# Helical close-packing of anisotropic tubes

Benjamin R. Greenvall & Gregory M. Grason  Department of Polymer Science and
Engineering, University of Massachusetts, Amherst, MA 01003
<EMAIL_ADDRESS>
###### Abstract

Helically close-packed states of filaments are common in natural and
engineered material systems, ranging from nanoscopic biomolecules to
macroscopic structural components. While the simplest models of helical close-
packing, described by the ideal rope model, neglect anisotropy perpendicular
to the backbone, physical filaments are often quite far from circular in their
cross-section. Here, we consider an anisotropic generalization of the ideal
rope model and show that cross-section anisotropy has a strongly non-linear
impact on the helical close-packing configurations of helical filaments. We
show that the topology and composition of the close-packing landscape depends
on the cross-sectional aspect ratio and is characterized by several distinct
states of self-contact. We characterize the local density of these distinct
states based on the notion of confinement within a ‘virtual’ cylindrical
capillary, and show that states of optimal density vary strongly with the
degree of anisotropy. While isotropic filaments are densest in a straight
configuration, any measure of anisotropy leads to helicity of the maximal
density state. We show the maximally dense states exhibit a sequence of
transitions in helical geometry and cross-sectional tilt with increasing
anisotropy, from spiral tape to spiral screw packings. Furthermore, we show
that maximal capillary density saturates in a lower bound for volume fraction
of $\pi/4$ in the large-anisotropy, spiral-screw limit. While cross-sectional
anisotropy is well-known to impact the mechanical properties of filaments, our
study shows its strong effects to shape the configuration space and packing
efficiency of this elementary material motif.

  *

Keywords: filaments, helices, geometry of materials, packing

##  1 Introduction

Filamentous or fibrous structures are an essential building block of material
systems, ranging from the molecular to human scale. At the microscopic end,
natural biomolecules [1, 2, 3] and synthetic polymers [4, 5, 6] form the
building blocks of life and of many engineered consumer products,
respectively. At macroscopic scale, fibers, wires and ropes form structural
materials [7, 8, 9] with diverse applications ranging from textiles [10, 11,
12] to tension bearing components in marine mooring lines [13, 14] and in
architecture (e.g. cables in suspension bridges) [15, 16].

A filamentous material’s function, utility, or morphology is inherently
intertwined with its _packing_ , or space filling arrangement. Entanglements
[17], knots [18], knits [19] and inter-filament orientation [20] have been
shown to influence the mechanical properties of filament assemblies. Precisely
folded proteins inherit both function and stability from their ordered
arrangement [21]. Braiding string or tying our shoes requires specific spatio-
temporal control of the strand. In each of the above, the packing of the
material is essential.

Figure 1: Dense helical coils across length scales. (A.) $\alpha$-helix
protein shown using the “ribbon” representation; backbone and sidechain atoms
rendered in yellow and gray, respectivley (PDB structure from [22] rendered
using VMD [23]). (B.) Self-assembled ferrocene-dipeptide “screw” nanostructure
(scale bar = $1\mu m$). Adapted with permission from [24]. Copyright 2015
American Chemical Society. (C.) Nested mesoscale polymer ribbon with tunable
photocrease-mediated configuration (scale bar = $50\mu m$). Adapted from [25].
(D.) Twisted helical nanofiber yarn (scale bar = $500\mu m$). Adapted from
[26]. (E.) Twisted Graphene Oxide/Alginate hydrogel fiber (scale bar = $500\mu
m$). Adapted with permission from [27]. Copyright 2020 American Chemical
Society. (F.) ’Blue Bouquet’ Passion Flower Passiflora caerulea tendril self-
coiling, photographed at University of Massachusetts Durfee Conservatory. (G.)
Coiled telephone cable. (H.) Oriental Bittersweet (Celastrus orbiculatus)
vine, photographed in Northampton, Massachusetts.

Indeed, the packing of material components has been a long-standing and broad
interest in mathematics, science, and technology, dating at least as early as
Kepler’s 17th century conjecture on sphere packing [28, 29], though likely
beginning many centuries earlier, especially in the case of mosaics and
tilings [30]. Historically studies on packing have largely focused on
arrangements of _discrete_ , rigid components (e.g. tiles, spheres,
polyhedra). Recently, more attention has been given to the packing of
continua, i.e. soft, deformable and extended objects [31], such as flexible
sheets [32, 33, 34], membranes [35, 36, 37] and filaments [38, 39, 40, 41,
42]. However, despite the natural and technological relevance of filamentous
materials, their packing has received considerably less attention, in part due
to the complex and vast configuration space accessible to even a single
deformable high aspect ratio structure. Due to their flexibility, the packing
may be constrained by contact at any location in space and at either local or
non-local positions along the filament length [43].

Key progress in understanding filament packing geometry is based on what is
known as the Ideal Rope or Ideal Tube model of packing, which considers space
filling configurations based on a filament centerline and a rigid, circular
cross-section of uniform diameter $d$ swept-out normal to the backbone curve.
Early applications of this model have been applied to understand the geometry
of closed curves in optimally “tight” knots and links [44, 45, 46, 47, 48, 49,
50, 51, 52, 53] This model has also been applied to model the complex close
contact geometry in multi-filament clasps [54, 55], as well as plies and
bundles [56, 57, 58, 59, 60, 61], which have formed a basis for comparison to
experimental systems [62, 63, 64, 65, 66, 67].

A particular focus of the ideal tube model has been on helical close-packing
of single filaments, initially motivated as a generic, physical model of
condensed helical motifs in biomacromolecules, e.g. $\alpha$-helices,
nucleosomes and chromatids [68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. In a
seminal work, Przybyl and Pieranski determined the configuration space of
helical packing in ideal tubes, solving for the conditions for self-contact
that delineate allowed from self-overlapping states in terms of helical radius
and pitch of the tubular centerline [78]. Notably, they showed that contact is
defined by turn-to-turn stacking of consecutive pitches at large radius, while
for small radius contact constraints are local, defined by a minimal curvature
radius of the finite-thickness tube. These two contact conditions intersect in
configuration space at a particular point, which they consider to be a state
of ideal close-packing for helical tubes. This packing geometry is used to
rationalize the stability of similar helical states in models of entropic or
osmotically collapsed helical molecules [75, 79, 80].

While the ideal tube model is a natural starting point for consideration of
filamentary close-packing, it is important to recognize that most physical
realizations of helical packing present some measure of cross-sectional
anisotropy, deviating by some measure from locally circular shapes. Fig. 1
shows several examples of anisotropic tightly-packed helical structures,
across a range of size scales and materials systems. Some form of cross-
sectional asymmetry is present in flexible, densely-coiled filamentous
structures at the molecular and meso-engineered systems [81, 25], and in a
variety of organism structures, including (but certainly not limited to)
bacteria [82, 83], curly human hair [84], plant tendrils and seed pods [85,
86, 87, 88], climbing and swimming snakes [89, 90], elephant trunks [91], and
squid and octopus tentacles[92, 93]. Naturally, such filament assemblies lead
to the basic question, _how does cross-section geometry influence the states
of dense packing in helical filaments_?

In this article, we study an extension of the (circular) ideal tube model to
include filaments with _elliptical_ cross-section, and consider the states of
helical packing. In addition to the radius $R$ and pitch $P$ of the helical
backbone, anisotropic tubes are described by two additional parameters: the
elliptical aspect ratio $\epsilon$; and relative orientation, or tilt angle
$\alpha$, of the cross-section relative to local bending direction, as
illustrated in Fig. 2. Like the circular ideal tube model, we model the
packing configurations by assuming the filament to be perfectly flexible (i.e.
it does not require any energy to bend the filament) but inextensible and
unshearable (that is, the cross-section of the tube is everywhere elliptical
of specified dimensions). Based on these assumptions, we fully determine the
states of self-contact, which for a given anisotropy $\epsilon$ correspond to
a two-dimensional manifold in helical configuration space defined by $R$, $P$
and $\alpha$. We show that for any measure of anisotropy $(\epsilon\neq 1)$
close contact configurations include a new class of local curvature
singularities, corresponding to “outward folding” cusps on the surface of low-
radius packings. Moreover, we find that the nature of the self-contact
singularity that separates allowed from overlapping configurations is highly
sensitive to the degree of anisotropy as well as the orientation of the cross-
sectional axis.

