# A topologically stabilized metastable fluid in a system of cylindrically
confined hard spheres.

Mahdi Zarif<EMAIL_ADDRESS>Department of Physical and Computational
Chemistry, Shahid Beheshti University, Tehran 19839-9411, Iran.  Richard K.
Bowles<EMAIL_ADDRESS>Department of Chemistry, University of
Saskatchewan, SK, S7N 5C9, Canada.

###### Abstract

Metastability in soft condensed matter systems usually results from the
presence of a nucleation free energy barrier and/or slow dynamics caused by
high density jamming phenomena. Here, we use molecular dynamics and Monte
Carlo simulation to show that the interactions between topological defects
stabilizes a chiral helical fluid in a confined quasi-one-dimensional hard
sphere fluid dramatically slowing its decay toward the equilibrium achiral
fluid state. Analysis of thermodynamic, structural and dynamic properties of
the system show the equation of state bifurcates continuously at intermediate
pressures into two distinct branches that are accessed from different initial
conditions, but terminate at the same close packed single helix in the high
pressure limit. The equilibrium fluid, which forms the high pressure branch as
the system is compressed from low density, is characterized by helical
sections separated by randomly distributed topological defects that change the
helical twist direction giving rise to an achiral fluid. The low pressure
metastable branch, formed by decompressing the system from the perfect helix,
is characterized by the appearance of loosely paired defects that help retain
the chiral excess of the original state and stabliize the fluids until it
merges continuously with the equilibrium branch at intermediate pressures.

###### keywords:

Confined Fluid, Hard Spheres, Topological, Defects, Molecular Dynamics, Monte
Carlo

Centre for Quantum Topology and its Applications (quanTA), University of
Saskatchewan, SK S7N 5E6, Canada.

##  1 Introduction

Topological order and topologically protected states usually appear in
correlated quantum systems 1 but there is growing evidence to suggest that
they can also play an important role in the properties of classical systems 2,
3, 4. For example, Zygmunt et al. 5 found that dense packings of anisotropic
colloids form topological phases that retain near perfect order below close
packing, demonstrating a form of classical topological protection. The origin
of topological order in these systems differs from those of their quantum
counterparts and arises from the organization of particle contacts within the
unit cell of the packing which suggests other colloidal systems may exhibit
similar phenomena.

The geometric confinement of hard sphere particles to narrow quasi-one-
dimensional (quasi-1d) channels leads to the spontaneous formation of helical
structures 6, 7, 8 ranging from simple single helices through to multi-
stranded helices with slip or staggered structures, depending on the channel
diameter. When the channel diameter is sufficiently narrow, all the particles
in the dense packings contact the channel wall, which allows their structures,
and the transitions between them, to be described in terms of phyllotactic
disk arrangements on a plane 9, 10, 11, 12. A variety of new structures arise
as the channel becomes wide enough for particles to enter the core of the
packings 13, 14 and eventually, the appearance of bulk face-centred-cubic
(FCC) crystal packing arrangements leads to the formation of complex core-
shell structures 15. A similar array of structures have been found in confined
quasi-1d soft sphere systems 16, 17, and have been observed experimentally in
molecular nanotube systems 18, 19, 20, colloidal particles 21, 22, 14, 23 and
macroscopic, athermal systems 24. Introducing particle shape anisotropy then
broadens the range of structural motifs formed under cylindrical confinement
and can generate new chirality elements 25, 26.

Colloidal crystals have applications in photonics 27, 28, 29, 30 and these
helical packings of hard spheres exhibit chiral photonic properties 31.
However, quasi-one-dimensional systems with short ranged interactions
generally do not exhibit phase transitions 32, 33 because there is always an
entropic advantage to introducing a defect into the system that overcomes the
energetic cost of the defect in the thermodynamic limit 34. Randomly
distributed defects would break up the helical structure, leading to an
achiral fluid. Nevertheless, there are circumstances where phase transitions
can occur in quasi-one dimensional systems 35, 36 and phase transitions can
arise in 1d systems when the particle interactions become long-range and the
defects have a topological character 37, 38, 39.

The current work examines the role helical topology plays in the structural,
thermodynamic and dynamic properties of a system of hard spheres confined to a
narrow, quasi-1d channel, focusing on a system that has the simplest, perfect
single helix ground state 40, 41. To capture the topological properties of the
fluid, we use an analysis (see Methods) that identifies structural “defects"
in fluid where the twist direction of the helix changes, i.e. between left
($\mathcal{M}$) and right ($\mathcal{P}$) twist directions. The method
provides an approximate mapping of a fluid configuration to its local jammed,
or inherent structure 42, 43, 44, which helps us relate the thermodynamics and
dynamics of the fluid to its underlying topological structure through the
number and distribution of the helical defects.

Previously 40, 45, it has been shown that defects in the jammed helical
packing appear in pairs and the structure of the helix between two defects is
fundamentally changed to that of an asymmetric double helix with a pitch that
depends on their distance of separation. This affects the packing density of
the system as a function of position of the defects and leads to an effective,
entropically driven, long range attraction between isolated defect pairs. Such
a topological, long range interaction could, in principle, lead to phase
transition. Fu et al 14 found, using $N,P,T$ Monte Carlo (MC) simulations,
that initial low and high density starting conditions converged to a single
equation of state (EOS) at low pressure, but the two different starting
conditions did not converge at high pressure. Hu et al 46 used the transfer
matrix method to show that correlation lengths in the system remain finite,
ruling out the possibility of a phase transition at the bifurcation point of
the two branches of the EOS.

However, the topological properties of the two branches of the EOS have not
been examined. Here, we show that, contrary to expectations, the high pressure
branch of the EOS of this system represents the equilibrium fluid where the
defects in the helical structure are randomly distributed, ensuring the fluid
is achiral. The second, low pressure branch formed by decompressing the system
from a perfect helix at high density, maintains an excess helical twist in one
direction, exhibits slow relaxation times and remains stable for long
simulation times at densities where the higher pressure equilibrium fluid
rapidly relaxes. This suggests the topology of the helical twist plays a role
in the system’s properties, and provides a degree of stabilization for the
chiral state. The details of the model and simulations methods can be found in
the Methods section.

##  2 Results

Figure 1(a) shows the pressure EOS for the system over a wide range of
densities. Below $P_{L}/kT\approx 38$ both compression and decompression
simulations follow the same EOS, which exhibits a small shoulder centred at
$P_{L}/kT\approx 15$ ($\phi\approx 0.25$) and approaches ideal gas behaviour
in the low density limit. At higher densities, we see two distinct branches of
the EOS, a higher pressure branch for compression and a lower pressure branch
for decompression. Fig. 1(b) highlights that both branches of the EOS can be
reproduced, from their respective initial conditions, using different
simulation methods that follow distinct equilibration pathways. The MD
simulations move through a sequential series of equilibrium states,
essentially following the EOS as the system is equilibrated at one $\phi$
before being compressed, or decompressed, to the next. The MC simulation for
each state point evolves independently from its starting condition at its
assigned constant pressure. Notably, the EOS of the decompression branch is
slow to converge, and there is some evidence that the fluid structure still
evolves at a very slow rate (see Supporting Information for more details).
While we have never observed the system move from one branch of the EOS to the
other at $\phi>0.33$, where the bifurcation occurs, if the system is
decompressed below this point from either branch, and then recompressed, it
always follows the higher pressure branch.

