain of hard spheres in a harmonic confining potential: Numerical and analytical results for low and high compression

“…As an example we show in We have repeated such calculations for a range of values of κ (and thus ∆). Fig 11 shows that in the limit of small compression, θ max varies linearly with ∆, consistent with the analytic result of Eq (19) deduced from the solution of the First Reduced Equation, Eq (17).…”

“…where y n is the transverse displacement of the n th sphere from the central axis (for details of how the non-dimensional quantities are derived see [18]). Note that the model is confined to be planar because only planar structures are found in the regime that we explore, as may be confirmed by energy minimiation calculations not confined to two dimensions [19].…”

“…We intend to undertake this continuation in the future. It should complement previous investigations [19]) on the full range of threedimensional structures, which was based on energy minimisation.…”

Equilibrium configurations of hard spheres in a cylindrical harmonic potential

“…where as in [11] we have omitted the (constant) energy contribution due to the moment of inertia of the spheres. The compression ∆ from eq.…”

Densest-Packed Columnar Structures of Hard Spheres: An Investigation of the Structural Dependence of Electrical Conductivity

“…Prominent examples include the application of the face-centered cubic (fcc) and hexagonal close-packed (hcp) structures as models for bulk crystal structures of solids [4] and the application of random close packings as models for bulk amorphous structures of liquids [3,6]. In contrast to these examples for bulk systems, the past few decades have seen an uprising interest in the packings of particles in confined settings, such as those of particles confined within a two-dimensional box [7,8], within a parallel strip [9][10][11][12][13][14], within a spherical container [15,16], within a cylindrical container [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], onto a cylindrical surface [37], between parallel plates [38][39][40][41][42][43][44][45][46], within a wedge cell [47,48], or within a flexible conta...…”