# Janus helices: From fully attractive to hard helices

Laura Dal Compare<EMAIL_ADDRESS>Dipartimento di Scienze
Molecolari e Nanosistemi, Università Ca’ Foscari di Venezia Campus
Scientifico, Edificio Alfa, via Torino 155,30170 Venezia Mestre, Italy  Flavio
Romano<EMAIL_ADDRESS>Dipartimento di Scienze Molecolari e
Nanosistemi, Università Ca’ Foscari di Venezia Campus Scientifico, Edificio
Alfa, via Torino 155,30170 Venezia Mestre, Italy  European Centre for Living
Technology (ECLT) Ca’ Bottacin, 3911 Dorsoduro Calle Crosera, 30123 Venice,
Italy  Jared A. Wood<EMAIL_ADDRESS>ARC Centre of Excellence in
Exciton Science, School of Chemistry, University of Sydney, Sydney, New South
Wales 2006, Australia  The University of Sydney Nano Institute, University of
Sydney, New South Wales 2006, Australia  Asaph Widmer-Cooper  asaph.widmer-
<EMAIL_ADDRESS>ARC Centre of Excellence in Exciton Science, School of
Chemistry, University of Sydney, Sydney, New South Wales 2006, Australia  The
University of Sydney Nano Institute, University of Sydney, New South Wales
2006, Australia  Achille Giacometti<EMAIL_ADDRESS>Dipartimento
di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari di Venezia Campus
Scientifico, Edificio Alfa, via Torino 155,30170 Venezia Mestre, Italy
European Centre for Living Technology (ECLT) Ca’ Bottacin, 3911 Dorsoduro
Calle Crosera, 30123 Venice, Italy

###### Abstract

The phase diagram of hard helices differs from its hard rods counterpart by
the presence of chiral "screw" phases stemming from the characteristic helical
shape, in addition to the conventional liquid crystal phases also found for
rod-like particles. Using extensive Monte Carlo and Molecular Dynamics
simulations, we study the effect of the addition of a short-range attractive
tail representing solvent-induced interactions to a fraction of the sites
forming the hard helices, ranging from a single-site attraction to fully
attractive helices for a specific helical shape. Different temperature regimes
exist for different fractions of the attractive sites, as assessed in terms of
the relative Boyle temperatures, that are found to be rather insensitive to
the specific shape of the helical particle. The temperature range probed by
the present study is well above the corresponding Boyle temperatures, with the
phase behaviour still mainly entropically dominated and with the existence and
location of the various liquid crystal phases only marginally affected. The
pressure in the equation of state is found to decrease upon increasing the
fraction of attractive beads and/or on lowering the temperature at fixed
volume fraction, as expected on physical grounds. All screw phases are found
to be stable within the considered range of temperatures with the smectic
phase becoming more stable on lowering the temperature. By contrast, the
location of the transition lines do not display a simple dependence on the
fraction of attractive beads in the considered range of temperatures.

##  I Introduction

Guided by site-specific directional interactions, colloidal patchy particles
have been shown, both numerically Glotzer and Solomon (2007); Damasceno,
Engel, and Glotzer (2012) and experimentally Walther and Muller (2013), to
self-assemble into higher order structures that cannot be obtained by other
synthetic routes, thus paving the way to obtain unique morphologies for
various technological and biomedical applications. Among the key factors
controlling the phases and the morphologies of the self-assembled structure
are the location, number, and size of the patches, as well as the strength and
range of interactions. Together with thermodynamic conditions, such as the
effective temperature and pressure, the number of patches and their
arrangement form a relatively small set of parameters that can be tuned to
guide the self-assembly toward the desired structure.

Janus colloids Sciortino, Giacometti, and Pastore (2009, 2010); Chen  _et al._
(2011) are spherical particles having one attractive patch on an otherwise
repulsive particle surface that demonstrate how the heterogeneity and
directionality of interactions can be designed to achieve a prescribed bulk
material with specific properties. Notwithstanding the challenges in the
synthesis and functionalization of such colloids, several techniques have been
devised that demonstrate the ability to control these processes. A spectacular
example of this is provided by the experimental self-assembly of spherical
triblock Janus particles into a two-dimensional kagome lattice Chen, Bae, and
Granick (2011), matched by numerical analysis Romano and Sciortino (2011).

Within the general class of Janus colloids where the coverage, i.e., the
fraction of attractive surface patch compared to the total surface, can be
tuned from $0\%$ (hard spheres) to $100\%$ (fully-attractive hard spheres),
the Janus limit of equal coverage with just two patches of equal size, one
attractive and one repulsive ($50\%$ coverage), has been shown to display a
particularly interesting behaviour Sciortino, Giacometti, and Pastore (2009,
2010); Giacometti  _et al._ (2014). For this system, a gas of micelles is
formed at low temperature and low density that has lower energy than the
coexisting homogeneous liquid formed at the same temperature and higher
density. The corresponding gas-liquid coexistence curve in the pressure-
temperature plane is negatively sloped, mimicking the anomalous behavior found
in water Gallo  _et al._ (2016).

While controlling the interactions plays a paramount role in guiding the self-
assembly process, the role of excluded-volume interactions provided by the
shape of the colloidal particles has been recognized to be equally important,
especially at high density Damasceno, Engel, and Glotzer (2012). Surprisingly
complex and remarkably diverse colloidal crystals can self-assemble driven
purely by particle shape anisotropy and the tendency to maximize the entropy,
with this somewhat counter-intuitive mechanism effectively behaving as an
entropic bond Harper, van Anders, and Glotzer (2019). The combination of
directional interactions and shape anisotropy is clearly possible, and
examples in the literature include colloidal dumbbells Avvisati, Vissers, and
Dijkstra (2015), "Mickey Mouse" shaped colloids Wolters  _et al._ (2015),
Janus dumbbells Munaò  _et al._ (2015); O’Toole, Giacometti, and Hudson
(2017); O’Toole  _et al._ (2017), rods Chaudhary  _et al._ (2012), ellipsoids
Liu  _et al._ (2012); Xu, Wang, and He (2015) and lobed particles Paul and
Vashisth (2020a, b), and polyhedral nanoparticles Henzie  _et al._ (2012).

Colloidal helices are another example of colloids that can be synthesized Oh
_et al._ (2019) and designed from first principles Fejer  _et al._ (2014).
Simple hard helices formed by spherical fused beads (colloids) arranged into a
helical shape with tunable pitch and radius represents a paradigmatic example
of entropically driven self-assembly providing a rich and unconventional
behaviour Frezza  _et al._ (2013); Kolli  _et al._ (2014a, b); Frezza  _et
al._ (2014); Dussi  _et al._ (2015); Kolli  _et al._ (2016); Cinacchi  _et
al._ (2017) that can also find biological representation in colloidal
suspensions of semiflexible virus particles Dogic and Fraden (1997, 2006);
Grelet (2014). Elongated hard helices with large aspect ratio form liquid
crystal phases that are similar to their hard rods counterpart. Unlike hard
rods, however, hard helices are chiral objects and hence can form chiral
phases (e.g. a chiral nematic also known as a cholesteric phase) whose origin
can be traced back to the shape and the chirality of the helices. More
specifically, it was found that the nematic phase was either cholesteric,
where the main director is rotating about an axis perpendicular to the
original plane, or screw-nematic, where the secondary director is rotating
about an axis parallel to the main director Frezza  _et al._ (2014); Dussi
_et al._ (2015); Cinacchi  _et al._ (2017). Even the smectic phase was found
to display screw-like behaviour under some circumstances Kolli  _et al._
(2014a, b, 2016). The entropy-driven formation of chiral nematic phases is not
limited to helical shapes but includes other interesting examples, such as
particles with twisted polyhedral shape Dussi and Dijkstra (2016) and curled
hard-rods Wensink and Morales-Anda (2015).

The aim of the present work is to study Janus helices, which combine enthalpic
bonding guided by directional interactions with the tendency of helices to
maximize the number of available microstates driven by entropy. This is a
natural extension of previous work on the phase behavior of Janus spheres
Sciortino, Giacometti, and Pastore (2009, 2010) and hard helices Frezza  _et
al._ (2013); Kolli  _et al._ (2014a, b, 2016); Cinacchi  _et al._ (2017). As
in the case of Janus spheres, the attractive sites represents interactions
induced by the presence of the solvent on different moieties of the helix.
Starting from the case of hard helices, we progressively add a short-range
attraction, in the form of a square-well tail, to some of the beads forming
the helices up to the case where all beads are attractive. The main aim of the
present study is to discuss the effect on the liquid-crystal phases displayed
by hard helices with the addition of a weak and short-range attraction to some
of the beads forming the helices; and at the same time to extend previous
Monte Carlo simulations based on hard core interactions with the addition of
Molecular Dynamics simulations that allow for faster equilibration and more
extensive analysis at high volume fraction. The Janus helices considered in
the present study have not been studied previously, but some comparison can be
made with the behaviour of fully attractive Gámez and Lago (2017) and Janus
rods Tripathy and Schweizer (2013) in the limit of very slender helices. The
case of a weak and highly localized directional attractive site is also of
experimental interest and has already been studied both experimentally and
numerically Chaudhary  _et al._ (2012); Repula  _et al._ (2019); Jack and
Millett (2021).