Figure 2(a) shows the heat capacity exhibits distinct peaks for the
compression and decompression branches of the EOS, as well as a peak at low
pressure, that coincide with changes in the defect fraction, $\theta=n_{d}/N$
(see Fig. 2(b)), where $n_{d}$ is number of defects, suggesting the fluid goes
through a number of structural changes as a function of pressure. We now
examine the relationship between the thermodynamic, dynamics and structure in
the three distinct regions of the EOS.

Figure 1: Equation of state. (a) MD compression and decompression over full
range of $\phi$ and (b) MD and MC compression and decompression in the high
$\phi$ regime. Figure 2: Comparing thermodynamics, defects and helical excess
(a) Constant pressure heat capacity, $C_{p}/Nk$, (b) defect fraction $\theta$
(inset: low pressure regime), (c) $f_{\mathcal{P}}$, the excess fraction of
$\mathcal{P}$ tetrahedra, as a function of $P_{L}/kT$.

###  2.1 Low density branch: $\phi<0.33$, $P/kT<38$

In the ideal gas limit, the system is expected to sample basins in the
inherent structure landscape near the maximum of the distribution 44 where
$\theta=(5-\sqrt{5})/10\approx 0.276$ 40. The inset for Figure 2(b) shows that
when we perform a direct count of all the defects in the system, highlighted
as the MC results, $\theta\approx 0.35$. However, if we assume that the
unstable environment associated with neighbouring defect pairs leads to their
annihilation as the system is compressed to its local inherent structure, then
the number of defects decreases and $\theta\approx 0.3$ (MD results), which is
only marginally higher than the expected value. With a high fraction of
defects, the number of spheres between the defects is small and the particles
tend to adopt linear or zig-zag arrangements. As the pressure increases, the
number of defects decreases rapidly before reaching a plateau at
$\theta\approx 0.25$ where the average defect separation is four particles,
which corresponds to the smallest section capable of forming part of a helical
twist. The number of unstable neighbouring defect environments also becomes
small. The helical sections alternate between left ($\mathcal{M}$) and right
($\mathcal{P}$) twist directions and with no preference for one twist
direction over the other the excess fraction of ($\mathcal{P}$) tetrahedra,
$f_{\mathcal{P}}$, remains zero.

###  2.2 Compression branch: $\phi>0.33$, $P/kT>38$

The EOS of the compression branch obtained by our MD and MC simulations
follows that obtained by Hu et al 46 using the transfer matrix method. The
structure of the fluid along the rest of the compression branch consists of
loosely organized sections of helix, separated by defects, where all the
particles in a given helical section have the same local twist direction and
the twist alternates between $\mathcal{M}$ and $\mathcal{P}$ directions. With
increasing pressure, $\theta$ begins to decrease again as the fluid continues
to move to basins on the inherent structure landscape associated with jammed
structures with higher $\phi_{J}$, maximizing its total entropy by trading
configurational entropy for increased vibrational entropy. However, eventually
the system finds its way toward the bottom of the landscape where there are
very few basins, characterized by a small number of defects, and $\theta$
plateaus again and slowly tends to zero in the limit
$P_{L}/kT\rightarrow\infty$, where it will eventually form the perfect helix.
The loss of configurational entropy associated with the decrease in $\theta$
leads to the appearance of the $C_{p}$ maximum located at $P_{L}/kT\approx
100$ in a process similar to a Schottky anomaly 47.

The topological nature of the fluid structure means that defects are
eliminated in pairs leading to the formation of larger helical sections.
Furthermore, as there is no preference for one helical twist direction over
another, the defects eliminate randomly and Fig. 2(b) shows $f_{\mathcal{P}}$
is zero over the entire compression branch of the EOS. Figure 3(a) shows that
$P(n)$, the probability distribution for helical section sizes, decays
exponentially with helical section size at intermediate pressures, before the
$C_{p}$ maximum. This is consistent with the general form predicted for a
random distribution of helical sections, with the exception of the appearance
of the single particle section, which is not accounted for in the random
model. As the $C_{p}$ maximum is approached, the form of the distribution
changes, developing a shoulder at $n\approx 10$ ($P_{L}/kT=90$) that evolves
into a maximum at higher pressures, above the $C_{p}$ maximum. We also see an
oscillation in the probabilities for odd and even sized helical sections, with
the even helical sections being preferred. The amplitude of the oscillation
increases for smaller $n$ and higher pressure. The same trend was observed in
the jammed states of the model and results from the more efficient packing of
the even sections45.

Figure 3: Helical section length distribution. Log plots of $P(n)$ as a
function of $n$ for (a) Compression and (b) Decompression at different
pressures. The insert shows a log–log plot of $P(n)$ for small $n$ at
$P_{L}/kT$ on the decompression branch.

Figure 4(a) shows that the longitudinal pair correlation function decays
exponentially, beyond the small $n$ region, as expected for a quasi-1D fluid
with no long range translational order, which also allows us to extract a
translational correlation length, $\xi$, by fitting the peaks. The insert to
Fig. 4(b) shows that $\xi$ initially grows along the compression branch, but
then plateaus above $P_{L}/kT\approx 50$. Similar high pressure plateaus have
been observed in the correlation functions of the 2D hard discs next-nearest-
neighbour model 46.

Figure 4: Longitudinal correlation function and correlation length. Log plot
of $|g(z)-1|$ as a function of $z$ at $P_{L}/kT=52$ for (a) Compression and
(b) Decompression simulations. Insert shows the translational correlation
function $\xi/\sigma$ as a function of pressure for compression (circles) and
decompression (diamonds).

Figure 5(a) shows that along the compression branch of the EOS, the local
twist correlation function, $g_{0}$, decays to zero on the time scale of our
MD simulations over a wide range of $\phi$, including those densities at and
above the high pressure heat capacity maximum ($\phi\approx
0.37,P_{L}/kT\approx 100$), highlighting the fluid-like nature of the system.
The time dependence of $g_{0}$ also fits the Kohlraush-Williams-Watts (KWW)
function 48, 49, $\exp\left[-(t/t_{r})^{\beta}\right]$, where $t$ is time,
$t_{r}$ is a time constant and $\beta$ is an exponent that characterizes the
nature of the dynamics in a variety of supercooled liquids 50, 51. At low
$\phi$, we find $\beta>1$ (see insert Fig. 5(b)), which indicates the
relaxation follows a compressed exponential decay, suggesting the fluid
relaxes through a combination of ballistic and diffusive dynamics 52. When the
particles are well separated along the channel, they tend to collide with the
channel wall, reversing direction, before particle-particle collisions occur.
This generally results in a reversal of the sign of $v_{tet}$ as the particles
cross the channel relative to the other particles that form the tetrahedron of
a given particle, and $g_{0}$ at low density exhibits negative correlations
after a short time (not shown), similar to those observed in the velocity
auto-correlation function of bulk hard spheres at low $\phi$. With increasing
$\phi$, particle-particle collisions tend to cage a given particle,
maintaining the sign of $v_{tet}$, leading to diffusive behaviour and we see
$\beta$ decreases linearly. Interestingly, we see a change of slope in $\beta$
at $\phi\approx 0.33$, which coincides with density where the helical
structure of the liquid begins to develop, and the unstable neighbouring
defect environments disappear. At high $\phi$, $\beta<1$, plateauing at
$\beta\approx 0.65$, which indicates a stretched exponential relaxation that
is characteristic of slow glassy dynamics.

