Densest helical structures of hard spheres in narrow confinement: An analytic derivation

The emergence of helicity from the densest possible packings of equal-sized hard spheres in narrow cylindrical confinement can be understood in terms of a density maximization of repeating microconfigurations. At any cylinder-to-sphere diameter ratio D ∈ (1 + √ 3/2, 2), a sphere can only be in contact with its nearest and second nearest neighbors along the vertical z-axis, and the densest possible helical structures are results of a minimized vertical separation between the first sphere and the third sphere for every consecutive triplet of spheres. By considering a density maximization of all microscopic triplets of mutually touching spheres, we show, by both analytical and numerical means, that the single helix at D ∈ (1 + √ 3/2, 1 + 4 √ 3/7) corresponds to a repetition of the same triplet configuration and that the double helix at D ∈ (1 + 4 √ 3/7, 2) corresponds to an alternation between two triplet configurations. The resulting analytic expressions for the positions of spheres in these helical structures could serve as a theoretical basis for developing novel chiral materials. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5131318., s


I. INTRODUCTION
Helical structures emerge in many different areas of science, from condensed matter physics [1][2][3][4][5][6][7] all the way to structural biology. [8][9][10][11] A fundamental understanding of the nature and origin of helical structures would help advance the development of new molecules or materials, where notable examples reported in recent years include chiral photonic crystals, 12 supramolecular helical systems, 13 and chiral nematic liquid crystals. 14 An ideal platform for a comprehensive study of helicity is a model system of equalsized hard spheres in cylindrical confinement, where many of the densest possible structures discovered from computer simulations are helical [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] (Fig. 1). Of great experimental relevance to such a model system, helical structures have been observed for a variety of systems in cylindrical confinement, where notable examples include nanotube-confined fullerenes, 4 nanochannel-confined copolymers, 5,6 colloidal crystal wires, 7 glass-confined thermoresponsive microspheres, 32 capillary-tube-confined microbubbles, 31,33 fluid-driven assembly of polymeric beads, 34 cylindrically confined chromosomelike polymers, 11 and packings of fruits inside a cylinder (Fig. 2). Although these systems exhibit interactions of different types and scales, in many cases their densest possible structures resemble those of the above-mentioned model system of equalsized hard spheres. This suggests that geometric confinement could be an important contribution to the helicity of a physical system, even at the molecular level, and that we could achieve and manipulate such helicity through confinement if we master the geometric interplay between the confining space and the confined entities. For such confined systems, the densest possible structures, which depend solely on the ratio D ≡ cylinder diameter/sphere diameter (i.e., cylinder diameter expressed in units of the sphere diameter), were, in most cases, discovered computationally, without any rigorous proof that they are indeed the densest possible ones. Therefore, a long-standing goal has been to derive those densest possible structures analytically and to understand why they change from one to another at specific diameter ratios. Some of the previous theoretical attempts and their corresponding limitations are described as follows: An analytic theory 20,24,25,27 that maps the problem to a twodimensional scenario of disk packing has been worked out to explain the emergence of many of those helical structures. In this theory, each helical structure is thought of as a line slip of macroscopic regions of close-packed disks, as a result of the periodicity imposed by the cylindrical confinement. The theory works for a particular range of narrow confinement at 2 ≤ D ≤ 1 + 1/sin(π/5), where all spheres of any densest possible packing are in contact with the cylindrical boundary. Yet, it cannot be used to explain the emergence of the single-and double-helix structures at D < 2, where no two spheres can be placed at the same vertical z-position in cylindrical polar coordinates, nor does it apply generally to cases of wider cylindrical space at D > 1 + 1/sin(π/5), where some spheres of densest possible packings are located away from the boundary. 29,30 On the other hand, it has been shown numerically that the densest possible structures at 1 ≤ D < 1 + 1/sin(π/5) can all be constructed microscopically through a method of sequential deposition. 21,26 In this microscopic approach, spheres are dropped one by one onto their lowest possible positions to guarantee a local maximization of the packing fraction, and each columnar structure is optimized to be the densest possible through a finetuning of its underlying template. While this method shows great promises from a practical point of view, the original papers 21,26 do not provide physical insights into the microscopic mechanism behind the successful construction of those densest possible structures.
The aim of this work is to examine, from a novel microscopic perspective, how the densest possible single-and doublehelix structures at D < 2 emerge, where there has not been any comprehensive theory that explains the emergence of these two helical structures. By means of both numerical calculations and analytic derivations, we show that these structures are results of a density maximization of repeating local microconfigurations of spheres, thus giving rise to a global maximum of the packing fraction. Neighboring microconfigurations are coupled in a way that the same type of handedness (right or left) is propagated across the whole system, resulting in a uniform helicity across the columnar structure.
This paper is organized as follows. In Sec. II, we present a microscopic model of three mutually touching spheres and, also, the corresponding definitions of chirality for such systems. In Sec. III, we show, by both numerical and analytical means, that, at D ∈ (1 + √ 3/2, 1 + 4 ARTICLE scitation.org/journal/adv configuration of the triplet of spheres and that the densest possible single-helix structure corresponds to a repetition of this optimal triplet configuration. In Sec. IV, we show, also both numerically and analytically, that, at D ∈ (1 + 4 √ 3/7, 2), there exist two different density-maximized configurations of the triplet and that the densest possible double-helix structure corresponds to a repeated alternation between these two densest possible triplet configurations. For both densest possible helical structures, the exact positions of spheres, as well as the allowed ranges of D, are derived analytically from the corresponding optimal triplet configurations. Our results are summarized and discussed in Sec. V.