To set the stage, it is important to identify the regimes to be probed. For
spherical Janus colloids, the interesting regime occurs at very low
temperatures and low densities where self-assembly competes with the
possibility of observing gas-liquid and liquid-solid transitions. For athermal
hard helices, the interesting regime occurs at high pressures (and hence high
densities) where the isotropic phase is progressively replaced by liquid
crystal phases, having positional disorder but orientational order. Before
probing the challenging combined regime of low temperature and high pressure,
it is desirable to understand the effect of including a temperature dependence
(through the addition of an attractive part to the potential) on the hard
helices liquid crystal phases, as well as the effect of the coverage. This
will be systematically done in the present study by considering coverages
$\chi$ ranging from hard helices ($0\%$ $\chi=0$) to fully attractive square-
well (SW) helices ($\chi=100\%$), and includes the single site attractive case
($\chi=6.7\%$ coverage), and the Janus limit ($\chi=50\%$). Particular
emphasis will be devoted to understanding the stability of the peculiar screw-
like nematic phase that, so far, has only been found for hard helices. The
complementary limit of low temperature and low density (or pressure) will be
discussed in a companion paper Wood  _et al._ (2023) that will focus on Janus
rods, thus removing the additional complexity stemming from the chirality of
the helices. The shape dependence is also known to play an important role in
the case of hard helices Kolli  _et al._ (2014a, b); Dussi  _et al._ (2015);
Cinacchi  _et al._ (2017). Hence, the present study will be confined to the
case of slender helices that can be more easily contrasted with results from
the Janus rods.

The outline of the paper is as follows. Section II presents the model (Section
II.1) and the required order parameters and correlation functions (Section
II.2). In order to identify the temperature regime far from a possible gas-
liquid transition and clearly study competition between self-assembly and
liquid crystal formation, a preliminary calculation of the second virial and
the corresponding Boyle temperature (the temperature of vanishing second
virial coefficient) is carried out in Section III.1 as a function of the
coverage. Most of the successive analyses presented in Section III will then
refer to temperatures well above the Boyle temperatures, where entropic
effects due to excluded volume dominate, with a special focus on the location
of the liquid crystal phases and the effect of the temperature and increasing
attraction on the equation of state. Finally, in Section IV we summarise the
key findings from this work and discuss directions for future work.

##  II Theory and Methods

###  II.1 Model

Figure 1: From hard helices to fully attractive square-well helices. Red beads
have HS potential, green beads SW potential. The helices shown have $R=0.2D$
and $p=8D$ with a train of $n_{s}=15$ spherical beads arranged in a helix with
contour length $L=10D$, where $D$ is the diameter of each bead, which is the
case considered in the present study. Intermediate cases correspond to 1/15
($\approx 6.7\%$) and 8/15 ($\approx 50\%$) attractive sites. The $50\%$ case
corresponds to the Janus limit.

Hard helices have been implemented as a line of 15 fused hard spherical beads
of diameter $D$ (the unit of length) rigidly arranged into a helicoidal shape,
with a contour length fixed to $L$. On changing the radius $R$ and pitch $p$
at fixed total contour length $L$, the shape of the helical particle can be
tuned from a straight rod to very wound coils.

We define the coverage $0\leq\chi\leq 1$ as the fraction of attractive sites,
with $\chi=0$ ($0\%$ coverage) representing the HS case, $\chi=0.5$ ($50\%$
coverage) representing the Janus limit, and $\chi=1.0$ ($100\%$ coverage) the
SW fully attractive case. Figure 1 shows the progressive increase of the
coverage in the specific case of a slender helix with $R=0.2$ $p=8D$.

Following past studies Kolli  _et al._ (2014a, b, 2016), we can organize the
shapes into three different classes - small, medium, and large curliness -
identified by the radius $R$ and pitch $p$ of the helical shape, that are
related at a fixed total contour length Frezza  _et al._ (2013). In the
present study, we focus on the case of small curliness corresponding to
$R=0.2D$ and $p=8D$, as it can be easily related to its Janus rod counterpart.
The contour length has been fixed to $L=10D$, which is sufficiently large to
probe all of the liquid crystal phases for both hard spherocylinders Bolhuis
and Frenkel (1997) and hard cylinders Lopes  _et al._ (2021). As in our
previous studies Kolli  _et al._ (2014a, b, 2016), we have assumed $n_{s}=15$
spherical beads forming the helices, some of which will be made attractive as
detailed below.

Two complementary methods have been used to study the phase behaviour of this
system. The first is isobaric-isothermal (NPT) Monte Carlo (MC) simulations
using a shape-adapting rectangular box with periodic boundary conditions,
which is an extension of the approach used in previous investigations of hard
helices Frezza  _et al._ (2013); Kolli  _et al._ (2014a, b, 2016). Here, the
excluded volume between different beads belonging to different helices
separated by a distance $r$ is modelled as a pure hard-sphere (HS) potential

$\phi_{\text{HS}}\left(r\right)=\begin{cases}+\infty\,,\quad\,r<D&\\\ 0,\qquad
r\geq D&\end{cases}$ (1)

and this was the only interaction appearing in past work. When attraction is
added, we assume that two attractive beads belonging to different helices will
interact via a square well (SW) interaction

$\phi_{\text{SW}}\left(r\right)=\begin{cases}+\infty\,,\quad\,r<D&\\\
-\epsilon/n_{s}^{2},\quad\,D\leq r<R_{c}&\\\ 0,\qquad r\geq R_{c}&\end{cases}$
(2)

where an attraction having strength $-\epsilon/n_{s}^{2}$ is added in the
presence of a favourable bond and no corresponding overlap. The range of
attraction $R_{c}$ clearly plays a fundamental role, especially for non-convex
objects such as helices: when $R_{c}$ is large, all attractive beads on
different helices are able to interact, whereas only single pairs of beads
interact in the presence of very short-range attraction. As in the case of
Janus spheres Sciortino, Giacometti, and Pastore (2009, 2010), we expect an
intermediate regime given by $R_{c}=1.5D$ to be the most representative, and
this value will be used in all NPT-MC simulations of the present study.

It is worth stressing that each interaction is normalized by the maximum
possible number of pair-wise interacting spheres $n_{s}^{2}$ so that for a
$100\%$ (SW) case, $\epsilon$ is the energy scale of the problem. This choice
allows to more easily compare helices with different number of sites.

In NPT-MC simulations, at constant temperature $T$ and constant pressure $P$,
each MC move consists of $N$ attempts to move a randomly-selected particle and
a single volume change attempt. A particle move attempt consists of
simultaneous random translation of the center of mass and a random rotation
about it. We have used two sets of simulations with different numbers of
helices to test for finite size effects. In the "small-size" simulations, we
used $N=972$ helices with $3\times 10^{6}$ steps for equilibration and an
additional $10^{6}$ steps as a production run, in line with previous
simulations of hard helices Frezza  _et al._ (2013); Kolli  _et al._ (2014a,
2016). In the "large-size" simulations, we used $N=2400$ helices with an
equilibration time of $6\times 10^{6}$ MC steps and an additional $10^{6}$
steps to collect statistics. The initial condition was a low-density array of
parallel helices, with all the attractive parts (if any) initially pointing
along the $+z$ "up" directions. We also performed Molecular Dynamics (MD)
simulations at constant volume and temperature (NVT) for $N=4068$ helices
using the software package LAMMPS Plimpton (1995) employing the Nosé-Hoover
thermostat Hoover (1985) to keep the temperature fixed. For MD simulations, we
used a combination of a Weeks-Chandler-Anderson (WCA) potential Weeks,
Chandler, and Andersen (1971) to mimic hard-core repulsion and a pseudo-
square-well (PSW) potential Zerón, Vega, and Benavides (2018) to mimic
attraction:

$\phi(r)=\begin{cases}\phi_{\text{PSW}}(r)\quad\text{between attractive
sites}\\\ \phi_{\text{WCA}}(r)\quad\text{otherwise}\end{cases}$ (3)

where

$\displaystyle\phi_{WCA}\left(r\right)$ $\displaystyle=$ $\displaystyle
4\epsilon\left[\left(\frac{D}{r}\right)^{12}-\left(\frac{D}{r}\right)^{6}+\frac{1}{4}\right]$
(4)

which vanishes for $r\geq\sqrt[6]{2}D$ and acts on the repulsive-repulsive and
repulsive-attractive bead pairs, and

$\displaystyle\phi_{\text{PSW}}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\left[\left(\frac{D}{r}\right)^{n}+\frac{1-e^{-\lambda\left(r/D-1\right)\left((r-R_{c})/D\right)}}{1+e^{-\lambda\left(r/D-1\right)\left((r-R_{c})/D\right)}}-1\right]$
(5)

which act only among the attractive beads, where $n$ and $\lambda$ are
parameters that tune the shape of the well. Here we have used $\lambda=50$ and
$n=200$ so that $\phi_{PSW}(r)$ approximates a square-well tail while allowing
for efficient equilibration. This is the same potential used in the companion
paper Wood  _et al._ (2023), where it is shown to reproduce the phase behavior
of the SW-line model Gámez and Lago (2017) to within a small ($<5\%$) shift of
the phase boundaries to higher volume fraction. Slightly different potentials
have been used to model similar systems De Braaf  _et al._ (2017); Liu and
Widmer-Cooper (2019); Liu  _et al._ (2022), but we expect the results to
depend only weakly on the details of the model potential. Natural reduced
units are $T^{*}=k_{B}T/\epsilon$ for temperature and for pressures
$PD^{3}/k_{B}T$ in the case of HS interactions and $PD^{3}/\epsilon$ in the
case with attractions. We will also use the fraction of the total volume
occupied by the helices, $\eta=V_{occ}/V$, as a measure of their
concentration. Here $V_{occ}=v_{0}N$, where $v_{0}$ is the volume of a single
helix and $N$ is the number of helices Frezza  _et al._ (2013). Our time unit
in the NVT-MD analysis is $(m/k_{B}T)^{1/2}D$, where $m$ is the mass of each
bead that is taken so that the mass of the helix has a unit value. As in the
MC case, the initial configuration was a crystal phase with all helices
perfectly aligned in an AAA stack with all of the attractive patches directed
along the positive $z$ direction. To test for possible dependence on the
initial conditions, we also considered a situation where the attractive beads
were aligned up in one layer and down in the immediately successive layer, so
that the attractive beads on the helices belonging to the two layers were
initially in contact.

