# Heat Transport with a Twist

Ethan Abraham<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
Mohammadhasan Dinpajooh<EMAIL_ADDRESS><EMAIL_ADDRESS>Physical and Computational Sciences Directorate, Pacific Northwest National
Laboratory, WA 99352 USA  Clàudia Climent  Department of Chemistry, University
of Pennsylvania, Philadelphia, Pennsylvania 19104, USA  Abraham Nitzan
Department of Chemistry, University of Pennsylvania, Philadelphia,
Pennsylvania 19104, USA  School of Chemistry, Tel Aviv University, Tel Aviv
69978, Israel

###### Abstract

Despite the desirability of polymers for use in many products due to their
flexibility, light weight, and durability, their status as thermal insulators
has precluded their use in applications where thermal conductors are required.
However, recent results suggest that the thermal conductance of polymers can
be enhanced and that their heat transport behaviors may be highly sensitive to
nanoscale control. Here we use non-equilibrium molecular dynamics (MD)
simulations to study the effect of mechanical twist on the steady-state
thermal conductance across multi-stranded polyethylene wires. We find that a
highly twisted double-helical polyethylene wire can display a thermal
conductance up to three times that of its untwisted form, an effect which can
be attributed to a structural transition in the strands of the double helix.
We also find that in thicker wires composed of many parallel strands, adding
just one twist can increase its thermal conductance by over $30\%.$ However,
we find that unlike stretching a polymer wire, which causes a monotonic
increase in thermal conductance, the effect of twist is highly non-monotonic,
and certain amounts of twist can actually decrease the thermal conductance.
Finally, we apply the Continuous Chirality Measure (CCM) in an attempt to
explore the correlation between heat conductance and chirality. The CCM is
found to correlate with twist as expected, but we attribute the observed heat
transport behaviors to structural factors other than chirality.

##  I Introduction

For many years, the ability to control heat transport on a macroscopic scale
has defined important technologies such as refrigeration devices, heating
systems, and thermal insulators. One would also expect that the future ability
to control heat transport on the nanoscale will play a pivotal role in the
success of technologies such as nanoelectronics and nanoscale energy devices
that require thermal management.Hoogeboom-Pot  _et al._ (2015) In this regard,
polymers are materials that may be desirable for use in many such products due
to their flexibility, light weight, and resistance to corrosion, yet their
status as thermal insulators has precluded their use in products where thermal
conductors like metals and ceramics are required.Xu  _et al._ (2018); Peplow
(2016); Sperling (2005) This makes the nanoscale control and improvement of
heat transport in polymers an important engineering application.Xu  _et al._
(2018)

It is now known that the heat transport properties of polymers are highly
sensitive to external conformational factors. For example, while the thermal
conductivity of polymers in bulk amorphous form is often quite small, these
same polymers when aligned in crystalline form can display, in the chain
direction, a thermal conductivity that begins to approach that of a
conductor.Dinpajooh and Nitzan (2020); Henry  _et al._ (2010) Multiple recent
studies have investigated and found effects of mechanical strain on the heat
conduction in aligned polymer materials,Henry  _et al._ (2010); Liu and Yang
(2010); Xu  _et al._ (2018); He  _et al._ (2018); Liu and Yang (2012); Liu
_et al._ (2023) and previous work in our group has found that stretching a
thin polymer wire can increase its thermal conductivity by up to an order of
magnitude.Dinpajooh and Nitzan (2020, 2022) Another less explored
conformational factor is twist. Controlled twisting has recently been used as
a way to study chirality effects on molecular transport properties,Kopp and
Genack (2003, 2011); Kumar  _et al._ (2023) and in a recent study Wang et al.
have used twist to engineer fibers that cool either during stretch or stretch
release.Wang  _et al._ (2019) This observation of twist-based cooling, which
they attributed in the case of polyethylene to entropy changes associated with
deformation-driven monoclinic-to-orthorhombic phase conversions, shows promise
as a novel and potentially sustainable approach to cooling.Wang  _et al._
(2019) All of this invites one to explore the general effect of twist on the
heat transport properties of molecular chains.

While MD simulations have been used to compute the heat transport across a
polymer wire as a function of stretch, the same has yet to be performed when
the wire is twisted.Dinpajooh and Nitzan (2020); Liu and Yang (2012) In an
analogous fashion to previous work in our group that controlled stretch by
changing the end-to-end distance across a polyethylene wire of fixed
size,Dinpajooh and Nitzan (2020) here we control the twist in such a wire. The
effect of twisting a single polyethylene chain is null due to the invariance
of the C-C sigma bonds under complete rotation, so the simplest model system
for the present study is a two-stranded polyethylene wire, which forms the
familiar double helix when twisted. In addition to its direct relevance for
heat transport applications, this study is also motivated by recent
theoretical interest in the way chirality is manifested in molecular and
condensed phase optical and transport phenomena,Gutierrez  _et al._ (2012);
Göhler  _et al._ (2011); Naaman and Waldeck (2015); Daplan, Terranova, and
Turin (2022); Chen  _et al._ (2018); Juraschek, Neuman, and Narang (2022); Ray
_et al._ (1999); Li  _et al._ (2022); Xu  _et al._ (2023) as exemplified by
recent discussions of possible correlations between chirality and heat
conduction in carbon nanotubes,Feng and Zhu (2015); Feng, Zhu, and Tang
(2014); Nan (2012); Zhang and Li (2005); Osman and Srivastava (2001); Zhang
_et al._ (2004); Qiu  _et al._ (2012) and by recent observations that
implicate nuclear motions of chiral molecules in the observed temperature
dependence of some of these phenomena.Alwan  _et al._ (2023); Li  _et al._
(2022); Osman and Srivastava (2001)

We note that chirality, as a symmetry property, has traditionally been viewed
as a binary classification of molecules as either chiral or achiral. Yet in
the past three decades, quantification of chirality has been considered.Fowler
(2005); Dryzun and Avnir (2009, 2011); Pinsky  _et al._ (2008); Sidorova  _et
al._ (2021); Alvarez, Alemany, and Avnir (2005); Zahrt and Denmark (2019);
Carreras  _et al._ (2019) Here and in a follow-up paper, we will show that the
control parameter of twist in our system does correlate with these chiral
observables, and so the model we develop in this paper provides a system with
a continuous variation of chirality that can be used as a laboratory for
studying chiral effects on molecular transport properties.

It should be pointed out, as already suggested in Ref. (29), that the twist-
dependence is not necessarily associated with the chiral nature of the twisted
structures and may be better attributed to mechanical factors such as
torsional strain or interactions between the chains. Like in many other
systems, analysis of the heat transport mechanisms in polymer wires has
revealed a competition between diffusive and ballistic transport mechanisms.
Diffusive transport generally occurs when the length scale of molecular
interactions is much smaller than the length of the transport process of
interest and is stochastic in nature. On the other extreme, ballistic
transport occurs by the free motion of phonons, and such dynamics are
dominated by the harmonic part of the molecular force field (FF). Polymer
wires are of such a size and connectivity that the relative contribution of
each can be sensitive to external variables. For example, the study by
Dinpajooh’s et al. showed characteristics of mostly ballistic transport when
the single polymer is stretched, and characteristics of diffusive transport
when more slack is present.Dinpajooh and Nitzan (2020) Note that ballistic
transport by delocalized phonons is free from the stochastic back-scattering
events that characterize diffusive transport and is therefore associated with
a higher thermal conductance. We have seen that stretching can increase the
ballistic nature of the heat transport, and we will see here (Sec. III) that
to some extent twisting has a similar effect, albeit in a more complex and
non-monotonous fashion.

In Sec. II, we present the technical details of our simulations. In Sec. III,
we discuss our results pertaining to the effect of twist on molecular heat
conduction in polymer wires for different chain lengths, molecular force
fields, and number of chains. We also discuss the possible correlation of
these results with other physical observables such as chirality. We conclude
in Sec. IV.