Figure 2: Anisotrpic helical tubes with helical radius $R=0.6d_{0}$ and
helical pitch $P=1.5d_{0}$. (A.) Tube surface with helical pitch $P$ and
helical radius $R$; together, these parameters define the helical angle,
$\theta=\tan^{-1}{\big{(}}\frac{2\pi R}{P}{\big{)}}$. Everywhere along the
helical backbone, the cross-sectional elements are ellipses. (B.) Tilt angle
($\alpha$) of the cross-section, where $\alpha$ measures the rotation of the
material short axis away from $\hat{{\bf r}}$, see B. (C.) cross-section
anisotropy. The (shorter) semi-minor elliptical axis has length $a$ while the
(longer) semi-major axis has length $b$. $\epsilon$ defines the aspect ratio,
$\epsilon=a/b$.

To understand conditions for “optimal” helical close-packing, we compute a
measure of local-density that we call capillary density, $\phi$, which is
simply the occupied volume fraction within a cylindrical volume enclosing the
helical tube (of unlimited length) [94]. Previous studies of ideal tubes have
employed a variety of metrics to define “maximally-dense” helical packing,
including minimizing the local radius of gyration [78, 69] and maximizing
excluded volume in a bath of spherical particles [75]. Our use of capillary
density is motivated by the fact that helical packing under cylindrical-like
confinement is also prominent in a variety of natural and engineered systems,
ranging from DNA coiled within viral phage capsids [95, 96], biopolymers
during synthesis [97, 98], translocation [99, 100], and incorporated within
membranes [101], as well as in variety of confined synthetic systems including
colloids [102, 103, 104, 105, 106, 107, 108], block copolymers [109, 110,
111], and liquid crystals [112, 113, 114, 115]. Moreover, as we show, the
states of optimal capillary density for isotropic versus anisotropic helical
tubes are both qualitatively and quantitatively distinct, and capillary close-
packing exhibits a strongly non-linear sensitivity to the degree of cross-
sectional anisotropy.

While the densest configuration of an isotropic tube under capillary
confinement is trivially a straight rod, we show that once cross-sectional
isotropy is broken, the densest configurations within _any size_ capillary are
helical, with a geometry that depends on the level of anisotropy. When the
tube is moderately anisotropic, the densest configuration occurs under high
confinement (i.e. in a narrow capillary) while increased anisotropy prefers a
larger capillary. The densest structures within these two regimes exhibit
strikingly different packing motifs both in terms of the helical angle and
cross-section orientation; light anisotropy prefers a “tape-like” motif
reminiscent of Fig. 1E while high anistropy prefers a “screw-like” motif
reminiscent of Fig. 1B. We determine that these packing regimes crossover when
the cross-sectional aspect ratio is approximately $4:1$. While the densest
structure (for a given cross-section) never exhibits scrolled or nested
packing (i.e. intermediate tilt, like Fig. 1C), we find such structures do
optimize density at intermediate levels of confinement, and over a range of
confinement that increases with asymmetry. This analysis reveals that the
densest structures for any degree of anisotropy are bounded, exhibiting a
packing fraction bounded between $1\geq\phi\geq\pi/4$.

The remainder of this article is as follows. In Section 2, we describe the
geometric model used to represent our helical tube, the method for finding the
boundary between expanded and self-contacting helical states and the
_capillary packing fraction_. In Section 3, we describe the self-contacting
manifold and the packing motifs that constrain different regimes of the
configuration space. In Section 4, we assess the capillary packing density of
contacting structures, and the dependence on maximally dense structures on
cross-section anisotropy. We discuss the potential implications of these key
results in Section 5 for physical and mechanical properties of filamentary
systems.

##  2 Parameterization and Modes of Contact

###  2.1 3D Helical Geometry

We model helical configurations of anisotropic tubes using a centerline (or
backbone) described by a _circular helix_ (i.e. one with constant radius, $R$
and pitch, $P$):

${\bf x}(s)=R\hat{\bf{r}}(s)\pm\frac{P}{\ell}s\hat{z}$ (1)

where $\hat{\bf{r}}(s)=\cos(2\pi s/\ell)\hat{x}+\sin(2\pi s/\ell)\hat{y}$, $s$
is an arclength coordinate along the backbone and
$\ell=\sqrt{4\pi^{2}R^{2}+P^{2}}$ is the arclength per helical turn 111Note
that in this parameterization, the chirality is controlled by the sign of $P$,
where $P>0$ corresponds to right-handed coils while $P<0$ refers to left-
handed coils. The contact geometry and packing density are, of course,
invariant under changes in chirality. In the current manuscript, all
schematics shown are left-handed; therefore, listed values of $P$ throughout
the text can be take to be negative.. Here, we consider a filament backbone of
unlimited length, or in effect, neglect contributions to packing from the ends
of the structure, appropriate when the total length is much larger than the
tube diameter. The geometry of the helix is independent of backbone position
up to screw motions, so the overall configuration can be described by a single
ratio, the helical angle, $\theta=\tan^{-1}(\frac{2\pi R}{P})$ bounded between
$\theta=0$ (a straight line) and $\theta=\pi/2$ (a closed circle).

To describe the material body _anisotropically distributed_ around this
backbone (Fig. 2A), we employ an orthonormal material frame, {${\bf
T}(s),\hat{{\bf e}}_{1}(s),\hat{{\bf e}}_{2}(s)$} where ${\bf T}(s)$ is the
backbone tangent vector, ${\bf T}(s)={\bf
x}^{\prime}(s)=\sin\theta\hat{\bf{\phi}}(s)+\cos\theta\hat{z}$ and
$\hat{\bf{\phi}}(s)$ is the azimuthal unit vector. We constrain the
(undeformable) cross-section to lie within the $\hat{{\bf e}}_{1}(s)$,
$\hat{{\bf e}}_{2}(s)$ plane, defined in terms of the Frenet-Serret frame
(where {${\bf T}(s),{\bf N}(s),{\bf B}(s)$} form an orthonormal triad, see A
and Ref [116])

$\displaystyle\hat{{\bf e}}_{1}(s)=\cos\alpha{\bf N}(s)+\sin\alpha{\bf B}(s),$
(2a) $\displaystyle\hat{{\bf e}}_{2}(s)=\cos\alpha{\bf B}(s)-\sin\alpha{\bf
N}(s)$ (2b)

Thus, the backbone tangent is normal to the cross-section at all points along
the curve. The parameter $\alpha$ describes the “tilt” (angle) of the material
frame, $\hat{{\bf e}}_{1}(s)$ and $\hat{{\bf e}}_{2}(s)$ relative to the
normal (${\bf N}(s$)) and binormal (${\bf B}(s)$) components of the Frenet-
Serret frame. For helices, the normal points toward the pitch axis (i.e. ${\bf
N}(s)=-\hat{\bf{r}}$), and hence, $\alpha$ is easy to visualize as the local
orientation of the material frame in the $\hat{\bf{r}}$, $\hat{z}$ plane, as
shown in Fig. 2B.

We model the tubular cross-section as an ellipse with semi-minor axis of
length $a$ oriented in the direction of $\hat{{\bf e}}_{1}(s)$ and semi-major
axis of length $b$ oriented in the direction of $\hat{{\bf e}}_{2}(s)$, see
Fig. 2C. The 3D filament _surface_ is then described by

${\bf X}(s,\psi)={\bf x}(s)+a\cos\psi\hat{{\bf e}}_{1}(s)+b\sin\psi\hat{{\bf
e}}_{2}(s)$ (2c)

where $\psi$ parameterizes the boundary of the elliptical cross-section at a
backbone position $s$. The material body around the helical backbone is then
defined by the aspect ratio of the cross-section,

$\epsilon\equiv a/b$ (2d)

and the tilt or _rotation of the material frame_ , $\alpha$ (Fig. 2B, C).
Intuitively, when $\alpha=0$, the short material axis is oriented in the
direction of curvature (i.e. towards the core of the coil), while
$\alpha=\pi/2$ corresponds to the long axis oriented in the curvature
direction. Importantly, here we extend the “Ideal Tube” model, and therefore
treat the filaments as inextensible (fixed length), unshearable (rigid cross-
section), and perfectly flexible. In the following, we consider filaments with
fixed cross-section area, $A_{\rm fil}$. We define the effective diameter of
filaments as,

$d_{0}=\sqrt{\frac{4A_{\rm fil}}{\pi}}=2\sqrt{ab}$ (2e)

that is independent of $\epsilon$, and we consider all other length scales
relative to this microscopic dimension.

###  2.2 Sectional geometry

Figure 3: 3D and 2D helical configurations for $\epsilon=0.64$ all with
$R/d_{0}=0.5$. (A.) Non-contacting structure with $P>P_{\rm min}$ and
$\alpha=5\pi/16$. (B.) Locally contacting structures obtained from (A.) by
_increasing_ tilt ($\alpha=3\pi/8$) and decreasing $P$ until contact, see also
Fig. 13 (C.) Non-locally contacting structures obtained from (A.) by
_decreasing_ tilt ($\alpha=\pi/4$) and decreasing $P$ until contact.