Snapshots are color-coded according to the helix orientation in the case of
hard (HS) helices (as in previous work) Kolli  _et al._ (2014a, b, 2016);
Cinacchi  _et al._ (2017), as well as for fully attractive (SW) helices, and
according to the coverage (green attractive, red repulsive sites) in the case
of partially attractive helices, in analogy with the spherical counterpart
Sciortino, Giacometti, and Pastore (2009, 2010).

###  II.2 Order parameters and screw-like liquid crystal phases

(a)

(b)

(c)

(d)

Figure 2: (a) The local frame for the helical model used in this work Frezza
_et al._ (2013). Here $\widehat{\mathbf{u}}$ and $\widehat{\mathbf{w}}$
identify the main and secondary helical axes; (b) The screw-like phase where
the secondary director $\widehat{\mathbf{C}}$ rotates in a helical fashion
about the main director $\widehat{\mathbf{N}}$; (c) The cholesteric phase
where the main director $\widehat{\mathbf{N}}$ rotates in a helical fashion
about an axis $\widehat{\mathbf{H}}$ that is perpendicular to the original
plane; (d) The coupling between translation along the main director
$\widehat{\mathbf{N}}$ and rotation of the secondary director
$\widehat{\mathbf{C}}$ characteristic of the screw-like phase. The three
helices have a relative vertical displacement of $z=0$, $z=p/2$, and $z=3p/2$
respectively, where $p$ is the pitch of the helix and hence their
$\widehat{\mathbf{w}}$ directors rotate by $\pi/4$ about the main director
$\widehat{\mathbf{N}}$ every $p/2$. Note that in this case their main axes
$\widehat{\mathbf{u}}$ are all parallel to the phase main director
$\widehat{\mathbf{N}}$. All cases shown here refer to the most slender helix
considered with $R=0.2D$ and $p=8D$.

The presence of different phases with specific features requires the
definition of order parameters able to distinguish between them. To this aim,
we have used a combination of global order parameters and specific correlation
functions that can act as local order parameters Lopes  _et al._ (2021) that
have been introduced recently in the literature. In the following, we will be
using capital letters ($\widehat{\mathbf{N}}$ and $\widehat{\mathbf{C}}$), to
identify principal directions of the liquid crystal phases, and lower case
letters for local axes identifying the orientation of the helices. The
position and orientation of each helix in space is identified by its center of
mass $\mathbf{r}$ and by three unit vectors $\widehat{\mathbf{u}}$,
$\widehat{\mathbf{v}}$, and $\widehat{\mathbf{w}}$. Here,
$\widehat{\mathbf{u}}$ is identifying the helix main axis, whereas
$\widehat{\mathbf{w}}$ and $\widehat{\mathbf{v}}$ are two unit vectors,
perpendicular to each other and to $\widehat{\mathbf{u}}$, that identify the
azimuthal orientation of the helix in this perpendicular plane. As only one of
the two is required for this aim, we will focus on $\widehat{\mathbf{w}}$ in
the following (see Figure 2(a)). The onset of a nematic phase is best
identified by evaluating the Veilliard-Baron tensor Vieillard-Baron (1974)

$\displaystyle\mathbf{Q}_{\widehat{\mathbf{u}}}$ $\displaystyle=$
$\displaystyle\left\langle\frac{1}{N}\sum_{j=1}^{N}\left[\frac{3}{2}\widehat{\mathbf{u}}_{j}\widehat{\mathbf{u}}_{j}-\frac{1}{2}\mathbf{I}\right]\right\rangle$
(6)

where $\widehat{\mathbf{u}}_{i}$ is the unit vector identifying the
orientation of the $i-$th helix in space. The maximum eigenvalue
$\Lambda_{u_{1}}$ gives the nematic order parameter $\langle P_{2}\rangle$,
$\approx 1$ for a nematic phase and $\approx 0$ for an isotropic phase, and
the corresponding eigenvector gives the nematic director
$\widehat{\mathbf{N}}$ (see Figure 2(b)). Because of the uniaxial symmetry of
the helix and the traceless character of tensor
$\mathbf{Q}_{\widehat{\mathbf{u}}}$, the other two eigenvalues
$\Lambda_{u_{2}}=\Lambda_{u_{3}}$ will be identical and negative. For hard
helices, an additional chiral nematic phase was recently unveiled using both
numerical simulations Kolli  _et al._ (2014a, b, 2016) and density functional
theory Dussi  _et al._ (2015); Frezza  _et al._ (2014); Cinacchi  _et al._
(2017). In this phase, henceforth denoted as screw-like Kolli  _et al._
(2014a, b, 2016), the secondary director $\widehat{\mathbf{C}}$ rotates
perpendicularly to the main director $\widehat{\mathbf{N}}$ (defined as the
eigenvector of $\Lambda_{u_{1}}$) in a helical fashion as a screw in a cork
(see Figure 2(b)). It is important to stress that this chiral nematic phase is
specific for helical particles and is different from the cholesteric phase,
common to general chiral particles, where it is the main director
$\widehat{\mathbf{N}}$ that revolves with its tip forming a helix around an
axis $\widehat{\mathbf{H}}$ that is perpendicular to the plane of the original
direction of $\widehat{\mathbf{N}}$ (see Figure 2(c)). One of the interesting
outcomes of the analysis of the screw-like nematic phase for hard helices
stems from the coupling between translation of the helix along the main
director $\widehat{\mathbf{N}}$ and a rotation of the secondary director
$\widehat{\mathbf{C}}$ as depicted in Figure 2(d). This coupling has been
observed in both simulations Kolli  _et al._ (2014a, b); Frezza  _et al._
(2014); Kolli  _et al._ (2016); Cinacchi  _et al._ (2017) and in experimental
studies of helical flagella Barry  _et al._ (2006); Yardimci  _et al._ (2023).

In order to identify this particular screw-like phase, it is convenient to
introduce a new Veilliard-Baron tensor

$\displaystyle\mathbf{Q}_{\widehat{\mathbf{w}}}$ $\displaystyle=$
$\displaystyle\left\langle\frac{1}{N}\sum_{j=1}^{N}\left[\frac{3}{2}\widehat{\mathbf{w}}_{j}\widehat{\mathbf{w}}_{j}-\frac{1}{2}\mathbf{I}\right]\right\rangle$
(7)

which is the counterpart of $\mathbf{Q}_{\widehat{\mathbf{u}}}$ given in
Eq.(6). Again for the two-fold degeneracy of the uniaxial symmetry of the
nematic phase, there will be two positive identical eigenvalues
$\Lambda_{w_{1}}=\Lambda_{w_{2}}$ and a third negative eigenvalue
$\Lambda_{w3}$ due to the traceless nature of the tensor. Hence, the screw-
like phase is characterized by a breaking of the uniaxial symmetry and the
onset of a characteristic secondary director $\widehat{\mathbf{C}}$ along
which all secondary axes $\widehat{\mathbf{w}}$ of the helices preferentially
align. This, in turn, is signalled by a removal of the degeneracy
($\Lambda_{w_{1}}\neq\Lambda_{w_{2}}$) which can be used as a order parameter
for the onset of the screw-like phase.

An alternative, already used in past work Kolli  _et al._ (2014a, b), is given
by the screw-order parameter $\left\langle P_{1c}\right\rangle$ (see
Supplementary Section SLABEL:supp-sec:additional for this and additional order
parameters). Here, the basic idea is that if all the
$\widehat{\mathbf{w}}_{i}$ align along the secondary director
$\widehat{\mathbf{C}}$ then $\langle P_{1c}\rangle\approx 1$, whereas $\langle
P_{1c}\rangle\approx 0$ for randomly oriented $\widehat{\mathbf{w}}_{i}$. In
the present formulation, we have implemented an additional layering procedure
that avoids the artificial unscrew procedure used in past work Kolli  _et al._
(2014a, b, 2016), but it has its own shortcomings. The main one is that it is
very sensitive to the choice of the width of the layering.

A global parameter $\left\langle\tau_{1}\right\rangle$ can also be implemented
to identify the onset of the smectic phase SmA as defined in Ref. 42 (see also
Supplementary Section SLABEL:supp-sec:additional). Again
$\left\langle\tau_{1}\right\rangle\approx 1$ in the smectic phase and
$\left\langle\tau_{1}\right\rangle\approx 0$ otherwise. While very useful, it
requires preliminary identification of an optimal width, which makes its
implementation very time consuming when analysing many different cases.

Finally, the SmA-SmB transition can be identified by the use of the hexatic
order parameter $\left\langle\psi_{6}\right\rangle$ Lopes  _et al._ (2021)
(see also Supplementary Section SLABEL:supp-sec:additional). With this
definition, $\langle\psi_{6}\rangle\approx 1$ for hexagonal in-plane ordering
and $\langle\psi_{6}\rangle\approx 0$ otherwise. A bead is defined to be a
nearest-neighbor of a bead belonging to another helix if the distance between
the two beads is within the first minimum of the perpendicular correlation
function (defined further below). As this value clearly depends (albeit not
very strongly) on the specific considered state point (i.e. temperature and
pressure or volume fraction), even this calculation is not free from
ambiguities.

In addition to global order parameters, local order parameters in the form of
suitably defined correlation functions can also be exploited to pin down the
exact location of the transition lines. We briefly recall them here and refer
to recent past literature for further details (see e.g. Ref. Lopes  _et al._ ,
2021 and references therein).