##  II Methods

Molecular Model. For the majority of our calculations, we have followed
Dinpajooh et al. in representing our polymers using the Transferable
Potentials for Phase Equilibria (TraPPE) United Atom (UA) model, which treats
each CHx unit as a single particle.Dinpajooh and Nitzan (2020); Chen and
Siepmann (1999); Keasler  _et al._ (2012) This model is useful because it
greatly reduces the computational cost, yet it has been shown to still perform
with high accuracy for calculations of thermal conductance in
hydrocarbons.Sharony, Chen, and Nitzan (2020) It is based on a force field
(FF) that represents bonds with harmonic potentials and includes also angle
(3-body), dihedral (4-body), and Lennard Jones (LJ) potentials. The
Hamiltonian is given by

$\begin{split}H_{\rm{molecule}}&=\sum_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i}k_{bi}(l_{i}-l_{0i})^{2}+\sum_{i}k_{\theta
i}(\theta_{i}-\theta_{0i})^{2}+\\\
&\sum_{i}\sum_{n_{i}}^{4}\frac{C_{ni}}{2}\left[1+(-1)^{n{{}_{i}}-1}{\rm{cos}}(n_{i}\phi_{i})\right]+\\\
&\sum_{i}\sum_{j}4\epsilon_{ij}\left[(\frac{\sigma_{ij}}{r_{ij}})^{12}-(\frac{\sigma_{ij}}{r_{ij}})^{6}\right],\end{split}$
(1)

where $p_{i}$ and $m_{i}$ are the momentum and mass of a given particle
respectively, $l_{i}$,$l_{0i}$, $\theta_{i}$,$\theta_{0i}$, are the actual and
equilibrium bond lengths and angles respectively, $\phi_{i}$ are the dihedral
angles, and $r_{ij}$ are the inter-particle separations. Accordingly, the
parameters $k_{bi}$ and $k_{\theta i}$ are the spring constants for the bonds
and angles, and the $C_{ni}$ are the constants that define the dihedral
potentials. As in Ref. (5), the above parameter values were chosen to fit
observed physical properties.

We have compared results with two other models: i) an Explicit Hydrogen (EH)
model and ii) a simplified force field inspired by the Freely Joint Chain
(FJC) model.Rubinstein and R. H. Colby (2003) The Explicit Hydrogen model uses
the same Hamiltonian as above, except that the Hydrogen and Carbon atoms are
treated as separate bodies and different parameter values are chosen as
appropriate, which are available in the Supplementary Information (SI). The
FJC force field is also a united atom model that conventionally omits the
angular potentials, dihedral potentials, and Leonard Jones potentials. In the
present application, since our calculations pertain to wires with multiple
strands, the omission of the LJ potentials would create a non-physical
situation with no repulsive force between two chains. We, therefore, modify
the FJC field to include the LJ potential and denote this force field as FJC∗.
Hence the Hamiltonian becomes

$H_{\rm{FJC^{*}}}=\sum_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i}k_{bi}(l_{i}-l_{0i})^{2}+\sum_{i}\sum_{j}4\epsilon_{ij}\left[(\frac{\sigma_{ij}}{r_{ij}})^{12}-(\frac{\sigma_{ij}}{r_{ij}})^{6}\right],$
(2)

using the same parameter values as the TraPPE-UA FF. It should be noted that
this FJC∗ model allows the angles to relax from $\theta_{0}=114^{\circ}$ to
$180^{\circ}$, changing the natural length of the polymer. Therefore we report
two separate sets of results for the FJC∗ FF, one that keeps the same end-to-
end distance as the previous calculations (FJC∗ Original Length), and one that
is adjusted to the new natural length, which is the same as the contour length
of the polymer (FJC∗ Contour Length).

Calculation of Thermal Conductance. Our simulation consists of a polymer wire
held at either end by parallel walls of specified temperature (see Fig. 1).
This is obtained by designating, at each end of the wire, a segment of the
chain (typically 3 atoms long) to be fixed, followed on either side by a
segment (roughly $6$ Å long) in which the atoms are set to move as Markovian
Langevin thermostats. We note that the designation of internal atoms as
thermostats is justified since we are concerned not with junction effects but
with the intrinsic conduction of the polymer. The rest of the wire’s length is
then divided into segments (typically 20-30 segments of equal length) for
computation of the coarse-grained temperature along the wire. The (classical)
time evolution of the system is computed according to the above-described
Hamiltonian using LAMMPS (March 2018 version 16 with corrections for heat
flux).Plimpton (1995); Boone, Babaei, and Wilmer (2019) The timestep used was
1.25 fs for the UA models and 0.65 fs for the Explicit Hydrogen model.111A
shorter timestep is preferred for the Explicit Hydrogen model due to the
higher frequencies of the C-H bond vibrations. The steady state heat current
is then computed as the average over the two sides of the energy taken out
$(\Delta E_{\text{hot}})$ of the hot thermostat region and added $(\Delta
E_{\text{cold}})$ to the cold thermostat region per unit time

$I_{z}=\frac{1}{2}\left(\left|\frac{\Delta E_{\rm{hot}}}{\Delta
t}\right|+\left|\frac{\Delta E_{\rm{cold}}}{\Delta t}\right|\right).$ (3a)

The thermal conductance is then obtained from

$K=I_{z}/\Delta T.$ (3b)

In addition, temperature profiles are obtained by monitoring the kinetic
energy in each of the segments mentioned above to plot temperature as a
function of position across the wire.

Figure 1: Top: a schematic of a double-helical polyethylene wire with 10
twists in a non-equilibrium MD simulation of steady state heat transport. The
atoms in the slabs at each end are fixed to act as a wall, and those in the
adjacent slabs are treated as Langevin thermostats, which set the imposed
temperature difference. Bottom: the steady-state temperature profile obtained
for the same polymer when the left and right temperatures are set to TL = 320
K and TR = 280 K.

These heat transport simulations were then carried out as in Ref. (5; 48)
using the standard velocity-Verlet time integrator. The starting
configurations were first equilibrated under canonical NVT (constant number of
particles, volume, and temperature) for several nanoseconds using the
Nose–Hoover thermostat at 300 K for all atoms. The outputted equilibrated
configurations were then used as the starting configurations for the non-
equilibrium Molecular Dynamics (NEMD) simulations, in which Langevin
thermostats were used only for the hot and cold regions. The target
temperatures for the hot and cold regions were 320 K and 280 K. We used a
damping parameter of $50$ fs-1 for the thermostated regions, and we observed
that the results have minimal dependency on the damping parameters ranging
from $10$ to $100$ fs-1 (results not shown). After relaxation to the steady
state for about 1 ns, the simulations were continued for 40 ns to obtain data
for statistical averaging. Five independent simulations were used to further
reduce statistical uncertainties. Note the heat transport behavior is reported
in terms of heat conductance (Eq. 3b) because, as discussed in Ref. (5) and
also below, heat transport on the nanoscale is not necessarily diffusive,
making the intensive concept of conductivity difficult to define.

Generation of Twisted Structures. Twisted configurations were prepared by
applying a torque to one end of the wire while keeping the other end fixed.
The LAMMPS command addtorque was used to apply the torque. Untwisted wires
sampled from equilibrium room temperature simulations were used as the
starting configuration for these twist-generating dynamics. The number of
twists was found to grow continuously from 0 at the start of the simulation
until a point where bonds began to break as a result of the torque.222We found
that for the purposes of generating these twisted structures, applying a large
torque for a short time was more practical than applying a small torque and
waiting for equilibrium. The number of twists at any MD step was computed as
half the number of crosses between the two chains in the wire333When twisting
is done about the z axis, a “cross” is defined as a change of sign of
$x_{j}-x_{j}^{\prime}$ (or $y_{j}-y_{j}^{\prime})$ as $j$ runs from $1$ to $N$
along the chain, where $x_{j}$ and $x_{j}^{\prime}$ (or $y_{j}$ and
$y_{j}^{\prime}$) refer to the coordinates of atom $j$ on different chains.
and verified by visual inspection of the structures using VMD
software.Humphrey, Dalke, and Schulten (1996) The coordinate files from the MD
steps corresponding to the desired number of twists were then used as the
starting configurations for the heat transport simulations described above.
For further details and proof of principle regarding these twist-generating
simulations, see SI.