Based on the parameterization of anisotropic tubes defined in eqn. 2c, we then
determine helical configurations of self-contact by solving for two types of
singularities in the tubular surface. The first correspond to over-curvature
resulting in cusps or creases in the tubular surface, which we denote as local
contact. The second correspond to overlap of the tube at two distinct surface
points, which we denote as non-local contact

For the purposes of illustrating and analyzing these singular structures it is
particular useful to consider the geometry of 2D sections perpendicular to the
pitch axis $\hat{z}$, as shown in Fig. 3. We refer to these perpendicular cuts
as “croiss-sections” due to their crescent-like shape (see Fig. 3A). These are
defined by solutions to fixed-height equation ${\bf
X}(s,\psi)\cdot\hat{z}={\rm const.}$, which requires a parameteric relation
$s_{0}(\psi)$ between surface coordinates, from which define the 2D curves

${\bf X}_{\perp}(\psi)={\bf X}\big{(}s_{0}(\psi),\psi\big{)}.$ (2f)

For a given helical tube, all croiss-sections are identical up to rigid
rotations by screw symmetry. Fig. 3A shows an example for a non-contacting
helical configuration. Fig. 3B shows an example of a local contact singularity
for higher tilt and minimal pitch, which appears as an “outward” cusp in ${\bf
X}_{\perp}(\psi)$ for this example. Fig. 3C shows an example of a non-local
contact singularity for lower tilt and minimal pitch, which appears as two
distinct locations on the curve ${\bf X}_{\perp}(\psi)$ meeting at single
point in space.

###  2.3 Local Contact

Local contact occurs at a single $\psi$ along the tube surface where the
surface fails to be smooth; this discontinuity manifests as either an inward
or outward crease in the ${\bf X}(s,\psi)$ surface (or a cusp of the same
”direction” in the ${\bf X}_{\perp}(\psi)$ slice, see Fig. 3B, Fig. 13).
Operationally, these singularities can be selected by locations where the
_differential surface element_ of the tube vanishes. Using the surface metric,
$\rm g$, the magnitude of the surface normal can be expressed as

$\sqrt{\rm det(g)}=|\partial_{s}{\bf X}(s,\psi)\times\partial_{\psi}{\bf
X}(s,\psi)|$ (2g)

which yields the surface area of a differential tube element as $dA=\sqrt{\rm
det(g)}\,ds\,d\psi$.

The conditions for local contact are defined by the solutions to $\rm
det(g)=0$ (see eqns. 2ab – 2ad in B). Notably, the Ideal Tube case
($\epsilon=1$) corresponds to a singular limit of these equations in which
local-contact takes the form on an inward crease, at a minimum pitch derived
by Przybyl and Pieranski [78]

$P_{\rm local}=2\pi\sqrt{\frac{d_{0}}{2}R-{R}^{2}},\ \ \ \ {\rm for}\
\epsilon=1$ (2h)

More generally for $\epsilon\neq 1$ there are two solutions, corresponding to
local singularities at distinct surface locations, given by

$\displaystyle P_{\rm local(-)}=2\pi\sqrt{a\cos\alpha R-{R}^{2}},\ \ {\rm
for}\ \ \epsilon\neq 1$ (2i) $\displaystyle P_{\rm
local(+)}=2\pi\sqrt{b\sin\alpha R-{R}^{2}},\ \ {\rm for}\ \ \epsilon\neq 1$
(2j)

which also varies with orientation of the tubular cross-section .

While these local-contact solutions share similarities with the isotropic
(Ideal Tube) case, there are notable differences in _which_ elliptical axis
sets the curvature constraint. The first solution (eqn. 2i) is constrained by
the cross-section semi-minor axis ($a$) and is maximized when this length is
oriented in the radial ($\hat{{\bf r}}$) direction (i.e. $\alpha=0$). On the
other head, the second solution (eqn. 2j) is constrained by the cross-section
semi-major axis ($b$) and is maximized when the cross-section is rotated by
$\alpha=\pi/2$, such that this length is oriented in the $\hat{{\bf r}}$
direction. These two regimes have distinct geometrical structure and
consequences of constraining close-packed geometries, discussed in Section 3.

###  2.4 Non-local Contact

The tube surface makes contact non-locally when two distinct elements of the
croiss-section occupy the same point in space (Fig. 3C), which like the local-
contact case, implies contact along a 1D helical curve where two parts of the
same tube meet along its length.

In searching for non-local contact, we compute the _distance of closest
approach_ between these distinct points along the tube surface. When the
cross-section is isotropic, it is sufficient to calculate the distance between
points on the backbone and require that they be separated by a distance equal
to the tube diameter $d_{0}$ [78, 48]. Because of the asymmetry in the cross-
section, we must instead compute distances between surface elements. Our
approach employs several coupled criteria which determine points of distance
of closest approach along the tube surface (e.g. that the separation is normal
to the tube surface at both points ) then vary the configuration parameters
(i.e. $R$, $P$ and $\alpha$) until this closest distance is zero (i.e. the
points are in contact, see B). In practice, we fixed combinations of $R$ and
$\alpha$, and vary $P$, numerically solving for $P_{\rm non-local}(R,\alpha)$.

While it is straightforward to see that for sufficiently large $P$ any
combination of $R$ and $\alpha$ will be embeddable (i.e. non-overlapping), in
general, it is not a priori known what type of contact first limits the
reduction of pitch at a given $R$ and $\alpha$ (i.e. local or non-local
contact). We determine the pitch at which the tube first makes contact at a
given given $R$ and $\alpha$, i.e. minimum non-overlapping pitch, as the
largest pitch at which the tube either makes local or non-local contact,
$P_{\rm min}(R,\alpha)={\rm Max}\big{[}P_{\rm local(\pm)}(R,\alpha),P_{\rm
non-local}(R,\alpha)\big{]}$.

###  2.5 Capillary Confinement

For illustrative purposes, consider a expanded, loosely coiled filament. If
this filament is compressed until self-contact is made, then a confining
capillary is constricted around assembly until the capillary is in contact
with the filament surface. In a sense, the capillary contact constrains the
structure from dilating radially, while the filament self-contact prevents
further radial compression; this conceptual paradigm is presented in Fig 4A.

Operationally, we numerically determine the smallest cylinder (with radius
$R_{C}$) which could confine the structure, shown in Fig 4B,C. Because each
planar section of the packing is identical up to a rigid rotation about the
pitch axis, the occupied volume fraction in the capillary is identical to the
occupied area fraction of the croiss-section within the circular boundary of
the cylinder. Because the filament cross-sections are unshearable, and by
Pappus’s theorem the enclosed volume per unit backbone length is constant
[117], it follows that $A_{{\rm croiss}}P=A_{{\rm fil}}\ell$, where $A_{{\rm
croiss}}$ is the area enclosed by the cross-section normal to the pitch axis
and $A_{{\rm fil}}$ is the area of (elliptical) cross-section normal to the
backbone. Using the fact that $A_{{\rm croiss}}/A_{{\rm fil}}=\sec\theta$ we
compute the capillary packing fraction as

$\phi=\frac{A_{{\rm
croiss}}}{\pi{R_{C}}^{2}}=\frac{d_{0}^{2}\sec\theta}{4R_{C}^{2}}$ (2k)

where $R_{C}={\rm max}_{\psi}|{\bf X}_{\perp}(\psi)|$. Rationale for the use
of the minimal size capillary is discussed in C.

Figure 4: Dense capillary packing. (A.) Hypothetical filament constrained by
capillary- and self-contact. Schematic of uniform filament confined in a (B.)
3D and (C.) 2D cylindrical pore of radius $R_{C}$

##  3 Close-packed configuration landscapes

In this section, we analyze the manifold self-contacting helical geometries,
specifically, analyzing the minimal non-overlapping pitch $P_{\rm
min}(R,\alpha)$for for a given helical backbone radius and cross-sectional
tilt $\alpha$ for a sequence of cross-sectional anisotropy $\epsilon$.

###  3.1 Isotropic tubes

When the cross-section is isotropic (e.g. circular, $a=b$), the packing
landscape is determined only by helical radius (e.g. it is invariant to tilt).
Przybyl and Pieranski observed that contact is delineated into two regimes
[78]; when $R$ is small, the structure is curvature-limited and makes self-
contact locally, or along an inward crease at the center of helix (analogous
to overbending a garden hose) as illustrated in Fig. 5A i.-iii. Packing at
large $R$ is limited by (non-local) contact between successive turns of the
tube (analogous to coil stacking of a garden hose), illustrated in Fig. 5A
v.-vi. Here, the single boundary between local and non-local contact occurs at
$R/d_{0}\simeq 0.431$ and $P/d_{0}\simeq 1.083$ (Fig. 5A iv.); a single local
maximum in $P_{\rm min}/d_{0}=\pi/2$ occurs at $R/d_{0}=0.25$. Together, these
contact criteria generate the minimum pitch bounding hull, reproduced in the
style of [78] in Fig. 5B, where structures below $P_{\rm min}(R)$ are
forbidden (due to self-overlap) and structures above $P_{\rm min}(R)$ are said
to be “expanded”.