A general tendency to form local order can be inferred from the radial
distribution function

$\displaystyle g\left(r\right)$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r^{2}N}\left\langle\frac{1}{\rho}\sum_{i\neq
j}\delta\left(r-r_{ij}\right)\right\rangle$ (8)

where $r_{ij}=|\mathbf{r}_{ij}|=|\mathbf{r}_{j}-\mathbf{r}_{i}|$. In the case
of an anisotropic object such as the helices presented in this study, two
additional correlation functions prove particularly useful to identify liquid
crystal phases. The first is the perpendicular correlation function

$\displaystyle g_{\perp}\left(r_{\perp}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi Lr_{\perp}N}\left\langle\frac{1}{\rho}\sum_{i\neq
j}\delta\left(r_{\perp}-r_{ij}^{\perp}\right)\right\rangle$ (9)

where $\mathbf{r}_{\perp}$ is the projection of $\mathbf{r}$ in the plane
perpendicular to the nematic director $\widehat{\mathbf{N}}$, and $r_{\perp}$
is its magnitude. An oscillatory behavior of $g_{\perp}\left(r_{\perp}\right)$
with specifically located peak points indicates hexagonal orientational order.

The second useful function is the parallel correlation function

$\displaystyle g_{\parallel}\left(r_{\parallel}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2L^{2}N}\left\langle\frac{1}{\rho}\sum_{i\neq
j}\delta\left(r_{\parallel}-r_{ij}^{\parallel}\right)\right\rangle$ (10)

where
$\mathbf{r}_{\parallel}=(\mathbf{r}\cdot\widehat{\mathbf{N}})\widehat{\mathbf{N}}$
and $r_{\parallel}$ is its magnitude. An oscillatory behavior of the
$g_{\parallel}\left(r_{\parallel}\right)$ is indicative of the presence of SmA
that becomes SmB in the presence of a simultaneous oscillatory behavior of
$g_{\perp}\left(r_{\perp}\right)$. The peak distance in
$g_{\parallel}\left(r_{\parallel}\right)$ is a measure of the distance between
layers. Note that while $g(r)$ is normalized in such a way that it tends to
$1$ as $r/D\to\infty$, $g_{\perp}\left(r_{\perp}\right)$ and
$g_{\parallel}\left(r_{\parallel}\right)$ are normalized such that they tend
to $0$ as $r_{\perp}/D\to\infty$ and $r_{\parallel}/D\to\infty$.

The screw-correlation function has also proven useful in this framework. It is
defined as

$\displaystyle g_{1w}(r_{\|})$ $\displaystyle=$
$\displaystyle\left\langle\frac{\sum_{i\neq
j}\delta(r_{\|}-\mathbf{r}_{ij}\cdot\hat{\mathbf{N}})(\widehat{\mathbf{w}}_{i}\cdot\widehat{\mathbf{w}}_{j})}{\sum_{i\neq
j}\delta(r_{\|}-\mathbf{r}_{ij}\cdot\hat{\mathbf{N}})}\right\rangle$

and it highlights the tendency of the helices to have their secondary axes
$\widehat{\mathbf{w}}$ aligned.

For hard helices it has been argued, based on numerical evidence, that the
nematic phase can be either screw-nematic NS or chiral nematic (cholesteric)
N, and not both Cinacchi  _et al._ (2017). Whether this will also be true for
the Janus helices remains to be seen, and for this reason we shall use the
general terminology of nematic N for indicating a phase that is not screw-
nematic, and which is associated with structureless behavior of
$g_{1,\|}^{\widehat{\mathbf{w}}}(r_{\|})$. Note that here and above the short-
hand notation $\sum_{i\neq j}\equiv\sum_{i=1}^{N}\sum_{\overset{j=1}{j\neq
i}}^{N}$ has been used.

In general, we found the use of the global screw-nematic order parameter
$\langle P_{1c}\rangle$, the smectic order parameter $\langle\tau_{1}\rangle$,
and the hexatic order parameter $\langle\psi_{6}\rangle$ (see Supplementary
Section SLABEL:supp-sec:additional) to be less informative than the local
correlation functions, so they were all monitored but not reported here.

###  II.3 Diffusion coefficients

Diffusion was also monitored via simulations. The common definition of the
diffusion coefficient $D$ is:

$\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{1}{6t}\left\langle
R^{2}\left(t\right)\right\rangle$ (12)

where the factor $6$ stems from having $2$ possible directions in each of $3$
dimensions. In systems where there is a preferred direction, it is useful to
monitor the parallel and perpendicular diffusion coefficients

$\displaystyle{\cal D}_{\parallel}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{1}{2t}\left\langle
R_{\parallel}^{2}\left(t\right)\right\rangle$ (13) $\displaystyle{\cal
D}_{\perp}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{1}{4t}\left\langle
R_{\perp}^{2}\left(t\right)\right\rangle$ (14)

Here, $\langle R^{2}(t)\rangle$ is defined as the mean-square displacement of
the center of mass of the helix at time $t$ with respect to the initial time
$t=0$, and $\langle R_{\parallel}^{2}(t)\rangle$ and $\langle
R_{\perp}^{2}(t)\rangle$ are their counterpart projections along the parallel
(to $\widehat{\mathbf{N}}$) and perpendicular directions, respectively. Notice
that this initial time is the initial monitoring time after equilibration,
unless stated otherwise.

###  II.4 Virial expansion and Boyle temperature

The virial expansion for pressure $P$ is given by

$\displaystyle\frac{\beta P}{\rho}$ $\displaystyle=$ $\displaystyle 1+\rho
B_{2}\left(T\right)+\rho^{2}B_{3}\left(T\right)+\ldots$ (15)

The Boyle temperature $T_{B}$ is defined by the condition $B_{2}(T_{B})=0$
Hansen and McDonald (2006). For $T>T_{B}$, $B_{2}(T)>0$ and repulsions
dominate; For $T<T_{B}$, $B_{2}(T)<0$ and attractions dominate. Hence $T_{B}$
represents the transition temperature between entropically and energetically
dominated regimes.

##  III From hard to fully attractive helices

We have carried out ramps in both pressure (for NPT-MC simulations) and volume
fraction (for NVT-MD simulations). In the first case, we considered 10
different reduced pressures ranging from $0.2$ to $2.0$ for each of the
$N=2400$ NPT-MC simulations and ranging from $0.1$ to $1.0$ for the $N=972$
NPT-MC simulations. In the second case, we considered 12 different volume
fractions from $\eta=0.1$ to $\eta=0.65$. In both cases, the simulations were
repeated for three different (reduced) temperatures $T^{*}=1.0,0.5,0.1$, and
four different coverages, from the case of hard helices to the case of fully
attractive helices, passing through the single attractive bead case (out of
15) and the Janus limit of approximately half attractive beads. The three
temperatures were selected on the basis of the analysis of the second virial
coefficients reported below.

###  III.1 Second virial coefficient $B_{2}$ and Boyle temperature $T_{B}^{*}$
at different coverages.

It proves convenient to compute the reduced second virial coefficient black
$B_{2}^{*}=B_{2}/D^{3}$ as a function of the reduced temperature $T^{*}$, as
well as the reduced Boyle temperature $T_{B}^{*}$, where
$B_{2}^{*}(T_{B}^{*})=0$, as a function of the coverage $\chi$. To this aim,
we have followed past suggestions Yethiraj and Hall (1991); Heyes  _et al._
(2015) and recall the definition of the second virial coefficient

$\displaystyle B_{2}\left(T\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int d\mathbf{r}\left\langle
f_{12}\left(\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}\right)\right\rangle_{\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}}$
(16)

where we have introduced the Mayer function Hansen and McDonald (2006)

$\displaystyle
f_{12}\left(\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}\right)$
$\displaystyle=$ $\displaystyle
e^{-\beta\phi\left(\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}\right)}-1$
(17)

as well as the average over the angular orientations
$\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}$ of the two helices

$\displaystyle\left\langle\ldots\right\rangle_{\widehat{\mathbf{u}}}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int
d\widehat{\mathbf{u}}~{}\ldots$ (18)

Here $\phi(\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2})$ is
the sum of all potentials between two sites on different helices, $\mathbf{r}$
being the displacement between the center of mass of the two helices and
$\widehat{\mathbf{u}}_{1}$ and $\widehat{\mathbf{u}}_{2}$ defining their
orientations in space. This integration was performed numerically by
generating a large number $N_{c}$ of independent configurations of two
identical helices in a cubic box of volume $V$.  $B_{2}$ can then be computed
as Yethiraj and Hall (1991); Munaò  _et al._ (2015)

$\displaystyle B_{2}\left(T\right)$ $\displaystyle=$
$\displaystyle-\frac{V}{2N_{c}}\left\langle
f_{12}\right\rangle_{\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}}$
(19)

Note that the average over all possible $\mathbf{r}$ has been included in the
last average appearing in Eq.(19).

For calculation speed, note that the Mayer function can assume three different
types of values depending on the relative position
$\mathbf{r}_{12}\equiv\mathbf{r}$ of the two closest beads between the two
helices:

$f_{12}\left(\mathbf{r},\widehat{\mathbf{u}}_{1},\widehat{\mathbf{u}}_{2}\right)=\begin{cases}-1&\mbox{if
helices overlap i.e.}~{}r<D\\\ e^{n_{c}/n_{s}^{2}T^{*}}-1&\mbox{if no overlap
and }D<r<R_{c}\\\ 0&r>R_{c}\end{cases}$ (20)

where $r$ is the smallest distance between two spheres on different helices,
and $n_{c}$ is the number of square-well contacts ($D<r<R_{c}$ between
attractive spheres on two different helices).

In the case of a hard overlap, the Mayer function can be immediately
evaluated, with $\phi\to\infty$ forcing the value of the corresponding Mayer
function to $-1$, which we then add to the accumulated average. In the case of
no overlaps, but at least one interaction, care must however be exercised in
doing this calculation in the presence of multiple sources of attraction. As
$-\epsilon/n_{s}^{2}$ is the energy contribution from each favourable
interaction (see Eq.(2)), the energy must first be summed with a double loop
over the beads belonging to the two helices, adding the $-\epsilon/n_{s}^{2}$
energy contribution from each favourable interaction, before evaluating the
Mayer function.

