# Inherent Structure Landscape of Hard Spheres Confined to Narrow Cylindrical
Channels.

Mahdi Zarif<EMAIL_ADDRESS>Department of Physical and Computational
Chemistry, Shahid Beheshti University, Tehran 19839-9411, Iran.  Raymond J.
Spiteri  Department of Computer Science, University of Saskatchewan,
Saskatoon, Saskatchewan S7N 5C9, Canada.  Richard K. Bowles
<EMAIL_ADDRESS>Department of Chemistry, University of Saskatchewan,
SK, S7N 5C9, Canada.  Centre for Quantum Topology and its Applications
(quanTA), University of Saskatchewan, SK S7N 5E6, Canada.

###### Abstract

The inherent structure landscape for a system of hard spheres confined to a
hard cylindrical channel, such that spheres can only contact their first and
second neighbours, is studied using an analytical model that extends previous
results [Phys. Rev. Lett. 115, 025702 (2015)] to provide a comprehensive
picture of jammed packings over a range of packing densities. In the model, a
packing is described as an arrangement of $k$ helical sections, separated by
defects, that have alternating helical twist directions and where all spheres
satisfy local jamming constraints. The structure of each helical section is
determined by a single helical twist angle, and a jammed packing is obtained
by minimizing the length of the channel per particle with respect to the $k$
helical section angles. An analysis of a small system of $N=20$ spheres shows
that the basins on the inherent structure landscape associated with these
helical arrangements split into a number of distinct jammed states separated
by low barriers giving rise to a degree of hierarchical organization. The
model accurately predicts the geometric properties of packings generated using
the Lubachevsky and Stillinger compression scheme ($N=10^{4}$) and provides
insight into the nature of the probability distribution of helical section
lengths.

##  I Introduction

Hard sphere particle packings play a fundamental role in understanding a broad
range of problems in material science. For example, mixtures of hard sphere
particles of different sizes give rise to a variety of stable crystal
structures [1, 2, 3, 4], and amorphous packings are used as the basis for
describing the properties of liquids and glasses [5, 6, 7, 8, 9, 10, 11, 12],
as well as colloidal systems [13] and granular materials [14, 15]. Sphere
packing problems also appear in applications for computer science and
information technology [16]. However, rigorous results for sphere packings are
difficult to obtain. The most dense jammed packing of single component hard
discs in two dimensions (2D) is the triangular packing, with an occupied
volume fraction $\phi=\pi/\sqrt{12}$. Recently, it was shown that the face-
centred-cubic (FCC) crystal, with $\phi=\pi/\sqrt{18}\approx 0.74$, is the
most dense jammed packing of spheres in 3D [17], but results for amorphous
structures are generally obtained using numerical and molecular simulation
techniques.

Confining the fluid to small pores or narrow channels limits the number of
contacts a particle can make, reducing the type of local structures that can
be formed and simplifying the inherent structure landscape [18, 19]. As a
result, it is sometimes possible to obtain analytical results describing the
nature of particle jamming. The 2D system of hard discs trapped between two
lines, where the confinement only allows contacts up to the second nearest
neighbours, has been studied extensively because its local packing
environments are highly ordered and the jammed states, from least to most
dense, can be characterized in terms of defects [20, 21]. The comprehensive
description of the inherent landscape and the relationship to the properties
of the fluid have facilitated a detailed analysis of glassy dynamics [22, 23,
24, 25, 26, 27], hyperuniformity in amorphous packings [28], the statistical
mechanics of granular materials [29], and the possibility of a Gardner
transition in hard particle systems [30]. Recent studies of this system have
also revealed the existence of novel asymptotically crystalline states [31].

Both analytical [32, 33, 34] and simulation [35, 36, 37, 38] studies have
shown that spheres confined to quasi-one-dimensional cylindrical channels
spontaneously form a variety of structures including, single, double, and
staggered helices, as well as some achiral packings, depending on the diameter
of the channel [39, 40, 41]. These helical structures have also been observed
experimentally in molecular [42, 43], colloidal [44, 45], and athermal [46]
systems. Most studies have focused on the formation of the most dense packing
with a perfect structure, but to develop connections between the
thermodynamics and dynamics of fluids and the inherent structure landscape or
to establish a statistical mechanics of athermal systems, it is necessary to
understand the nature of defect and amorphous packings in these systems.

The goal of the current work is to develop a comprehensive picture of the
inherent structure landscape for a system of hard spheres confined within a
narrow cylindrical channel, where particle–particle contacts up to the nearest
second neighbours are possible. This builds on our previous analysis [47] by
removing the need to use assumptions concerning the nature of the defect
states, identifying additional packing environments, and studying the
distribution of packings for small systems. As a result, we find that the
inherent structure landscape for this system has a degree of hierarchical
organization, with basins formed from the arrangements of helical sections
splitting into sub-basins, associated with distinct jammed states, separated
by small barriers. We also use the model to understand features that appear in
the probability distribution of helical section lengths in simulation
generated packings.

The remainder of the paper is organized as follows: Section II describes the
model under study. Section III describes the geometry of perfect and defect
helical packings. Section IV explores inherent structure landscape of the
packing model for a small system before the model predictions are compared to
the results of large system jammed states produced by simulation in Section V.
Sections VI and VII contain our discussion and conclusions, respectively. The
Appendix outlines the analytical calculation of the random probability
distribution for helical section lengths.

##  II Model

The model studied here consists of N three–dimensional hard spheres, with
diameter $\sigma$, confined in a cylindrical narrow channel of length L with
channel diameter $H_{d}$ in the range of $1+\sqrt{3/4}<H_{d}/\sigma<2$, which
ensures spheres can only contact their first and second neighbours in either
direction along the channel. The particle–particle and particle–wall
interaction potentials are given by,

$U(r_{ij})=\left\\{\begin{matrix}0\;\;\;&r_{ij}\geqslant\sigma\;\;\;\;\;\;\;\;\\\
\infty\;\;\;&r_{ij}<\sigma\;\;\;\;\;\;\;\;\end{matrix}\right.,$ (1)
$U_{w}(r_{i})=\left\\{\begin{matrix}0\;\;\;&|r_{xy}|\leqslant\left|H_{0}/2\right|\\\
\infty\;\;\;&\textup{otherwise}\end{matrix}\right.,$ (2)

respectively, where $r_{ij}=\left|\mathbf{r_{j}-r_{i}}\right|$ is the distance
between particles, $|r_{xy}|$ is the magnitude of the position vector for a
particle perpendicular to the wall where the center of the cylinder is located
at $x=y=0$ and the longitudinal direction of the channel extends in the $z$
direction. The volume accessible to the particles centers is $V_{0}=\pi
L(H_{0}/2)^{2}$, where $H_{0}=H_{d}-\sigma$, and the occupied volume is
$\phi=2N\sigma^{3}/\left(3LH^{2}_{d}\right)$.

##  III Geometric Model for Helical Packings

###  III.1 Perfect Helical Packings

This section provides improved details of our geometric analysis presented in
Ref. [47], which focused on the case of $H_{d}/\sigma=1.95$. The most dense
packing of the current system is a helix [41] where each sphere has four
sphere-sphere contacts, formed with its first and second neighbours and a
single sphere wall contact. Figure 1 shows the geometric construction used to
determine the properties of the helix. Particle one is placed in the channel
at a fixed point, then particle two is placed along the channel at a distance
$z_{1}$ and angle $\alpha_{1}$ in a anticlockwise direction looking from the
top, such that it contacts sphere one and the wall. Particle three, which is
placed at a distance $z_{2}$ and $\alpha_{2}$ along the channel is then
constrained to contact the wall as well as particles one and two. Subsequent
particles are added, alternating between increments of $(z_{1},\alpha_{1})$
and $(z_{2},\alpha_{2})$, maintaining the contact constraints until particles
$N-1$ and $N$ are forced to contact particles one and two to ensure that the
helical periodic boundary conditions are enforced.

Figure 1: Geometric construction for perfect helix. (a) The top view with
black points indicating the sphere centers of particles 1, 2, 3 and the blue
point marking the central axis of the cylinder. The heavy dotted line outlines
the volume accessible to the particle centers and the solid grey line
indicates the cylinder wall. (b) The side view showing central accessible
volume. The hypotenuses of the colored triangles indicate sphere contacts with
lengths $\sigma$. Spheres and external cylinder wall have been remove for
clarity.

Using this geometric construction yields the following relationships:

$\sigma^{2}=a^{2}+z_{1}^{2}\mbox{,}$ (3) $\sigma^{2}=b^{2}+z_{2}^{2},$ (4)
$\sigma^{2}=c^{2}+(z_{1}+z_{2})^{2}\mbox{,}$ (5)
$a=H_{0}\sin(\alpha_{1}/2)\mbox{,}$ (6) $b=H_{0}\sin(\alpha_{2}/2)\mbox{,}$
(7) $c=H_{0}\sin(\alpha_{3}/2)\mbox{,}$ (8)

where $a$, $b$, and $c$ form the projected triangle connecting the particle
centers. Expressions for $z_{1}$ and $z_{2}$ can be obtained by using Eqs. 6
and 7 in Eqs. 3 and 4 to give respectively,

$z^{2}_{1}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha_{1}\right]=\sigma^{2}$ (9)

and

$z^{2}_{2}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha_{2}\right]=\sigma^{2}\mbox{.}$
(10)

Substituting these results into Eq. 5 and using the constraint,
$\alpha_{1}+\alpha_{2}+\alpha_{3}=2\pi$ in Eq. 8 gives,

$\sigma^{2}=\left(\sqrt{\sigma^{2}-\frac{H_{0}^{2}}{2}+\frac{H_{0}^{2}}{2}\cos\left[\frac{\alpha_{1}}{2}\right]}\right.\\\
+\left.\sqrt{\sigma^{2}-\frac{H_{0}^{2}}{2}+\frac{H_{0}^{2}}{2}\cos\left[\frac{\alpha_{2}}{2}\right]}\right)^{2}\\\
-\frac{H_{0}^{2}}{2}\left(\cos\left[\alpha_{1}+\alpha_{2}\right]-1\right)\mbox{,}$
(11)

which can be solved numerically to provide values of $\alpha_{2}$ as a
function of $\alpha_{1}$.