Figure 5: Packing landscape for isotropic tube, $\epsilon=1$, originally
calculated in ref. [78]. (A.) Maximal pitch structures for increasing values
of $R/d_{0}$. (B.) Minimal pitch contact hull indicating the limiting mode of
contact.

###  3.2 Slightly anisotropic tubes

Figure 6: Packing landscape for $\epsilon=0.95$. (A.) Contact type phase
diagram and (B.) minimal pitch contact hull. (C.) Maximal pitch structures for
increasing values of $\alpha$.

With apparently any perturbation to the cross-section shape (or at least with
as small as $0.5\%$ deviation, $\epsilon=0.995$), we find that the modes of
contact change significantly; notably, now with a vital dependence on cross-
section tilt. These large changes for weak anisotropy illustrate that the
isotropic tube case ($\epsilon\to 1$) is in fact a singular limit of the more
general variable anisotropy behavior. As shown in Fig. 6B, the large $R$
behavior is largely unchanged (albeit with a slight tilt-bias which
intuitively grants large $\alpha$ structures the ability to pack at lower
pitch). However, when $R$ is small in the regime corresponding to local
contact of the isotropic filament, we observe three distinct contact regimes,
shown in Fig. 6A. Two of these regimes are determined by local contact
corresponding to solutions of vanishing area element of the metric tensor(eqn.
2i and 2j), however these represent distinct types of contact, in particular
driven by the long material axis ($b$, at high $\alpha$) and the short
material axis ($a$, at low $\alpha$). Surprisingly, the low $R$ \-
intermediate $\alpha$ behavior exhibits non-local contact; this seemingly
stems from variation in tube-surface curvature (resulting from cross-section
anisotropy) thereby smoothing what was otherwise an inward local kink. This
packing paradigm allows for a slight softening of the previous global maximum
in $P_{\rm min}$; in other words, non-local contact at intermediate tilt
allows for a lower pitch configuration. While these changes are challenging to
recognize visually at the 3D scale (structures along the $P_{\rm
min}(R,\alpha)$ ridge are shown in Fig. 6C), their modes of contact are
distinct when magnified; we note that the low-$\alpha$ local contact creases
“inward” in the structure, manifesting as an _intruding groove_ (and hence
labeled “Local (-)”) while the high-$\alpha$ local contact creases “outward”,
manifesting as a _protruding seam_ (and hence labeled “Local (+)”). It is also
important to note that despite the minuscule differences in contact geometry
and fairly modest shifts in the values of $P_{\rm min}$ at a given $R$, the
boundaries between distinct contact topologies exhibit a large shift in then
$R$-$\alpha$ plane, even when the cross-section anisotropy is small. These
shifts in the contact manifold become even more pronounced as the tubular
cross-section becomes more anisotropic, as we discuss next.

###  3.3 Highly anisotropic tubes

When the cross-section is symmetric (with respect to tilt), the packing
landscapes are also symmetric (see Fig. 5B). Introducing asymmetry in the
cross-section results in asymmetry in the landscape; the Local (+) regime
occupies slightly more area than the Local (-) regime in Fig. 6A and the
corresponding maxima in Fig. 6B is more pronounced.

Figure 7: Packing landscape for $\epsilon=0.8175$. (A.) Contact type phase
diagram and (B.) minimal pitch contact hull. (C.) Maximal pitch structures for
increasing values of $\alpha$.

As the filament becomes more anisotropic (see Fig. 7A,B for
$\epsilon=0.8175$), the asymmetry between + and $-$ local contact grows. With
decreasing $\epsilon$, the Local (+) and Non-local regions become more
prominent while the Local (-) shrinks, and the differences in $P_{\rm min}$
between large- and small-$\alpha$ become significant. The minimal pitch also
exhibits a stronger tilt-dependence; while two maxima separated by a saddle
still dominated the low-$R$ behavior, the magnitude of the maxima
(corresponding to structures Fig. 7C.i and C.vii now differ by approximately
20 percent (with $P_{\rm min}\simeq 1.42d_{0}$ and $1.73d_{0}$, respectively).
At this scale, the 3D structures exhibit some perceptible asymmetry;
structures with intermediate tilt (Fig. 7C.ii-vi) display uneven protrusions
in the radial direction, while structures constrained by Local (+) curvature
(Fig. 7C.v-vii) display a noticeable seam.

When the cross-section is increasingly asymmetric (below $\epsilon\simeq
4/5$), the Local (-) branch ceases to constrain the packing manifold, at which
point, the entire low-tilt regime is limited by non-local turn-to-turn contact
exclusively. When the cross-section has _at least_ this degree of anisotropy,
it is not possible to construct a viable configuration that is more curved
than the short material length scale would allow because when the structure is
in the low-tilt regime, the curvature of the filament is primarily in the
direction of the short material axis. There simply is not enough material in
this direction to curve such that a kink arises, while in the pitch-direction,
the cross-section is wide which obstructs larger regions of turn-to-turn
configurations.

An example of the “two-phase” contact topology for highly-anisotropic tubes
($\epsilon=0.25$) is shown in Fig. 8. As the filament becomes more
anisotriopic, the global maxima of $P_{\rm min}(R,\alpha)$ at $\alpha=\pi/2$
become more pronounced (Fig. 8B). The secondary maxima at low-$R$ for
$\alpha=0$ persists, despite the change between modes of contact (from Local
(-) to Non-local), though the feature continues to softens relative to the
global maximum (Fig. 8C.i and C.v have $P_{\rm min}\simeq 2.25d_{0}$ and
$\simeq 3.15d_{0}$ at $\alpha=0$ and $\alpha=\pi/2$, respectively). In this
asymmetry regime, these extrema remain separated by a saddle which spans
intermediate tilt; however, the span is not smooth (i.e. $\partial P_{\rm
min}/\partial\alpha$ is discontinuous), owing to the boundary between modes of
contact. In these significantly asymmetric cases, the corresponding structures
exhibit exotic morphologies; a selection that spans the $P_{\rm min}$ saddle
are shown in Fig. 8C. In particular, the global maxima of $P_{\rm min}$ Fig.
8C.v., corresponding to Local (+) contact at small-$R$, $\alpha=\pi/2$
presents a relatively large $P_{\rm min}$ characterized by elaborate outward
cusps at the high curvature edge of the cross-section. This behavior is in
stark contrast to the Non-local large-$R$, $\alpha=\pi/2$ packing, in which
$P_{\rm min}$ is minimal (i.e. $P_{\rm
min}(R\to\infty,\alpha=\pi/2)\rightarrow 2a$, or “small-axis stacking”).
Intuitively, as the radius of curvature is compressed beyond the width of
cross-section in radial direction, the packing must expand correspondingly
into the $\hat{z}$ dimension (resulting in larger $P_{\rm min}$).

Figure 8: Packing landscape for $\epsilon=0.25$. (A.) Contact type phase
diagram and (B.) minimal pitch contact hull. (C.) Maximal pitch structures for
increasing values of $\alpha$.

Przybyl and Pieranki considered the transition point between local and non-
local contact branches to correspond to a state of optimally dense helical
packing. In the next section, we consider the density of these much richer
close-packed landscapes, and in particular, the behavior in proximity to
packing motif coexistence branches. Additionally, a more complete analysis of
contacts along the coexistence branch are provided in C.

##  4 Capillary packing

Figure 9: Multiplicity of states within a given capillary of $R_{C}$ shown for
$\epsilon=0.25.$. Traces of (A.) density and (B.) helical angle for _fixed_
tilt angle as a function of confining radius. (C.) Dense and loose structures
within the same capillary $R_{C}/d_{0}=0.79$ (i.), $0.84$ (ii, iii), $0.89$
(iv, v), $0.94$ (vi, vii), $0.99$ (viii, ix). (D.) $R_{C}$ evaluated along the
contact hull; the region at low-$R$ low-$\alpha$ corresponds to bifurcated
regions, (E.) Landscape of maximum with variable $R_{C}$ and $\alpha$. A plot
of the maximal density at fixed $R_{C}$ is overlaid.

Having found that helical geometry of filament self-contact is strongly
sensitive to the anisotropy of the cross-section, we now consider its effects
on capillary packing fraction, $\phi$.