Note that a multi-strand polymer wire of a given length is characterized by
the maximum number of twists that can be imposed on it before bonds start
breaking. We infer that bonds are breaking when the end of the wire receiving
the torque continues to rotate, yet the number of twists does not increase. We
denote this maximal number of twists, which depends on the chain length $N$,
by $N_{T}^{\text{max}}$, and we define
$\lambda_{\text{twist}}(N)=N_{T}^{\text{max}}(N)/N$. Results below indicate
that $N_{T}^{\text{max}}$ is linear in $N$ (at least to a very good
approximation), so $\lambda_{\text{twist}}$ does not depend on $N$ and is a
property of the wire’s constitution.

Other Calculations.

In order to gain further insight into the relationship between the structures
and the heat transport behavior, more numerical analysis was performed on the
wire configurations. The normal modes and corresponding frequencies were
obtained from energy minimized structures. Energies were minimized using a
simple “steepest descent minimizer” such that the minimization was converged
when the maximum force was smaller than $25$ kJ mol-1 nm-1), and normal modes
were then obtained by diagonalizing the Hessian matrix. These calculations
were carried out using GROMACS software.Abraham  _et al._ (2023) A canonical
localization measure for normal modes is the participation ratio ($P_{k}$) of
mode $k$, varying from $1$ when the vibrations are localized on a single atom
to $N$ if the mode is equally distributed on all atoms. The participation
ratio is computed as follows: let the coefficients $u^{\alpha}_{k,i}$ denote
the expansion of normal mode $k$ in the atomic coordinates where $\alpha$
denotes the Cartesian coordinate and $i$ denotes the atom number. Defining
$p_{k,i}=\sum_{\alpha}|u_{k,i}^{\alpha}|^{2}$, the participation ratio is then
defined as

$P_{k}=1/\sum^{N}_{i}p^{2}_{k,i}.$ (4)

Note that since we have taken the modes to be normalized, we have
$\sum_{i}\sum_{\alpha}|u^{\alpha}_{k,i}|^{2}=\sum^{N}_{i}p_{k,i}=1$.Feldman
_et al._ (1993); Allen and Kelner (1998) We have also used the Savitzky–Golay
filter method to filter the set of participation ratio data points $P_{k}$ for
the purpose of smoothing the data,Savitzky and Golay (1964) which in our
experience is a way to avoid averaging over thousands of frames to get the
participation ratios. The smoothing method involves fitting successive sub-
sets of adjacent data points with a low-degree polynomial by the method of
linear least squares, and we found that the difference between the average
$P_{k}$ sum from several frames and the smoothed $P_{k}$ sum from one frame
was less than $0.1\%$.

Finally, noting that the double-helical wire is a model chiral system, we have
applied the Continuous Chirality Measure (CCM), a mathematical tool for
measuring chirality that returns 0 for an achiral structure and 1 for a
structure that is very different than its enantiomer.444In many of the
mathematical texts where the CCM is developed, the CCM is said to range from 0
to 100. We as physical scientists, prefer our dimensionless quantities to be
on the order of unity. The only difference between the two approaches is the
factor of 100 which we omit. For mathematical details of the CCM, see SI and
Ref. (9; 10). These CCM calculations were implemented by the gsym function of
the cosymlib library in Python.Alemany  _et al._ (2021) The CCM was computed
using real-time configurations from our MD calculations rather than energy-
minimized structures, raising the possibility that random thermal fluctuations
could slightly affect the CCM. To account for this, the reported CCM values
are thermal averages taken from at least five configurations.

##  III Results & Discussion

In studies of polymer wires, the simplest system to examine is typically a
single chain of polyethylene. However, twisting across the sp3 sigma bonds has
no persistent effect on the structure, rendering such a model ineffective when
the control is twist. Therefore, most of the results presented below take our
model system to be a two-stranded polyethylene chain, which forms the familiar
double helix when twisted. Studies of DNA usually define twist as the total
number of complete helical turns in a given DNA segment,Lionnet  _et al._
(2006) which is mathematically equivalent to Gauss’s linking number evaluated
between curves representing the two strands.Ricca and Nipoti (2011) We extend
this same definition to our polyethylene double helix and refer to it as the
number of twists ($N_{T}$) to avoid ambiguity. A left-handed helix has been
used for all of our reported results, but we performed several trials to
confirm the intuition that the right-handed helix exhibits the same thermal
conductance behavior. Unless otherwise specified, we choose the end-to-end
distance of the polymer wire to be its natural length, that is the maximal
extension that can be obtained without stretching any bonds or angles away
from their equilibrium values. In the case of polyethylene, this corresponds
to a fraction $R_{\rm{frac}}=0.83$ of the chain contour length. Here
$R_{\rm{frac}}=R_{\rm{ete}}/R_{\rm{contour}}$ where $R_{\rm{contour}}$ is the
contour length of the polymer and $R_{\rm{ete}}$ is the given end-to-end
distance. Note that the contour length as defined here is equivalent to the
equilibrium bond length times the number of bonds.

Figure 2: (A) Cartoon visualizations of an $N=98$ double-helical polyethylene
wire containing various numbers of twists. The atoms shown in red at the ends
of the wires are fixed in position throughout the simulations. (B) The
temperature profiles along the direction of the wire ($z$ coordinate) at
selected levels of twist. (C) Thermal conductance results for the wire as a
function of the number of twists. The statistical uncertainties (standard
deviations) are shown when larger than the symbol sizes.

In what follows, we report our results for heat conduction and temperature
profiles of polymer wires subject to various amounts of twist. We also look
for correlations between the observed heat conduction behaviors and other
properties such as chirality, potential energy, and mode localization for the
twisted structures. In these NEMD simulations, the thermostats operating at
the two ends of the molecular chain were set to 280 K and 320 K. Unless
otherwise specified, the TraPPE-UA force field has been used to model the
polymers.

Figure 2A shows cartoon representations of double helices composed of two
$N=98$ polyethylene chains containing various levels of twists. The atoms
marked in red are fixed throughout the simulations. Figure 2B shows average
temperature profiles for the NEMD simulations at five selected numbers of
twists, and Fig. 2C the results of our thermal conductance calculations. For
this chain length, the maximal number of twists that could be obtained before
bonds began to break was 17, so we refer to this number as the twist
saturation point and report results only for levels of twist less than or
equal to this amount. The statistical uncertainties marked in Fig. 2C are
obtained using five independent NEMD simulations of 40 ns each.

Apart from the quantitative thermal conductance results, there are several key
qualitative observations. The most striking is that when the point of twist
saturation is approached, the thermal conductance spikes to roughly three
times the value of the untwisted wire. Also noteworthy is the oscillating
pattern for lower levels of twist. The conductance increases as the number of
twists increases from 0 to 2 twists at which point a relative maximum is
obtained, but then decreases as the number of twists increases from 2 to 5.5,
at which point a relative minimum is obtained. Another oscillation with a
similar period and amplitude occurs between 5.5 and 11 twists before the final
spike occurs. The temperature profiles show that for levels of twist where the
conductance is high, the majority of the temperature bias falls on the edges
whereas for levels of twist where the conductance is lower, we see more of the
temperature bias spread out over the length of the chain. This is best seen by
comparing the temperature profile for 15 twists (green line in Fig. 2B), which
has a high conductance and shows temperature bias mostly near the interfaces,
to that of 5 twists (purple line), which has a low conductance and thus a
profile that falls more uniformly across the junction. As discussed in earlier
work,Dinpajooh and Nitzan (2020) a large temperature drop at the interfaces
and a flatter temperature profile in the interior of the junction indicates a
more ballistic character that involves less arbitrary back-scattering events,
making the heat transport more efficient.

Figure 3: Representation of the bonds in PE polymer structures for selected
number of twists. Snapshots are shown (from left to right) for 2 twists, 4
twists, 8 twists, and 12 twists. The two chains are visually distinguished by
different shades of blue.