All the spheres in the helix satisfy the three-dimensional local jamming
condition, where each particle has at least four contacts that are not all
contained within the same hemisphere. However, this does not guarantee the
system is collectively jammed because the concerted motion of particles can
lead to unjamming [48]. In the limit that $H_{d}/\sigma\rightarrow
1+\sqrt{3/4}$, Eq. 11 has a single solution, with $\alpha_{1}=\alpha_{2}=\pi$,
that corresponds to the formation of the expected single zig–zag chain of
jammed particles. For wider channels, there is a continuous range of solutions
where the variation of $\alpha_{1}$ produces a concerted twisting motion of
the helix that compresses the overall structure. To locate the most dense
jammed state of the system, we then minimize the length per particle,

$\frac{L}{N}=\frac{1}{2}\left(z_{1}+z_{2}\right)\mbox{.}$ (12)

Figure 2 shows that $L/N$ exhibits a single minimum as a function of
$\alpha_{1}$ in the range $1+\sqrt{3/4}<H_{d}/\sigma<1+4\sqrt{3}/7$, where
$\alpha_{1}=\alpha_{2}$ and $z_{1}=z_{2}$, confirming that the most dense
packings are single helices. In the region $1+4\sqrt{3}/7<H_{d}/\sigma<2$,
$L/N$ $\alpha_{1}$ no longer equals $\alpha_{2}$ and, the resulting jammed
structure consists of two staggered helices, which leads to the appearance of
two identical minima corresponding to the two possible ways of alternating
between $\alpha_{1}$ or $\alpha_{2}$. It is also important to note that both
the right ($\mathcal{P}$) and left ($\mathcal{M}$) handed helices can be
constructed by incrementing the angle in clockwise or anticlockwise directions
respectively. The results from our geometric construction are consistent with
the simulation results obtained by Picket et al. [41]. It is also important to
note that Fig. 2 essentially reproduces the results of Chen et al. [34], who
used a similar approach to study the most dense perfect helical packings for
$1+\sqrt{3/4}<H_{d}/\sigma<2$, and we include this figure here for
completeness and to highlight the importance of the minimization of $L/N$ with
respect $\alpha_{1}$, which plays an expanded role in determining the nature
of jammed structures containing defects.

Figure 2: Length per particle, $L/N$, as a function of $\alpha_{1}$, for
perfect helical sphere packings with channels diameters in the range
$H_{d}/\sigma=1.87-1.999$. Red points indicate local minima associated with
jammed packings.

###  III.2 Defect Helical Packings

To generate lower density jammed states, we introduce topological defects into
the helix that reverse the direction of the helical twist, noting that these
must be introduced in pairs in order to maintain the helical periodic boundary
conditions. As a result, a packing with $k$ defects consists of $k$
alternating left and right twisting sections of helix that can be
characterized by a list $\left\\{n^{(1)},n^{(2)},\ldots,n^{(k)}\right\\}$,
where $n^{(j)}$ specifies the number of spheres in a helical section $j$ so
that $N=\sum_{j=1}^{k}n^{(j)}$. Figure 3 shows the geometric construction used
to calculate the properties of a defect located between two sections of helix,
denoted with superscripts $a$ and $b$ respectively. Spheres 1 and 2, which are
in contact, represent the last two particles of helix section $a$ twisting in
an anti-clockwise direction (from the top) and the angle $\alpha^{a}_{i}$ can
be either $\alpha_{1}$ ($i=1$) if $n^{(a)}$ is even or $\alpha_{2}$ ($i=2$) if
$n^{(a)}$ is odd. Spheres 3 and 4 are also in contact with each other and are
the first two particles in helix section $b$, twisting in a clockwise
direction. The defect is then located between particles 2 and 3 and is
characterized by the longitudinal length $z_{d}$ and angle $\alpha_{d}$. In
order to satisfy local jamming constraints, all the particles must contact the
walls, sphere 3 must also contact sphere 1, and sphere 2 must contact sphere
4, but a contact between spheres 2 and 3 is not necessary.

Figure 3: Geometric construction for helical defects (a) The top view, and (b)
the side view.

Using the geometric construction shown in Fig. 3, we obtain the following
relations for the particle separations along the longitudinal $z$-axis:

$(z^{(a)}_{i})^{2}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha^{(a)}_{i}\right]=\sigma^{2}\mbox{,}$
(13)
$(z^{(b)}_{1})^{2}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha^{(b)}_{1}\right]=\sigma^{2}\mbox{,}$
(14)
$(z_{e})^{2}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha_{e}\right]=\sigma^{2}\mbox{,}$
(15)
$(z_{f})^{2}+\frac{H_{0}^{2}}{2}\left[1-\cos\alpha_{f}\right]=\sigma^{2}\mbox{,}$
(16)

where $\alpha_{e}$ and $\alpha_{f}$ are the in-plane angles between particles
one and three and particles two and four, respectively. Substituting Eqs.
13–16, along with the angular constraints
$\alpha_{e}=2\pi-\alpha^{(a)}_{i}-\alpha_{d}$ and
$\alpha_{f}=\alpha_{d}-\alpha^{(b)}_{1}$ into
$(z_{e})^{2}=(z^{(a)}_{i}+z_{d})^{2}$ and
$(z_{f})^{2}=(z^{(b)}_{1}+z_{d})^{2}$ yields two relations,

$z_{d}^{2}+\frac{H_{0}^{2}}{2}\cos\alpha^{(a)}_{i}+2z_{d}\sqrt{\sigma^{2}-\frac{H_{0}^{2}}{2}\left[1-\cos\alpha^{(a)}_{i}\right]}\\\
-\frac{H_{0}^{2}}{2}\left[\cos(\alpha^{(a)}_{i}+\alpha_{d})\right]=0\mbox{,}$
(17)

and

$\frac{H_{0}^{2}}{2}\left[1-\cos(\alpha_{d}-\alpha_{f})\right]-\sigma^{2}\\\
+\left(z_{d}+\sqrt{\sigma^{2}-\frac{H_{0}^{2}}{2}\left[1-\cos\alpha^{(b)}_{1}\right]}\right)^{2}=0\mbox{,}$
(18)

that can be solved numerically to obtain $z_{d}$ and $\alpha_{d}$. Equations
17 and 18 highlight the fact that the geometric properties of the defect are a
function of the properties of the helical sections on either side through
their dependence on $\alpha^{(a)}_{i}$ and $\alpha^{(b)}_{1}$. This represents
an improvement on our original analysis in Ref. [47], where it was assumed
$\alpha_{d}=\pi$, a condition that is only true if the helices on both sides
of the defect are the same, i.e., for what we referred to as helical defect
crystals because they represent repeated units of identical left– and
right–handed helical sections so the defects appear at regular spacings along
the packing.

The length per particle of a configuration of spheres satisfying the contact
conditions and containing $k$ defects is then given by,

$\frac{L}{N}=\frac{1}{N}\sum_{j=1}^{k}\left[\left(\frac{n^{(j)}-A}{2}\right)z_{1}(\alpha_{1}^{(j)})\right.\\\
+\left.\left(\frac{n^{(j)}-A}{2}-B\right)z_{2}(\alpha_{1}^{(j)})+z_{d}(\alpha_{i}^{(j)},\alpha_{1}^{(j+1)})\right]\mbox{,}$
(19)

where $A=0\mbox{ or }1$ if $n^{(j)}$ is even or odd respectively, and
$B=1\mbox{ or }0$ if $n^{(j)}$ is even or odd respectively. Helical periodic
boundaries are imposed for the first and last helical sections, noting that in
determining $z_{d}$, $i=1$ for even $n^{(j)}$ and $i=2$ for odd $n^{(j)}$. A
jammed packing can then be obtained by minimizing $L/N$ in Eq. 19 with respect
to the set of $k$ angles, $\alpha_{1}^{(j)}$, that characterize each helical
section and the jamming density,
$\phi_{J}=2N\sigma^{3}/\left(3LH^{2}_{d}\right)$.