By gradually varying the temperature $T$ until $B_{2}$ vanishes, the Boyle
temperature $T_{B}$ was obtained.

We explicitly checked that the limiting behaviour at high temperatures is
consistent with Onsager’s celebrated limit for hard spherocylinders Onsager
(1949); Heyes  _et al._ (2015)

$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\pi\left(\frac{2D^{3}}{3}+LD^{2}+\frac{L^{2}D}{4}\right)$ (21)

(a)

(b)

Figure 3: (a) The second virial coefficient $B_{2}$ as a function of the
reduced temperatures $T^{*}$ at different coverages $\chi$ for $R=0.2D$ and
$p=8D$; (b) The reduced Boyle temperature $T_{B}^{*}$ as a function of the
coverage $\chi$ for the different shapes of the helix. The curve labelled as
"RODS" depicts the Boyle temperature derived by the vanishing of $B_{2}$ for
rods with length $L/D=10$.

As a representative calculation, Figure 3(a) shows the results of the reduced
second virial coefficient $B_{2}^{*}$ as a function of the reduced temperature
$T^{*}$ for helices with $R=0.2D$ $p=8D$. Figure 3(b) reports the
corresponding Boyle temperature as a function of the coverage $\chi$ for all
of the helical shapes considered in this study. A comparison with the straight
rod limit $p\to\infty$ is also displayed. Supplementary Table LABEL:supp-
tab:boyle reports the numerical values of the Boyle temperatures in the case
of helices with $R=0.2D$ and $p=8D$.

Two main insights can be extracted from this analysis. First, there appear to
be no major differences between the three shapes and the straight rod
counterpart at this first approximate level, thus justifying our choice of
focusing on one specific shape. Second, the Boyle temperature varies from
$T_{B}^{*}\approx 0.04$ for the fully attractive SW case to $T_{B}^{*}\approx
0.0015$ in the $6.7\%$ limit where only one bead is attractive. As the Boyle
temperature signals the transition from an entropy-dominated regime ($T^{*}\gg
T_{B}^{*}$) to an energy-dominated one ($T^{*}\ll T_{B}^{*}$), this means that
significantly low temperatures are necessary to observe competition between
the liquid crystal phases observed in the purely repulsive hard helices case
Frezza  _et al._ (2013); Kolli  _et al._ (2014a, b); Frezza  _et al._ (2014);
Cinacchi  _et al._ (2017), and a possible gas-liquid transition akin to that
observed in the spherical Janus counterpart Sciortino, Giacometti, and Pastore
(2009, 2010). In the present study, this low $T^{*}$ regime will not be
addressed as its analysis will be the subject of a companion paper on Janus
rods Wood  _et al._ (2023). Rather, we will focus our attention on the effect
of a relatively weak attraction to the liquid-crystal phase behaviour by
probing temperatures in the range $0.1\leq T^{*}\leq 1.0$. As $T^{*}=0.1$ is
larger then $T_{B}$ by a factor $\approx 2$ for fully attractive helices but
by a factor $\approx 100$ for helices with a single attractive site ($\approx
6.7\%$ case), a correct interpretation of the results must be drawn
accordingly.

###  III.2 The hard helices case

Before discussing the case of attractive helices, we now briefly recall the
hard helices case discussed in previous work Kolli  _et al._ (2014a, a, 2016)
with a two fold aim. First, it will allow a quantitative comparison in terms
of finite size within the same isobaric-isothermal NPT-MC simulations, from
$N=972$ of past work to $N=2400$ of the present study. Second, it will allow
to test for consistency by a comparison with isochoric-isothermal NVT-MD
calculations using $N=4068$ helices. In all cases, the standard initial
condition was taken as a system of initially parallel helices assembled at
different densities. Supplementary Figure SI shows the approach to equilibrium
for both NPT-MC and NVT-MD simulations. This shows that the NPT-MC simulations
have difficulty reaching complete equilibration at the highest considered
pressures despite the relatively long equilibration times ($5\times 10^{6}$ MC
steps). In contrast, the NVT-MD simulations appear unaffected by this
shortcoming, and so results on partially attractive helices mainly rely on
them. It is reassuring, however, that the two sets of simulations provide
compatible results when both are equilibrated.

(a)

(b)

Figure 4: (a) Equation of state $PD^{3}/k_{B}T$ as a function of the packing
fraction $\eta$ for the case of hard helices. The three sets of points refer
to Monte Carlo simulations with $N=972$ helices (black circle) and $N=2400$
(red squares), and to MD simulations with $N=4068$ (green triangles). Parallel
dotted lines indicate the transitions identified in past work ( see e.g.
Cinacchi  _et al._ (2017)) and refer to the I-N isotropic-nematic transition
(blue line), N-Ns nematic-screw-nematic transition (green line), Ns-
SmA${}_{\text{s}}$ screw-nematic-screw-smectic A transition (magenta line),
SmA${}_{\text{s}}$-SmB screw-smectic A-polar-smectic B transition (orange
line); (b) The $\langle P_{2}\rangle$ order parameter as a function of the
packing fraction $\eta$ for the same three sets of simulations as in part (a).

Figure 4(a) reports the equation of state (reduced pressure $PD^{3}/k_{B}T$
versus packing fraction $\eta$) from both NVT-MD with $N=4068$ helices (green
triangle) and NPT-MC calculations with $N=2400$ (red squares), and contrasted
with the results of the smaller size NPT-MC calculations with $N=972$ (black
spheres) used in previous studies Kolli  _et al._ (2014a, b, 2016)). Color
coded vertical lines indicate the putative transitions from isotropic I to
nematic N (blue line), nematic N to screw-nematic Ns (green line), screw-
nematic Ns to screw-smectic A SmAs (magenta line), screw-smectic A SmAs to
polar-smectic B SmB (orange line) as identified in previous work Kolli  _et
al._ (2014a, b, 2016); Cinacchi  _et al._ (2017). Our present estimates
confirm these findings. As anticipated, the isotropic I to nematic N (blue
line) can be located by considering the nematic order parameter $\langle
P_{2}\rangle$ that also coincides with the maximum eigenvalue
$\Lambda_{u_{1}}$ of $\mathbf{Q}_{\widehat{\mathbf{u}}}$ (Eq.6). This is shown
in Fig. 4(b) for both NVT-MD and NPT-MC calculations, updated and original, as
in Figure 4(a). The upswing of the curve at $\eta\approx 0.25$ marks the end
of the isotropic phase and the onset of the nematic one.

In Supplementary Figure SV we show results for the equation of state from both
NPT-MC and NVT-MD simulations at different temperatures. The closest match
between the simulated state points appears to be when $T^{*}=1.0$ as one could
have expected from the outset. We will then use this temperature value when
contrasting the two different techniques hereafter.

Supplementary Figure SVII and SVIII report the equilibrium behaviour of the
three eigenvalues $\Lambda_{u_{i}}$ ($i=1,2,3$) of
$\mathbf{Q}_{\widehat{\mathbf{u}}}$ (Eq.6) as a function of $\eta$, as
obtained from the NPT-MC and NVT-MD calculations. This shows that the maximum
eigenvalue $\Lambda_{u_{1}}$ progressively increases from $\approx 0$ for
$\eta$ small to $\approx 1$ for large $\eta$. Accordingly, the other two
eigenvalues $\Lambda_{u_{2}}$ and $\Lambda_{u_{3}}$ also move from $\approx 0$
for $\eta$ small to $\approx-0.5$ for large $\eta$. Their equality is an
indication of the uniaxial symmetry of the system at this stage.

Figure 5: Representative snapshots (top and side views) of state points
appearing in Figure 4 as obtained by MD $N=4068$. Different helices are color-
coded according to the orientation $\widehat{\mathbf{u}}$ of the helix main
axis, and both top and side views are depicted in all cases, with tick red
dotted lines separating different phases. (a) I state point $\eta=0.15$; (b)
Ns state point $\eta=0.40$; (c) SmAs state point $\eta=0.45$; (d) SmB state
point $\eta=0.65$. Screw phases are not easily identifiable here as they are
best illustrated in Figure 6. The light colored backgrounds that appear behind
the phase identification letters are consistent with the phases in Figure 4(a)
and thick red dashed lines separate different phases. Visualizations here and
below were done using the Ovito Package Stukowski (2010). Figure 6: Same as in
Figure 5 but now with the beads colored according to the local tangent as in
Ref. Kolli  _et al._ (2014a, b). The color changes as the tangent moves along
the helix and thus the periodicity of the color pattern is equal to the pitch
of the helix. Supplementary Figure SII shows how the tip of the local tangent
performs a conical path along the helix. Note that this modality mirrors the
experimental one used in Ref. Barry  _et al._ (2006). Here only side views are
presented and the light colored background is color-coded as in the phase
diagram of Figure 4(a). The periodic stripes appearing here in the Ns and SmAs
phases are indicative of screw-like phases. The stripe periodicity of the
screw-nematic Ns phase is equal to the pitch $p$ of the helix. Figure 7: A
visual representation of the screw-like mechanism within a screw-nematic phase
Ns at $\eta=0.40$. Helices colored in dark magenta in a) are originally
aligned in a layer and then eventually dispersed within the computational box
by the dynamics. Within this framework, the white helix is shown to diffuse
down from b) to c) to d) via a translation (along the $\widehat{\mathbf{N}}$
axis) coupled with a rotation (about the main axis $\widehat{\mathbf{n}}$ of
each helix) as a screw in a bolt, in close analogy with the experiments in
Refs. Barry  _et al._ (2006); Yardimci  _et al._ (2023).