In this section, we first briefly summarize the approach to computing
optimally dense configurations. We then describe features of the packing
density manifold and the underlying structural motifs as a function of the
size of the confining capillary, $R_{C}$. Finally, we analyze and compare the
overall densest configurations as a function of tube aspect ratio, $\epsilon$
and classify the distinct packing motifs selected for variable filament
asymmetry and confinement.

###  4.1 Measuring local density by capillary confinement

In the prior sections, it was most convenient to parameterize packing in terms
of the purely geometric parameters that describe the configuration, in
particular, the radius $R$ and pitch $P$ of the tubular “centerline”. This
description is sufficient for capillary confinement of isotropic tubes
($\epsilon=1$) because the tightest capillary radius for a given helical
radius is simply $R_{C}=R+d_{0}/2$. For anisotropic tubes ($\epsilon\neq 1$),
the mapping between helical centerline radius $R$ and capillary radius, is a
more complex and non-linear function $R_{C}(R,P,\alpha)$. In practice, this is
computed from the maximal radial distance of the tubular surface from the
pitch axis, i.e. $R_{C}={\rm max}_{\psi}|{\bf X}_{\perp}(\psi)|$ for a given
$R,P$ and $\alpha$. That is, the capillary radius is the smallest possible to
enclose the helical filament, or other words, the minimal possible value of
$R_{C}$ is chosen so as to maximize the packing fraction for that $R,P$ and
$\alpha$.

For all close-packed (i.e. self-contacting) structures for a given $\epsilon$,
that is, all combinations of $R$, $\alpha$, and the resulting minimal pitch
$P_{\rm min}$, we calculate $\phi(R,\alpha)$ and $R_{C}(R,\alpha)$222Note that
by constraining our optimization to close-packed structures, $\phi(R,\alpha)$
and $R_{C}(R,\alpha)$ have an implicit dependence on $P_{\rm min}$, where
$P_{\rm min}$ = $P_{\rm min}(R,\alpha)$ . By inverting the mapping
$R_{C}(R,\alpha)$ (to yield $R(R_{C},\alpha)$), we analyze the relationship
between confinement size for a close-packed structure and its packing
fraction. As an example, we show several traces of $\phi(R_{C},\alpha)$ for
fixed $\alpha$ and $\epsilon=0.25$ in Fig. 9A-B. Importantly, when both
$\alpha$ and $R_{C}$ are small, $\phi(R_{C})$ is multivalued since there are
multiple $R$ values with the same $\alpha$ that _perfectly_ fit within a
specific capillary (see schematic examples in Fig. 9C.).

As we are presently interested in the densest packing, we take only the higher
$\phi_{C}$ configuration in this multi-valued region in our analysis,
neglecting the features of the “loose” (larger pitch) branch of capillary
confinement.

For $\epsilon=0.25$ the map between helical radius and capillary radius is
shown in Fig. 9D, where the “loose” structures occupy a region at low-$R$
low-$\alpha$. The boundary of this region (white dashed line in Fig. 9)
corresponds to the minimal capillary radius for a given $\alpha$ in this
regime. For all confined packings along this boundary between single- and
multi-valued helical radius, we observe helical geometry ($\theta>0$, marked
with black dots in Fig. 9B), which is not the case for larger tilt and lower
asymmetry, where the tightest capillary radius is achieved for $R\to 0$. For
example, at values of tilt larger than the bifurcation region for
$\epsilon=0.25$, $\alpha\gtrsim 0.6$, the minimal capillary size is
independent of $\alpha$; all structures exhibit a minimal $R_{C}$ governed by
$\theta=0$ “rod-like” packing, in which $R_{C}$ is exactly the semi-major
cross-sectional axis, $b$. Regardless of aspect ratio (and tilt), these rod-
like packings exhibit a density $\phi=A_{{\rm fil}}/\pi{R_{C}}^{2}=\pi ab/\pi
b^{2}=\epsilon$.

Taken together, these effects lead to complex landscapes of capillary packing
density as a function of cross-section tilt and capillary radius, as shown for
example for $\epsilon=0.25$ in Fig. 9E. Here, $\phi$ spans
$0.8\gtrsim\phi\gtrsim 0.1$. Within the parameter range shown, $\phi$ exhibits
a minimum when $R_{C}/d_{0}\simeq 1.5$, $\alpha=\pi/2$; structures in this
neighborhood correspond to high-$P_{\rm min}$ Local (+) structures, like Fig.
8C.v. Of course, these configurations only correspond to a local minima;
$\phi\rightarrow 0$ as $R_{C}/d_{0}\rightarrow\infty$, resulting from the
center of the coil opening.

In the following, we next consider $\alpha$ as a configurational degree of
freedom (like $R$ and $P$) and determine the optimal tilt and helical geometry
for dense packing of a tube with a given anisotropy $\epsilon$ and in a given
capillary radius $R_{C}$. For the example of $\epsilon=0.25$ shown in Fig. 9E,
the optimal $\alpha$ value for each given $R_{C}$ is shown as blue curve,
which transitions from $\alpha=0$ for small $R_{C}$ to $\alpha=\pi/2$ for
large $R_{C}$, a generic feature for all $\epsilon$ as we discuss below.

###  4.2 Variable-Radius Capillary Packing

Fig. 10A shows plots of the optimal capillary packing density and
configurations as function of capillary radius ($R_{C}$) and for a sequence of
increasing cross-sectional asymmetry. For all aspect ratios of the filament,
packing fraction $\phi$ exhibits a non-monotonic dependence on the confining
radius. When the tube is isotropic ($\epsilon=1$), the packing exhibits four
extrema (Fig. 10A,top):

  1. 1.

when the capillary is the same size as the tube, the tube perfectly packs
within the capillary ($\phi=1$). We consider this to be a “rod-like” packing
motif.

  2. 2.

when the capillary size is intermediate to the tube radius and tube diameter,
the structure is helical with lower packing fraction, corresponding to
high-$P_{\rm min}$ local contact structures.

  3. 3.

when the capillary size is comparable to the tube diameter, the packing
fraction reaches a local maxima of $\phi=\pi/4$; based on the density, we
consider this to be a “toroid-like” packing motif 333Toroid-like packing
exhibits a packing density $\phi=\pi/4$, equivalent to that of a horn-torus
within a cylinder of radius and height equal to the tube diameter;
interestingly, the helical capillary exhibits the same ratio of size (a
cylinder of radius equal to the tube diameter).

  4. 4.

when the $R_{C}$ is larger than the tube diameter, the core of the helical
structure begins to open, leading to a monotonic decrease in density with
increasing capillary size ($\phi\rightarrow 0$ with $R_{C}\rightarrow\infty$).

The helical angle $(\theta=\tan^{-1}(\frac{2\pi R}{P}))$ exhibits a monotonic
dependence on $R_{C}$. When the filament is isotropic and the cylinder is in
the small limit ($R_{C}=d_{0}/2$), the tube is straight (i.e, $R=0$) thus
$\theta=0$. In the large limit ($R_{C}\rightarrow\infty$), the tube approaches
a toriodal geometry (i.e, a helix with $P\rightarrow 0$) thus $\theta=\pi/2$.

Figure 10: Densest configurations within variable sized capillaries. (A.)
Maximal density ($\phi$, black, left axis), tilt angle ($\alpha$, red, right
axis), helical angle ($\theta$, blue, right axis) within capillary
$R_{C}/d_{0}$ for selected values of $\epsilon$. Global maxima of $\phi$
marked with solid dots; local maxima marked with open dots. (B.) Structures
for $\epsilon=0.316$ within increasing $R_{C}/d_{0}$, displaying increasing
$\alpha$ and $\theta$.

Helical packing of asymmetric tubes ($\epsilon\neq 1$) leads to new features
of the capillary dependent packing, shown in Fig. 10A. With any degree of
anisotropy, we observe that the minimal capillary radius no longer corresponds
to a local maxima in density; this maxima shifts to larger capillaries and
decreases in density with increasing anisotropy, first very slightly for low
anistropy (e.g. $\epsilon=0.95$) and then more obviously with larger
anisotropy $(\epsilon=0.49)$ The minimum helical angle (which occurs within
the smallest possible capillary radius) increases with anisotropy, never
limiting to the isotropic value of $\theta=0$; in other words, the straight,
non-helical tube is _never_ a density maximizer (for an aniostropic tube),
regardless of size of the capillary. For any degree of cross-sectional
anisotropy, capillary density favor helical over straight configurations.

Relative to small-$R_{C}$, the large-capillary behavior is qualitatively
insensitive to anisotropy; the density exhibits a local maxima when the
capillary radius is equal to twice the semi-major tube axis. Naturally, this
value of $R_{C}$ grows as with increasingly tubes anisotropy.