The number of possible helical arrangements for a given defect fraction,
$\theta=k/N$, can be found by considering the number of ways we can distribute
$N$ indistinguishable spheres into $k$ helical sections. The geometric contact
restrictions of the defects require each section of helix to contain at least
two spheres, which leaves $N-2k$ spheres to be placed without restrictions,
yielding the number of helical section arrangements with $k$ defects,

$N_{h}(k)=\frac{2(N-k-1)!}{(k-1)!(N-2k)!}\mbox{,}\\\ $ (20)

where the factor for $2$ appears because we can choose to begin twisting the
first helical section in either a clockwise or anticlockwise direction. Taking
the natural logarithm of Eq. 20 and using Stirling’s approximation then gives,
in the thermodynamic limit,

$\frac{\ln
N_{h}(\theta)}{N}=(1-\theta)\ln(1-\theta)-\theta\ln\theta-(1-2\theta)\ln(1-2\theta)\mbox{,}\\\
$ (21)

where $0\leq\theta\leq 0.5$. The total number of helical arrangements can be
obtained by summing over all possible values of $k=0,1,\ldots,N/2$ in Eq. 20,
to obtain,

$\lim_{N\rightarrow\infty}\frac{\ln
N_{h}}{N}=\ln\left(\frac{1+\sqrt{5}}{2}\right)\approx 0.481\mbox{.}\\\ $ (22)

Equation 21 is the same distribution of structures found in the 2D model of
hard discs confined between two lines where only nearest neighbour contacts
are allowed [20]. However, in the 2D model, $\theta$ is directly linked to the
jamming density, $\phi_{J}$, which is independent of the way the defects are
arranged, and there is only one jammed state associated with each defect
arrangement. As we will see, this is not the case for the current model, so
$N_{h}(\theta)$ becomes a lower bound on the number of jammed states, and
$\phi_{J}$ for a packing is not only determined by $\theta$, but it also
depends on how the defects are distributed.

##  IV Small System Jammed Packings

To understand how the presence of defects affects the overall structure of the
packings, we begin by studying a system of $N=50$ spheres containing just two
defects and two sections of helix, denoted $a$ and $b$, for a case where
$H_{d}/\sigma=1.95$. Figure 4a shows a contour plot of $L/N$, from Eq. 19, as
a function of $\alpha^{(a)}_{1}$ and $\alpha^{(b)}_{1}$ for a system
$\\{n^{(a)},n^{(b)}\\}=\\{15,35\\}$. When the defects are well separated and
both helix sections contain an odd number of spheres, the minimum occurs where
$\alpha^{(a)}_{1}=\alpha^{(b)}_{1}$, at a value where $\alpha_{1}=\alpha_{2}$
within both sections of helix. As a result, the jammed packing consists of two
sections of a perfect single helix, identical to the most dense structure,
with opposite $\mathcal{P}$ and $\mathcal{M}$ twists, separated by a large
volume defect.

Figure 4: Contour plots of $L/N$ as a function of $\alpha_{1}^{(a)}$ and
$\alpha_{1}^{(b)}$ for packings (a) $\\{15,35\\}$ and (b) $\\{14,36\\}$, with
$H_{d}/\sigma=1.95$. Contours are separated by 0.001, the red dashed lines
represent the contact constraint between particles 2 and 3 in the defects, and
the red points denote the minima.

When the two helical sections contain an even number of spheres, such as
$\\{n^{(a)},n^{(b)}\\}=\\{14,36\\}$ (see Fig. 4b),
$\alpha^{(a)}_{1}\neq\alpha^{(b)}_{1}$ so that the two sections no longer have
the same structure. Furthermore, within each section,
$\alpha_{1}\neq\alpha_{2}$, indicating the structure has changed to that of a
double helix. In this case, the presence of a single defect has disrupted the
entire global structure of the packing. Figure 4 also shows that when the
defects are well separated (large helical sections) the minima are located
away from the contact constraint between particles 2 and 3 in the defect
indicating there is a gap between the particles. As the defects are brought
closer together, by decreasing the size of one of the helical sections, the
surface of the minimum elongates along the $\alpha^{(a)}_{1}$ axis before
eventually splitting in two when $n^{(a)}\approx 10$. Figure 5 shows that the
system becomes jammed when particles 2 and 3 in the defect come into contact,
locking the structure in place. Now, $\alpha^{(a)}_{1}\neq\alpha^{(b)}_{1}$
for odd–numbered helical sections, but the two minima have the same $L/N$ and
result from a switching between the values of $\alpha_{1}$ and $\alpha_{2}$
within a section. Nevertheless, the two packings are distinct structures
because they are separated by a barrier, and it requires the concerted motion
of the relative position of all the particles to move from one state to the
other. The jamming densities of the two states are not the same for the
even–number helical sections (Fig. 5).

Figure 5: Contour plots of $L/N$ as a function of $\alpha_{1}^{(a)}$ and
$\alpha_{1}^{(b)}$ for packings (a) {3,47} and (b) {2,48}, with
$H_{d}/\sigma=1.95$. Contours are separated by 0.001, the red dashed lines
represent the contact constraint between particles 2 and 3 in the defects and
the red and green points denote a global minimum and a local minimum,
respectively.

Figure 6a shows $\phi_{J}$, where Eq. 19 has been minimized, as a function of
$n^{(a)}$ for a system containing two defects, with $H_{d}/\sigma=1.95$, and
we have only identified the global density maxima ($L/N$ global minima) for
each $n^{(a)}$. For $N=100$, as $n^{(a)}$ is varied, the structure
$\\{n^{(a)},N-n^{(a)}\\}$ oscillates between structures where both sections of
helix are odd-sized and then both sections are even-sized. The jamming density
of the even-sized sections increases as $n^{(a)}$ decreases, bringing the
defects closer together, whereas $\phi_{J}$ for the structures with odd-sized
sections remains constant until $n^{(a)}<10$. When $N=99$, $n^{(a)}$, the
smaller of the two helical sections, oscillates between odd and even, and we
see a similar trend in $\phi_{J}$, where $\phi_{J}$ is lower when $n^{(a)}$ is
odd. Figure 6b shows the same analysis for a system with $H_{d}/\sigma=1.99$,
where the most dense state is a double helix. The behaviour of $\phi_{J}$ for
the even helical sections remains the same as observed for the narrower
channel and increases as $n^{(a)}$ decreases. However, we now see $\phi_{J}$
for the odd helical sections increasing over the entire region. In addition,
$\alpha_{i}$ in both odd helical sections differ from those of the perfect
helix.

Figure 6: The jamming occupied volume fraction, $\phi_{J}$, for packings
$\\{n^{(a)},N-n^{(a)}\\}$ as a function of $n^{(a)}$ for systems with $N=100$
(filled circles) and $N=99$ (open circles) for (a) $H_{d}/\sigma=1.95$ (b)
$H_{d}/\sigma=1.99$. Only the most dense structure for each
$\\{n^{(a)},N-n^{(a)}\\}$ is included.

The analysis of a system with just two defects clearly shows that the way the
defects are distributed within a packing influences $\phi_{J}$, suggesting it
is worth examining packings with an increased number of defects. The total
number of jammed packings increases exponentially with $N$, but the simplicity
of the packing model described in Section III means that it is possible to
generate all the possible model packings for small systems and examine the
full jamming landscape. The challenge of finding all the jammed structures is
simplified by recognizing that all the helical arrangements, under periodic
boundaries, can be generated from a series of canonical arrangements by a
rotation,
$\\{n^{(1)},n^{(2)},\ldots,n^{(k)}\\}\rightarrow\\{n^{(k)},n^{(1)},\ldots,n^{(k-1)}\\}$
or a reversal of the order,
$\\{n^{(1)},n^{(2)},\ldots,n^{(k)}\\}\rightarrow\\{n^{(k)},n^{(k-1)},\ldots,n^{(1)}\\}$,
then any replicated states can be removed. For example, the canonical
arrangement $\\{2,2,3,2,2,9\\}$, generates six distinct arrangements by
rotation, but the operation of reversal generates six identical arrangements,
which are then removed to leave the original six. However, for
$\\{2,2,2,3,2,9\\}$, the rotation and reversal operations generate a total of
12 distinct arrangements.

All structures related to the same canonical arrangement maintain the same
relative position of all the helical sections and minimize to a set of jammed
structures with the same respective geometric properties, with the same set of
$\alpha^{(j)}_{1}$ and jamming density. In addition, each helical arrangement
actually represents two possible arrangements that can be formed by winding in
the opposite direction, but the arrangements will again minimize to structures
with the same geometric properties. This mechanical approach lets us count the
number of states associated with each canonical arrangement and reproduces the
same total number of helical arrangements as predicted by Eq. 20. Table 1
compares the number of canonical arrangements, $N_{c}$, with the total number
of helical arrangements, $N_{h}$, for a series of small system sizes and
highlights the significant reduction in the number of arrangements that need
to be analysed.

Table 1: The number of canonical helical arrangements compared to the total number for small systems.  $N$ | $N_{h}$ | $N_{c}$
---|---|---
16 | 306 | 54
20 | 2091 | 238
24 | 14330 | 1206
32 | 673135 | 38889

Here, we study systems of $N=20$ spheres with $H_{d}/\sigma=1.95$ and $1.99$,
respectively, by searching for the jammed packings associated with all of
their 238 canonical helical arrangements. Since the basin for a given helical
arrangement can split into sub–basins, as shown in Fig. 5, we use the local
optimization, Nelder-Mead algorithm in Mathematica [49] and perform five
minimizations of Eq. 19 for each helical arrangement with randomized initial
starting points. While this does not ensure we find the global minimum for
each helical arrangement, and a larger number of trials for each arrangement
could reveal additional jammed states, it allows us to explore the general
features affecting the appearance of sub–basins. We will also focus our
analysis on arrangements that start with a single direction of twist for the
first helical section as the other starting twist direction will yield the
same results for $\phi_{J}$.

