# One-dimensionally confined ammonia molecules: A theoretical study

Maksym Druchok<EMAIL_ADDRESS>Institute for Condensed Matter Physics,
National Academy of Sciences of Ukraine, Svientsitskii Street 1, 79011 L’viv,
Ukraine  Volodymyr Krasnov<EMAIL_ADDRESS>Institute for Condensed
Matter Physics, National Academy of Sciences of Ukraine, Svientsitskii Street
1, 79011 L’viv, Ukraine  Taras Krokhmalskii<EMAIL_ADDRESS>Institute for
Condensed Matter Physics, National Academy of Sciences of Ukraine,
Svientsitskii Street 1, 79011 L’viv, Ukraine  Oleg Derzhko
<EMAIL_ADDRESS>Institute for Condensed Matter Physics, National Academy
of Sciences of Ukraine, Svientsitskii Street 1, 79011 L’viv, Ukraine
Professor Ivan Vakarchuk Department for Theoretical Physics, Ivan Franko
National University of L’viv, Drahomanov Street 12, 79005 L’viv, Ukraine

###### Abstract

We examine a single-file chain of ammonia molecules in a carbon nanotube. To
this end, we use i) molecular dynamics simulations (combined with the charges
for ammonia nitrogen and hydrogen obtained from quantum chemistry) and ii)
lattice-model calculations [M. Druchok et al., J. Chem. Phys. 158, 104304
(2023)]. Our findings demonstrate the occurrence of the orientational
quasiorder of the ammonia dipoles, which become parallel to the tube axis, at
intermediate temperatures below $100$ K.

single-walled carbon nanotubes, single-file ammonia molecules, orientational
quasiorder, quasiphase transition

The authors have a great pleasure to dedicate this paper to the upcoming
eighties birthday of Professor M. F. Holovko, whose broad research activity,
inspiring talks and lectures, relevant comments at various occasions, and
simply good advises have strong influence on a few generations of researches
in statistical physics and soft matter theory in L’viv.

##  I Introductory remarks

Confinement of molecules to pores of nanometer diameters, when they form a
single-file chain, results in essentially new properties of such a substance.
Single-walled carbon nanotubes (CNTs) of a corresponding diameter provide an
excellent experimental setup to study the one-dimensionally confined substance
Cambre2010 .

Recently, X. Ma et al. Ma2017  have reported temperature-dependent
photoluminescence spectroscopy data for single-walled (6,5) CNTs. While empty
CNTs exhibit a linear temperature-dependent photoluminescence spectral shift
as expected, the water-filled CNTs show a stepwise photoluminescence spectral
shift centered at about $150$ K, which is superimposed on the anticipated
linear temperature-dependent one. X. Ma et al. Ma2017  assumed that the origin
of the observed additional spectral shift is related to a significant change
in the orientation of the water dipoles. They performed molecular dynamics
(MD) simulations and indicated three different regimes: 1) traditional
hydrogen-bonded chains (below $\sim 40$ K), 2) predominantly bifurcated
hydrogen bonds, where hydrogen bond from a single oxygen atom is distributed
over both hydrogen atoms of a single neighboring water molecule (around $\sim
70$ K), and 3) disordered chains (for $T>200$ K). The effective total dipole
moment of the structures dominated as temperature grows agrees with the
direction of the measured photoluminescence spectral shift. Several
theoretical studies have been inspired by experiments reported in Ref. Ma2017
. Thus, the ground-state and finite-temperature properties of one-
dimensionally confined water molecules were discussed in Refs. Sahoo2021 ;
Serwatka2022a ; Serwatka2022b ; Serwatka2023 ; Druchok2023 .

In the present study, we address a question whether quasiphase transitions,
observed for the water molecules H2O encapsulated in single-walled (6,5) CNT,
can be expected for other molecules. To this end, we consider the ammonia
molecules NH3 and the same, yet fillable, CNT. Ammonia molecule has a dipole
moment that makes it polar and it has ability to form hydrogen bonds Bauer2007
. Similarly to Refs. Ma2017 ; Druchok2023 , we carry out i) quantum chemistry
calculations to obtain charges of H and N inside the (6,5) CNT, ii) MD
simulations to obtain the dipole moment dependence on temperature for
illustration of quasiphases as well as to obtain reference values for a
lattice model (dipole moment and intermolecular distance), and iii) lattice-
model calculations. Our theoretical analysis gives evidence that the ammonia
molecules encapsulated in the (6,5) CNT may show a temperature-driven
orientational quasiordering at intermediate temperatures below 100 K.