Figure 5: Orientational structural relaxation in the compression branch. (a)
Log-Log plot of $g_{0}$ for the compression branch as a function of time for
different $\phi$ from MD simulation (solid lines) and fits of the data to the
KWW function (dashed lines). (b) Compression branch relaxation time, $\tau$,
as a function of $\phi PV/NkT$ with fits to the VFT, parabolic and Arrhenius
equations. Insert: KWW exponent, $\beta$, obtained from fits to data in (a),
as a function of $\phi$.

Supercooled liquids exhibit a range of behaviour for the temperature
dependence of their relaxation times, $\tau$ 53. In a strong liquid, $\tau$
has an Arrhenius temperature dependence, $\tau\sim\exp(A/T)$ where $A$ is
constant. Fragile liquids exhibit super Arrhenius behaviour that can be
described by a number of models, such as the Vogel-Fulcher-Tammann (VFT)
equation 54, 55, 56, $\tau\sim\exp\left[A/(T-T_{0})\right]$ which predicts a
divergence at a finite temperature $T_{0}$, and the parabolic law 57, 58,
$\tau\sim\exp[A/T^{2}]$, which predicts no divergence and arises from a
facilitated dynamics description of glassy fluids. A 2D system of hard discs
confined to a narrow channel 59 even exhibits a crossover from fragile to
strong fluid behaviour, located at the $C_{p}$ maximum, where the fragile
behaviour at low $\phi$ occurs because defect-defect annihilation create
irreversible particle rearrangements that form more stable states. As $\phi$
increases, the defects become rare and structural relaxation proceeds through
the simple hopping events of isolated defects, leading to strong fluid
behaviour.

To examine the nature of the dynamics in the current system, $\tau$ is defined
as the time required for $g_{0}$ to decay to 0.2. For hard spheres, $\phi PV$
is a constant along an isobar so $\phi PV/NkT$ varies as $\sim 1/T$, and a log
plot of $\tau$ vs $\phi PV/NkT$ represents an effective Arrhenius plot for the
relaxation times (see Fig. 5(b)). The VFT and parabolic equations are fit to
the data obtained from the fragile fluid densities, between the low and high
pressure $C_{p}$ maxima on the compression EOS. The Arrhenius equation is fit
over the data from $\phi$ at and above the high pressure $C_{p}$ maximum. The
VFT and Arrhenius equations fit the data over the fragile and strong regions
respectively, and extending the data fitting region for either equation leads
to a decrease in the quality of the fits, suggesting this system may well
exhibit a fragile-strong dynamic crossover, similar to that observed in the 2D
confined fluid. However, the parabolic law, which does not contain a fragile-
strong crossover, describes the relaxation times over the entire range
studied, despite only being fit using the data from below the $C_{p}$ maximum.

A direct visualization of a trajectory, showing the structure and dynamics of
the fluid, can be achieved by plotting the local twist direction of each
particle as a function of time (Fig. 6). At low $\phi(P_{L}/kT<33)$, the high
defect fraction ensures that defects are generally closer than three
particles, preventing the formation of the helix structure, and the fast
particle dynamics means the tetrahedron associated with a particle frequently
changes sign. With increasing density and pressure $(P_{L}/kT=70,90)$, along
the compression branch of the EOS, the helical sections increase in length. We
also see the diffusion of the defects, as well as the spontaneous creation and
annihilation of defect pairs. However, at the highest pressure
$(P_{L}/kT=120)$ the defects are well separated and diffuse slowly, suggesting
the system is in a glassy state.

Figure 6: Time evolution of fluid helical structure. Structural evolution of a
section of the fluid (particles 1-150) as a function of time with spheres
described by a $\mathcal{M}$ tetrahedron (red) and $\mathcal{P}$ tetrahedron
(blue), for the compression branch (Left Column) and the decompression branch
(Right Column).

###  2.3 Decompression branch: $\phi>0.33$, $P/kT>38$

The thermodynamics, structure and dynamics of the fluid along the
decompression branch of the EOS are distinct from those of compressed fluid.
The initial state of the system consists of a perfect helix with a
$\mathcal{P}$ twist direction, but as there is no bias in the system that
would preferentially select one twist direction over the other, we obtain the
same general results when decompressing from a perfect $\mathcal{M}$ helix. As
the system is decompressed, the heat capacity exhibits a broad feature at high
pressures where $\theta$ is essentially zero and $f_{\mathcal{P}}\approx 1$
(See Fig. 2). At $P_{L}/kT\approx 70$, $\theta$ rapidly increases before
plateauing as the decompression branch joins the compression branch when
$\theta\approx 0.25$, leading to the sharp maximum in the $C_{p}$ located at
$P_{L}/kT\approx 50$. With increasing $\theta$, we also see $f_{\mathcal{P}}$
decrease, but it remains non-zero. This implies the defects, which are
generated in pairs, remain at least loosely paired so that the sections of
$\mathcal{M}$ helix created between the defects are small, allowing the
$\mathcal{P}$ tetrahedra to remain in excess until the equation of state
finally reaches the compression branch and $f_{\mathcal{P}}$ falls to zero.

Figure 3(b) shows that $P(n)$ in the decompression branch also exhibits
important differences compared to that of the compression branch. At
$P_{L}/kT=60$, the distribution decays as a power law for small helical
sections before eventually crossing over to an exponential decay for helical
section sizes larger than $n\approx 100$ (see insert Fig. 3(b). With
decreasing pressure, the crossover size moves to smaller $n$ so the
distribution begins to resemble that of the low pressure region of the
compression branch where the entire distribution is exponential. It is not
possible to measure the distribution at higher pressures as the defect
fraction effectively goes to zero. While $P(n)$ suggests there are non-trivial
correlations in the helical structure of the system along the decompression
branch, Figure 4(b) shows that the translational correlation decays
exponentially, as expected. For pressures in the region, $38<P_{L}/kT<70$,
$\xi$ is smaller in the decompression branch than in the compression branch,
going through a minimum before it grows at higher pressure to become the
largest translational correlation in the system. However, at these high
densities, it is difficult to accurately measure the correlation lengths.

The trajectories for the decompression branch, pictured in Fig. 6, exhibit
differences from those of the compression branch. At $P_{L}/kT=70$ there are
essentially no defects. As the pressure decreases, defects appear in pairs
that remain closely paired as they diffuse until they annihilate. The pairing
of the defects ensures the excess of $\mathcal{P}$ tetrahedra.

Finally, Fig. 7 shows that $g_{0}$ no longer decays to zero, in contrast to
the behaviour observed in the compression branch, but instead plateaus at a
finite value of $g_{0}$ that increases with increasing $\phi$. It is also
interesting to note how quickly the difference in relaxation behaviour between
the decompression and compression branches develops. At $\phi=0.33$ the two
branches exhibit the same behaviour and have the same pressure. However, at
$\phi=0.331$, where the equations of state have only just separated, the
decompression branch no longer decays to zero on the time scale of our
simulation. Furthermore, the structural relaxation in the compression branch
continues to decay to zero at even higher densities, despite being at a higher
pressure where the fluctuations required for particle rearrangement should be
more difficult because of a reduced free volume.

Figure 7: Orientational structural relaxation in the decompression branch.
Log-Log plot of $g_{0}$ for the decompression branch as a function of time for
different $\phi$.