To begin, it is useful to examine some of the properties of a few particular
packings. The perfect helix arrangement, $\\{20\\}$, i.e., with $\theta=0$,
has a single canonical structure that leads to one and two jammed structures
for $H_{d}/\sigma=1.95$ and $1.99$, respectively, as discussed earlier. The
$\\{10,10\\}$ canonical arrangement represents a single arrangement that
minimizes to a single jammed structure, where $\alpha^{j}_{1}$ is the same for
all helical sections, which is also true for $\\{2,2,2,2,2,2,2,2,2,2\\}$
($\theta=0.5$). For the $\\{5,5,5,5\\}$ arrangement, we see a more complex set
of jammed states. With $H_{d}/\sigma=1.95$, we find two distinct packings
where $\alpha^{(j)}_{1}$ is the same in all four helical sections, one where
the list of angles $\\{\alpha^{(j)}_{1}\\}=\\{2.2943,2.2943,2.2943,2.2943\\}$
and the other with $\\{\alpha^{(j)}_{1}\\}=\\{2.8104,2.8104,2.8104,2.8104\\}$.
We can understand the appearance of these packings by noting the helical
sections containing an odd number of spheres have an even number of
alternating angles, $\alpha^{(j)}_{1}$ and $\alpha^{(j)}_{2}$ (see Eq. 19),
where the second angle is a function of the first. In the first packing,
$\alpha^{(j)}_{1}=2.2943$ and $\alpha^{(j)}_{2}=2.8104$, so it is possible to
construct the second packing by simply interchanging the two angles. For
$H_{d}/\sigma=1.99$, these uniform packings, where $\alpha^{(j)}_{1}$ is the
same in all helical sections, are replaced by packings that alternate
$\alpha^{(j)}_{1}$; i.e.,
$\\{\alpha^{(j)}_{1}\\}=\\{1.7639,2.7512,1.7639,2.7512\\}$ and
$\\{\alpha^{(j)}_{1}\\}=\\{2.7512,1.7639,2.7512,1.7639\\}$. In addition, we
also find that the $\\{5,5,5,5\\}$ helical arrangement, for both channel
diameters, jams in a series of asymmetrical structures, where two of the
helical sections are the same, but the remaining two are different; i.e., for
$H_{d}/\sigma$=1.95 we find
$\\{\alpha^{(j)}_{1}\\}=\\{2.5378,3.0837,2.5378,2.1206\\}$, which functions as
a canonical structure for four additional jammed states. As a result, the
inherent structure basin associated with the $\\{5,5,5,5\\}$ helical
arrangement splits into a total of six distinct jammed states. These sets of
structures illustrate the complexity of the packings that arise, despite the
simplicity of the model, and show that while Eqs. 20–22 provide a count of the
number of basins, or meta–basins, they represent a lower bound to the number
of jammed states. Most of the remaining canonical structures minimize to at
least two distinct jammed states, and it is possible additional structures
would be found with a more extensive search.

A number of factors affect the $\phi_{J}$ of a structure, including the number
of defects and how the defects are distributed. To capture some of the more
generic features of the jammed states, we plot (Fig. 7a) the distribution of
most dense jammed states obtained for each canonical structure, weighted by
the number of helical arrangements it represents, which gives rise to a
distribution of 2091 structures for $N=20$, for the $H_{d}/\sigma=1.95$ model.
Similar results were obtained for the $H_{d}/\sigma=1.99$ case (see the
Supplemental Material (SM) [50]) . Each defect fraction, $\theta$, exhibits a
broad distribution of $\phi_{J}$, again indicating that the jamming density is
sensitive to the way the defects are organized and not just to their
concentration. Notably, there is a significant gap between the $\phi_{J}$ of
the most dense packing of the perfect helix and the packings with a single
defect, but this is the result of the small system size and the distribution
fills the density gap as $N$ increases. Figure 7a also shows that the
$\theta=0.5$ jammed structure, which is the maximum defect fraction possible
in the model, is not the least dense structure. Structurally, this packing is
made up of two parallel zig-zag chains [47]. It becomes the least dense
packing in the limit $H_{d}/\sigma\rightarrow 1+\sqrt{3/4}$, where the
particles can only touch their first nearest neighbours, but as
$H_{d}/\sigma\rightarrow 2$,
$\phi_{J}(\theta=0.5)\rightarrow\phi_{J}(\theta=0)$ and the two structures,
($\theta=0.5;\theta=0$) become equivalent, leading to the formation of an
achiral most dense packing of alternating “doublets” [41]. However, the full
distribution of jammed states is not contained between these two apparent
limits, and the $\theta=0.5$ structure moves through the distribution as the
channel becomes wider.

Figure 7: (a) The cumulative number distribution of jammed states, $N_{J}$,
with different $\theta$, as a function of $\phi_{J}$, for $H_{d}/\sigma=1.95$.
The $\phi_{J}$ for the jammed structure with $\theta=0$ and $\theta=0.5$ are
highlighted with blue and brown circles, respectively. The bin size for the
distribution $\Delta\phi_{J}=0.001$. (b) Fraction of odd helical sections,
$f_{odd}$, for each $\theta$ from the same distribution. Solid lines provide a
guide to the eye.

The underlying double helix structure of the packings means that sections of
helix containing odd and even numbers of spheres have different jamming
properties. To examine this effect in general, we calculate the fraction of
helical sections in a jammed state containing an odd number of spheres,
$f_{odd}$, for each $\theta$, as a function of $\phi_{J}$. Figure 7b shows
that increasing $f_{odd}$ tends to decrease the jamming density, with the
effect becoming greater for larger $\theta$. This suggests that smaller
sections, with odd numbers of particles, tend to pack less efficiently because
they generate large defects. For the current small system studied here, the
lowest density packings are formed from structures containing a mixture of
helical sections with two and three particles.

##  V Large System Simulated Jammed Packings

The model developed here suggests that the jammed packings of this system
adopt a “poly-helical” type structure, containing a mixture of well formed
sections of single or double helix, where the properties of each helical
section, such as helical pitch, depend on the distribution of the defects. To
test this, we generate jammed packings by continually compressing a system of
$N=10^{4}$ spheres from low density, $\phi=0.01$, using a modified version of
the Lubachevsky and Stillinger (LS) molecular dynamics (MD) scheme [51], with
compression rates $d\sigma/dt=1\times 10^{-3}$, $5\times 10^{-4}$, and
$5\times 10^{-5}$ in reduced units. Packings formed in this way have
previously been shown to be jammed and follow the free volume equation of
state near their jamming densities, with slower compression rates leading to
increased $\phi_{J}$, and decreased $\theta$ [47]. The average properties of
the packing generated for this study are listed in Table 2. Figure 8 shows the
cylindrical angle, $\alpha_{i}$, between successive spheres, as a function of
sphere position along the channel, for a typical section of a jammed packing.
The well formed sections of double helix can be clearly identified, with their
alternating $\alpha_{1}$, $\alpha_{2}$ angles, as can the presence of the
defects separating the helical sections. The configuration also shows the
presence of a section of single helix where $\alpha_{1}=\alpha_{2}$,
containing an odd number of particles, as predicted by the model.

Table 2: Average properties of simulation generated packings, $N=10000$ and $H_{d}/\sigma=1.95$, using compression rate $d\sigma/dt$. The standard deviation in the last digit is given in brackets. $d\sigma/dt$ |  $n^{\circ}$ packings | $\left<\phi_{J}\right>$ | $\left<\theta\right>$
---|---|---|---
$1\times 10^{-3}$ | 65 | 0.4117(1) | 0.044(1)
$5\times 10^{-4}$ | 65 | 0.4129(1) | 0.033(1)
$5\times 10^{-5}$ | 153 | 0.4151(2) | 0.011(2)
Figure 8: The cylindrical angle between neighbouring spheres, $\alpha_{i}$, as
a function of the position of the sphere along the $z-$axis of the channel.

As a check on the validity of the packing model, we compare the geometric
parameters, $z_{1}$, $z_{2}$, and $\alpha_{1}$, obtained from the simulation,
for each helical section, with the predictions of Eqs. 9–11. For almost all
the spheres in the generated packings, we find $\Delta
z_{i}=z_{i}-z_{i}^{\prime}<5.0\times 10^{-6}$, where $z_{i}$ $(i=1,2,d)$ is
measured from the configurations obtained by simulation and $z_{i}^{\prime}$
is calculated from the model using the measured $\alpha_{i}$. However, we do
find configurations, (Fig. 9), that show the presence of particles adopting
spacings not predicted by the model at small values of $\alpha_{1}$, which
suggests there may be some packing environments not included in the model.

Figure 9: Geometric properties, $z_{1}$ (circles) and $z_{2}$ (triangles), of
helical sections in a jammed packing, formed using $d\sigma/dt=0.001$ as a
function of measured $\alpha_{1}$, compared to predictions of the model (solid
and dashed lines respectively).