The rest of the paper is organized as follows. In Section II, quantum-chemical
computations and MD simulations are described. In Section III, we report the
lattice-model calculations to further illustrate a temperature-driven dipole
quasiordering. Then, we conclude with a brief discussion and summary in
Section IV.

##  II Molecular simulations

###  II.1 Quantum chemistry and ammonia molecule model

To examine the ammonia molecules encapsulated in (6,5) CNT by MD simulations,
the realistic charges for the ammonia nitrogen and hydrogen atoms are
primarily required. According to Ref. Eckl2008 , for the bulk ammonia case
$q_{\rm N}=-0.9993e$ and $q_{\rm H}=-q_{\rm N}/3$, where $e$ is the elementary
electric charge. The OPLS (Optimized Potentials for Liquid Simulations) force
field Jorgensen1996  gives $q_{\rm N}=-1.026e$ and $q_{\rm H}=-q_{\rm N}/3$.
However, these charge values do not account the presence of CNT and therefore
are of limited applicability for MD simulations of ammonia molecules inside
CNT.

Figure 1: Quantum chemistry predictions for one-dimensionally confined $N=10$
ammonia molecules. Top panel: Visualization of AM1 data. Bottom panel:
Visualization of B3LYP data (only the 4th, 5th, 6th, and 7th ammonia molecules
are shown).

To obtain the charges for the nitrogen and hydrogen atoms in ammonia in the
(6,5) CNT, we used the GAMESS package Schmidt1993  and performed semi-
empirical (AM1, PM6), first-principle (STO-2G, MINI), and density-functional-
theory (B3LYP) calculations. Within these calculations, we examined ammonia
molecules inside a (6,5) CNT (AM1, PM6 – see the top panel of Fig. 1) and
ammonia molecules restricted to one dimension but without CNT (AM1, PM6,
STO-2G, MINI, B3LYP – see the bottom panel of Fig. 1). To model the (6,5) CNT,
we used the structure with 272 carbon atoms and 22 hydrogen atoms added to
saturate free carbon bonds on the edges of CNT (see the top panel of Fig. 1).
The coordinates of carbon atoms were frozen while for the coordinates of
hydrogen atoms the optimal positions were found. In our study, we consider 10
ammonia molecules and search for optimal positions of the nitrogen and
hydrogen atoms. To avoid surface effects, i.e., to minimize the effects of
molecules at terminal positions, only charges of the four inner ammonia
molecules were averaged for further molecular dynamics simulations.

Mean charges obtained at the AM1/PM6 level are: $q_{\rm N}=-0.3965/-0.6127$
(in units of $e$) for nitrogen atoms and $q_{\rm H}=0.1322/0.2042$ for
hydrogen atoms. In the presence of CNT we obtained $q_{\rm N}=-0.3913/-0.6281$
and $q_{\rm H}=0.1289/0.2100$ at the AM1/PM6 level, respectively. Only small
changes in atomic charges allow us to examine the former case alone while
performing more demanding density-functional-theory calculations. Furthermore,
the B3LYP method yields $q_{\rm N}=-0.7814$ (Löwdin population analysis) or
$q_{\rm N}=-0.9582$ (Mulliken population analysis) for nitrogen atom and
$q_{\rm H}=0.2704/0.2540$ (Löwdin) or $q_{\rm H}=0.3761/0.2918$ (Mulliken) for
hydrogen atoms. Here, doubled values of $q_{\rm H}$ correspond to the charge
of hydrogens forming hydrogen bonds (higher value before the slash) and the
charge of two dangling hydrogens (lower value after the slash), see the bottom
panel in Fig. 1. Semi-empirical AM1/PM6 calculations also provide some hints
for forming the hydrogen bonds, however, the difference in two values of
charges $q_{\rm H}$ is smaller. It should be stressed that the values of
$q_{\rm N}$ (and $q_{\rm H}$) are lower than the bulk ones, found in Ref.
Eckl2008  to fit to experimental vapor-liquid equilibrium data ($q_{\rm
N}=-0.9993$) or from OPLS ($q_{\rm N}=-1.026$).