Close-up snapshots of the representative state points are reported in Figure
5, with different helices color-coded according to their directions. Both top
and side views are depicted in all cases. The four depicted snapshots refer to
states with $\eta=0.15$ (isotropic I phase), $\eta=0.40$ (screw-nematic Ns
phase), $\eta=0.45$ (SmAs phase), and $\eta=0.65$ (smectic B SmB phase). The
presence of the screw-nematic phase is not obvious in Figure 5(b) and can be
more easily seen in Figure 6, where the same snapshots have been color-coded
according to the local tangent rather than the local director
$\widehat{\mathbf{u}}$. Particularly evident are the stripes in the
$\eta=0.40$ (screw-nematic Ns phase), $\eta=0.45$ (SmAs phase), with the
screw-like nature of the SmAs originating from the screw-like nature of the Ns
phase). Also the smectic B SmB phase presents a striped pattern, indicating
that all $\widehat{\mathbf{w}}$ within a layer are in phase. However, in
contrast with its SmAs counterpart, stripes in different layers are
uncorrelated. This was rationalized Kolli  _et al._ (2014a, b) on the basis
that the driving force originating the SmB phase stems mainly from the
requirement of minimizing excluded volume at the expenses of orientational
ordering.

Further insights on the presence of the screw-nematic Ns phase can be seen in
Figure 7 where a subset of helices initially equilibrated at $\eta=0.40$ and
confined within a given layer have been highlighted via a different color
(dark magenta) and they can be seen to visibly diffuse up and down along the
main director $\widehat{\mathbf{N}}$ (as is also visible in Movie1 reported in
Supplementary Information). The three snapshots of Figure 7 show one
particular helix colored in white that is seen to perform a screw-like motion
akin to that found for helical flagellae by Barry  _et al._ Barry  _et al._
(2006) and more recently by Yardimci  _et al._ Yardimci  _et al._ (2023). As
helices tend to align due to an increasing concentration, they loose
rotational entropy. This can be compensated by corkscrewing up and down along
the main director $\widehat{\mathbf{N}}$.

(a)

(b)

(c)

(d)

Figure 8: Fully repulsive helices represented by a WCA model at $T^{*}=1.0$.
(a) Radial correlation functions $g(r)$ as a function of the reduced distance
$r/D$ between the centers of the helices at different state points; (b)
Perpendicular correlation function $g(r_{\text{pe}})$ as a function of the
reduced perpendicular distance $r_{\text{pe}}/D$ where $r_{\text{pe}}\equiv
r_{\perp}$; (c) Parallel correlation function $g(r_{\text{pa}})$ as a function
of the reduced parallel distance $r_{\text{pa}}/D$ where $r_{\text{pa}}\equiv
r_{\parallel}$. Color coding of the different state points here is the same as
in Figure 4; (d) Overall diffusion coefficient $D$ as a function of the
packing fraction $\eta$, contrasted with its perpendicular $D_{pe}\equiv
D_{\perp}$ and parallel $D_{pa}\equiv D_{\parallel}$ components.

The exact location of the phase boundaries can be conveniently obtained by
considering the correlation functions discussed in Section II.2. Figure 8(a)
displays the radial distribution function of the center-of-masses of the
helices, as a function of $r/D$ for the same state points reported in Figure
5. The color coding of the different curves is the same as the phases reported
in Figure 4(a). Starting from $\eta\approx 0.4$, the first and second shell
peaks begin developing before additional crystal-like peaks appear at
$\eta=0.65$. The location of the peaks along the $r$ axis nearly matches those
of the perpendicular correlation function $g_{pe}\equiv g_{\perp}(r_{\perp})$
along the $r_{pe}\equiv r_{\perp}$ axis (Figure 8(b)), thus locating parallel
neighbouring helices with an in-plane hexatic ordering characteristic of the
SmB phase (see below).

The onset of smectic phases is signalled by the parallel correlation function
$g_{pa}\equiv g_{\parallel}(r_{\parallel})$ (see Figure 8(c)) that develops
marked oscillating from $\eta=0.45$ onward, in agreement with Figure 4(a).
Note the significant range difference of the $r_{pa}\equiv r_{\parallel}$ axis
compared to the case of $r$ (Figure 8(a)) and $r_{pe}$ (Figure 8(b)) that is
clearly due to the large aspect ratio of the helices that require an
asymmetric computational box elongated along the $\mathbf{N}$ axis. As the
helices were originally in a screw-like phase, the obtained phase is a SmAs,
as indicated by the magenta vertical line of Figure 4(a). At $\eta\approx 0.5$
the system undergoes a SmAs to SmB transition (see orange line in Fig. 4(a)),
with in-plane hexagonal symmetry combined with alignment of all secondary
vectors $\widehat{\mathbf{w}}$ within the plane. Here, however, entropic gain
in off-setting parallel alignment between consecutive layers disfavors AAA
stacking as well as the screw-like ordering, so that consecutive layers have
secondary directors $\widehat{\mathbf{C}}$ (see Fig. 2(b)) that become
uncorrelated, mirroring the loss of AAA alignment for the positional ordering
Kolli  _et al._ (2014b). The in-plane hexagonal ordering is clearly visible in
the perpendicular correlation function $g_{\perp}(r_{\perp})$ displayed in
Figure 8(b), with the well-developed peaks at $\eta=0.65$ showing a
characteristic $1:\sqrt{2}$ periodicity.

Another interesting point stems from the analysis of the equilibrium values of
the three eigenvalues $\Lambda_{w_{i}}$ ($i=1,2,3$) of
$\mathbf{Q}_{\widehat{\mathbf{w}}}$ (Eq.7) as a function of $\eta$. This
analysis mirrors the same analysis on the eigenvalues of
$\mathbf{Q}_{\widehat{\mathbf{u}}}$ discussed earlier. This is reported in
Supplementary Figure SVII and SVIII for NPT-MC and NVT-MD respectively.
Complementary to that analysis, here the two largest eigenvalues
$\Lambda_{w_{1,2}}$ are positive $\approx 0.25$ and identical (i.e.
degenerate), with the third eigenvalue negative and equal to $\approx-0.5$ for
unscrew phases. Once again, the degeneracy of the first two eigenvalues stems
from the uniaxial symmetry of the nematic phase and breaks down at the onset
of the screw phases, where $\Lambda_{w_{1}}\neq\Lambda_{w_{2}}$ for volume
fractions from $\eta\approx 0.40$ to $\eta\approx 0.55$, in good agreement
with the phase diagram of Figure 4(a) and with evidence from the corresponding
correlation function previously discussed. It is important to stress that in
all cases, the entries of the tensor matrices are averages over different
configurations, and hence fluctuations from one calculation to another are
certainly possible.

The use of MD simulations also makes it possible to calculate the overall
diffusion coefficient $D$ (defined in Eq. 12), the parallel diffusion
coefficient $D_{\parallel}$ (Eq. 13), and the perpendicular diffusion
coefficient $D_{\perp}$ (Eq. 14). These were calculated after equilibration
and are plotted in Figure 8(d) as a function of the packing fraction $\eta$.
As expected on physical grounds, all three diffusion coefficients markedly
decrease as the system transitions from the isotropic phase into the nematic
phase at $\eta\approx 0.25$. The diffusion coefficients are still significant
in the screw-nematic Ns, as expected from Figure 7), eventually becoming
negligibily small as the system enters the smectic phase ($\eta\geq 0.45$).
Interestingly, the lateral diffusion appears to be more pronounced than the
longitudinal diffusion (i.e. along the main director $\widehat{\mathbf{N}}$)
after entering into the nematic phase, whereas it is subdominant just before
the I-N transition. No evidence of either intralayer or interlayer diffusion
is observed in any of the smectic phases Cinacchi and De Gaetani (2009).

Figure 9: (NPT-MC results with $N=2400$ helices for $g_{1w}(r_{pa})$ as a
function of $r_{pa}\equiv r_{\parallel}$ for the same volume fractions as in
Figure 8. Note that we have used the same color coding though the mapping
between reduced pressures and volume fraction is approximate as inferred from
the equation of state (see Figure 4(a)).

The final point concerns a further quantitative evidence of the screw-like
phases. In line with previous observations, this can be also highlighted using
the $g_{1w}(r_{\parallel})$ function defined in Eq.(10). This is reported in
Figure 9 for NPT-MC simulations with $N=2400$ helices at the same state points
as in Figure 8 where the mapping between reduced pressures and volume
fractions is obtained using the equation of state Figure 4(a). Here clear,
regular oscillations are visible starting at a reduced pressure $P^{*}=1.0$,
roughly corresponding to $\eta=0.40$ and hence to the onset of the screw-
nematic phase. The period also coincides with the pitch $p=8D$, in agreement
with past studies Kolli  _et al._ (2014a, b). We also found this calculation
to be less accurate when carried out at constant volume.

###  III.3 Hard helices with a single attractive bead ($\chi=6.7\%$)

(a)

(b)

(c)

(d)

Figure 10:  Reduced pressure $P^{*}=PD^{3}/\epsilon$ as a function of the
packing fraction $\eta$ for the case of a single attractive bead ($\chi=0.67$)
at reduced temperatures $T^{*}=k_{B}T/\epsilon$ corresponding to (a)
$T^{*}=1.0$, (b) $T^{*}=0.5$, and (c) $T^{*}=0.1$. Representative snapshots of
each phase are shown for specific state points. Attractive beads are colored
in green, hard (repulsive) beads in red. Results are shown for both small
($N=972$ black solid circles) and large ($N=2400$ red solid squares) NPT-MC
simulations, as well as for even larger NVT-MD simulations ($N=4068$ green
solid triangles). (d) Comparison of the NVT-MD results at different reduced
temperatures $T^{*}=1.0,0.5,0.1$.