The packings can be further characterized in terms of the probability, $P(n)$,
of finding a helical section of length $n$, where we calculate the
distribution over packings formed with the same compression rate because they
have similar $\theta$ and $\phi_{J}$. The location of the defects in the
simulation generated packings can be identified using the local geometric
properties of the particles (see SM [50] for details), which includes
calculating a volume, $v_{tet}(i)$ for each particle as the volume of the
tetrahedron formed by particles, $i-1$, $i$, $i+1$, and $i+2$. The sign of
$v_{tet}(i)$ captures the local twist direction of the helix so the particles
in the same helical section have the same sign, and in most cases, the sign
changes at a defect. The probability distribution, $P(|v_{tet}(i)|)$ (Fig.
S1), also shows the defects all have small volumes compared to those in the
helical sections, providing an additional tool for locating and counting
defects. To compare this to the random distribution, $P_{R}(n)$ (see the
Appendix for details), we also calculate a non-equilibrium potential of mean
force (PMF),

$\mathrm{PMF}=-\ln\frac{P(n)}{P_{R}(n)}\mbox{,}\\\ $ (23)

which will be zero if the distributions are the same and negative if helical
section lengths are more probable than random. Figure 10 shows the
distributions of odd and even helical sections exhibit similar properties at
larger $n$, but at small $n$, the probability of finding an odd-sized helical
section rapidly decreases. We also see the presence of sections with $n=1$
that are not predicted by the model, although only with a low probability. At
the fastest compression rate, $P(n)$ for the even-sized sections decreases
monotonically with increasing $n$, with the exception of the $n=2$ section,
but as $d\sigma/dt$ decreases, the distribution develops a shoulder and then
becomes bimodal at the slowest compression rate. As a result, the PMF exhibits
two minima, one at small $n$ and another at larger $n$, separated by a
barrier.

Figure 10: (Top Row) Probability, $P(n)$, of finding a helix section
containing $n$ spheres for jammed packing with $H_{d}=1.95$ and $N=10000$.
formed with (a) $d\sigma/dt=1\times 10^{-3}$, (b) $d\sigma/dt=5\times 10^{-4}$
and (c) $d\sigma/dt=5\times 10^{-5}$. (Bottow Row) Corresponding potential of
mean force.

##  VI Discussion

The inherent structure paradigm applied to hard particle systems describes the
statistical mechanics of the fluid by grouping together all the configurations
that map the same jammed state into a local basin of attraction. The
thermodynamics and dynamics of the fluid and glassy behaviour can then be
described in terms of the properties of the accessible basins and how the
system moves between them. The goal of the current work is to develop an
accurate model of the jammed states for this quasi-one-dimensional model, in
terms of helical structures, that improves on the defect crystal model
developed previously. In particular, the new model now includes odd-sized
helical sections and does not assume a regular arrangement of the defects,
which helps account for a broader range of helical arrangements and accurately
reproduces the helical geometries found in large packings generated by
simulation. The model does miss some possible arrangements, such as the
presence of neighbouring defects observed in simulation, but this state
appears to be rare and further decreases in probability as the compression
rate is decreased.

The model for jammed packings also reveals that the inherent structure
landscape has some degree of hierarchical organization, where the basins of
configuration space associated with a given arrangement of helical sections
split into sub-basins for distinct jammed states. A single helical arrangement
represents a basin on the inherent structure landscape because, at high
density, there is a significant barrier to moving a particle from one section
of helix, which is twisting in one sense, to the next helical section, which
is twisting in the opposite sense. For example, particles 3 and 4 in Fig. 3
would need to swap places in terms of the anticlockwise rotation so that the
defect becomes located between particles 3 and 4, and particle 3 moves from
one helical section to another. Figure 5 then shows a basin can split further
into smaller sub-basins, each associated with distinct jammed states that are
separated by a small barrier. As the number of helical sections increases, we
see an increasingly complex set of minima located within the basin of a given
helical arrangement.

A highly successful mean field theory [52] of the bulk hard sphere glass
transition in high dimensions also predicts the presence of a Gardner
transition [53] between stable and marginally stable glasses that is
associated with the repeated fracturing of inherent structure basins into sub-
basins at high densities. There is growing evidence to suggest the Gardner
transition persists down to three dimensions [54, 55], making it relevant to
real glassy materials. However, it has also been argued that the physics
observed in these lower dimensions results from an avoided transition [30].
The quasi-one-dimensional nature of our system suggests that it is unlikely to
exhibit a Gardner transition and the types of particle rearrangements involved
in the transition between sub-basins are globally concerted motions rather
than the highly localized motion thought to be important in bulk glasses. For
example, Fig. 5 shows that the angle characterizing the structure changes in
both helical sections, even though the change in the longer helical section is
small and within the $\\{5,5,5,5\\}$ helical arrangement, the transition from
the jammed packing,
$\\{\alpha^{j}_{1}\\}=\\{2.2943,2.2943,2.2943,2.2943\\}\rightarrow\\{2.8104,2.8104,2.8104,2.8104\\}$,
requires significant changes in all helical sections. However, the properties
of the sub-basins still need to be studied in larger systems. The connectivity
of the sub-basins also remains to be explored because it is not clear whether
the sub-basins show repeated hierarchical organization as expected in bulk
glasses. Nevertheless, access to a detailed model of the inherent structure
landscape for this system opens up the possibility of understanding how the
presence and organization of sub-basins influence the properties of materials
at high density.

Our study of jammed packings in small systems shows that $\phi_{J}$ depends on
more than the defect fraction. In particular, the presence of small odd-sized
packings tends to lower the packing density. Figure 7 shows that $\phi_{J}$
varies over a large range for any given $\theta$ and that increasing the
fraction of odd sections leads to lower densities, with the least dense
packing consisting of alternating sections of two and three particles.
Similarly, Fig. 6 shows that small, odd-sized helical sections lead to a
decrease in $\phi_{J}$ relative to similar sized, even-sized sections. This
inefficiency of packing for small, odd-sized sections has a significant effect
on the distribution of helical sections lengths of the large system packing
formed in simulation. The LS molecular dynamics scheme continually expands the
particles during the trajectory, allowing the system to relax during
compression and move to denser basins on the inherent structure landscape that
have greater vibrational entropy until the fluid eventually falls out of
equilibrium and finally becomes jammed. Figure 10 shows that the smaller, odd-
sized, helical sections are preferentially eliminated from $P(n)$, leading to
a non-monotonic distribution with a maximum that moves to larger $n$ as the
compression rate is slowed, allowing the system more time to find increasingly
dense basins.

Earlier [47], it was shown, using a defect crystal model, that this system
exhibits a long range defect-defect attraction that results from an increase
in $\phi_{J}$, and hence the vibrational entropy, as two defects approach.
Figure 6 shows that the same picture persists for a single pair of defects
with the improved model, although the inclusion of the odd-sized helical
sections complicates the details by adding an oscillation to the attraction.
Furthermore, Fig. 10 clearly shows the PMF for the even-sized sections
develops two minima at slow compression rates, one at small $n$ and a second
one at larger separations, suggesting the defect-defect attraction leads to a
degree of defect pairing. One-dimensional and quasi-one-dimensional systems
with particles interacting via short–ranged potentials cannot exhibit a phase
transition because there is an entropic advantage in the thermodynamic limit
associated with introducing defects that break up long–range order [56, 57,
58]. However, if the interactions between defects become sufficiently
long–ranged, phase transitions can occur [59, 60]. Hu et al. [61] recently
obtained numerically exact results using the transfer matrix method that shows
a transition does not occur in this system, suggesting the configurational
entropy, which is related to the number of defects, still dominates at high
density, despite the defect-defect attraction driven by the vibrational
entropy. The fact that the PMF for the odd-sized helical sections only
exhibits a single minimum at large $n$, and that the PMF for the even-sized
sections has a large barrier at $n=2$, reducing defect-defect annihilation,
may work to counter the attractive interaction and help prevent a transition.
Finally, we also note that $P(n)$, and hence the PMF, are obtained from a non-
equilibrium compression of the fluid. It is likely that the jammed states
retain some properties of the equilibrium fluid when they fall out of
equilibrium, but kinetic effects, such as defect creation and annihilation
mechanisms and the ability of defects to diffuse through the structure, could
play a role in determining the properties of the packings.

##  VII Conclusions

A model for the jammed packings of hard spheres confined to a narrow
cylindrical channel, allowing only next–nearest contacts, has been developed.
It assumes all particles satisfy local packing constraints and describes the
structure in terms of helical sections, with alternating twist direction,
separated by defects. While it is possible that additional local packing
environments could be found, the model provides an improved description of the
defects in the packings, as well as packings with odd-sized helical sections,
compared to the defect crystal model developed earlier, and accurately
reproduces the geometries of jammed packings created by molecular dynamics
simulation. The model also confirms the possibility of an entropically driven
defect-defect attraction that appears to influence the structure of the
particle packings.

###### Acknowledgements.

We would like to acknowledge NSERC grants RGPIN–2019–03970 (R. K. B) and
RGPIN-2020-04467 (R. J. S.) for financial support. Computational resources
were provided by the ICT at the University of Saskatchewan, WestGrid, and
Compute Canada.