Lower charges result in a reduced dipole moment of the ammonia molecule in
CNT. As MD simulations, besides $q_{\rm N}$ and $q_{\rm H}$, use other
parameters of atoms constituting ammonia molecules, we combine the obtained
charges with the OPLS force field geometry. In particular, the valence angles
and intramolecular distances were controlled with harmonic terms
$k_{\alpha}\cdot(\alpha_{\rm H-N-H}-\alpha_{0})^{2}$ and $k_{r}\cdot(r_{\rm
N-H}-r_{0})^{2}$, where $\alpha_{0}=106.4^{\circ}$ and $r_{0}=1.01$ Å
Rizzo1999 . Because of such flexibility, the dipole moment of ammonia
molecules was found to vary within the range of $1.26\ldots 1.36$ D (Löwdin)
or $1.32\ldots 1.43$ D (Mulliken) along the span of simulated temperatures,
see the upper panel of Fig. 3 and Fig. 8 below. For the sake of comparison,
the higher dipole moment of $\mu=1.94$ D for bulk ammonia is reported in Ref.
Eckl2008 . Significantly lower dipole moment of water molecules confined in
nanotubes ($\mu=1.105$ D) in comparison to the bulk value ($\mu=2.351$ D) is
also known in literature, see, e.g., Refs. Mann2003 ; Ma2017 . More details
about quantum chemistry calculations are given in Appendix A.

As in the case of confined water in (6,5) CNT Ma2017 ; Druchok2023 , the
quantum chemistry predictions are rather diverse and strongly depend on a
calculation scheme (choice of the basis set, the quantum-mechanical method,
the population analysis method and the choice of the geometry of the ammonia
molecule), see, e.g., Ref. Ertural2019 . The ambiguity related to the ammonia
molecule model, which is required for further MD studies, can be avoided while
performing ab initio MD simulations Marx2009 , however, such calculations are
far beyond the focus of the present paper. In our MD simulations we use the
B3LYP data, i.e., two sets: $q_{\rm N}=-0.7814$, $q_{\rm H}=-q_{\rm N}/3$
(Löwdin charges, see Sec. II.2) and $q_{\rm N}=-0.9582$, $q_{\rm H}=-q_{\rm
N}/3$ (Mulliken charges, see Appendix B). Both sets yield qualitatively
similar results, although larger Mulliken charge values result in larger
dipole-dipole interaction strength pushing the temperature-driven
orientational quasiordering to higher temperatures.

###  II.2 Molecular dynamics simulations and results

We use the DL-POLY molecular simulation package Todorov2006  to perform a
series of MD simulations of ammonia molecules encapsulated inside the CNT with
chirality of (6,5) at different temperatures. The (6,5) CNTs have a diameter
of $\approx 7.4$ Å. This value denotes the diameter of the circle over the
centers of carbon atoms constituting the CNT openings. The actual interior,
available for ammonia molecules, is smaller due to van der Waals sizes of
carbons. Such a small-sized nanopore allows only for a single-file arrangement
of molecules. Furthermore, we consider $N=35$ ammonia molecules inside a CNT
of $\approx 170$ Å. We ran a set of simulations over a temperature range of
$10\ldots 240$ K. The results around the lower temperature boundary should be
taken with caution because of quantum effects which are not accounted for in
our MD simulations.

We generated starting configurations consisting of CNTs and ammonia molecules.
The intramolecular geometry and short-range interactions for ammonia were
taken from the OPLS force field, while the charges for nitrogens and hydrogens
were optimized within the B3LYP level of approximation (see Sec. II.1 and
Appendix A). The CNT model was taken from Ref. Huang2006 , namely, the
Lennard-Jones parameters for carbons of nanotube sidewalls. Usually, the CNT
simulations also imply a set of harmonic bonds and angles maintaining the CNT
geometry, however, since ammonia molecules are in the spot of interest, we
froze the ideal CNT in vacuum to cut the computational costs. We mention that
CNT sidewall carbons are neutral. The Lennard-Jones parameters for unlike
sites were calculated using the Lorentz-Berthelot mixing rules. This
combination of interaction parameters was successfully utilized in our recent
studies Druchok2017 ; Druchok2018 ; Druchok2019 ; Druchok2023  of nanotubes
interacting with SPC/E water. Finally, two additional carbon atoms were placed
at the centers of CNT openings, which play a role of obstacles, to assure that
ammonia molecules stay inside the nanotube interior during the whole set of
simulations.