##  3 Discussion

Our results show the quasi-one-dimensional system of hard spheres confined to
a narrow, cylindrical channel with diameter $H_{d}/\sigma=1.95$ has two
branches to its EOS above $P_{L}/kT\approx 38,(\phi\approx 0.33)$ with
distinct thermodynamic, structural and dynamic properties. It is not unusual
for quasi-one-dimensional hard sphere systems to exhibit metastability and
hysterisis 17, 14 because at a given channel diameter the system may support
structurally distinct ideal helical structures with different limiting
densities. However, for channel diameters $1+\sqrt{3/4}/7<H_{d}<1+4\sqrt{3}/7$
there is only one perfect, single helical packing so the two branches of the
EOS necessarily terminate at the same most dense packed structure.

Hu et al. 46 used the transfer matrix method to study the equation of state
and correlation lengths of confined hard spheres with next nearest neighbour
interactions up to $P_{L}/kT\approx 50$, and showed they did not exhibit a
phase transition, consistent with the expectations for one and quasi-one-
dimensional systems with short-ranged interactions. The analysis also found
that the largest eigenvalue for the system, which determines the equilibrium
properties, consists of two conjugate eigenvalues at low pressure that split
into two distinct real eigenvalues at an intermediate pressure. The
correlation length associated with the largest eigenvalue continues to grow
after the split, while the second largest correlation length goes through a
minimum before growing again at higher pressure. The translational correlation
lengths obtained in our simulations follow the same trend, allowing us to
identify the compression branch of the EOS as the equilibrium state of the
system. The pressure measured in our simulations along the compression branch
is also the same as that predicted by the largest eigenvalue of the transfer
matrix method. Furthermore, $\xi$, measured along the decompression branch
follows the same evolution as the correlation length obtained for the smaller
eigenvalue of the transfer matrix, which suggests the states are related, and
implies the decompression branch represents a non-equilibrium or metastable
state.

However, the EOS pictured in Fig. 1 then leads to a thermodynamic paradox. If
there is no phase transition along either branch of the equation of state and
both pressures vary continuously as a function of $\phi$, then the relative
stability of the two states can be determined by comparing their Gibbs free
energies, $G/NkT=PV/NkT-S/Nk$, at the same pressure. Figure 8 shows the
difference in Gibbs free energies between the compression and decompression
branches, $\Delta G/NKT=G_{c}-G_{d}$, where the entropy along each branch is
calculated relative to the ideal gas at the same $N,V$ and $T$,

$\Delta S(\phi)/Nk=\Delta
S(\phi_{r})/Nk-\int_{\phi_{r}}^{\phi}(PV/NkT-1)d\ln\phi^{\prime}\mbox{,}\\\ $
(1)

and the reference state occupied volume fraction, $\phi_{r}<0.33$, is chosen
to be below the bifurcation point in the EOS. The predicted free energy is
positive, which would suggest the decompression branch is more stable,
contradicting the transfer matrix results. One way to resolve the paradox
would be to note that, as the system is compressed from below $\phi\approx
0.33$, it always follows the higher pressure branch. To make the system fall
onto the lower pressure branch, it is necessary to impose a constraint that
ensures defects are eliminated in a way that leads to an excess in
$f_{\mathcal{P}}$. This would reduce the entropy of the decompression branch
making $\Delta G/NKT<0$. If the entropy loss is large enough, then the
decompression branch will be metastable over the entire pressure range, with
$\Delta G/NKT\rightarrow 0$, from below only as $P_{L}/kT\rightarrow\infty$
and the two branches approach the same most dense jammed packing. The low
pressure/low density point where the two branches meet, then represents the
low pressure limit of stability for the metastable state.

Figure 8: Gibbs free energy. The free energy difference between the
compression and decompression branches, $\Delta G/NkT$, as a function of
$P_{L}/kT$ (solid line). Error estimates based on standard deviations in the
EOS for both branches (dashed lines).

The way defects are organized within the fluid determines the topological
properties of the system. It also has important consequences for the
configurational and vibrational contributions to the entropy. An earlier study
40 of the jammed states of this system found that the packing density,
$\phi_{J}$, of the system increases as two isolated defects approach each
other. At a fixed $\phi$ of the fluid, this leads to an increase in the
vibrational entropy of the system that induces an effective long range
attraction between the two defects. Along the compression branch, the defects
in the fluid are arranged randomly, giving rise to the exponential decay of
$P(n)$, and even at high densities, where the peaks in $P(n)$ suggests there
is at least some short-ranged attraction between defects, there is no
preference for either helical twist direction, which ensures $f_{\mathcal{P}}$
is always zero. This suggests the fluid is stabilized by configurational
entropy because the defects can be arranged in a large number of different
ways.

On the other hand, the structure of the system along the decompression branch
is intrinsically topological. The helical twist direction of the system is set
by the initial condition. Defects in the helix appear in pairs that must
remain loosely bound in order to preserve the excess twist. The defect pairs
can appear though out the system, which helps retain a degree of
configurational entropy, but there are still fewer available configurations
than would be possible for a system with unpaired defects, at the same defect
fraction. This suggests the decompression branch is stabilized through the
increased vibrational entropy gained by keeping the defect close together. In
fact, the degree of stabilization is such that the chiral excess phase
persists down to low densities, where in principle the system has sufficient
free volume to rearrange as demonstrated by the fast relaxation times of the
higher pressure compression branch at the same $\phi$, and only terminates
when there are enough defects ($\theta=0.25$) to effectively eliminate the
helical sections altogether.

An attraction between topological defects is a key feature of the physics that
leads to the KTHNY transition and hexatic phase in 2D hard discs 60, 61, 62,
which raises the question of whether or not the decompression branch of our
quasi-1D system exhibits similar phenomena. Figure. 2(b) shows that $\theta$
goes continuously to zero around $P_{L}/kT\approx 70$, which might suggest
there is an ordering transition at high pressure along the decompression
branch, even if it is metastable with respect the equilibrium branch. The
helical correlations along the channel, captured by $P(n)$, also exhibit a
power law decay over small $n$, where the crossover to exponential decay moves
to larger $n$ with increasing pressure. However, these may also simply reflect
the difficulties associated with establishing the true metastable equilibrium
along the decompression branch, particularly as the fraction of defects
becomes small. In particular, the crossover to an exponential decay in $P(n)$
confirms that the system remains fluid at the pressures studied. A similar
crossover was recently observed in a quasi-one dimensional system of hard
disks 63, 64, 65. Studies with larger numbers of particles, along with
transfer matrix calculations performed at higher pressures, would be required
to fully resolve the behaviour of the system at densities close to the most
dense jammed state.

##  4 Conclusion

Our results clearly show that the decompression branch remains metastable,
with an excess twist direction, on the time scale of our simulation, despite
being in a lower pressure state relative to the equilibrium branch. We also
show that the topological properties of the fluid play an important role in
maintaining its stability down to low densities where the two branches of the
EOS meet, suggesting the decompression branch is in fact a topologically
stabilized state. However, the fluid along the decompression brach is is
metastable and so will eventually decay over time. Simulation time scales are
short in comparison to experimental time scales, so it remains to be seen if
such a topologically stabilized state can be observed real systems, and
whether the phenomena can be found in other quasi-one-dimensional systems that
have helical structure.