##  Appendix A Random Distribution of Helical Section Lengths

Here, we derive the random distribution of helical section lengths to obtain
an expression for the probability, $P_{R}(n)$, of finding a helical section of
length $n$ in a system of $N$ spheres containing $k$ defects. It is more
convenient to develop the analysis in terms of the clusters of the “bonds”
connecting the neighbouring spheres in contact, so that a cluster of $l$ bonds
represents a helical section of $n=l+1$ spheres. If $n_{0}$ is the total
number of non-defect bonds and $q_{l}$ is the number of clusters of with $l$
bonds then we have the following constraints:

$n_{0}=\sum_{l=1}^{\infty}lq_{l}=N-k\mbox{,}\\\ $ (24)

and

$k=\sum_{l=1}^{\infty}q_{l}\mbox{,}\\\ $ (25)

where the conservation condition, $N=n_{0}+k$, was used in the right hand side
of Eq. 24 and Eq. 25 arises from the requirements that each cluster be
separated by a defect and defects cannot be adjacent to each other.

The random equilibrium distribution of clusters is obtained by finding the
cluster distribution $\\{q_{l}\\}$ that maximizes the entropy
$S/k_{B}=\ln\Omega$, subject to the constraints, Eq. 24 and Eq. 25, where

$\Omega=\sum_{\\{q_{l}\\}}g\mbox{,}\\\ $ (26)

and $g$ is the number of distinct ways to distribute the clusters. To obtain
an expression for $g$, it is useful to begin by treating all clusters as
distinguishable and then remove the over counting of any indistinguishable
clusters [62]. In a system with periodic boundaries, such that particle $N$
contacts particle 1, there are $N$ places to put the first cluster and
$(k-1)!$ ways to place the remaining $k-1$ distinguishable clusters. The
distinguishability is then removed by dividing by $q_{l}!$ for each $l$,
yielding,

$g=\frac{N(k-1)!}{\prod_{l=1}^{\infty}q_{l}!}\mbox{.}\\\ $ (27)

Figure 11: Probability, $P_{R}(n)$, of finding a helical section containing
$n$ spheres in a random distribution of defects with $\theta=0.0333$, from
simulation (open circles) and exact theory (solid line) given by Eq. 33.

The maximum can be obtained by introducing the underdetermined multipliers,
$\lambda_{1}$ and $\lambda_{2}$, for the two constraints and setting the
derivative with respect to $q_{m}$ to zero, for each $m$,

$\frac{\partial}{\partial
q_{m}}\left[\ln[\\{q_{l}\\}]+\lambda_{1}\left(N-k-\sum_{l=1}^{\infty}lq_{l}\right)\right.\\\
\left.+\lambda_{2}\left(k-\sum_{l=1}^{\infty}q_{l}\right)\right]=0\mbox{.}$
(28)

The only terms to survive are those where $m=l$, which gives,

$q_{l}=e^{-\lambda_{1}l}e^{-\lambda_{2}}=S_{1}^{l}S_{2}\mbox{,}\\\ $ (29)

then using Eq. 29 in the constraints, Eq. 24 and Eq. 25, and solving for the
unknowns yields,

$S_{1}=\frac{N-2k}{N-k}\mbox{,}$ (30)

and

$S_{2}=\frac{k^{2}}{N-2k}\mbox{.}\\\ $ (31)

Using Eqs 30 and 31 in Eq. 29 and introducing the fraction of defects,
$\theta=k/N$, then gives

$\frac{q_{l}}{N}=\frac{\theta^{2}}{1-2\theta}\left[\frac{1-2\theta}{1-\theta}\right]^{l}\mbox{.}\\\
$ (32)

Finally, to be consistent with our analysis in the main text, we normalize
$P_{R}(n)$ with respect to the number of defects rather than $N$ and note
$n=l+1$, which gives,

$P_{R}(n)=\frac{1}{\theta}\frac{q_{n-1}}{N}=\frac{\theta}{1-2\theta}\left[\frac{1-2\theta}{1-\theta}\right]^{n-1}\mbox{.}\\\
$ (33)

Figure 11 compares the exact result for $P_{R}(n)$ (Eq. 33), which is valid in
the thermodynamic limit ($N\rightarrow\infty$), with the results obtained from
simulations for a system with $N=10000$, where $5\times 10^{6}$ configurations
were generated with randomly placed defect bonds such that there were no
neighbouring defects bonds and $\theta=0.0333$. The finite–sized system
accurately reproduces the exact results for the small helical sections but
begins to systematically underestimate $P_{R}(n)$ for larger sections.

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# Detecting Endangered Marine Species in Autonomous Underwater Vehicle Imagery
Using Point Annotations and Few-Shot Learning

Heather Doig1, Oscar Pizarro1,2, Jacquomo Monk3 and Stefan Williams1 1H. Doig,
O. Pizarro and S. Williams are with the Australian Centre for Robotics,
University of Sydney, Australia. (heather.doig, oscar.pizarro,
<EMAIL_ADDRESS>Pizarro is also with the Marine Technology
Department, Norwegian University of Science and Technology, Trondheim,
Norway.3 J. Monk is with the Institute for Marine and Antarctic Studies,
University of Tasmania, Australia<EMAIL_ADDRESS>
###### Abstract

One use of Autonomous Underwater Vehicles (AUVs) is the monitoring of habitats
associated with threatened, endangered and protected marine species, such as
the handfish of Tasmania, Australia. Seafloor imagery collected by AUVs can be
used to identify individuals within their broader habitat context, but the
sheer volume of imagery collected can overwhelm efforts to locate rare or
cryptic individuals. Machine learning models can be used to identify the
presence of a particular species in images using a trained object detector,
but the lack of training examples reduces detection performance, particularly
for rare species that may only have a small number of examples in the wild. In
this paper, inspired by recent work in few-shot learning, images and
annotations of common marine species are exploited to enhance the ability of
the detector to identify rare and cryptic species. Annotated images of six
common marine species are used in two ways. Firstly, the common species are
used in a pre-training step to allow the backbone to create rich features for
marine species. Secondly, a copy-paste operation is used with the common
species images to augment the training data. While annotations for more common
marine species are available in public datasets, they are often in point
format, which is unsuitable for training an object detector. A popular
semantic segmentation model efficiently generates bounding box annotations for
training from the available point annotations. Our proposed framework is
applied to AUV images of handfish, increasing average precision by up to 48%
compared to baseline object detection training. This approach can be applied
to other objects with low numbers of annotations and promises to increase the
ability to actively monitor threatened, endangered and protected species.

##  I Introduction

Figure 1: Handfish in imagery captured by _AUV Sirius_ and _AUV Nimbus_.
Manually identifying a small and cryptic species like handfish against a
complex background is time-consuming and can lead to missed observations. The
top image shows a complete image from _AUV Nimbus_ with a single handfish.
There were 7 images with handfish identified out of 11,171 images captured
during the mission. The bottom row shows cropped examples of handfish. Images
downloaded from IMOS-UMI Squidle+, squidle.org.

Handfish (Brachionichthyidae) are a rare and iconic species of bony fish found
in southern waters of Australia, with a diversity hotspot for the species
around Tasmania. Of the 14 confirmed species of handfish, half of the species
are listed as either Endangered or Critically Endangered by the International
Union for Conservation of Nature (IUCN), making them the most threatened
family of bony fishes in the world [1]. One of the species, the red handfish
(Thymichthys politus), is thought to have less than 100 adults left in the
wild. Many of these handfish species are believed to be confined to shallow
waters, where impacts from human activities (such as pollution, habitat
destruction and terrestrial runoff) and climate change are at their greatest.
However, in October 2021, the endangered and very rare pink handfish
(Brachiopsilus dianthus) was seen for the first time since 1999, in footage
from a baited remote underwater video at a depth of 150 m [2]. The discovery
has prompted researchers to reconsider the depth range for some of these
handfish species, giving optimism that deeper waters may provide some refuge
from the impacts causing their population declines.

The low abundance and elusive nature of threatened species, like handfish,
make it challenging to monitor their populations. Monitoring of handfish
populations, like many other shallow water species, has historically been
undertaken using SCUBA-diver-based methods and more recently using eDNA [2].
The recent detection of listed handfish in deeper waters highlights the need
for remote tools to collect high-resolution seafloor imagery such as
Autonomous Underwater Vehicles (AUVs). AUVs have been widely used to assess
and monitor changes in benthic organisms [3, 4, 5].

AUV imagery has historically relied on manual annotation to identify organisms
in the images. However, this process is slow and time-consuming. Additionally,
most handfish species are less than 150mm long and exhibit highly cryptic
behaviour, further complicating their detection and identification as shown in
Figure 1. If handfish are present in an AUV mission, they only appear in
around 0.2% of images captured. A deep-learning object detector could
significantly reduce the effort required to identify handfish species within
AUV imagery. This approach could also improve data quality, helping gather
fundamental population data such as spatial distributions and trends in
abundance, critical for the conservation of handfish species.

Object detectors use deep learning models to locate an object with a bounding
box and classify the object using a percentage confidence level. Bounding box
annotations with the object class are used to train the detector. Ideally, the
detector is trained with large amounts of data to provide high performance [6,
7, 8, 9], but this is often not available for underwater images [10, 11].
Annotations of marine species are limited due to the time and expertise
required but also because the species may be very rare with low populations in
the wild [12, 13]. For example, the top image of Figure 1 was one of seven
images found with handfish (single individual) out of 11,171 images captured
on a mission by _AUV Nimbus_ in 2023.