Though we could not generally pinpoint the microscopic origin of this
difference in back-scattering behavior when the number of twists is varies,
direct visualizations of PE polymer chains show features of the structures
that change categorically where the high-twist spike begins. Figure 3 shows
representations of the bonds in our model double-helix with a selected number
of twists. As can be seen, at low levels of twist, the bonds make a local zig-
zag pattern that is expected for an ethylene chain. This persists even at 8
twists although the zig-zag pattern becomes a bit more complex. However, at 12
twists, the local structures almost completely transition to the helical
curve. This categorical smoothing of the structure appears to coincide
perfectly with the observed increase in thermal conductance, and it is
therefore likely that the ultra-high conductance at high twist is related to a
decrease in phonon back scattering associated with this structural transition.
In SI, Fig. S4 shows that a similar structural transition can be induced by
holding the number of twists constant and varying the LJ intermolecular
interactions between the chains. This suggests that the interchain
interactions are critical to the observed increase in thermal conductance at
high twist.

Figure 4: (A) Twist-generating MD simulations before performing the thermal
conductance measurements shown in the right panel: the number of twists in a
double-helical polyethylene wire in an MD simulation where one end of the
polymer is subject to a constant torque at 300K. A torque of $3\times
10^{-7}~{}$kcal/mol was used for $N=36$ (red), and a torque of $6\times
10^{-7}~{}$kcal/mol was used for $N=98$ (blue). Note that the number of twists
for each point in this figure is computed naively based on the number of times
the chains crossed, leading to some baseline fluctuations. (B) The thermal
conductance for the production MD simulations plotted against the twist
fraction $(N_{T}/N_{T}^{\text{max}})$ for the polymer wires generated in the
left panel. The statistical uncertainties (standard deviations) are shown when
larger than the symbol sizes.

Figure 4 shows that the effect of twist on the conductance behavior persists
across wires of different lengths. Panel 4A shows how the number of twists
increases with time when a torque is applied at one end of a polymer wire
along the axis of the wire. At first $N_{T}$ grows quadratically as predicted
by classical rotational mechanics, until a point of twist saturation
$N_{T}^{\text{max}}$ is reached. No further increase in twist is registered
beyond this point, indicating practically free rotation due to bond breakages.
An important observation is that
$\lambda_{\rm{twist}}=N_{T}^{\text{max}}(N)/N$ does not depend on $N$,
implying that $\lambda_{\rm{twist}}$ is a characteristic property of a type of
a polymer wire that is independent of the wire’s length. When the fixed atoms
at either end of the simulation box that do not contribute to the twist are
accounted for, it turns out that the polyethylene double-helical wire with
this force field has a value of $\lambda_{\rm{twist}}\approx 0.18$ twists per
degree of polymerization. Importantly, note that the quantity
$N_{T}^{\text{max}}$ (and hence $\lambda_{\rm{twist}}$) does not depend on the
magnitude of the torque used. Also, because the torque was applied only to
generate the twisted starting configurations and then turned off, the thermal
conductance calculations for the twisted structures do not depend on the
torque used. For comparison of different torque values and further discussion
of these twist-generating simulations, see SI.

In Fig. 4B we show the heat conduction computed for configurations of
different $N(=36,98)$ but plotted against the fractional quantity
$N_{T}/N_{T}^{\text{max}}.$ Clearly, the heat transport behavior observed in
Fig. 2 is better understood as a function of this twist fraction than as a
function of the absolute number of twists. It is worth noting that the heat
conductance seems to begin its spike when the twist fraction reaches $\sim
0.75.$ This resembles the result of previous work which concluded that the
heat conductance begins to spike when a polymer wire is stretched to a
fraction $R_{\rm{frac}}\approx 0.75$ of its contour length.Dinpajooh and
Nitzan (2020) This suggests a curious analogy between linear and torsional
strain.

Figure 5: (A) The thermal conductance of double-helical polyethylene wire with
$N=98$ shown as functions of the number of twists for various force field
models. The purple crosses correspond to the TraPPE United Atom force field
and shows the same line as Fig. 2 and Fig. 4. The red circles correspond to
the Explicit Hydrogen model, which treats Hydrogens as separate bodies, unlike
the other models which coarse grain each CHx into a single interaction site.
The blue squares correspond to the more flexible FJC model with the addition
of Leonard Jones potentials to account for inter-chain repulsion. Since the
FJC model omits angular potentials, the CCC bond angles can relax to
$180^{\circ}$. As such, the natural length is longer than in our original
force field; this adjusted length is used in the model corresponding to the
green triangles. (B) Average temperature profiles along the direction of the
wire (z coordinate) for selected levels of twist, where the colors and shapes
have the same meaning as panel A.

Figure 5 shows similar results for several different force field models for
length $N=98$. The TraPPE-UA FF used in the previous figures, which coarse
grains each CHx monomer into a single interaction site, is indicated by purple
crosses. The Explicit Hydrogen model, which does not make this united atom
simplification, is indicated by red circles. The other two models refer to the
FJC∗ model which omits the angular and dihedral potentials. For the
simulations indicated by the blue squares, the end-to-end distance is the same
as the first two models. For the green triangles, the end-to-end distance of
the polymer has been increased to account for the bond angle relaxation
allowed by the FJC∗ model as discussed in Sec. II. The following observations
are noteworthy:

> (a) The results for the TraPPE-UA and Explicit Hydrogen models are quite
> similar for 0 twists. Differences increase with twist, suggesting that
> modeling twisted or coiled structures could be a relative weakness of UA
> models. This could be due to the detailed steric interactions of the
> hydrogen atoms between chains that are not accounted for when the UA
> simplification is made. Note that the TraPPE-UA and Explicit Hydrogen models
> used in this paper are developed independently from each other; previous
> works such as Ref. (43) comparing the UA FF with the same all-atom FF
> results have found agreements in the thermal conductances within the
> statistical uncertainties for single, untwisted molecules. Our purpose here
> is not to assess the accuracy of UA models but to assess qualitatively
> whether the trends observed persist with these models.
> (b) Despite the deviations with increased twist, the main qualitative
> observations from our discussion of Fig. 2 persist in both the TraPPE-UA and
> Explicit Hydrogen models. These are the presence of oscillations as we
> increase twist, and a large spike as we approach twist saturation.
> (c) The FJC∗ models strongly deviates from the other two. This suggests that
> angular and dihedral contributions to the force field are important in twist
> simulations. Nevertheless, in these models as well, the heat conductance
> shoots up at high levels of twist.
> (d) For untwisted wires, the thermal conductance in the FJC∗ model without
> making the length adjustment (blue squares) is quite low. This is consistent
> with our previous finding that slack greatly diminishes the heat conductance
> in polymer wires compared to their natural length.Dinpajooh and Nitzan
> (2020)
> (e) Here we have used 300 K as the average temperature of the polymer wire,
> but we find that the trends observed persist when the same calculation is
> performed with different average temperatures (results shown in SI).

Overall, we see that the high-twist increase is present in all models, but the
pattern of oscillation in the low-twist regime varies. This raises the
possibility that two competing factors affect the thermal conductance as a
polymer wire is twisted: (i) increase in torsional strain which could increase
the thermal conductance in all force fields, and (ii) damping of normal modes
important for ballistic transport caused by the interlocking of the two
chains, which could decrease the thermal conductance. The latter could be
highly FF dependent, and the oscillations in the low-twist regime could be
attributable to subtleties of the competition between these two factors. Some
evidence for this is seen in the delocalization-weighted density of modes
plots discussed below.

Thicker wires.

Figure 6: (A) The thermal conductance per chain plotted as a function of the
number of twists for wires consisting of 2 (green triangles), 3 (blue
squares), 4 (purple crosses), and 12 (red circles) polyethylene chains of
$N=98$. Results are shown for up to 5 twists; note that green triangles are
points plotted in Fig. 2 and Fig. 4 (except scaled by a factor of 0.5 here).
(B) The average temperature profiles along the direction of the wire ($z$
coordinate) for the same wires containing 0 twists (top) and 5 twists
(bottom).