We now consider the case where some of the beads are attractive and apply the
same machinery discussed so far for hard helices. We start by considering in
detail the case of a single attractive bead, corresponding to approximately
$\chi=6.7\%$ coverage (see Figure 1). Figure 10 reports the reduced pressure
$P^{*}=PD^{3}/\epsilon$ as a function of the volume fraction $\eta$ (the
equation of state) for different reduced temperatures $T^{*}=k_{B}T/\epsilon$:
(a) $T^{*}=1.0$, (b) $T^{*}=0.5$, and (c) $T^{*}=0.1$. Colored vertical bars
identify the phase boundaries as was done for the hard helices case (Figure
4(a)), and representative snapshots are included to highlight specific state
points. Note that, at variance with the hard helices case displayed in Figure
5, here different beads are color-coded according to their interactions, with
green for attractive beads and red for purely repulsive ones. In all cases,
the initial conditions have been taken as a set of parallel helices with all
attractive tips aligned "up". A different choice for the initial condition
will be discussed later on in Section III.4. It is worth noticing that the
introduction of a fraction of attractive beads breaks the "up-down" symmetry
of the original hard helices case, and introduces a preferred sense. As in the
hard helices case, we report results from small ($N=972$ black solid circles)
and large ($N=2400$ red solid squares) NPT-MC simulations, along with
extensive NVT-MD ($N=4068$ green solid triangle up). The "small" NPT-MC
simulations are used to contrast with the original simulations with hard
helices. It is important here to stress once more that all these temperatures
are well above the corresponding Boyle temperature $T_{B}(\chi=6.7\%)\approx
0.0022$ (see Section III.1) that is nearly 2 orders of magnitude smaller
compared with the minimum temperature considered here. The goal here is to
study the stability of the various liquid crystal phases – and specifically
the screw phases found in the case of hard helices, against the introduction
of a weak attraction.

In Figures 10(a)-10(c), we observe the same sequence of liquid crystal phases
with nearly the same location of the phase boundaries as in the case of hard
helices (compare with Figure 4(a)). As a general rule, we observe a shift of
the smectic phases to lower $\eta$ upon decreasing the temperature. In
particular the onset of the SmAs phase is shifted from $\approx 0.45$ at
$T^{*}=1.0$ (Figure 10(a)) to $\approx 0.37$ at $T^{*}=0.1$ (Figure 10(c)).
Likewise the SmB phase at $T^{*}=0.1$ starts at $\approx 0.45$ (Figure 10(c))
as opposed to at $\approx 0.50$ at $T^{*}=1.0$ (Figure 10(a)). The actual
effect of the temperature can be inferred from Figure 10(d) that summarizes
the behavior of the reduced pressure $P^{*}=PD^{3}/\epsilon$ as a function of
the volume fraction $\eta$ for all three considered temperatures. Here, we
note a decrease of the pressure at a given packing fraction $\eta$ on lowering
the temperature, a trend that can be ascribed to the increasing contribution
of the attractive interactions. Here the phase boundaries were located using
the same methodology previously presented for fully repulsive hard helices.

As in the fully repulsive hard helices, the SmAs phase is screw, in the sense
that successive layers have in-plane positional disorder of the center-of-mass
of the helices in that plane, but aligned helix secondary axes
$\widehat{\mathbf{w}}$ along the phase secondary axis $\widehat{\mathbf{C}}$
that rotate for successive layers along $\widehat{\mathbf{N}}$ (see Figure
2(b)). Hence, the nematic region is split in two sub-regions, a conventional
nematic N – presumably a cholesteric one when seen at a larger scale, at lower
volume fraction, and a screw-nematic Ns at higher volume fraction whose
boundary is only mildly dependent on the temperature.

Supplementary Movie2 shows evidence of the screw-nematic phase for the single
attractive bead along the lines used in the case of fully repulsive hard
helices (Movie1), and Supplementary Figure SVI provides the corresponding
relevant snapshots, the counterpart of Figure 7.

Also worth noting is the increasing tendency of the NPT-MC simulations to
become kinetically trapped. This is especially visible at $T^{*}=0.1$ (Figure
10(c)), where the pressure in both the $N=972$ and $N=2400$ NPT-MC simulations
deviates substantially from the NVT-MD results at high volume fraction. This
deviation is likely related to the difficulty of the NPT-MC runs in reaching
full equilibration and thermalization (see Supplementary Figure SIII). This
issue becomes even more severe as the fraction of attractive beads increases,
and hence NPT-MC results will not be discussed further in this study.

This single attractive bead case is particularly interesting because at very
low temperatures (lower that the Boyle temperature $T_{B}(\chi=6.7\%)\approx
0.0022$), we expect the formation of micelles at low densities replacing the
isotropic phase. This is indeed what occurs for Janus rods with a single
attractive site (see companion paper Wood  _et al._ (2023)). In contrast, at
the temperatures considered in the present study, the liquid crystal phases
are modified but not destabilized as the attractive energy is insufficient to
compensate for the higher entropy of the thermotropic liquid crystal phases.
Hence, the phase behavior of these systems is still mainly determined by
entropy. At lower temperature, however, there should be a transition region in
which these two opposite tendencies compete and give rise to interesting
effects. This can in fact be seen in the companion paper on Janus rods Wood
_et al._ (2023).

###  III.4 Stability with respect to the initial conditions

Figure 11: Test of the stability with respect to the initial conditions. Top
panel left: final equilibrated conformation at $\eta=0.35$ (nematic phase)
obtained with all helices initially parallel (i.e. with attractive tips all
pointing along the same direction); Top panel right: Same but with
antiparallel initial conditions (attractive tips up for odd layers and down
for even layers); Bottom panel left: Same as before with $\eta=0.65$ (SmB
phase); Bottom panel right: Same with antiparallel initial conditions. In all
cases, the lowest temperature $T^{*}=0.1$ has been considered.

In principle, it would be desirable to have a final configuration that is
fully independent of the initial conditions. In practice, however, this is
hardly achievable for the most compact configurations, even in the absence of
attractive interactions Kolli  _et al._ (2014b) or for hard spherocylinders
Bolhuis and Frenkel (1997). We therefore explicitly checked for this
dependence and report it in Figure 11 by considering two different initial
conditions. In the first case, used in all of the cases discussed so far, all
helices are oriented with the attractive beads initially pointing up (i.e.
along the $+z$ direction). We denote this as the parallel initial condition.
In the second case, the odd layers have initially helices oriented with the
attractive beads pointing up, whereas in the even layers all helices have
their attractive beads initially pointing down (along the $-z$ direction). We
denote this as antiparallel initial condition.

In the top panels of Figure 11, we show two equilibrated configurations
obtained at the same volume fraction of $\eta=0.35$ and at the same
temperature $T^{*}=0.1$ (corresponding to a nematic N phase), but starting
with parallel (left) and antiparallel (right) initial conditions. Very
reassuring, the two final configurations are essentially indistinguishable and
hence equivalent from a statistical viewpoint. However, this turns out not to
be the case for the most demanding case of $\eta=0.65$ (bottom panel)
corresponding to a SmB phase, where it is clear that initial parallel (left)
and antiparallel (right) configurations are essentially preserved upon
equilibration to the local thermodynamically stable states. As mentioned, this
is a very common feature of particles with a significant aspect ratio, and it
should not come as a surprise, since we have also considered the most
challenging situation of $T^{*}=0.1$. We further note that the total energy of
the antiparallel conformation (right) is lower than the parallel one (left) at
temperature $T^{*}=0.1$, thus indicating that the antiparallel configuration
becomes more stable than the parallel one as the temperature decreases. In our
companion paper Wood  _et al._ (2023) on Janus rods, the antiparallel layered
configuration appears spontaneously upon compression. These findings suggest
that the screw-nematic phases that we discussed so far should not be affected
by the energetic preference of the antiparallel conformation, but the smectic
ones might.

###  III.5 Janus ($\chi=50\%$) and fully attractive ($\chi=100\%$) helices

(a)

(b)

Figure 12: Reduced pressure $PD^{3}/\epsilon$ as a function of packing
fraction $\eta$ from MD-NVT simulations with $N=4068$ helices at $T^{*}=0.1$.
(a) Case of Janus helices (8/15 attractive beads, $\chi=50\%$). Green =
attractive beads, Red = hard core beads. (b) Case of fully attractive (SW)
helices ($\chi=100\%$). Here helices are colored according to their
orientations. Representative snapshots at indicated state points are also
displayed.

It is instructive to inspect what happens to the phase behavior upon
increasing the number of attractive beads (i.e. the coverage $\chi$). Two
cases appear to be particularly interesting.

The first case is when approximately half of the beads of the helices are
attractive (8 out of 15), which we refer to as the Janus limit ($\chi\approx
50\%$, see Figure 1). The $\chi=50\%$ limit has shown a particularly rich
phenomenology both, in the case of single colloids Sciortino, Giacometti, and
Pastore (2009, 2010) and in the case of dumbbells O’Toole, Giacometti, and
Hudson (2017); O’Toole  _et al._ (2017). At very low temperatures – lower than
the corresponding Boyle temperature $T_{B}^{*}(\chi\approx 50\%)\approx 0.015$
– one might expect the formation of bilayers where two attractive halves of
two helices bind to form a non-covalent bond. This is indeed what happens for
Janus rods (see the companion paper Wood  _et al._ (2023)). However, as in the
single attractive bead case ($\chi=6.7\%$) discussed earlier in Section III.3,
the range of temperatures considered in this work, $0.1\leq T^{*}\leq 1.0$, is
too high to destabilize the entropically dominated phases found in the hard
helices counterpart. This is clearly visible in Figure 12(a)), which reports
the equation of state at $T^{*}=0.1$. Compared to the single attractive bead
counterpart (see Figure 10(c)), the phase boundaries are all at very similar
positions. Supplementary Figure SIX shows the potential energy per helix
$E_{p}/N$ as a function of the volume fraction $\eta$ from the NVT-MD
simulations at $T^{*}=0.1$. As expected, for nearly all coverages $\chi$ this
potential energy is positive and increases with increasing $\eta$. Not
surprisingly, in the case of fully attractive helices $\chi=100\%$ case, the
energy is slightly negative and decreases on increasing $\eta$.