Reduced performance due to domain shift is another issue for deep learning
models trained with underwater images [14]. AUV surveys provide high-quality
images taken under consistent operating conditions during a mission, such as
altitude, camera, lighting and water conditions. Images are captured from a
birds-eye camera pose at a relatively constant altitude, ensuring that species
are shown from similar angles and scales. While the images are captured with
consistent operating parameters in the same mission, images for training and
at test time are often from different missions and may potentially be from
different vehicles or imaging systems. Domain shift can occur where the
detector’s performance during testing is reduced compared to training due to
the differences in the conditions in which the images were captured.

In this work, we present a framework for detecting rare and cryptic marine
species to address the issues of low numbers of annotations and domain shift,
inspired by few-shot learning. This technique trains a model to detect both
common base classes with large amounts of examples as well as rare or novel
classes with only a few examples [6, 15]. We use six common marine species as
base classes to pre-train the detector before fine-tuning with the novel class
of handfish to compensate for the low number of novel class annotations.

In addition, the annotations of the base classes are used in a copy-paste
operation to augment the training data based on [16]. This operation either
copies and pastes the segmented handfish to base class images or copies the
segmented base class instances to handfish images. This two-way operation is
randomly applied and aims to reduce domain shift by adding the images with the
base classes to training while also addressing the low number of annotations.

Squidle+111squidle.org is a powerful marine image data management platform for
exploring and annotating underwater imagery. With thousands of images and
annotations, it provides a rich archive of underwater training data. Squidle+
is used to source the training data for the base and novel classes of marine
species from AUV images captured off the coastline of Tasmania. Existing
annotations of marine species are commonly in point format as this is the
scientific approach often used to measure the presence and abundance of
species by labelling randomly placed points on an image [17]. The Segment
Anything semantic segmentation model [18] is used through the Squidle+
platform to provide a segmentation boundary around the object under the point
annotation. The boundary is then used to create bounding boxes for training
and image masks for the copy-paste augmentation operation.

Our framework’s aim is to train an object detector for a rare or novel class
using images from base classes to pre-train the backbone of the object
detector and augment the training data. Existing point annotations are
transformed efficiently into segmentation masks and bounding boxes for
training. Applying the framework to the example of handfish detection in AUV
imagery demonstrated improved detection performance for objects with low
annotations compared to training without the framework.

The contribution of this paper is a framework to improve detection performance
for objects with low annotations by:

  * •

using pre-training of the object detector backbone with annotated images of
common base classes followed by fine-tuning with novel class data to create
discriminative features

  * •

augmenting training data during fine-tuning by applying a copy-paste operation
in two directions to address both low numbers of annotations and domain shift

  * •

efficiently generating bounding box annotations for object detection training
using point annotations and Segment Anything segmentation model

  * •

demonstrating improved performance for one-stage and two-stage detectors by
presenting results when applied to an image dataset of handfish and common
marine species taken by two AUVs around the coastline of Tasmania

The datasets, including images, point annotations and segmentations, are
publicly available on the Squidle+ marine imaging platform.

The remainder of this paper is organised as follows. Section II presents an
overview of work related to the training of object detectors, with a
particular focus on tools that accommodate low numbers of training examples.
Section III describes our framework and how it uses pre-training and copy-
paste augmentation during training. Section IV describes the application of
the proposed framework to the detection of a rare species of fish captured in
AUV imagery offshore of Tasmania and the results of the study, while Section V
provides concluding remarks and directions for further study.

##  II Related Work

###  II-A Object Detector models

Deep learning object detectors can locate and classify an object after
supervised training. The main architectures for object detectors are one-stage
and two-stage detectors. Two-stage detectors such as Faster R-CNN [19] have an
initial stage for locating proposed objects for detection called a Region
Proposal Network (RPN), followed by a second stage that classifies the object
and refines the location. A single-stage detector uses a single network to
both locate and classify the object. Examples of single-stage detectors
include the series of You Only Look Once (YOLO) detectors [20] and the Fully
Convolutional One-Stage (FCOS) [21] object detector. The architectures for
both one- and two-stage detectors include a backbone for feature extraction
and a final network layer to classify the detected object and locate the
bounding box.

Training the detector is usually performed in a supervised manner with
bounding boxes and classes. However, generating bounding box annotations can
be time-consuming [22, 23]. Furthermore, there may be existing point
annotation data available that may be useful for training classifiers. This is
particularly the case in the seafloor imaging context where a lot of
historical effort has focused on generating point labels for classes of
interest [24]. There has been some research into using point annotations to
weakly supervise the training of the object detector [22, 25]. Our work uses a
semantic segmentation model, Segment Anything [18], to efficiently generate
broadly accurate bounding boxes using the existing point annotations.

###  II-B Few-shot learning

Few-shot learning aims to train an object detector to detect novel classes
with few examples by leveraging base classes with large numbers of
annotations. Kang et al. used a meta learner with feature reweighting by
training with abundant base classes to create features that can generalise to
novel classes [6]. Pre-training of the detector’s backbone on an adjacent task
can improve object detection as shown with earlier object detector networks
[26]. More recently, a simple two-stage method uses a pre-training and fine-
tuning step to further improve detection performance [15]. Base class data is
used for pre-training the object detector to provide a backbone that produces
rich features for unseen novel classes. Our framework is based on this
approach.

###  II-C Data augmentation

Data augmentation methods have been shown to improve object detection
performance. In YOLOv4 [20], the augmentations included photometric
distortions adjusting brightness, contrast and colour and geometric
distortions like random rotation and flipping. Another successful augmentation
operation is copy-paste where either the segmentation mask or the bounding box
of the class instance is copied to another image. Dvornik et al. used a method
to merge the copied example into the background so it appeared seamless [27].
Ghiasi et al. simplified the process for segmentation models by simply copying
and pasting the segmented mask of the object without any blending with the
background context [16]. We use the same approach by copying positive examples
to other images as well as copying negative examples to the novel class
images.

##  III Detector Training with Few Labelled Examples

Our work introduces a framework that aims to train a detector with low numbers
of positive training annotations. This is undertaken in the particular context
of detecting rare or unusual species but is applicable to a broader range of
problems where low numbers of positive training examples are available. We
adopt terms from few-shot learning for our methodology. The target species
will be called the novel class, and the common marine species will be called
the base classes. Differing from other few-shot learning tasks, we aim to
detect only one novel class to support monitoring the presence of an
endangered species or other object of interest. We also use fewer base classes
compared to other work, with only 6 base classes compared to 20 or more in
related work [6, 15]. The number of base classes was limited to those with
available annotations but could be increased.

There are three main parts to the framework. The first is preparing a dataset
for training the object detector derived from images with point annotations of
the novel and base classes. Second, an adaptation of the few-shot learning
method by Wang et al. is used to pre-train the detector’s backbone using the
base class dataset [15]. Finally, an augmentation method using copy-paste in
two directions based on Ghiasi et al. is applied to increase the variety of
images and reduce domain shift [16]. Figure 2 provides an overview of the
framework. These three parts will be described in the next sections.

Figure 2: Detector training for one-stage detector, FCOS. The pre-training
step trains the object detector using a base class dataset. In the fine-tuning
step, training begins with the pre-trained detector with the box classifier
and box regressor layer replaced with a newly initialised layer that only
detects the novel class (boxes in green). The first three layers of the
backbone are frozen. The base class dataset is used in the copy-paste
operation to augment the dataset during fine-tuning.

###  III-A Annotations and training data

The training data includes one set of images with point annotations of the
novel class and another with point annotations of the base class. The
annotations had already been collected for a statistical coverage survey using
the methodology described in [3]. To generate bounding boxes for object
detection training, the Segment Anything semantic segmentation model [18]
creates a segmentation mask for the object under the point annotation, which
is then converted to a segmentation boundary. The segmentation boundary is
then used to create a bounding box for object detection training and to create
a mask for copying the instance to another image.

Figure 3 gives examples of successful and failed segmentation boundaries of
both novel and base class instances. Failed or poor quality boundaries
occurred in around 1% of base class annotations and around 15% of handfish
examples. Errors in the handfish data were manually corrected to ensure a
high-quality mask for the copy-paste operation. The most common error was the
exclusion of either the distinctive hand-like fins or the full tail. No
corrections were applied to the base class segmentation boundaries leading to
a small amount of noisy training data for the pre-training step and the
pasting of base class instances.

Figure 3: Examples of semantic segmentation boundaries generated from point
annotations. The first row for each dataset (A, B, E, F) show successful
segmentations. The row below (C, D, G, H) shows failed or poor-quality
segmentation. (C) incorrectly includes the seafloor around the handfish, while
(D) removes the distinctive hand-like fins of the handfish. Only errors in the
handfish boundaries were manually corrected to ensure high-quality masks for
the copy-paste operation.

###  III-B Pre-training and Fine-Tuning

Training starts with an initial pre-training step with the base class dataset,
followed by fine-tuning with the novel class dataset. In pre-training, the
backbone layers are all updated with the goal of generating features that
adequately describe a range of marine species for classification.

In the fine-tuning step, the final layer of the object detection network is
replaced with a randomly initialised classifier and box regression layer. This
replacement layer is only for the novel class and no longer detects the base
classes. Fine-tuning is performed with the novel class dataset, with 1/20 of
the learning rate used during pre-training as in [15]. During this step, we
freeze the layers of the first three levels of the backbone so the deeper
layers remain fixed from the pre-training step, as shown in Figure 2.

Figure 4: Examples of the two-way copy-paste augmentation operation. Each pair
of cropped images shows without (left) and with (right) copy-paste using the
segmentation boundary as the mask. The top row (A, B) contains images with
handfish with a base class instance added. The bottom row (C, D) is base class
images with a handfish instance added.