II. MODEL OF A TRIPLET OF MUTUALLY TOUCHING SPHERES
Microconfigurations of triplets of mutually touching spheres exist for the densest possible structures at D ∈ (1 + √ 3/2, 2). We begin our discussion by defining, in cylindrical polar coordinates (ρ, ϕ, z), the chirality of such a microconfiguration as follows. At D < 2, there is enough space for any sphere to be placed in contact with its nearest and second nearest neighbors along the vertical z-direction of the cylindrical polar coordinates, but there is not enough space for any two spheres to be placed at the same vertical z-position. Any triplet of mutually touching spheres can then be indexed as {1, 2, 3} in ascending order of their vertical z-positions, z 1 < z 2 < z 3 . When viewed vertically from the top of the coordinate system ( Fig. 3), any microconfiguration is described as right handed if the sequence {1, 2, 3} follows a counterclockwise rotation, whereas it is described as left handed if the sequence of indices follows a clockwise rotation, with Δϕij ∈ (0, 2π) and Δϕij ∈ (−2π, 0) denoting, respectively, a counterclockwise and a clockwise angular displacement of magnitude |Δϕij| from the center of sphere i to the center of sphere j. With the signs of Δϕij defined to be handedness-dependent, the magnitudes of Δϕij must generally be allowed to exceed π such that all possible arrangements of spheres could be considered. The corresponding angular displacements of the triplet of spheres are given by Δϕ 13 ≡ Δϕ 12 + Δϕ 23 ∈ (0, 2π) with Δϕ 12 ∈ (0, 2π) and Δϕ 23 ∈ (0, 2π) for any right-handed configuration, and Δϕ 13 ≡ Δϕ 12 + Δϕ 23 ∈ (−2π, 0) with Δϕ 12 ∈ (−2π, 0) and Δϕ 23 ∈ (−2π, 0) for any left-handed configuration.
Let Δzij ≡ zj − zi be the vertical displacement from the center of sphere i to that of sphere j. For a triplet of mutually touching spheres at any given diameter ratio, our method is to work out the relation between Δϕ 13 and Δϕ 12 and then to find out the triplet configurations that correspond to a minimum possible value of Δz 13 . The two densest possible helical structures considered in this study belong to a regime of D where all spheres of any densest possible packing are in contact with the cylindrical boundary, i.e., all at a radial position of ρ = (D − 1)/2. This allows for a simpler mathematical procedure in our analysis, where the equation that describes two spheres i and j in mutual contact is simplified to 21,24 where all length quantities are expressed in units of the sphere diameter. The angular displacements among a triplet of mutually touching spheres are then interrelated as follows: (2) can be used to work out the dependence of Δϕ 13 on Δϕ 12 as well as the dependence of Δz 13 on Δz 12 . Taking into account the extremum condition for a minimization of Δz 13 , a differentiation of both sides of Eq. (2) with respect to Δϕ 12 leads to a "symmetric" relation between Δϕ 12 and Δϕ 23 ≡ Δϕ 13 − Δϕ 12 , where Multiplying each side of Eq. (4) by itself and using the trigonometric identity cos 2 (x/2) ≡ 1 − sin 2 (x/2), we obtain a quadratic relation between Δϕ 12 and Δϕ 23 , as well as a nontrivial solution of where these solutions correspond, respectively, to the densest possible single-helix and double-helix structures at D < 2.