The second case is the extension of the attractive well to all beads (SW
$\chi=100\%$). This leads to the equation of state displayed in Figure 12(b),
again for $T^{*}=0.1$. Unlike the partially attractive cases of $\chi=6.7\%$
and $\chi\approx 50\%$, but in line with purely hard helices, the inserted
snapshots representing selected state points are here colored according to the
direction $\widehat{\mathbf{u}}$ of the main axis compared with the main
director $\widehat{\mathbf{N}}$. This case is particularly interesting because
of its dual nature. On the one hand, it represents a smooth extension of the
previous $\chi=6.7\%$ and $\chi\approx 50\%$ cases in terms of coverage.
However, in contrast to the $\chi=6.7\%$ and $\chi\approx 50\%$ cases, each
helix can be flipped upside-down with no effect on the mesophase, thus
restoring the up-down symmetry. A comparison with the Janus case (Fig. 12(a))
shows that all transitions have shifted to slightly higher volume fraction,
increasing their region of stability and nearly reproducing the same results
of the fully repulsive hard helices (see Figure 4(a)). This "re-entrant"
behaviour can likely be ascribed to restoring the up-down symmetry as
remarked.

###  III.6 Temperature and coverage dependence

Figure 13: Reduced pressure $P^{*}=PD^{3}/\epsilon$ as a function of the
packing fraction from MD simulations at different reduced temperatures
$T^{*}=k_{B}T/\epsilon=1.0$ (solid circles), $T^{*}=k_{B}T/\epsilon=0.5$
(solid squares), and $T^{*}=k_{B}T/\epsilon=0.1$ (solid triangles). Different
coverages $\chi=6.7\%$ single attractive bead (black), $\chi=50\%$ Janus case
(red), and $\chi=100\%$ SW case (green) are also shown. For reference, the
hard helices case $\chi=0\%$ (solid circle magenta) is also reported. In this
case, the natural unit $P^{*}=PD^{3}/k_{B}T$ has been used for the pressure.

We can now summarize our findings on the equation of state, and the location
of the transitions, in terms of the temperature and coverage dependence.
Figure 13 reports such a comparison for coverages from hard to fully
attractive and temperatures from $T^{*}=1.0$ to $T^{*}=0.1$. This shows that
the pressure increases upon increasing the temperature (at a given volume
fraction and coverage) or upon decreasing the coverage $\chi$ (at a fixed
temperature). Again, this can be rationalized on the basis that decreasing the
temperature and/or increasing the fraction of attractive beads increase self-
attraction and hence decreases the pressure. How the phase boundaries depend
on coverage is more complex. At the lowest considered temperature $T^{*}=0.1$,
the phase boundaries of the partial coverage cases (single attractive bead and
Janus helices) do not display any dependence on the coverage. Somewhat
surprisingly, the phase boundaries of the fully attractive case appear to have
nearly identical locations as the case of hard helices. A summary of the
coverage dependence of the various transitions can be found in Supplementary
Figure SX.

A progressive destabilization of the isotropic I phase upon cooling was
previously observed in a system of square-well prolate spherocylinders with a
continuous line of interaction sites Gámez and Lago (2017). Note however that
in that case the I boundary does shift, but not because the nematic N and
smectic Sm phases become more stable. Their phase diagram shows that the Sm
phase disappears at low $T$ and that the shift in the I boundary is due to
growth in size of a coexistence region rather than the N phase. We further
note that study cannot be compared quantitatively with the present one as
$L/D=5$ in that case. This point is further discussed in the companion paper
Wood  _et al._ (2023), which considers Janus rods with $L/D=5$.

##  IV Conclusions

In this study we considered the phase behavior of a system of helices formed
by a set of fused beads arranged into a prescribed helical shape. Unlike
previous studies that focused on hard helices Kolli  _et al._ (2014a, b,
2016); Cinacchi  _et al._ (2017), here some of the beads on different helices
attract one another, with the fraction of attractive sites ranging from
$100\%$ coverage to $0\%$ (pure hard helices). Using this system, we attempted
to address the following two questions: (1) How does attraction change the
phase behavior from that of hard helices?; and (2) What regions of the
parameter space are worth studying in more depth?

Starting from the known phase diagram of the hard helices case, we studied the
effects of coverage and temperature on the phase behavior. This was done in
regimes well above the Boyle temperature but at sufficiently low temperature
(in some cases) for attraction to compete with the purely entropic effects
that determine the ordering of hard helices. As the Boyle temperature for low
coverage (e.g. $T_{B}^{*}\approx 0.0015$ in the case of a single attractive
site) is considerably smaller than at high coverage (e.g. $T_{B}^{*}\approx
0.04$ for a system of fully attractive hard helices), the results obtained at
the considered temperatures ($T^{*}=0.1,0.5,1.0$) probe different multiples of
$T_{B}^{*}$, but with the common denominator of being all in the entropically
dominated regime.

In this work, we focused on a slender helical shape with radius $R=0.2D$ and
pitch $p=8D$ on the basis that this is nearly a rod-like particle but still
chiral. We also extended past studies of hard helices in terms of the number
of particles, the maximum volume fraction, and by including a study of the
particle dynamics via the use of molecular dynamics simulations. This allowed
us to check previous results and revealed possible kinetic trapping in MC
simulations stemming from the combined effects of hard helical interactions,
high aspect ratio, low temperature and high packing fraction. We found that
while fully consistent for hard helices and high temperatures, NPT-MC
calculations become problematic at low temperatures and very high densities
because of slow equilibration and kinetic trapping. For this reason, in the
present study we mainly used molecular dynamics simulations with slightly
softer potentials, which allow for parallel simulation, in addition to
providing access to particle dynamics.

In the case of hard helices, we confirmed previous results Frezza  _et al._
(2013, 2014); Kolli  _et al._ (2014a, b, b); Cinacchi  _et al._ (2017) and
provided further evidence on the presence of screw-like phases, both nematic
and smectic, originating from the helical shape. We then extended the model to
include attraction between a fraction of the beads forming each helix, and
studied the stability of the various phases upon lowering the temperature and
increasing the fraction of attractive beads. In all cases, the range of
temperatures selected were well above the corresponding Boyle temperatures,
meaning that the attractive interactions add an energetic component to the
free energy but that the phase behaviour is still dominated by entropy. As a
result, we found a pressure decrease at fixed volume fraction on lowering the
temperature and/or increasing the fraction of attractive beads, a fact that
can be easily rationalized with the progressive increase of the relative
balance between attraction and repulsion, but only small changes in the phase
behavior. While the location of the nematic phases are only mildly affected,
we observed a shift of the screw and the smectic phases to lower volume
fraction. However, this shift appears to be non-monotonic, with the fully
attractive helices behaving differently from the partially attractive
counterparts. For the case of a single attractive bead, we observed no
tendency of the liquid crystal phases to compete with the formation of
micelles at the considered temperatures. We do expect the onset of micelles
below the Boyle temperature at sufficiently low densities, as this is what
occurs for Janus rods studied in the companion paper Wood  _et al._ (2023).
Using the same rationale, we expect Janus helices where half of the beads are
attractive to display competition between the formation of lamellar phases and
the liquid crystal phases observed in this work at sufficiently low
temperature, again discussed in the companion paper on Janus rods Wood  _et
al._ (2023).

There are many avenues that the present study (as well as the companion paper
on Janus rods Wood  _et al._ (2023)) open for future analyses. As this study
focused on very slender helices, it would be interesting to check what happens
to much curlier helices where the in-plane locking of neighboring parallel
helices is expected to be much more effective at reducing the rotational
entropy, thus further promoting the screw-like mechanism which stabilizes the
screw-like phases observed here. Another point which deserves further
attention is the effect of attraction on the cholesteric phase observed for
hard helices Frezza  _et al._ (2014); Cinacchi  _et al._ (2017). Finally, a
recent study by some of the current authors Liu  _et al._ (2022) has
highlighted the important role that helical shape and chirality has on the
twisting of monolayer assemblies of rod-like or helical particles. It would
therefore be interesting to extend the low temperature analysis carried out
for Janus rods to Janus helices, thus probing the complementary regime for
helices. We plan to pursue these investigations in a future study.

## Supplementary Material

See the supplementary material for additional results in the case of fully
repulsive hard helices and hard helices with a single attractive beads and for
representative movies highlighting the screw-like nematic phase in these two
cases.

###### Acknowledgements.

The present study builds on a past collaboration that was initiated together
with Alberta Ferrarini and Giorgio Cinacchi. The use of the SCSCF
multiprocessor cluster at the Università Ca’ Foscari Venezia is gratefully
acknowledged. The authors acknowledge financial support by MIUR PRIN-COFIN2022
grant 2022JWAF7Y (AG), the Australian Research Council Grants CE170100026 and
FT140101061 (AWC and JW), the Galileo Project 2018-39566PG (AG), and the
Erasmus+ International Mobility Program.

## Data Availability Statement

The data that support the findings of this study are available within the
article [and its Supplementary Material].

## Author declaration

The authors have no conflict to disclosure

## Author contributions

LDC: software, formal analysis, investigation, methodology, review $\&$
editing. FR: methodology, review $\&$ editing. JAW: software, formal analysis,
methodology. AWC: conceptualization, funding acquisition, review $\&$ editing.
AG: conceptualization, software, formal analysis, methodology, funding
acquisition, review $\&$ editing.

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