###  III-C Copy-paste data augmentation

Data augmentation compensates for the low number of annotations and domain
shift. Basic horizontal and vertical flipping is applied to all images in pre-
training and fine-tuning. The copy-paste operation is used as an additional
augmentation operation to reduce domain shift by introducing images taken
under different conditions to the novel class images. The simple copy-paste
method involves copying the selected segmentation masks to a new image [16].

First, an image from the novel and base class datasets is randomly selected.
One of these images is randomly cropped to provide more variety in the scaling
of the image while ensuring that the instance of the novel class is kept
within the cropping region. Next, the instances from one of the images are
copied onto the other image. Figure 4 shows examples from this operation for
handfish as the novel class and common marine species as the base classes.
While increasing the variation of images with the novel class, the copy-paste
operation also increases the number of negative examples in the training data
when copying the base classes to the novel class images. The base classes
include examples that could be false positives for the novel class.

##  IV Experiments and Results

We present results when applying the proposed framework to the problem of
detecting an endangered marine species, handfish found off the coast of
Tasmania. Very few training examples of this species are available due to the
low frequency of it being observed in the wild and the challenges of finding
this cryptic species in the imagery of its natural habitat (see Figure 1).

###  IV-A Dataset

All images and annotations were extracted from the Squidle+ marine image data
management platform. The novel and base class dataset images were taken by
_AUV Sirius_ and _AUV Nimbus_ around Tasmania’s coastline between 2009 and
2023. The datasets are publicly available on Squidle+222’ACFR Handfish
Detection - IROS 2024’, IMOS-UMI Squidle+, squidle.org.

For the handfish class, there are 284 images with a single handfish present in
each image. A test set of the most recent images is used for evaluation
containing 42 images with a single handfish captured between June 2021 to
February 2023 by _AUV Sirius_ and _AUV Nimbus_. The remaining 242 images were
randomly divided into training sets with 50, 100 and 200 sample sizes and a
validation set with 42 images.

The base class dataset includes 275 images captured by _AUV Sirius_ and _AUV
Nimbus_ with 1904 point annotations for six frequently occurring species.
These base classes were chosen because they had the highest number of existing
annotations in Squidle+ and they also appear frequently in AUV images with
handfish. The images were annotated to capture all instances of the six
species in the dataset. There are between 100 and 550 annotations per class.
Examples of the species can be seen in Figure 5, with each class distinguished
by a colour border.

Figure 5: Examples of the six common marine species and the number of annotations used in the base class dataset. The base class annotations are used in pre-training the backbone and for the two-way copy-paste augmentation operation. TABLE I: Average Precision of Handfish detection with training sample sizes of 50, 100 and 200 for a two-stage and one-stage object detector. Average result and standard deviation from 5 runs are shown with the best values in red and second best in blue. Pre- | Copy-Paste Mask | Two-Stage - Faster R-CNN | One-Stage - FCOS
---|---|---|---
Train | Novel Class | Base Class | 50 | 100 | 200 | 50 | 100 | 200
N | None | None | $13.8\pm 2.5$ | $18.7\pm 1.0$ | $19.1\pm 1.9$ | $16.7\pm 1.8$ | $19.5\pm 1.7$ | $21.0\pm 1.3$
N | None | Segment. | $11.8\pm 2.8$ | $15.2\pm 2.1$ | $20.1\pm 3.5$ | $18.2\pm 3.1$ | $19.8\pm 2.9$ | $22.8\pm 3.8$
N | Bounding Box | Segment. | $11.5\pm 2.5$ | $16.0\pm 2.1$ | $19.1\pm 2.4$ | $17.2\pm 4.1$ | $22.5\pm 2.6$ | $21.6\pm 3.8$
N | Segment. | Segment. | $7.7\pm 1.7$ | $9.9\pm 2.9$ | $18.0\pm 2.0$ | $19.5\pm 4.8$ | $17.5\pm 3.8$ | $24.4\pm 2.5$
Y | None | None | $15.9\pm 1.6$ | $19.6\pm 3.5$ | $21.1\pm 3.6$ | $17.5\pm 3.1$ | $18.2\pm 2.7$ | $23.4\pm 2.3$
Y | None | Segment. | $16.4\pm 5.8$ | $17.4\pm 3.8$ | $22.0\pm 3.3$ | $24.8\pm 4.8$ | $24.9\pm 3.9$ | $21.0\pm 3.2$
Y | Bounding Box | Segment. | $10.3\pm 4.2$ | $13.7\pm 3.4$ | $20.3\pm 3.0$ | $21.1\pm 3.2$ | $16.5\pm 4.5$ | $22.8\pm 3.6$
Y | Segment. | Segment. | $11.3\pm 3.3$ | $14.0\pm 5.3$ | $19.0\pm 3.3$ | $18.7\pm 6.8$ | $14.7\pm 3.8$ | $26.0\pm 2.3$

###  IV-B Implementation Details

Similar to [15], we use a Faster R-CNN object detector [19] as the two-stage
detector. In addition, we apply our framework to a one-stage detector, FCOS
[21]. Both detectors use a ResNet-50 backbone with Feature Pyramid Network
[28] and we use the PyTorch implementation of the detector networks. Also
following [15], pre-training uses a learning rate of 0.001 and all layers of
the backbone are updated. The fine-tuning step uses a learning rate of 0.0005,
which is 1/20 of the pre-training learning rate. During fine-tuning, the first
three layers of the backbone are frozen so that only the higher-level features
are updated. Each training run is performed for 40 epochs with 1000 iterations
per epoch and a batch size of 1. In all cases, training starts with a warmup
phase for the first 1000 iterations, linearly increasing the learning rate
from 1/1000 of its value to the learning rate, then decayed to 0.1 of its
current value at 90%, 95% and 99.5% of total iterations as in [16]. All models
were trained using an SGD optimiser with momentum of 0.9 and weight decay of
0.0005, and images were resized to 1000 pixels on the shorter side. The
training was performed on an NVIDIA A10G GPU.

###  IV-C Results

The framework is applied using novel datasets with a sample size of 50, 100
and 200 on a two-stage and one-stage detector. The model from the final
iteration of training is used for evaluation using Average Precision (AP) with
an Intersection over Union (IoU) of 0.5. Following other work on few-shot
learning [29], the results are averaged over five training runs, each with a
different random seed due to the small size of the training and test dataset.

The results are shown in Table I for training with and without the pre-
training step and with variations on the copy-paste augmentation. The first
line of the table is used as the baseline case where no elements of the
framework are applied. In addition to results without any copy-paste
augmentation, three variations of the copy-paste operation were evaluated with
changes in whether the novel instances were used and, if they were, whether
they used a bounding box mask or a segmentation boundary mask. When the base
class was used for copy-paste, the segmentation boundary was used.

###  IV-D Discussion

The framework’s goal was to improve detector performance that utilised point
annotations and a dataset of base class images and annotations for pre-
training and data augmentation. After being applied to a one-stage and two-
stage detector with varying novel class sample sizes, the performance after
pre-training alone increased in all but one case. The pre-training step on
more commonly available base class data appears to improve performance by
extracting features that are discriminative for the unseen novel marine
species. The base class annotations were derived from the Segment Anything
algorithm and included noisy data. Despite sometimes inaccurate segmentations,
the pre-trained backbone was able to improve performance (See Figure 3, (G) &
(H)). Greater performance gains from pre-training may be possible with
improved segmentation boundaries and also increasing the number of base
classes and annotations as used in Wang et al. [15].

Overall, the highest performance for 4 of the 6 cases was achieved using pre-
training and one-way copy-paste of base class instances to novel class images.
This included the largest AP increase of 8.1 or 48% for the one-stage detector
with a novel sample size of 50. The combination of the pre-trained backbone
that could extract discriminative features for marine species as well as
increased instances of negative examples from copy-paste of base class
instances resulted in the highest performance. Copy-paste was more beneficial
for the one-stage detector, FCOS. The framework is flexible enough to be
applied to other detector architectures with a backbone and final classifier
and box regression layer. It could be successful if applied to other one-stage
detectors.

##  V Conclusion

We have proposed a framework for detector training that increases performance
for marine species that have low numbers of annotations. The overall framework
is demonstrated on a dataset of endangered handfish producing increases in
detection performance of up to 48% with a sample size of 50. Pre-training the
detector with more commonly available marine species increased performance in
most cases. The copy-paste operation was most successful when used to add
instances of a base class or negative examples to a novel class image during
the fine-tuning step. The framework has the flexibility to be applied to other
one-stage detectors offering the opportunity to detect rare species in real-
time on the AUV. Future work could use the detector for adaptive planning to
explore the local habitat of rare species when found. Our framework provides a
promising start to increasing the ability to efficiently identify threatened,
endangered, and protected species from AUV images and was successfully used to
identify previously unrecorded instances of handfish in AUV imagery,
increasing the amount of data available to scientists interested in
characterising the distribution of these critically endangered species.

## ACKNOWLEDGMENT

Images and annotations from _AUV Sirius_ and _AUV Nimbus_ were sourced from
Australia’s Integrated Marine Observing System (IMOS) through the Squidle+
online platform. IMOS is enabled by the National Collaborative Research
Infrastructure Strategy (NCRIS), Australia.

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