##  5 Methods

###  5.1 Model

We study a system of $N$ hard spheres with diameter $\sigma$, confined in a
cylindrical narrow channel of length L with channel diameter
$H_{d}/\sigma=1.95$, 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,

$U(r_{ij})=\left\\{\begin{matrix}0\;\;\;&r_{ij}\geqslant\sigma\;\;\;\;\;\;\;\;\\\
\infty\;\;\;&r_{ij}<\sigma\;\;\;\;\;\;\;\;\end{matrix}\right.,$ (2)
$U_{w}(r_{i})=\left\\{\begin{matrix}0\;\;\;&|r_{xy}|\leqslant\left|H_{0}/2\right|\\\
\infty\;\;\;&\textup{otherwise}\end{matrix}\right.,$ (3)

respectively, where $r_{ij}=\left|\mathbf{r_{i}-r_{j}}\right|$ is the distance
between particles, $|r_{xy}|$ is the magnitude of position vector for a
particle perpendicular to the wall where the centre 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’ centres is $V_{0}=\pi
L(H_{0}/2)^{2}$, where $H_{0}=H_{d}-\sigma$, and the occupied volume is
$\phi=2N\sigma^{3}/\left(3LH^{2}_{d}\right)$.

###  5.2 Molecular Dynamics Simulation

Our study uses both molecular dynamics (MD) and Monte Carlo (MC) simulations
with systems containing $N=10^{4}$ particles and $H_{d}/\sigma=1.95$. The MD
simulations are performed, in the canonical ensemble ($N,V,T$) using a
modified version of the Lubachevsky and Stillinger event-driven algorithm 66
that compresses the system by expanding the particles and channel diameter
such that $H_{d}/\sigma$ remains constant. The unit of time for the simulation
is given by $\sigma\sqrt{m/kT}$, where $k_{\text{B}}$ is Boltzmann’s constant
and $m$ is the mass of a particle, which is set to unity. Particles are
assigned random velocities at the beginning of the run, scaled to ensure
$kT=1$ and velocity rescaling is used to maintain the temperature. Depending
on $\phi$, $200N-10^{6}N$ collisions are used to reach equilibrium before data
is collected over the next $400N-10^{7}N$ collisions.

Periodic boundary conditions for a helical system are described by 14;

$\boldsymbol{r}_{i\gamma}=\boldsymbol{r}_{i}+n_{\gamma}\boldsymbol{\lambda}\mbox{,}$
(4)

where $\boldsymbol{r}_{i\gamma}$ is the position of particle $i$ in the
$\gamma^{\text{th}}$ unit cell, $n_{\gamma}$ is an integer and
$\boldsymbol{\lambda}(\lambda_{|r_{xy}|},\lambda_{\alpha},\lambda_{z})$ is the
lattice vector for the radial, angular, $\alpha$, and longitudinal, $z$,
components of the cylindrical coordinates, respectively. For the MD
simulations, we use translational periodic boundaries with $\lambda_{z}=L$ and
$\lambda_{|r_{xy}|}=\lambda_{\alpha}=0$, so there is no twist between cells.

At the start of each MD compression simulation, particles are placed in a
linear lattice with $\phi=0.01$. The system is equilibrated and data collected
at each $\phi$ before it is compressed to the next density at a compression
rate of $d\sigma/dt=0.001$. The MD decompression simulations begin at high
density, $\phi=0.40$, with the particles arranged in the perfect single helix
packing containing a pair of defects so there are two helical sections
containing $N=9998$ and $N=2$ particles, respectively. This is necessary
because of the inherent twist associated with the perfect helical structure
and the fact that we use translational periodic boundaries. The close packed,
perfect helix has $\phi=0.421$.

###  5.3 Monte Carlo Simulation

Our MC simulations are carried out in the isobaric-isothermal ensemble where
$N$, $T$ and $P_{L}$, the longitudinal pressure applied to the ends of the
channel, are held fixed. We also employ helical periodic boundaries
characterized by vector components $\lambda_{z}=L$, $\lambda_{|r_{xy}|}=0$ and
$\lambda_{\alpha}=\alpha$. Three types of MC move are used: particles are
moved using the standard Metropolis algorithm, the volume is sampled in a
logarithmic random walk 67 and we allow twist MC moves that uniformly sample
the periodic boundary twist angle $\alpha$. The step size for each type of MC
move is adjusted to ensure an acceptance rate of approximately $40\%$. In the
simulations, a single MC cycle involves $N$ particle moves, $0.02N$ volume
moves and $0.02N$ twist moves. Our simulations are performed in blocks of
$5\times 10^{6}$ MC cycles, so we can follow the system converging to
equilibrium, and the results and errors are reported for the final block. The
MC compression simulations compress the system at their final $P_{L}$,
starting from a linear lattice with $\phi=0.01$, so that each state point
evolves independently from the others. The system is equilibrated for three MC
blocks ($1.5\times 10^{7}$ MC cycles), before data is collected over the
fourth MC block. The MC decompression simulations, performed at a fixed
$P_{L}$, start from a perfect helix, with single right ($\mathcal{P}$) twist
direction, and the $Z$-coordinate scaled to the lower density. Here, the use
of helical boundary conditions with a variable twist angle means there is no
need to introduce a defect. The system is equilibrated for 12 MC blocks and
the results are reported for the 13th block. More details concerning the
equilibration of the decompression branch are provided in the Supporting
Information. We also note that our results for the decompression branch are
not dependent on the choice of twist direction in the perfect helix.

###  5.4 Heat Capacity

We calculate a number of thermodynamic and structural properties of the fluid.
The constant pressure heat capacity for the system is given by,

$\frac{C_{p}}{Nk}=\left(\frac{\partial H}{\partial
T}\right)_{P_{L}}=\frac{3}{2}+\frac{Z}{1+\left(\frac{\partial\ln
Z}{\partial\ln\phi}\right)_{T}}\mbox{,}\\\ $ (5)

where $H$ is the enthalpy, $Z=P_{L}AL/NkT$ is the compressibility factor and
$A$ is the cross sectional area of the cylinder.

###  5.5 Helical Structure

To examine the helical structure of the fluid, we identify a local helical
twist direction for each atom $i$ based on the signed volume the tetrahedron
given by;

$v_{tet}(i)=\frac{\mathbf{a}\cdot\mathbf{b}\times\mathbf{c}}{6}\mbox{,}\\\ $
(6)

where $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are the position vectors for
particles $i-1$, $i$ and $i+1$, relative to particle $i+2$, respectively.
Successive particles with the same sign for $v_{tet}(i)$ have same helical
twist direction, which allows us to identify the length of a helical section.
Defects, located between helical sections with opposite twist directions,
occur when $v_{tet}$ changes sign. The method has been used to study helical
structure in the jammed structures of this system 45, where the method also
uses the magnitude $|v_{tet}|$, which is distinguishably small for particles
in the defect, to confirm the location of the defect. In the fluid, particles
adopt configuration with a broad distribution of $|v_{tet}|$ so this is no
longer possible. The current work identifies the location of the defects just
using the sign changes of $v_{tet}(i)$.

Our analysis of the helical structure in the fluid represents an approximate
mapping of a fluid configuration to a nearby jammed structure, effectively
providing an inherent structure landscape 42, 43, 44 description for the
system. However, we find cases where a pair of defects are located on
neighbouring particles, which represents an unstable environment that leads to
defect annihilation when compressed to jamming 40. Both MD and MC simulations
produce the same defect properties so in the results section we report the
fraction of defects, $\theta$ for the MC results including the neighbouring
defects, and the MD results with the neighbouring defects eliminated.