The fact that thermal conductance in polymer wires seems to be sensitive to
twist raises the possibility that twist can be used to engineer polymer wires
with improved thermal conduction. Therefore, it is worth exploring the extent
to which the thermal conductance behavior found in the double helix is
generalizable to wires of arbitrary thickness. Using the TraPPE-UA force
field, Fig. 6 shows the per-chain conductance for twisted $N=98$ polyethylene
wires composed of 2 (green triangles), 3 (blue squares), 4 (purpled crosses),
and 12 (red circles) chains, as well as selected temperature profiles. It
should be noted that the issue of twist saturation described above is more
complex in multichain ($>$2) wires as partial bond-breaking starts to occur
before a total softening of the structure under twist is observed (see SI).
For this reason, we report results here for only up to 5 twists, corresponding
to the first oscillation in Fig. 2.

The main observation of Fig. 6A is that the first oscillation observed for two
chains does persist for chains of various thicknesses, and the oscillation
actually seems to increase in amplitude for the thicker wires. Remarkably,
when just one twist is applied to the 12-chain wire, the conductance increases
by over 30$\%$. Since applying slightly more twist actually decreases the
thermal conductance below its original value, this raises the possibility that
precisely tuning the amount of twist in a wire could be consequential.

Correlations with Other Observables.

In order to further understand the structural effects of twist on polymer
wires, we looked at the effect of twist on several other physical observables
using the two-stranded wire. Figure 7A shows the total potential energy in the
polymer wire as a function of twist. The quadratic nature of the PE-twist
relationship suggests that the wire stores energy in a spring-like manner as
it is twisted. Figure 7B shows the linear tension in the wire, obtained from
$d(PE)/dL$ where $dL$ is a small stretch ($L\rightarrow L+dL)$ from the
natural length $L.$ In Fig. 7B, the most notable feature is that while the
tension increases roughly linearly in the low-twist regime, there comes a
point in the high-twist regime when increasing the twist actually reduces the
tension. It is interesting to note that this transition from increasing
tension to decreasing tension, and the steep increase in potential energy,
both occur in the twist regime where the thermal conductance spikes (see Fig.
2). Also note that this counterintuitive effect of increased twist decreasing
the tension is reminiscent of an observation by Lionnet et al.Lionnet  _et
al._ (2006) that overtwisting a DNA strand causes it to lengthen.

Figure 7: (A) The potential energy in a double-helical polyethylene wire of
$N=98$ plotted as a function of the number of twists. (B) The linear tension,
calculated as the derivative of the potential energy upon infinitesimal
stretching, in the same polymer wire. Figure 8: Spectral densities $g(\omega)$
defined as the density of modes (top), and the product $P(\omega)g(\omega)$
with $P(\omega)$ being the participation ratio (bottom) plotted against the
frequency $\omega$. The structures are $N=98$ double-helical polyethylene
wires using the TraPPE-UA force field and containing various numbers of
twists. The quantity $g(\omega)$ is the number of modes in each bin
(unnormalized). $P(\omega)$ is $\Sigma_{k}P_{k}$ in the bin divided by the
number of modes in the bin.

In an attempt to gain more insight into the origin of the heat conductance
behaviors described above, we have also examined the spectral densities and
localization properties of the normal modes for some structures. At room
temperature, phonons of low frequency are thought to play the greatest role in
heat transport both because they are thermally populated and because they are
less subjected to local scattering events. The top row of Fig. 8 shows the
spectral density $g(\omega)$ of normal modes as a function of frequency
$\omega$ for selected amounts of twist. Note the gap in the mid-frequency
range of the spectral density that disappears at high twist. As we discuss in
the SI, the disappearance of this gap is related to the intermolecular
interactions between the chains: the top panel of Fig. S6 shows that when the
well depth of the LJ interaction is turned down for $10$ twists, gaps begin to
appear in the spectral density. In the bottom row of Fig. 8, we scale the
spectral density by the average participation ratio in each bin (see Eq. 4).
We see that the product $P(\omega)g(\omega)$ does seem to predict the
oscillatory pattern of heat conductance in the low twist regime. In the lowest
frequency part of the spectrum ($\omega<250$ cm-1), the area under the curve
grows between 0 twists and 2 twists as the thermal conductance rises and then
falls at 6 twists where the thermal conductance is lower (compare with Fig.
2). However, this analysis of the spectrum above does not seem to predict the
high thermal conductance obtained in the most twisted structures.
Specifically, the thermal conductance is by far the highest for 16 twists, yet
the product $P(\omega)g(\omega)$ is by far the lowest (again referring to
$\omega<250$ cm-1). Further evidence that $P(\omega)g(\omega)$ does not always
correlate with thermal conductance is presented in Table S1 and Fig. S5 in the
SI, which show that when interchain interactions are turned down the low
frequency $P(\omega)g(\omega)$ increases but thermal conductance decreases.
Overall, we find that while analysis of the the $P(\omega)g(\omega)$ can
explain the low-twist oscillations, the smoothness of the helical curves (see
Fig. 3) is a better predictor of the high twist spike.

Combining stretch and twist. Previous work has shown an increase in thermal
conductance when a polymer wire is stretched,Dinpajooh and Nitzan (2020) while
here we have found interesting behavior when the same wire is twisted. Figure
9 shows the result of combining stretch and twist in our model $N=98$ double
helix. The left panel shows the thermal conductance as a function of twist for
various stretch levels $R_{\rm{frac}}=R_{\rm{ete}}/R_{\rm{contour}},$ where
$R_{\rm{contour}}$ is the contour length of the polymer and $R_{\rm{ete}}$ is
the given end-to-end distance. The right panel shows the dependence of thermal
conductance on the $R_{\rm{frac}}$ for various levels of twist. The two plots
can be viewed as transforms of each other with respect to the same data. For
instance, the red circles in the right panel are obtained from the column of
points highlighted by the red box (see above 0 twists) on the left panel. Note
that the natural length of the polymers corresponds to $R_{\rm{frac}}=0.83.$

Some observations include the fact that the high-twist spike is present for
all levels of stretch, though for the shorter $R_{\rm{frac}}$ values, kinks
and bond breakages seem to appear at high levels of twist causing the observed
drop-offs. Also the relative minimum at 11 twists persists for all levels of
stretch. In the low-twist regime, the oscillations are different but seem to
be correlated555The manner in which these curves seem to be related could be
described by a path homotopy. In topology, a path homotopy is a function that
continuously deforms a path $f$ into a path $g$ according to a parameter. Here
that parameter would be $R_{\rm{frac}}$ in a non-trivial manner. Munkres
(2000) Finally, we note that when comparing a highly twisted structure with a
moderate amount of stretch to an untwisted structure with a moderate amount of
slack, the thermal conductance can differ by well over an order of magnitude.

Figure 9: (A) The thermal conductance plotted as a function of twist in a
double-helical polyethylene wire with $N=98$ for selected values of
R${}_{\text{frac}}$ = R${}_{\text{ete}}$/R${}_{\text{contour}}$. (B) The
thermal conductance of the same wires as the left panel, plotted as a function
of R${}_{\text{frac}}$ = R${}_{\text{ete}}$/R${}_{\text{contour}}$ for
selected levels of twist. The points in the red box in (A) are the red circles
in (B).

Chirality Considerations.

Inspired by the recent interest in chirality effects on molecular transport
properties, we also applied a mathematical tool for quantifying chirality, the
Continuous Chirality Measure (CCM), in search for possible correlations
between thermal conductance and chirality. The CCM computes an abstract
distance of a structure from its nearest achiral structure, obtained by
averaging the original structure with its enantiomer. It ranges from 0 for an
achiral structure to 1 for a structure with high chirality. As we will discuss
further in future work, it makes most sense to apply the CCM locally, on a
length scale $\tau$, so that chirality depends on the intrinsic geometry of
the helix rather than the chain’s length. This approach is consistent with the
mathematical work of Raos,Raos (2002) who has explored how the CCM of helical
ribbons is affected by geometric parameters (radius $a$ and pitch $h$).