Given a tube anisotropy, optimal capillary density requires strongly non-
linear variation in the tilt angle ($\alpha$) of the cross-section. Near
maxima of $\phi(R_{C})$, we observe extremes in the tilt angle. When $R_{C}$
is small, the cross-section is oriented with the short axis pointing (inward)
in the curvature direction ($\alpha=0$). This general motif, observed at the
low $R_{C}$ density maxima, is reminiscent of a coiled tape, on for example,
bicycle handlebars, and shown for $\epsilon=0.316$ in Fig. 10B.i. When $R_{C}$
is larger, the dense structures exhibit a cross-section with the long axis
oriented (inward) in the curvature direction ($\alpha=\pi/2$; this motif is
reminiscent of the threads on a screw, see Fig. 10B.v; the onset of this
regime universally exhibits the toroidal density, $\phi=\pi/4$.

For large anisotropies ($\epsilon\lesssim 4/5$, these states are connected at
intermediate $R_{C}$ via a monotonic increase in the tilt angle, visualized in
Fig. 10B.ii-iv. The width of the transition broadens with anisotropy,
revealing an increasing number of dense, intermediate-tilt states that exhibit
a nested or “scroll”-like packing. Interestingly, these states with large but
sub-maximal tilt exhibit a packing density nearly degenerate to the
$\phi=\pi/4$ toroidal packing. When the tube is moderately anistropic, we also
note the existence of a set of low-$R_{C}$ nested states; these structures
correspond to a slight preference for non-local contact over Local (-)
contact, and persists until the Local (-) regime no longer constrains the
close-packing landscape ($\epsilon\simeq 0.81$). Given the slight amount of
anisotropy, these configurations are structurally similar to the untilted
structures; the packing benefit and resulting structural changes are slight in
this case.

We mark the locations of low-$R_{C}$ (tape-like) and high-$R_{C}$ (screw-like)
density maxima with black markers in Fig. 10A. For each value of $\epsilon$,
the global maximum is marked with a solid dot, while the secondary maximum is
marked with an open dot. When $\epsilon$ is high (i.e. nearly isotropic), the
low-$R_{C}$ maxima is most dense, and when $\epsilon$ is small (i.e. large
anisotropy), the high-$R_{C}$ maxima is most dense. This transition in the
geometry of maximally dense packing is examined in more detail in the next
section.

###  4.3 Optimal Capillary Packing

Figure 11: Competing dense packing motifs. (A.) Densest configurations for
selected values of $\epsilon$. (B. top) Density ($\phi$) global (“ _dense_ ”)
and local (“ _loose_ ”) maxima. Global maxima are shown as solid dots while
submaximal packings are shown as open dots. (B. bottom) Helical angle
($\theta$) for densest configurations

As shown in Fig. 10, $\phi(R_{C})$ is characterized by two local maxima, one
at low-$R_{C}$ with small $\theta$ and a second at large-$R_{C}$ and large
$\theta$, which are respectively closer to nearly-straight versus “gardenhose-
like” coiling. In Fig. 11, we track each of these two localy dense states (i.e
the $\phi$ maxima in Fig. 10). Again, global maxima (“dense”) are marked with
a solid point while the secondary maxima (“loose”) are marked with an open
point. When the tube is either isotropic or only moderately elliptical, the
density of the small-$R_{C}$ ($\alpha=0$, low $\theta$) state is higher than
that of the large-$R_{C}$ ($\alpha=\pi/2$, high $\theta$) state. However, the
small-$R_{C}$ density decreases with anisotropy, while the large-$R_{C}$
density remains constant; at a critical aspect ratio, $\epsilon_{\rm cr}\simeq
0.316$, the densest state transitions between solution branches. This
dependence is intuitive to understand by simply considering the capillary
fraction of a straight tube, $\phi(\theta=0)=\epsilon$, which obviously
decreases due to the inefficiently of filling a circular capillary with an
elliptic cross-section. Hence, secondary “gardenhose” maxima eventually
overtakes the nearly-straight state in density. This transition between
packing motifs shows a characteristic tilt- and helical angle dependence; the
densest high-$\epsilon$ structures pack with $\alpha=0$ and low $\theta$ (or a
_steep_ incline) while the low-$\epsilon$ structures with $\alpha=\pi/2$
orientation and high $\theta$ (or a _gradual_ incline); the globally optimal
configurations for assorted values of $\epsilon$ are shown in Fig. 11A. These
competing motifs are shown at approximately the transition point,
$\epsilon_{\rm cr}\simeq 0.316$, in Fig. 10B.i and B.v. Curiously, this
transition occurs when the aspect ratio is $\simeq\pi^{-1}$, at which point
the density of both the tape-like and screw-like structure exhibit
$\phi=\pi/4$.

Figure 12: Anisotropy-confinement landscapes. (A.) Density of packed filaments
with aspect ratio $\epsilon$ constrained to capillary with radius
$R_{C}/d_{0}$. (B.) Configurational phase diagram consists of spiral tapes
($\alpha=0$), spiral scrolls (($\pi/2>\alpha>0$)), and spiral screws
($\alpha=\pi/2$).

##  5 Discussion and concluding remarks

In this article, we have inventoried the dense-packing parameter space of
helical tubes with cross-sections of various aspect ratio (or _flatness_). We
find that once cross-section symmetry is broken, the tube configuration space
is constrained by novel and non-trivial variants of the local (curvature-
limited) and non-local (turn-to-turn) contact for isotropic filaments reported
in [78]. This complex spectrum of closed-packing geometries, and their non-
linear dependence on anisotropy has important implications for the “local
density” of helical close-packing, measured in terms of capillary packing
density. While the densest structure is straight for isotropic tubes, once
cross-section symmetry is broken, the densest configuration is either tape-
like ($\alpha=0$) when the tube is mildly anisotropic or screw-like
($\alpha=\pi/2$) if the tube is sufficiently anisotropic. Importantly, the
density of the weakly-anisotropic regime decreases with $\epsilon$ while the
latter is independent of $\epsilon$ (and displays a packing fraction of
$\phi=\pi/4$, equivalent to that of a horn torus confined to a tight
cylinder). Therefore, the densest configuration _for any filament_ is bounded
$1\geq\phi\geq\pi/4$, assuming an optimal capillary. Regardless of the
morphology, the “locally dense” packing motif required an increasingly larger
capillary as the tube became more anisotropic. These trends of optimal density
are summarized in Fig. 12A. Based on our highest aspect ratio studied here
($\epsilon=0.09$), it seems reasonable that the low $\epsilon$ preference for
$\alpha=\pi/2$ packing should hold in the limit of thin ribbon- or sheet-like
helices (i.e. $\epsilon\rightarrow 0$); perhaps, not surprisingly, these
packings seem to be abundant in biological membranes, including in the
endoplasmic reticulum [35] and plant (photosynthetic) thylakoid membranes
[36].

The strong influence of cross-sectional anisotropy on _optimal_ close-packing,
raises additional question about how (density and morphology) evolves under
sub-optimal confinement. Though the optimal structures always required
extremal (tape-like with $\alpha=0$ or screw-like with $\alpha=\pi/2$) tilt,
intermediate levels of confinement often preferred intermediately tilted,
nested scroll-like configurations, as shown in the morphological phase diagram
in Fig. 12B (see B for a discussion of the small tape/scroll region at low
$R_{C}$ and high $\epsilon$). Many of these nested configurations (especially
at large $\alpha$) exhibit a density nearly degenerate to the screw-like
structure (see the soft gradients in Fig. 12A at low $\epsilon$, intermediate
$R_{C}$). Further, the nested states occupy a larger portion of confinement-
parameter space as the cross-section becomes more asymmetric. Given a driving
force for densification (like some applied compressive force), these findings
hint at the potential to control the assembly morphology via the degree of
confinement.

In considering the anisotropic filament packing geometry, we account for more
complex material _shape_ , however we have thus far restricted ourselves to
very simple mechanical descriptions, namely, neglecting considerations of bend
and twist elasticity of filaments. A more complete mechanical model should
take into elastic costs of helical deformations, and specifically account for
elastic anisotropies that result from the geometric anisotropies [31, 118].
Based on simple Kirchoff rod theory, one should expect (for a filament
composed of an isotropic elastic medium) bending is stiffer in the wide axis
relative to the narrow axis by a factor of $\epsilon^{-2}.$ On these grounds,
and assuming that filaments are straight and untwised in their rest
configuration, it is reasonable to expect that the $\alpha=\pi/2$
configurations would impose a higher elastic cost (assuming a straight rest
configuration) than $\alpha=0$ packings. This suggests a basic antagonism
between geometries that favor optimal dense packing and those that require low
elastic energy. Further, we can imagine a class of situations for which the
rest state of filaments is helical [119, 66, 120, 121, 27], but the geometry
of close-packing may indeed frustrate the minimal elastic energy. One example
of this scenario is likely to be elastically programmed microfilaments that
target inaccessible values of helical pitch and radius [25]. Notably,
accounting for both density and elasticity in this situation is likely to
direct the structures towards the tilted, nested structures shown in Fig. 1C.