We also calculate the probability, $P(n)$, of finding a helical section
containing $n$ particles and the excess fraction of $\mathcal{P}$ tetrahedra,
$f_{\mathcal{P}}$, as the difference in the number of right (positive
$v_{tet}$) tetrahedra and left (negative $v_{tet}$) in a configuration
normalized by $N$. Assuming the number and distribution of the defects in the
fluid provide an effective instantaneous map of a configuration to its
inherent structure, we can follow the evolution of the system through the
inherent structure landscape (ISL) as a function of pressure and density.

###  5.6 Structural Relaxation

Structural relaxation in the system occurs through the creation, diffusion and
elimination of defects because these change the local direction of helical
twist. To measure structural relaxation, we define a self correlation
function,

$g_{0}=\left<\frac{v_{tet}(i,0)}{|v_{tet}(i,0)|}\frac{v_{tet}(i,t)}{|v_{tet}(i,t)|}\right>\mbox{,}\\\
$ (7)

where the average is taken over all particles and multiple time origins have
been used.

We would like to thank the Digital Alliance of Canada for computational
resources. RKB acknowledges NSERC grant RGPIN–2019–03970 for financial
support. This work was also supported by the Iran National Science Foundation
(INSF) and the Iranian National Foundation of Elites via Grant No. 4015274.

Supporting information provides details regarding the convergence of the
thermodynamic and structural properties of the system along the decompression
branch.