Figure 10: (A) Continuous Chirality Measure (CCM) plotted as a function of the
number of twists in a double-helical polyethylene wire of $N=98$ when the
molecular structure is considered to be a segment of the wire 10 atoms long
(blue), or one helical turn (red). See SI for mathematical details of the CCM.
(B) Thermal conductance as a function of CCM for the same two length scales as
above. The statistical uncertainties (standard deviations) are shown when
larger than the symbol sizes.

Figure 10A shows how the CCM is affected by the number of twists for two such
choices $\tau.$ The first choice is a fixed length corresponding to 10 atoms
along the chain (blue). The second choice for $\tau$ is the length of a
complete helical turn (red), which varies based on the wire’s amount of twist.
We see that with the latter approach, the CCM grows monotonically with the
number of twists, while with the former approach, the CCM obtains a maximum
and then decreases slightly for greater amounts of twist. Figure 10B plots
thermal conductance as a function of the CCM using both approaches. Note that
since the red line ($\tau$ = Turn length) in Fig. 10A appears linear, the red
circles in Fig. 10B display a similar pattern as when the horizontal axis is
the number of twists (compare to Fig. 2). The other approach ($\tau=N10$) does
not show any clear pattern. While our results clearly indicate (as expected)
that both chirality (as measured by the CCM) and heat conduction depend on the
level of twisting, our simulations do not show a clear correlation between
these two observables. While it is true that heat conductance depends on twist
and chirality depends on twist, this does not implicate chirality per se as a
causal agent in the former. Claims have been made that chirality affects
thermal conductance in Carbon nanotubes.Feng and Zhu (2015); Feng, Zhu, and
Tang (2014); Nan (2012); Zhang and Li (2005); Osman and Srivastava (2001);
Zhang  _et al._ (2004); Qiu  _et al._ (2012) However, these claims used
factors such as structural angles or optical activity as proxies for
chirality. These factors may correlate with both chirality and observed
transport behaviors, but this does not imply causation. It is likely that the
observed increase in thermal conductance was caused by structural factors
other than chirality (see Fig. 3).

##  IV Summary & Conclusions

We have used non-equilibrium MD simulations to study nanoscale control of
thermal conductance along polymer wires, and we have found a high sensitivity
to twist. For the two-stranded polyethylene wire, we found oscillations in the
thermal conductance at relatively low levels of twist, and a substantial rise
in thermal conductance at high levels of twist just before the breakage of the
wire occurs. These trends appeared in all force fields and levels of stretch
examined, though the details varied. The fact that the high-twist spike
persists across various force fields and levels of stretch while the
subtleties of the oscillations varied suggests two separate mechanisms by
which twist affects the thermal conductance, one that is more general and one
of which is more subtle and sensitive to details of the particular polymer. We
examined the normal mode spectrum and found that when mode delocalization was
considered, the oscillations in the low-twist regime could be predicted.
However, the normal mode spectrum in the high-twist regime did not always
correlate with the observed increase in thermal conductance. Nevertheless, the
high-twist spike turns out to be correlated to the structural transition from
zig-zag patterns to smooth helical curves, suggesting that twist can alter the
ordering of atoms in the polyethylene polymer wires, affecting the phase of
the propagating phonons and therefore the scattering center details. We found
that the threshold for the transition from the low-twist oscillations to the
high-twist spike appeared to be a fraction $\sim 0.75$ of twist saturation.
Interestingly, this was also the threshold fraction of contour length at which
heat conductance began to spike when end-to-end distance was controlled in a
previous study. Dinpajooh and Nitzan (2020) Finally, we have used the CCM, a
mathematical tool that treats chirality as a continuous rather than binary
property, to examine the chirality of the double-helical wire. Though we did
not find correlations between thermal conductance and chirality independent
from that with the number of twists, we demonstrate the possibility of using
quantitative methods to study molecular chirality, which has previously been a
rather qualitative and hand-wavy subject. Overall this study strongly suggests
that applying twists to polymer wires could be a useful strategy for
engineering wires with high thermal conductance.

## SUPPLEMENTARY INFORMATION

The details about twist-generating simulations, effects of intermolecular
interactions, temperature, and number of polymer chains on the thermal
conductance, and details of the explicit hydrogen model used are presented in
the Supplementary Information.

## Acknowledgements

The research of A.N. is supported This material is based upon work supported
by the Air Force Office of Scientific Research under award number
FA9550-23-1-0368 and the University of Pennsylvania. E.A. aknowledges the
support of the the University of Pennsylvania (GfFMUR Grant). M.D.
acknowledges the support from by U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences
Division, Condensed Phase and Interfacial Molecular Science program, FWP
16249. Pacific Northwest National Laboratory (PNNL) is operated by Battelle
for the U.S. DOE under Contract No. DE- AC05-76RL01830. Computing resources
were partly allocated by PNNL’s Institutional Computing program. C.C. was
supported by a postdoctoral fellowship from the Vagelos Institute for Energy
Science and Technology (VIEST). The authors are grateful to Zeyu Zhou and Abel
Carreras for technical assistance and discussions and also to Pere Alemany,
David Avnir and Inbal Tuvi-Arad for discussions on continuous chirality
measures.

## Data Availability

The data that support the findings of this study are available on github at:
https://github.com/eabes23/polymer_twist/.

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Supplemental Information: Heat Transport with a Twist

##  I Proof of Principle for Twist-Generating Simulations

The twisted structures used in this study were obtained from the structures of
untwisted polymer wires by performing twist-generating simulations in which
the atoms on one end of the wire were immobilized while the atoms on the other
end of the wire were treated as a rigid body subject to a constant torque. We
found that the twist at first increased quadratically as predicted by
classical rotational dynamics until a characteristic point of twist-per-length
saturation was reached. The coordinates at various snapshots that had the
desired number of twists could then be used as input files for our thermal
conductance simulations. Since this twist-generating simulation is both novel
and imperative to the validity of our study, we have included this appendix
for further justification.

Figure S1: The number of twists in a double-helical polyethylene $N=98$ wire
in an MD simulation where one end of the polymer is subject to a constant
torque at 300K. Results are shown when the applied torque is equal to $2\times
10^{-7}$ kcal/mol (purple), $6\times 10^{-7}$ kcal/mol (blue), $11\times
10^{-7}$ kcal/mol (green), and $16\times 10^{-7}$ kcal/mol (yellow).

Figure S1 shows that the same twist saturation is reached for several amounts
of applied torque. It should be noted that this is the case only because all
of the torques shown used for this figure are very strong relative to the
polymer wire and easily strong enough to break bonds. As implied by the
spring-like relationship between potential energy and twist shown in Fig. 7 of
the main text, there is a torque regime (perhaps orders of magnitude smaller
than those used in these simulations) where different equilibrium amounts of
twist would be reached. However, using a large torque to quickly twist-
saturate the wire was fast and allowed us to obtain structures of all levels
of twist from a single simulation.

Figure S2: Histogram of the number of times the two strands cross in the
x-direction (where the $z$ direction is the length of the polymer) binned by
the monomer number where the cross occurs in twist-generating simulations for
a double-helical polyethylene wire with torque of $6\times 10^{-7}$ at 300K.
The bin totals are aggregated from several such simulations in order to
generate enough data for the bins to be reasonably populated. Results are
shown for two distinct timesteps throughout the simulation, where the timestep
length is 1.25 fs. The timesteps are $1\times 10^{7}$ (top panel) and $2\times
10^{7}$ (bottom panel).

A second concern might be the asymmetry of the twist-generating simulation:
since torque is applied to only one end, the concern might be raised that the
result will be an asymmetric structure, with a higher density of twists and/or
higher particle kinetic energies on the side where the torque is applied.
Figure 11 and Fig. S2 support that this is not the case for the TraPPE-UA
model. Figure S1 shows that the twists are evenly distributed throughout the
chain at multiple MD steps throughout the simulation. The x-coordinate-crosses
refer to the number of times the difference in the x-coordinate values between
corresponding atoms on the two chains changes sign when stepping along the
length of the chain. The number of twists is then approximated as half the
number of x-coordinate-crosses. The atom number refers to the identity of the
atoms (between 0 and 98 for $N=98$ length chains) where such a cross occurs.
The data in Fig. S1 is binned over several independent twist-generating
simulations to aggregate enough data to make the histogram meaningful.