Of course, the current question of dense packings underlies a range of other
interesting phenomena, including the mechanical response of helical structures
under an applied load, the impact of incompatible coiling (e.g. geometric
frustration arising from an inaccessible elastic ground state) on the
resulting structure, multi-component (and size-disperse) interacting systems
including bundles [38, 122] and arrays [123]. In particular, it remains to be
explored how the mapping between packing of filaments in helically-twisted
structures and packing on non-Euclidean surfaces [56, 38] sheds useful light
on optimal states of packing anisotropic filaments. For example, for the
present model of helical filament configurations with elliptical cross-
sectional asymmetry, it can be expected that local and global geometric
constraints can be understood by packing of geodesic ellipses on positive
curvature surfaces [124].

The authors are grateful to K. T. Sullivan for helpful comments on this
manuscript. Numerical calculations were carried out on the Unity Cluster, a
collaborative, multi-institutional high-performance computing cluster managed
by the UMass Amherst Research Computing and Data team. This research was
supported by a National Science Foundation Graduate Research Fellowship
awarded to B.R.G.

## Conflict of Interest

The authors declare no conflict of interest.

## Data Availability

The analysis code that was used for this study is openly available in UMass
Amherst ScholarWorks at https://doi.org/10.7275/gr0k-fv13.

##  Appendix A Surface and Sectional Geometry

###  A.1 Frenet-Serret Formulas

To describe the material body _anisotropically inflated_ around the helical
backbone (Fig. 2A), we employ the orthonormal (geometric) Frenet-Serret frame,
composed of the tangent, normal, and binormal triad, {${\bf T},{\bf N},{\bf
B}$}:

${\bf T}(s)={\bf
x}^{\prime}(s)=\sin\theta\hat{\bf{\phi}}(s)+\cos\theta\hat{z}(s)$ (2l)

${\bf N}(s)={\bf T}^{\prime}(s)=-\hat{\bf{r}}(s)$ (2m)

${\bf B}(s)={\bf T}(s)\times{\bf
N}(s)=-\cos\theta\hat{\bf{\phi}}(s)+\sin\theta\hat{z}(s)$ (2n)

where {$\hat{\bf{r}},\hat{\bf{\phi}},\hat{z}\\}$ are the canonical cylindrical
coordinate basis vectors and $\tan\theta=2\pi R/P$:

$\hat{\bf{r}}=\cos(2\pi s/\ell)\hat{x}+\sin(2\pi s/\ell)\hat{y}$ (2o)

$\hat{\bf{\phi}}=-\sin(2\pi s/\ell)\hat{x}+\cos(2\pi s/\ell)\hat{y}$ (2p)

###  A.2 2D “Croiss-section” Derivation

We select the contents of the tube that lie in the $\hat{x}$, $\hat{y}$ plane
at fixed height, $z_{0}$

${\bf X}(s,\psi)\cdot\hat{z}=z_{0}$ (2q)

$\frac{P}{2\pi
l}s+a\cos\psi\sin\alpha\sin\theta+b\sin\psi\cos\alpha\sin\theta=z_{0}$ (2r)

This constraint allows us to eliminate one of the two parametric variable by
yielding the relationship between backbone ($s$) and surface ($\psi$)
coordinate (that satisfies the planar constraint).

$s_{0}(\psi)=-\frac{2\pi\ell\sin\theta}{P}(a\cos\psi\sin\alpha+b\sin\psi\cos\alpha+z_{0})$
(2s)

By substitution of this constraint, the full 3D model reduces to a 2D one-
coordinate function (see Fig. 3), where the (in-plane) boundary of the tube is
given by

${\bf X}_{\perp}(\psi)={\bf X}\big{(}s_{0}(\psi),\psi\big{)}$ (2t)

And the 2D tangent of the surface is given by

${\bf T}_{\perp}(\psi)=\frac{{\bf X}_{\perp}^{\prime}(\psi)}{|{\bf
X}_{\perp}^{\prime}(\psi)|}$ (2u)

##  Appendix B Determination of Self-Contact

###  B.1 Non-local Contact

To determine the distance between any two in-plane points, we define the
separation vector,

${\bf{\Delta_{\perp}}}(\psi_{1},\psi_{2})={\bf X}_{\perp}(\psi_{1})-{\bf
X}_{\perp}(\psi_{2})$ (2v)

The distance of closest approach is defined between surface coordinates
$(\psi_{1}$ and $\psi_{2})$ when ${\bf{\Delta_{\perp}}}(\psi_{1},\psi_{2})$ is
orthogonal to both surface tangents, ${\bf T}_{\perp}(\psi_{1})$ and ${\bf
T}_{\perp}(\psi_{2})$ (or, equivalently, ${\bf
T}_{\perp}(\psi_{1})\parallel{\bf T}_{\perp}(\psi_{2})$. In determining
instances of non-local contact, we seek to simultaneously satisfy ${\bf
T}_{\perp}(\psi_{1})\parallel{\bf T}_{\perp}(\psi_{2})$,
${\bf{\Delta_{\perp}}}(\psi_{1},\psi_{2})\rightarrow 0$, and
$\psi_{1}\neq\psi_{2}$.

Recalling that the ${\bf X}_{\perp}(\psi)$ croiss-section plane has $\hat{z}$
as a normal, we define three inequalities to capture the distance of closest
approach

${\big{|}}{\bf
T}_{\perp}(\psi_{1})\cdot{\bf{\Delta_{\perp}}}(\psi_{1},\psi_{2}){\big{|}}^{2}\geq
0$ (2w)

${\Big{|}}{\big{[}}{\bf T}_{\perp}(\psi_{1})\times{\bf
T}_{\perp}(\psi_{2}){\big{]}}\cdot\hat{z}{\Big{|}}^{2}\geq 0$ (2x)

${\Big{|}}{\big{[}}{\bf
T}_{\perp}(\psi_{2})\times{\Delta_{\perp}}(\psi_{1},\psi_{2}){\big{]}}\cdot\hat{z}{\Big{|}}^{2}\geq
0$ (2y)

When the first two quantities saturate $0$, $\psi_{1}$ and $\psi_{2}$
correspond to a location of distance of closest approach; these points are in
the neighborhood of non-local contact (given a sufficiently small $P$). The
third quantity extracts the separation distance between the points; when this
quantity _also_ equals $0$, non-local contact occurs. We numerically solve
this coupled system for saturation of all three constraints (in terms of the
contact pitch, $P_{\rm non-local}$ and the surface contact coordinates,
$\psi_{1}$ and $\psi_{2}$) for every combination of $R$, $\alpha$, and
$\epsilon$. In this procedure, we impose a constraint that
$\psi_{1}\neq\psi_{2}$; the angular constraints baked into our inequalities
then maintain that $\psi_{1}$ and $\psi_{2}$ are in fact non-local along the
tube surface.

###  B.2 Local Contact

Local contact occurs at a single location along the tube surface where the
surface fails to be smooth; this discontinuity manifests as a crease in the
surface or cusp singularity in the in-plane representation. Operationally,
these singularities can be selected by locations where the _differential
surface area_ of the tube vanishes, corresponding to the surface being
multivalued.

Using the surface metric, $\rm g$, the magnitude of the surface normal can be
expressed as

$\sqrt{\rm det(g)}=|\partial_{s}{\bf X}(s,\psi)\times\partial_{\psi}{\bf
X}(s,\psi)|$ (2z)

which yields the surface area of a differential element as

$dA=\sqrt{\rm det(g)}\,ds\,d\psi$ (2aa)

We select vanishing area elements via solutions to $\rm det(g)=0$, which
yields a sum of three squares (corresponding to component for each of the
orthonormal basis components),
${c_{1}}^{2}(\psi)+{c_{2}}^{2}(\psi)+{c_{3}}^{2}(\psi)=0$.

${c_{1}}^{2}(\psi)=b^{2}\cos\psi\sin\psi-a^{2}\cos\psi\sin\psi\\\ $ (2ab)

${c_{2}}^{2}(\psi)=ab\cos^{2}\psi\cos\alpha\sin\theta-b^{2}\cos\psi\sin\psi\sin\alpha\sin\theta-\ell
b\cos\psi\\\ $ (2ac)

${c_{3}}^{2}(\psi)=-ab\sin^{2}\psi\sin\alpha\sin\theta+a^{2}\cos\psi\sin\psi\cos\alpha\sin\theta-\ell
a\sin\psi$ (2ad)

Of course, satisfying
${c_{1}}^{2}(\psi)+{c_{2}}^{2}(\psi)+{c_{3}}^{2}(\psi)=0$ requires that each
term equal zero.