## References

  * Senthil 2014 Senthil, T. Symmetry-Protected Topological Phases of Quantum Matter. _Annu. Rev. Conden. Matt. Phys._ 2014, _6_ , 1–26
  * Kane and Lubensky 2013 Kane, C. L.; Lubensky, T. C. Topological boundary modes in isostatic lattices. _Nat. Phys._ 2013, _10_ , 39–45
  * Loehr et al. 2016 Loehr, J.; Loenne, M.; Ernst, A.; Heras, D. d. l.; Fischer, T. M. Topological protection of multiparticle dissipative transport. _Nat. Commun._ 2016, _7_ , 11745
  * Pedro et al. 2019 Pedro, R. P.; Paulose, J.; Souslov, A.; Dresselhaus, M.; Vitelli, V. Topological Protection Can Arise from Thermal Fluctuations and Interactions. _Phys. Rev. Lett._ 2019, _122_ , 118001
  * Zygmunt et al. 2019 Zygmunt, W.; Teich, E. G.; Anders, G. v.; Glotzer, S. C. Topological order in densely packed anisotropic colloids. _Phys. Rev. E_ 2019, _100_ , 032608
  * Erickson 1973 Erickson, R. O. Tubular Packing of Spheres in Biological Fine Structure. _Science_ 1973, _181_ , 705–716
  * Harris and Erickson 1980 Harris, W. F.; Erickson, R. O. Tubular arrays of spheres: Geometry, continuous and discontinuous contraction, and the role of moving dislocations. _J. Theor. Biol._ 1980, _83_ , 215–246
  * Pickett et al. 2000 Pickett, G. T.; Gross, M.; Okuyama, H. Spontaneous Chirality in Simple Systems. _Phys. Rev. Lett._ 2000, _85_ , 3652–3655
  * Mughal et al. 2011 Mughal, A.; Chan, H. K.; Weaire, D. Phyllotactic Description of Hard Sphere Packing in Cylindrical Channels. _Phys. Rev. Lett._ 2011, _106_ , 115704
  * Mughal et al. 2012 Mughal, A.; Chan, H. K.; Weaire, D.; Hutzler, S. Dense packings of spheres in cylinders: Simulations. _Phys. Rev. E_ 2012, _85_ , 051305
  * Mughal and Weaire 2014 Mughal, A.; Weaire, D. Theory of cylindrical dense packings of disks. _Phys. Rev. E_ 2014, _89_ , 042307
  * Fu et al. 2016 Fu, L.; Steinhardt, W.; Zhao, H.; Socolar, J. E. S.; Charbonneau, P. Hard sphere packings within cylinders. _Soft Matter_ 2016, _12_ , 2505–2514
  * Durán-Olivencia and Gordillo 2009 Durán-Olivencia, F. J.; Gordillo, M. C. Ordering of hard spheres inside hard cylindrical pores. _Phys. Rev. E_ 2009, _79_ , 061111
  * Fu et al. 2017 Fu, L.; Bian, C.; Shields, C. W.; Cruz, D. F.; López, G. P.; Charbonneau, P. Assembly of hard spheres in a cylinder: a computational and experimental study. _Soft Matter_ 2017, _13_ , 3296–3306
  * Zhu et al. 2021 Zhu, G.; Gao, L.; Xu, Z.; Dai, X.; Zhang, X.; Yan, L.-T. Entropy-Driven Unconventional Crystallization of Spherical Colloidal Nanocrystals Confined in Wide Cylinders. _Nano Lett._ 2021, _21_ , 8439–8446
  * Winkelmann et al. 2017 Winkelmann, J.; Haffner, B.; Weaire, D.; Mughal, A.; Hutzler, S. Simulation and observation of line-slip structures in columnar structures of soft spheres. _Phys. Rev. E_ 2017, _96_ , 012610
  * Mughal et al. 2018 Mughal, A.; Winkelmann, J.; Weaire, D.; Hutzler, S. Columnar structures of soft spheres: Metastability and hysteresis. _Phys. Rev. E_ 2018, _98_ , 043303
  * Mickelson et al. 2003 Mickelson, W.; Aloni, S.; Han, W.-Q.; Cumings, J.; Zettl, A. Packing C60 in Boron Nitride Nanotubes. _Science_ 2003, _300_ , 467–469
  * Khlobystov et al. 2004 Khlobystov, A. N.; Britz, D. A.; Ardavan, A.; Briggs, G. A. D. Observation of Ordered Phases of Fullerenes in Carbon Nanotubes. _Phys. Rev. Lett._ 2004, _92_ , 245507
  * Yao et al. 2016 Yao, Z.; Liu, C.-J.; Lv, H.; Liu, B.-B. Stable double helical iodine chains inside single-walled carbon nanotubes. _Phys. Lett. A_ 2016, _380_ , 2781–2785
  * Lohr et al. 2010 Lohr, M. A.; Alsayed, A. M.; Chen, B. G.; Zhang, Z.; Kamien, R. D.; Yodh, A. G. Helical packings and phase transformations of soft spheres in cylinders. _Phys. Rev. E_ 2010, _81_ , 040401
  * Jiang et al. 2013 Jiang, L.; de Folter, J. W. J.; Huang, J.; Philipse, A. P.; Kegel, W. K.; Petukhov, A. V. Helical Colloidal Sphere Structures through Thermo-Reversible Co-Assembly with Molecular Microtubes . _Angewandte Chemie Int. Ed._ 2013, _52_ , 3364–3368
  * Jiménez-Millán et al. 2021 Jiménez-Millán, S.; García-Alcántara, C.; Ramírez-Hernández, A.; Sambriski, E.; Hernández, S. Self-Aassembly of core-corona colloids under cylindrical confinement: A Monte Carlo study. _J. Mol. Liq._ 2021, _335_ , 116219
  * Meagher et al. 2015 Meagher, A.; García-Moreno, F.; Banhart, J.; Mughal, A.; Hutzler, S. An experimental study of columnar crystals using monodisperse microbubbles. _Colloids Surf A: Physicochem. Eng. Asp._ 2015, _473_ , 55–59
  * Jin et al. 2020 Jin, W.; Chan, H.-K.; Zhong, Z. Shape-Anisotropy-Induced Ordered Packings in Cylindrical Confinement. _Phys. Rev. Lett._ 2020, _124_ , 248002
  * Wan et al. 2022 Wan, S.; Xi, X.; Zhang, H.; Ning, J.; Zheng, Z.; Zhang, Z.; Long, Y.; Deng, Y.; Fan, P.; Yang, D.; Li, T.; Dong, A. Shape-Mediated Oriented Assembly of Concave Nanoparticles under Cylindrical Confinement. _ACS Nano_ 2022, _16_ , 21315–21323
  * Vlasov et al. 2001 Vlasov, Y. A.; Bo, X.-Z.; Sturm, J. C.; Norris, D. J. On-chip natural assembly of silicon photonic bandgap crystals. _Nature_ 2001, _414_ , 289–293
  * Zhao et al. 2014 Zhao, Y.; Shang, L.; Cheng, Y.; Gu, Z. Spherical Colloidal Photonic Crystals. _Accounts Chem. Res._ 2014, _47_ , 3632–3642
  * Lei et al. 2018 Lei, Q.-l.; Ni, R.; Ma, Y.-q. Self-Assembled Chiral Photonic Crystals from a Colloidal Helix Racemate. _ACS Nano_ 2018, _12_ , 6860–6870
  * Wan and Glotzer 2020 Wan, D.; Glotzer, S. C. Randomness in self-assembled colloidal crystals can widen photonic band gaps through particle shape and internal structure. _arXiv_ 2020,
  * Liu et al. 2023 Liu, T.; Chan, H.-K.; Wan, D. Chiral photonic crystals from sphere packing. _arXiv_ 2023,
  * Hove 1950 Hove, L. v. Sur L’intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension. _Physica_ 1950, _16_ , 137–143
  * Ruelle 1968 Ruelle, D. Statistical mechanics of a one-dimensional lattice gas. _Commun.math. Phys._ 1968, _9_ , 267–278
  * Landau and Lifshitz 1980 Landau, L.; Lifshitz, E. _Statistical Physics, Part 1. Course of Theoretical Physics Volume 5_ ; Oxford ; New York : Pergamon Press; 1980
  * Cuesta and Sánchez 2004 Cuesta, J. A.; Sánchez, A. General Non-Existence Theorem for Phase Transitions in One-Dimensional Systems with Short Range Interactions, and Physical Examples of Such Transitions. _J. Stat. Phys._ 2004, _115_ , 869–893
  * Saryal et al. 2018 Saryal, S.; Klamser, J. U.; Sadhu, T.; Dhar, D. Multiple Singularities of the Equilibrium Free Energy in a One-Dimensional Model of Soft Rods. _Phys. Rev. Lett._ 2018, _121_ , 240601
  * Dyson 1969 Dyson, F. J. Existence of a phase-transition in a one-dimensional Ising ferromagnet. _Commun.math. Phys._ 1969, _12_ , 91–107
  * Thouless 1969 Thouless, D. J. Long-Range Order in One-Dimensional Ising Systems. _Phys. Rev._ 1969, _187_ , 732–733
  * Kosterlitz 2017 Kosterlitz, J. M. Nobel Lecture: Topological defects and phase transitions. _Rev. Mod. Phys._ 2017, _89_
  * Yamchi and Bowles 2015 Yamchi, M. Z.; Bowles, R. K. Helical Defect Packings in a Quasi-One-Dimensional System of Cylindrically Confined Hard Spheres. _Phys. Rev. Lett._ 2015, _115_ , 025702
  * Chan et al. 2019 Chan, H.-K.; Wang, Y.; Han, H. Densest helical structures of hard spheres in narrow confinement: An analytic derivation. _AIP Advances_ 2019, _9_ , 125118
  * Stillinger et al. 1964 Stillinger, F. H.; DiMarzio, E. A.; Kornegay, R. L. Systematic Approach to Explanation of the Rigid Disk Phase Transition. _J. Chem. Phys._ 1964, _40_ , 1564–1576
  * Stillinger and Weber 1982 Stillinger, F. H.; Weber, T. A. Hidden structure in liquids. _Phys. Rev. A_ 1982, _25_ , 978–989
  * Ashwin et al. 2013 Ashwin, S. S.; Yamchi, M. Z.; Bowles, R. K. Inherent Structure Landscape Connection between Liquids, Granular Materials, and the Jamming Phase Diagram. _Phys. Rev. Lett._ 2013, _110_ , 145701
  * Zarif et al. 2021 Zarif, M.; Spiteri, R. J.; Bowles, R. K. Inherent structure landscape of hard spheres confined to narrow cylindrical channels. _Phys. Rev. E_ 2021, _104_ , 064602
  * Hu et al. 2018 Hu, Y.; Fu, L.; Charbonneau, P. Correlation lengths in quasi-one-dimensional systems via transfer matrices. _Mol. Phys._ 2018, _116_ , 3345–3354
  * Tari and Ahmet 2003 Tari,; Ahmet, _The specific heat of matter at low temperatures_ ; World Scientific, 2003
  * Kohlrausch and R. 1854 Kohlrausch,; R., Theorie des elektrischen Rückstandes in der Leidener Flasche. _Annalen der Physik_ 1854, _167_ , 179–214
  * Williams and Watts 1970 Williams, G.; Watts, D. C. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. _Trans. Faraday Soc._ 1970, _66_ , 80–85
  * Debenedetti and Stillinger 2001 Debenedetti, P. G.; Stillinger, F. H. Supercooled liquids and the glass transition. _Nature_ 2001, _410_ , 259
  * Wu and Jia 2016 Wu, J. H.; Jia, Q. The heterogeneous energy landscape expression of KWW relaxation. _Scientific Reports_ 2016, _6_ , 20506
  * 52 Bouchaud, J.-P. In _Anomalous Relaxation in Complex Systems: From Stretched to Compressed Exponentials, in Anomalous Transport Foundations and Applications_ ; Klages,, Radons, G., Sookolov, M., Eds.; Wiley-VCH Verlag GmbH & Co.: KGaA, Weinheim
  * Angell 1995 Angell, C. A. Formation of Glasses from Liquids and Biopolymers. _Science_ 1995, _267_ , 1924–1935
  * Vogel 1921 Vogel, D. H. Das Temperaturabhaengigkeitsgesetz der Viskositaet von Fluessigkeiten. _Phys. Zeit._ 1921, _22_ , 645
  * Fulcher 1925 Fulcher, G. Analysis of Recent Measurements of the Viscosity of Glasses. _J. Am. Chem. Soc._ 1925, _8_
  * Tammann and Hesse 1926 Tammann, G.; Hesse, W. Die Abhängigkeit der Viscosität von der Temperatur bie unterkühlten Flüssigkeiten. _Z. Anorg. Allge. Chem._ 1926, _156_ , 245–257
  * Elmatad et al. 2009 Elmatad, Y. S.; Chandler, D.; Garrahan, J. P. Corresponding States of Structural Glass Formers. _J. Phys. Chem. B_ 2009, _113_ , 5563–5567
  * Elmatad et al. 2010 Elmatad, Y. S.; Chandler, D.; Garrahan, J. P. Corresponding States of Structural Glass Formers. II. _J. Phys. Chem. B_ 2010, _114_ , 17113–17119
  * Yamchi et al. 2012 Yamchi, M. Z.; Ashwin, S. S.; Bowles, R. K. Fragile-Strong Fluid Crossover and Universal Relaxation Times in a Confined Hard-Disk Fluid. _Phys. Rev. Lett._ 2012, _109_ , 225701
  * Bernard and Krauth 2011 Bernard, E. P.; Krauth, W. Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition. _Phys. Rev. Lett._ 2011, _107_ , 155704
  * Kapfer and Krauth 2014 Kapfer, S. C.; Krauth, W. Two-Dimensional Melting: From Liquid-Hexatic Coexistence to Continuous Transitions. _Phys. Rev. Lett._ 2014, _114_ , 035702
  * Thorneywork et al. 2017 Thorneywork, A. L.; Abbott, J. L.; Aarts, D. G. A. L.; Dullens, R. P. A. Two-Dimensional Melting of Colloidal Hard Spheres. _Phys. Rev. Lett._ 2017, _118_ , 158001
  * Huerta et al. 2020 Huerta, A.; Bryk, T.; Pergamenshchik, V. M.; Trokhymchuk, A. Kosterlitz-Thouless-type caging-uncaging transition in a quasi-one-dimensional hard disk system. _Phys. Rev. Res._ 2020, _2_ , 033351
  * Hu and Charbonneau 2021 Hu, Y.; Charbonneau, P. Comment on “Kosterlitz-Thouless-type caging-uncaging transition in a quasi-one-dimensional hard disk system”. _Phys. Rev. Res._ 2021, _3_ , 038001
  * Trokhymchuk et al. 2021 Trokhymchuk, A.; Pergamenshchik, V. M.; Huerta, A.; Bryk, T. Reply to “Comment on ‘Kosterlitz-Thouless-type caging-uncaging transition in a quasi-one-dimensional hard disk system’ ”. _Phys. Rev. Res._ 2021, _3_ , 038002
  * Lubachevsky and Stillinger 1990 Lubachevsky, B. D.; Stillinger, F. H. Geometric properties of random disk packings. _J. Stat. Phys._ 1990, _60_ , 561–583
  * Frenkel and Smit 2002 Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications. _Academic Press, New York_ 2002,