Figure S3: The average temperature profiles obtained for the first and third
quarters of the simulation described in a twist-generating simulation for a
double-helical polyethylene wire with a torque of $6\times 10^{-7}$ kcal/mol
at 300K. The total simulation length is $32\times 10^{7}$ timesteps of 1.25 fs
each.

Figure S3 shows average temperature profiles of the first and third quarters
of a twist-generating simulation, obtained in the same way as in the main
figures of this paper. This figure indicates that the asymmetry of the applied
torque does not lead to an uneven energy distribution throughout the chain or
any departure from thermal equilibrium.

##  II Effect of Intermolecular Interactions on the Thermal Conductance

In this Section, we study the significance of intermolecular interactions
between the two chains in the thermal conductance behavior of double-helical
polymer wires as a way to get more insight into the relationship between
structure and heat conduction in the studied system. We note that the
structures of the twisted polymer chains are determined by a competition
between intermolecular and intramolecular interactions. We find (see details
below) that a main factor in the variation of heat conduction is a transition
from a zig-zag sub-structure charecteristic of a typical polyethelene chain to
a smoother helical structure. The zig-zag substructure, as compared to the
smoother helical structure, would have increase phononic backscattering and
consequently reduced heat conduction.

We focus on polymer chains of length $N=98$ with selected number of twists
($10$ and $16$ twists). Table 1 shows the thermal conductance values for two
polymer chains with $16$ number of twists as the well depth magnitudes change.
Fig. S4 also shows the snapshots of the conformations of the polymer chains at
three selected well depth magnitudes.

Table 1: Thermal conductance values, $K$ , for $2$ polymer chains with $16$ number of twists as the intermolecular interactions change. The intermolecular potential is the LJ potential with $\epsilon$ as the well depth and $\sigma$ as the parameter for the hard sphere diameter. For all calculations here, $\sigma=3.95$ Å while the $\epsilon$ values change. $\epsilon$, kcal mol-1 |  $K$, pW/K
---|---
$10^{-5}$ | $296\pm 5$
$10^{-3}$ | $798\pm 8$
$0.0912$ | $2315\pm 20$
$0.2$ | $876\pm 6$
$0.6$ | $832\pm 10$
$0.9$ | $762\pm 12$

As can be seen, the dependence on $\epsilon$ is not monotonic, and $K$
decreases when $\epsilon$ becomes considerably smaller or larger than the
value $\epsilon=0.0912$ kcal/mol which is the TraPPE well depth magnitude. A
visual inspection of the molecular structure (e.g., a snapshot shown on the
left side of Fig. S4) shows that in the small $\epsilon$ regime the molecular
double-helical chain develops a zig-zag substructure that may cause more
phonon (back)scattering events and consequently smaller thermal conductance.
In contrast, in the large $\epsilon$ regime, tight intermolecular interactions
results in increased average CC bond lengths disrupting the optimum helical
structure for enhanced thermal conductance (snapshot shown on the right side
of Fig. S4).

Figure S4: The snapshots of two polymer chains, each consisting of $98$ units
at $300$ K when the number of twists is $16$ when the intermolecular
interactions (LJ potential parameters) change. Note that $\sigma$ for all
intermolecular interactions is $3.95$ Å. When the well depth magnitudes are
very small ($\epsilon=0.00001$ kcal/mol) the polymer chain conformations are
more random resembling zig-zag structures. More uniform helical structures are
observed for well depth magnitudes of $0.0912$ and $0.2$ kcal/mol. The average
CC bond length increases when the well depth magnitude is $0.2$ kcal/mol as
compared to the one from the well depth magnitude of $0.0912$ kcal/mol. Note
that one polymer is shown in red and the other one is shown in cyan.

We have also calculated the spectral densities of modes of twisted pair
polymer chains with $16$ number of twists when the intermolecular interactions
change from very small well depth magnitudes to relatively large well depth
magnitudes.

Figure S5 shows the resulting density of states, $g(\omega)$, and
$P(\omega)g(\omega)$ for the aforementioned well depth magnitudes when the
number of twists is $16$. As the well depth $\epsilon$ increases, the density
of low frequency modes decreases while that of higher frequency modes
increases. In particular, for low frequency modes less than $250$ cm-1, the
magnitude of $P(\omega)g(\omega)$ generally decrease as the well depth
magnitude increase. However, we find that (generally in contrast to the work
by Feldman et. al.Feldman  _et al._ (1993) on the low $g(\omega)$ region in
their harmonic theory) trends in $P(\omega)g(\omega)$ are not clearly
correlated with the computed thermal conduction for the systems we study here.
For instance, when the well depth increases from $0.0912$ kcal/mol to $0.2$
kcal/mol, we observe a decrease in $P(\omega)g(\omega)$ at low frequencies as
the thermal conductance decreases. However, when the well depth increases from
$0.00001$ kcal/mol to $0.0912$ kcal/mol, $P(\omega)g(\omega)$ at low
frequencies decreases while the thermal conductance increases. Here we find
that the smoothness of the helices is well correlated with the thermal
conductance unlike $P(\omega)g(\omega)$. This may be attributed to the fact
that the relationship between the $P(\omega)g(\omega)$ and thermal conductance
is not direct and other factors such as phonon group velocities, mean free
paths, and scattering processes also play crucial roles in determining thermal
conductance.Feldman  _et al._ (1993); Allen and Kelner (1998)

Figure S5: The spectral densities of modes, $g(\omega)$, and
$P(\omega)g(\omega)$ for $16$ number of twists when the $\epsilon$ values of
LJ interactions (in kcal/mol) change while the $\sigma$ values are kept
constant as $3.95$ Å .

To further investigate the influence of intermolecular interactions on the
density of states ($g(\omega)$), we calculate $g(\omega)$ for two polymer
chains with $10$ and $16$ number of twists when the well depth varies. The top
panel of Fig. S6 shows that when the number of twists is $10,$ there is a gap
in the middle of frequency range of $g(\omega)$ that disappears as well depth
magnitude increases. When number of twists is $16$, however, the gaps
disappears in the middle frequency range for all well depth magnitudes
examined. Therefore, it appears that increased inter-chain interactions
diminish the gap in the middle of the frequency range of $g(\omega)$, since
increase in the number of twists as well as the increase in well depth
magnitudes are both factors that tend to diminish the gap.

Figure S6: The spectral densities of modes, $g(\omega)$ for $10$ and $16$
number of twists when the $\epsilon$ values of LJ interactions (in kcal/mol)
change while the $\sigma$ values are kept constant as $3.95$ Å .

Table 2 also shows how the thermal conductance for 2 polymer chains with $16$
number of twists change as $\sigma$ in the LJ parameters changes and non-
monotonic behavior is observed. We do not go to further analyses here because
we find the linear change in the well depth magnitudes is a more
straightforward control for the intermolecular interactions between the
polymer chains.

Table 2: Thermal conductance values, $K$ , for 2 polymer chains with $16$ number of twists as the intermolecular interactions change. The intermolecular potential is the LJ potential with $\epsilon$ as the well depth and $\sigma$ as the parameter for the hard sphere diameter. Here the $\epsilon$ values are kept constant to a value of $0.0912$ kcal mol-1 while the $\sigma$ values change. $\sigma$, Å |  $K$, pW/K
---|---
$2.95$ | $1568\pm 60$
$3.45$ | $2294\pm 11$
$3.95$ | $2315\pm 20$
$4.45$ | $1318\pm 11$
$4.95$ | $797\pm 10$

##  III Temperature Dependence of Thermal Conductance

Here we systematically investigate the temperature dependencies of heat
conductance for two cases. First, we show how the heat conductance values
change with zero number of twists for two polymer chains when the temperature
changes from $300$ K to $500$ K in the intervals of $50$ K.