Importantly, the first condition, ${c_{1}}^{2}$, exhibits a singular
dependence on the aspect ratio, $\epsilon$; when the cross-section is circular
($\epsilon=1$), the condition is automatically satisfied for any value of
$\psi$ and the minimal pitch ($P_{\rm local}$ takes the form of a helix
_“limited by it’s curvature”_ presented by Przybyl and Pieranski [78], see
eqn. 2h.

When $\epsilon<1$, satisfying the initial constraint (${c_{1}}^{2}(\psi)=0$)
requires that the product $\cos\psi\sin\psi=0$; naturally, this requires that
either $\cos\psi$ or $\sin\psi=0$ (i.e. $\psi={0,\pi/2,\pi,3\pi/2,2\pi}$).
Substitution of these constraints into ${c_{2}}^{2}(\psi)$ and
${c_{3}}^{2}(\psi)$ yields two physical solutions for $P_{\rm local}$, see
eqn. 2i and 2j:

These solutions present as ellipses in the $(R,P)$ plane, making it trivial to
identify their center and vertices. The ellipses are described by

$\frac{{P}^{2}}{(a\pi\cos\alpha)^{2}}+\frac{(R-\frac{1}{2}a\cos\alpha)^{2}}{(\frac{1}{2}a\cos\alpha)^{2}}=1$
(2ae)

and

$\frac{{P}^{2}}{(b\pi\sin\alpha)^{2}}+\frac{(R-\frac{1}{2}b\sin\alpha)^{2}}{(\frac{1}{2}b\sin\alpha)^{2}}=1$
(2af)

Perhaps more intuitively (or at least more graphically), this contact feature
can also be detected numerically by a diverging in-plane curvature; the
geometric significance of both operations are the same. In Fig 13, the in-
plane radius of curvature, $\kappa_{\perp}^{-1}=|{\bf
X}_{\perp}^{\prime}(\psi)|^{3/2}/|{\bf X}_{\perp}^{\prime}(\psi)\times{\bf
X}_{\perp}^{\prime\prime}(\psi)|$ is shown for the locally self-contacting
structure presented in Fig. 3B.

Figure 13: Radius of curvature ($\kappa_{\perp}^{-1}$) along ${\bf
X}_{\perp}(\psi)$ corresponding to Fig. 3B(top); In this structure,
$\kappa_{\perp}^{-1}$ spans 0 (cusp) to $\infty$ (inflection); for
visualization the color bar scales logarithmically and was truncated to
saturate at $\kappa_{\perp}^{-1}=10^{-5}$ (dark blue) and
$\kappa_{\perp}^{-1}=10^{5}$ (dark red).

###  B.3 The Local - Non-local Boundary

As reported in [78], the isotropic filament contact regimes meet a single
point, $R\simeq 0.431d_{0}$, $P\simeq 1.083d_{0}$. This structure, shown in
Fig 5A.iv simultaneously exhibits local and non-local. Perhaps intuitively,
these two modes occur at distinct locations along the tube surface (i.e
distinct $\psi$ coordinates); the local contact (or curvature constraint)
occurs inside the helical core (i.e in the direction of curvature) while the
non-local contact occurs between successive turns (i.e roughly in the vertical
direction). Several anisotropic filaments with local/non-local coexistence
were presented in the main text (Fig. 6C.ii, Fig. 6C.iv, Fig. 7C.ii, Fig.
7C.v, and Fig. 8C.iii), however all of these structures appear to exhibit
simultaneous contact at a single point in the croiss-section; the two sites of
contact are indistinct. This curious difference occurs due to the specific
pathway shown in Fig. 6C - 8C, in particular that the coexistent structure is
shown at a relatively low value for $R$. In Fig. 14, we show additional
structures along the local/non-local coexistence line. For both
$\epsilon=0.95$ and $\epsilon=0.25$, we find that the boundary exhibits
regions of distinct and indistinct coexistence, at large and small $R$,
respectively; transitions between these motifs can be determined by
inflections in the coexistence boundaries shown in Fig. 6B and 8B. It seems
then that this indistinct coexistence is then a consequence (or in fact, a
cause) of the non-local contact regime’s ability to persist to smaller helical
radii; this structural motif is an inherent feature of cross-sectional
anisotropy.

Figure 14: Coexistence between Local (L) and Non-local (NL) contact. (A.)
Single point of isotropic ($\epsilon=1$) L/NL coexistence; here, the local
contact points “inward” (hence classified as local(-)) and the contacts occur
at distinct locations along the tube surface. (B.) States along the L/NL
coexistence boundary for slightly anisotropic tubes ($\epsilon=0.95$); the
boundary pans both local(-) and local(+), as discussed in Section 3.2. Most
states (i.e. intermediate $\alpha$)coexistent contact at a indistinct (i.e.
the same) location along the tube surface exhibit; extreme values of $\alpha$
exhibit distinct contact. (C.) States along the L/NL coexistence boundary for
highly anisotropic tubes ($\epsilon=0.25$). States at smaller $\alpha$ exhibit
indistinct contact while states at larger $\alpha$ exhibit distinct contact.

###  B.4 Nested packings at low $R_{C}$ and high $\epsilon$?

Highly anistropic structures follow a straightforward progression, tape
$\rightarrow$ scroll $\rightarrow$ screw as $R_{C}$ increases, as shown in
e.g. Fig. 10A,bottom for $\epsilon=0.25$. Curiously, when $\epsilon$ is large,
we observe a second state of scroll-like packing, which is apparently more
dense than the corresponding tape-like packing. This phenomena is observed
under the entire $\epsilon$ scale where the Local(-) branch constrains the
contact manifold, $1>\epsilon\gtrsim 0.8175$, and corresponds for a preference
for non-local contact over local(-) contact; we therefore dub this transition
as the “Cusp Escape” regime.

Figure 15: Optimal packing at high $\epsilon$ yields two nested regimes. (A.)
$\epsilon=0.95$ contact mode phase diagram (black, see Fig. 6B), optimal
packing (red, corresponding to Fig. 10A), and the “Ideal” low $\epsilon$ tape
$\rightarrow$ scroll $\rightarrow$ screw progression (gray, dashed). The
preferred packing exhibits intermediate tilt to “escape” the local(-) regime.
(B. top) $\phi$ (black) and $\theta$ (blue) along the optimal branch, as well
as along the “Ideal” branch (gray, dashed). (B. bottom) Normalized deviation
of the quantities ($q$), $\phi$ and $\theta$.

Based on Fig. 15B, it is obvious that this preference for intermediate tilt at
low $R_{C}$ does not significantly influence the packing; for $\epsilon=0.95$,
the helical angle ($\theta$) and the packing density ($\phi$) only deviate by
at most $\simeq 2\%$; this largely is a consequence of the weak anisotropy.
While this secondary tilt motif persists down to $\epsilon\simeq 0.8175$
(where tilt should have a larger consequence), the local(-) regime decrease in
size with decreasing $\epsilon$ (see Fig. 7); as a result, the magnitude of
$\alpha$ (and therefore deviation from “Ideal” packing) needed for “Cusp
Escape” also decreases (compare, e.g. $\epsilon=0.95$ and $\epsilon=0.85$ in
Fig. 10A. As such, we consider these low $R_{C}$/high $\epsilon$ nested states
to be a relatively inconsequential deviation from the more intuitive tape
$\rightarrow$ scroll $\rightarrow$ screw progression with increasing capillary
size.

##  Appendix C Capillary Contact Maximizes Density

In the present work, we specifically report capillary density of dense (i.e.
self-contacting) helices within their tightest capillary. However, our
approach is in fact far more general; the capillary density, given in
expression Eqn. 2k can be applied to helices of arbitrary configuration (in
particular, “expanded helices” with $P>P_{\rm min}$) within arbitrary
capillaries (in particular, with $R_{C}\geq{\rm max}_{\psi}|{\bf
X}_{\perp}(\psi)|$).

While this larger ensemble of states may certainly be of interest in some
physical phenomena or systems, our current focus is on states that _maximize
density_.

It can be shown analytically that

$\frac{\partial\phi}{\partial P}\leq 0$ (2ag)

for $P\geq P_{\rm min}$ (independent of the values of $R$, $\alpha$,
$\epsilon$), so in surveying the densest configurations, it is appropriate to
only consider packings which simultaneously exhibit self- and capillary-
contact.

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