##  1 Supplemental Material: Convergence of Decompression Branch

Structural relaxation occurs rapidly along the high pressure equilibrium
compression branch of the EOS (see Fig 5 of main text) and the measurements of
the fluid properties converge quickly. However, structural relaxation along
the decompression is slow so to follow the convergence of a thermodynamic or
structural property we calculate its percentage change between MC cycles,

$\%\Delta X=100\left(1-\frac{X(i-1)}{X(i)}\right)\mbox{.}\\\ $ (S1)

Figures S1 and S1 show the evolution of $\phi$ and $\theta$, respectively, as
a function of MC cycle, starting from the perfect expanded helix. The largest
changes in both quantities occur at pressures near the bifurcation of the two
branches in the EOS ($P_{L}/kT\approx 40$) as the system moves to lower
density as it develops more defects. At high pressure, we actually see $\phi$
increase. Above $P_{L}/kT\approx 70$, the system does not develop defects and
the density increase is the result of a simple compression of the perfect
helix. However, at pressures just below this we see the system develop defects
while still increasing its density. This is somewhat surprising as the
equilibrium branch is located at lower density. The density of the system
appears to have converged, and is fluctuating around its equilibrium value, by
time we reached the conclusion of our simulation, but we notice the number of
defects continues to increase slowly, above the bifurcation point, although at
a very slow rate.

Figure S1: (a) Pressure, $P_{L}/kT$ as a function of $\phi$ at various MC
cycles, along the decompression branch (circles), compares to the equilibrium
compression branch pressure (squares). (b) $\%\Delta\phi$ (See Eq. S1) as a
function of MC cycle. Figure S2: (a) Defect fraction $\theta$ as a function of
$P_{L}/kT$ at various MC cycles, along the decompression branch (circles),
compares to the equilibrium compression branch pressure (squares). (b)
$\%\Delta\theta$ (See Eq. S1) as a function of MC cycle.


* [FM99] B. Farb, L. Mosher, _Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II._ , Invent. Math. 137 (1999), no. 3, 613–649.
  * [Fe20] Ferragut, Tom _Geodesics and Visual boundary of Horospherical Products_ , https://arxiv.org/abs/2009.04698.
  * [Fe22] Ferragut, Tom _Geometric rigidity of quasi-isometries in horospherical products_ , https://arxiv.org/abs/2211.04093.
  * [H74] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23–34.
  * [Hel78] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
  * [HS] G. Hochschild, J-P. Serre, _Cohomology of Lie algebras_ , Ann. of Math. (2) 57 (1953), 591–603.
  * [HSi90] W. Hebisch, A. Sikora, _A smooth subadditive homogeneous norm on a homogeneous group_ , Studia Math. Volume 96, Issue 3 (1990), 231–236.
  * [KLD17] V. Kivioja, E. Le Donne, Isometries of nilpotent metric groups, J. Ec. polytech. Math. 4 (2017), 473–482.
  * [LDNG21] E. Le Donne, S. Nicolussi Golo, Metric Lie groups admitting dilations, Ark. Mat. 59 (2021), no. 1, 125–163.
  * [LDPX22] E. Le Donne, G. Pallier, X. Xie, _Rough similarity of left-invariant Riemannian metrics on some Lie groups,_ https://arxiv.org/abs/2208.06510
  * [M62] G.D. Mostow, _Homogeneous spaces with finite invariant measure_ , Ann. of Math. (2) 75 (1962), 17–37.
  * [M70] G.D. Mostow, _Representative functions on discrete groups and solvable arithmetic subgroups,_ Amer. J. Math. 92 (1970), 1–32.
  * [P89] P. Pansu, _Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un,_ Ann. of Math. (2) 129 (1989), no. 1, 1–60.
  * [P11a] I. Peng, _Large scale geometry of nilpotent-by-cyclic groups_ , Geom. Funct. Anal. 21 (2011), no. 4, 951–1000.
  * [P11b] I. Peng.  Coarse differentiation and quasi-isometries of a class of solvable Lie groups I.  Geom. Topol., 15(4):1883–1925, 2011.
  * [P11c] I. Peng.  Coarse differentiation and quasi-isometries of a class of solvable Lie groups II.  Geom. Topol., 15(4):1927–1981, 2011.
  * [R02] V. Runde, _Lectures on amenability_ , Lecture Notes in Mathematics, 1774, Springer-Verlag, Berlin, 2002.
  * [R91] W. Rudin, _Functional analysis,_ Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991.
  * [S66] U. Shukla, _A cohomology for Lie algebras_ , J. Math. Soc. Japan 18 (1966), 275–289.
  * [S71] T. Sato, _The derivations of the Lie algebras_ , Tohoku Math. J. (2) 23 (1971), 21–36.
  * [SX12] N. Shanmugalingam and X. Xie, _A rigidity property of some negatively curved solvable Lie groups_ , Comment. Math. Helv. 87 (2012), no. 4, 805–823.
  * [SY05] H. Sawai, T. Yamada, _Lattices on Benson-Gordon type solvable Lie groups_ , Topology and its applications 149 (2005), 85–95.
  * [S78] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, In _Riemann Surfaces and Related Topics_ : Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 97 (1981), 465–496.
  * [T86] P. Tukia, _On quasiconformal groups._ _Journal d’Analyse Mathematique_ 46 (1986), 318–346.
  * [W00] B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2000), 195–237.