Table 3: The temperature dependencies of thermal conductance values for two PE polymer chains with chain lengths of $98$ for untwisted chains. $T$, K |  $K$, pW/K
---|---
$300$ | $738\pm 6$
$350$ | $670\pm 4$
$400$ | $597\pm 12$
$450$ | $550\pm 8$
$500$ | $500\pm 5$

Table 3 shows that as the temperature increases, the heat conductance
decreases for the junctions we study. This may be attributed to more jiggling
of the atoms in the polymers leading to more scattering events. Our
observation is consistent with the results from Muthaiah,Muthaiah and Garg
(2018) where they observed that the thermal conductance of stretched bulk
polymer chains decreases from $200$ K to $350$ K. It is worth noting that for
unoriented polyethylene, Luo et. al.Zhang and Luo (2016) observed that as the
temperature increases, the thermal conductivity initially increases, peaks
around 350 K, and then decreases.

Figure S7: The temperature dependencies of thermal conductance values from
classical MD simulations as the number of twists change for two polymer
chains.

Figure S7 shows how the thermal conductance values change for two PE polymer
chains with length $N=98$ as the number of twists changes for several average
temperatures. Each color corresponds to a different average temperature of the
polymer wire, where the two Langevin regions differ by 40 K. Overall, we see
that the primary qualitative trends of this study persist across various
temperature. Generally, for most of the twist numbers, the thermal conductance
values decrease as the temperature increases. Only in the high-twist regime
does this trend tend to deviate. It should be noted that a good understanding
of the temperature dependencies of the thermal conductance requires a rigerous
consideration of Bose-Einstein statistic, which awaits future work.Chen,
Dinpajooh, and Nitzan

##  IV Further Discussion of Polymer Wires with More than Two Chains

Figure S8: (A) The thermal conductance per chain plotted as a function of the
number of twists for wires consisting of 2 (green triangles), 3 (blue
squares), 4 (purple crosses), and 12 (red circles) polyethylene chains of
$N=98$. The part of this figure up to 5 twists is the same as Fig. 5. (B)
Number of twists in a polyethylene wire composed of 4 chains throughout the
equilibration of MD simulations to generate twists. The number of twists for
each point in this figure is computed naively based on the number of times the
chains crossed, leading to some baseline fluctuations in the measurement. (C)
Temperature profiles obtained during the production period of MD simulations
for wires composed of 2 (red), 3 (green), and 4 (blue) polyethylene chains
containing 10 twists during simulations used to compute (A).

In the main text, we have included only results for $\leq 5$ twists for
polymer wires composed of more than 2 strands. This is because the concept of
twist saturation was more difficult to understand for thicker wires. For
example, in Fig. 3A we saw that when torque was applied to the two-stranded
wire, its twist grew continuously and plateaued at twist-saturation due to
bond breakages. Figure S3B shows that this is not the case for the 4-stranded
wire, where there are multiple discontinuities in the twist increase over
time. We believe this is because when more than two chains are present, chains
can cross each other if bonds can break and reform. This can result in non-
trivial weaving patterns more conducive to twist increase. Furthermore, we
think that the way the non-reactive force fields (used in this study) model
these bond breakages and reformations might not be physically accurate because
the atoms which are bonded to each other are specified by the input topology
file. In reality, when an atom’s bond breaks, it should be able to form a new
bond with a different atom if the energy of that interaction is lowest. Here
we report the results of these calculations. Figure S3A shows such thermal
conductance results, and Figure S3C shows some corresponding temperature
profiles. Note, for instance, that 10 twists could not be obtained in a wire
that was 12 strands thick due to twist saturation and bond breakages.

##  V Details of the Explicit Hydrogen Model

For the Explicit Hydrogen model used in this work, the initial file containing
the atomic coordinates was first generated artificially from typical bond
lengths and angles. The structure was then allowed to relax at constant
temperature of 300K using the described force field, thereby generating a
realistic structure of a polymer wire.

The force field parameters used in the Explicit Hydrogen model were taken from
the literature as appropriate and are given in the following table.
Equilibrium bonds lengths and angles as well as angular spring constants were
taken from the standard TraPPE Explicit Hydrogen (EH) Force Field.Bock  _et
al._ (1979) Linear spring constants were computed from the accepted
frequencies of organic bonds. Leonard Jones parameters were taken from the
OPLS All-Atom Force Field.Jorgensen, Maxwell, and Tirado-Rives (1996)

Table 4: Explicit Hydrogen force field used in this work. Bond potentials: $U_{\rm{bond}}=k_{b}(l-l_{0})^{2}$ |
---|---
|  $k_{b}$, kcal mol-1 Å-2 |  $l_{0}$, Å
CHx–CHy | $450$ | $1.54$
CHx–H | $358$ | $1.10$
Angle potential: $U_{\rm{angle}}=k_{\theta}(\theta-\theta_{0})^{2}$ |
|  $k_{\theta}$, kcal mol-1 rad-2 |  $\theta_{0}$, deg
CHx–CHy–CHz | $58.4$ | $112.7$
CHx–CHy–H | $39.38$ | $110.7$
H–CHx–H | $40.54$ | $107.8$
Dihedral potential: $U_{\rm{dih}}=\Sigma_{n}^{4}\frac{C_{n}}{2}\left[1+(-1)^{n-1}{\rm{cos}}(n\phi)\right]$ |
|  $C_{i}$, kcal mol-1 | $C_{i}$
| $1.740$ | $C_{1}$
CHx–CHy–CHz–CHw | $-0.157$ | $C_{2}$
| $0.279$ | $C_{3}$
| $0$ | $C_{4}$
| $0$ | $C_{1}$
CHx–CHy–CHz–H | $0$ | $C_{2}$
| $0.366$ | $C_{3}$
| $0$ | $C_{4}$
| $0$ | $C_{1}$
H–CHx–CHy–H | $0$ | $C_{2}$
| $0.318$ | $C_{3}$
| $0$ | $C_{4}$
Non-bonded potential: $U_{\rm{LJ}}=4\epsilon\left[(\sigma/r)^{12}-(\sigma/r)^{6}\right]$ |
|  $\epsilon$, kcal mol-1 |  $\sigma$, Å
C, C (sp3) | $0.066$ | $3.5$
C, H (sp3) | $0.044$ | $3.0$
H, H (sp3) | $0.030$ | $2.5$

##  VI Mathematical Details of the Continuous Chirality Measure (CCM)

Let $Q=\ket{q_{1}},...,\ket{q_{N}}$ be a molecular structure where
$\ket{q_{k}}$ is the vector representing the atomic coordinate of the $k$-th
atom with respect to the center of mass. In this paper, we define the CCM of
the molecule $Q$ as

$CCM(Q)=1-\frac{\sum_{k=1}^{N}\bra{q_{k}}\hat{\mathds{1}}+\hat{\sigma_{n}}\ket{q_{k}}}{2\sum_{k=1}^{N}\braket{q_{k}}{q_{k}}},$
(5)

where $\sigma_{n}$ is the reflection operator over the plane defined by the
optimal unit normal $\ket{n}.$ The optimal unit normal $\ket{n}$, which
defines the plane of reflection that minimizes the CCM, can be found using
Lagrange multipliers.Pinsky  _et al._ (2008) In the mathematical literature,
the CCM typically optimizes over all permutations
$\ket{p_{k}}=\hat{P}\ket{q_{k}}$ in addition to optimizing the plane of
reflection. However, in the spirit of chemistry, we force the trivial
permutation in order to honor the information contained in the molecule’s bond
connectivity. We have checked that for the helical structures studied in this
work, the trivial permutation yields the lowest CCM value.

Note that the CCM can be understood intuitively as a structure’s distance from
its nearest achiral structure, where
$\frac{1}{2}\Sigma_{k}(\hat{\mathds{1}}+\hat{\sigma_{n}})\ket{q_{k}}$
represents the closest achiral structure to $Q$ (i.e. the average of the two
enantiomers). This appendix is not intended to be a thorough treatment of the
CCM. Readers who have further interest should consult Ref. (9; 10).

## References

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