III. DENSEST POSSIBLE SINGLE HELIX AT
In this section, we show that the densest possible structure at D ∈ (1 + where (Δz/ΔN) is the average vertical separation between adjacent spheres and its values are obtained from a conversion of existing packing-fraction data 20,24 via the relation 21 where VF is the packing fraction. The results imply that the density of every consecutive triplet of spheres is maximized, with all vertical separations between adjacent spheres being the same (Δz 12 = Δz 23 = Δz 34 . . .) as consistent with the definition of a single helix. Figure 4(b) shows a corresponding angular-displacement plot of |Δϕ 13 | vs |Δϕ 12 | for the same set of D values. According to Eq. (1), for each value of D, the minimum possible value of △z 13 as shown in Fig. 4(a) corresponds to the minimum possible value of |Δϕ 13 | as shown in Fig. 4(b), where the relation |Δϕ 13 | = 2|Δϕ 12 | holds for such a minimum. Note that the value of |Δϕ 12 | corresponding to this minimum possible value of |Δϕ 13 | exhibits a deviation from π. This implies that, for the density of the triplet configuration to be maximized, sphere 2 has to be shifted away from its lowest possible position |Δϕ 12 | = π with respect to sphere 1 such that there is space for sphere 3 to be placed at the above-mentioned optimal position (see Fig. 5). It can be shown analytically, based on our consideration of microconfigurations, that this densest possible single-helix structure emerges only at D ∈ (1 + , implying that, for sphere 3 to be placed at its lowest possible position, sphere 2 has to be shifted away from its lowest position |Δϕ 12 | = π with respect to sphere 1. existing results: 15,20,28 The trivial solution given by Eq. (7) implies Δϕ 13 = 2Δϕ 12 = 2Δϕ 23 and Δz 13 = 2Δz 12 = 2Δz 23 . Applying this result to all consecutive triplets of spheres, i.e., Δz 12 = Δz 23 = Δz 34 = Δz 45 = ⋯, we conclude that any pair of consecutive spheres shares the same angular and vertical separations, as consistent with the definition of an infinitely long single-helix structure. From Eq. (1), we obtain the following expressions for the relative positions between any two consecutive spheres: and where, as shown in Fig. 4(a) For Δϕ 13 to be a minimum as illustrated in Fig. 4(b), this second derivative has to be a real positive number with  What is different from the above-mentioned single-helix scenario is, corresponding to the two optimal triplet configurations, the existence of two allowed positions for sphere 2 with Δz 13 ≠ 2Δz 12 and |Δϕ 13 | ≠ 2|Δϕ 12 | for either configuration, as shown in Figs. 6(a) and 6(b), respectively. The alternation between the two triplet configurations across the helical structure implies that there exist two allowed relative displacements for pairs of adjacent spheres, taken on by alternating pairs: (a) (Δz 12  It can be shown analytically that the nontrivial solution given by Eq. (8) corresponds to the densest possible double-helix structure at D ∈ (1 + 4 √ 3/7, 2). We first show that this solution corresponds to the existence of a double-helix structure: The solution can be rewritten as we prove that this nontrivial solution is valid only for the regime of D ∈ (1 + 4 √ 3/7, 2). Using the definition Δϕ 13 ≡ Δϕ 12 + Δϕ 23 , the trigonometric identity sin(x + y) ≡ sin(x) cos(y) + sin(y) cos(x), as well as Eq. (8), we first express sin 2 (Δϕ 13 /2) as a function of sin 2 (Δϕ 12 /2) as follows: where and Taking square of both sides of Eq. (2) and incorporating Eqs. (8) and (20) into the new equation, we arrive at the following quadratic equation for sin 2 (Δϕ 12 /2): where . Equation (23) yields two quadratic roots for sin 2 (Δϕ 12 /2), where and The relative positions between any two consecutive spheres are then given by the double roots and such that we have 2(Δz/ΔN) = Δz 12,+ + Δz 12,− = 1 + where, as shown in Fig. 6(a), the numerical values computed from Eqs. (28) and (29) 7. Plots of (a) Δz 12 and (b) |Δϕ 12 |/π, of optimal triplet configurations, as a function of D. The bifurcation of these parameters at the critical diameter ratio D = 1 + 4 √ 3/7 corresponds to a transition of the densest possible structure from a single helix to a double helix. structure corresponds to the existence of two different real roots of sin 2 (Δϕ 12 /2) such that which implies D > 1 + 4 √ 3/7 as in agreement with the existing results. 15,20,21,24 As illustrated in Fig. 7, the critical diameter ratio D = 1 + 4 √ 3/7 can then be interpreted as a bifurcation point at which the number of density-maximized configurations in any triplet of mutually touching spheres changes from unity for a single helix to two for a double helix as the value of D increases.

V. DISCUSSION AND CONCLUSIONS
The work presented here is complementary to an aforementioned analytic theory 20,24,25,27 that accounts for the emergence of a range of densest possible helical structures at D > 2. Our theoretical analysis, which relates two densest possible helical structures of equal-sized hard spheres at D < 2 to a the density maximization of repeating microconfigurations, helps us understand why these helical structures can be constructed from a method of sequential deposition 21,26 where each sphere is dropped to its lowest possible position for a local maximization of the packing density: (1) A global maximum of the packing fraction is achieved when the densities of such local configurations are all maximized.
(2) The overlapping of each triplet configuration with its neighbors (e.g., the pair {2, 3} is shared between the adjacent triplets {1, 2, 3} and {2, 3, 4}) ensures that the same type of handedness is propagated across each helical structure. (3) A transition of helicity occurs at the critical diameter ratio D = 1 + 4 √ 3/7 because there is a change in the number of optimal microconfigurations for any triplet of mutually touching spheres. As to potential applications of our results, the analytic expressions obtained for the positions of spheres in these two helical structures could serve as a theoretical basis for developing novel rodlike chiral molecules of different lengths and helicities, where a subject of great interest would be the liquid-crystalline properties of such systems.