The simulation conditions were kept the same for all the runs except the
temperature variation. The temperature was controlled by means of the NVT
Nose–Hoover thermostat. Each simulation utilized the leapfrog integration
algorithm with the time step of 0.001 ps, covering 200 ps of equilibration and
then 600 ps of production runs. Smooth particle mesh Ewald technique was used
to calculate the electrostatic terms, while the short-range interactions were
calculated with the cut-off distance of 15 Å.

We also performed MD simulations for $N=35$ water molecules in the (6,5) CNT
of $\approx 170$ Å (the results are collected in Appendix C) for comparison
with the ones for the ammonia case.

Our MD simulation results for the ammonia molecules in the (6,5) CNT are
reported in Figs. 2, 3, and 4. These results i) illustrate the emergence of
the intermediate ordered quasiphase and ii) provide an input for the lattice
model to be considered in Sec. III.

Figure 2: Angles between CNT axis and dipole moments of ammonia molecules. Top
panel: Temperature dependence of mean angles averaged over time and ensemble
$\tilde{\theta}$. Bottom panel: $p(\theta)/\sin\theta$ for multiple
temperatures; $p(\theta)/\sin\theta$ has maximum at $\theta\approx 57^{\circ}$
for $T=10$ K.

Figure 2 illustrates temperature dependence of orientations of confined
ammonia molecules. Due to an arbitrary choice of the CNT axis direction, the
orientational order within the ammonia chain can be characterized in two
manners – “individual” and “group” ones. In general, the angle $\theta$
between the dipole moment of ammonia molecule and the CNT axis can take values
from the range of $0\ldots 180^{\circ}$. By adopting the same axis direction
for all dipoles we refer to this as the group orientation. The definition of
$\tilde{\theta}$ as $\min(\theta,180^{\circ}-\theta)$ has a meaning of an
individual angle. Both definitions make sense, as $\tilde{\theta}$ is more
focused on independent orientation of a dipole and is useful for our lattice
model, while $\theta$ distribution brings information about mutual orientation
of dipoles along the chain. Both definitions, in their own way, can serve as
indicators of temperature-induced rearrangement of dipoles. In particular, the
top panel of Fig. 2 demonstrates evolution of “individual” angles
$\tilde{\theta}$. Here and below in plots with MD results, besides the mean
values, we also show errorbars to indicate 25-th and 75-th percentiles of
values collected during the production runs of simulations. The mean value for
$\tilde{\theta}$ at $T=10$ K is about 57∘ and decreases down to $47^{\circ}$
at $T=50$ K, then increases with the temperature raise. One can also see that
the errorbars start to span over larger intervals with the temperature growth.
The bottom panel of Fig. 2 demonstrates the probability distribution density
of a “group” angles $\theta$ normalized by $\sin\theta$. Such a denominator is
needed to account for different number of redundant states for different
$\theta$. One can see that at low temperature ($T=10$ K) the distribution
$p(\theta)/\sin\theta$ is focused in vicinity of $\theta\approx 57^{\circ}$,
which corresponds to the value of $\tilde{\theta}\approx 57^{\circ}$ spotted
above. “Individual” and “group” angles coincide, thus, the chain demonstrates
a uniform orientation of dipoles. With the temperature rise, the distributions
$p(\theta)/\sin\theta$ at first spread to smaller angles, indicating the
intermediate quasiorder ($T=30$ and 60 K), then, become broader with
appearance of angles about 90∘, indicating that molecules can flip their
direction ($T=120$ and 180 K). Such a behavior of $p(\theta)/\sin\theta$ may
also yield a rough estimate for the temperature interval $T_{1}<T_{2}$ where
the intermediate quasiphase with dipole moments parallel to the CNT axis
exists.

Figure 3: Top panel: Temperature dependence of ammonia dipole moment $\mu$.
Bottom panel: Mean magnitudes of the perpendicular component of dipole moment
for ammonia molecule (red) and the tangential component of total dipole moment
of ammonia chain per ammonia molecule (green); both components are
additionally normalized by $\mu$.

Top panel of Fig. 3 shows temperature dependence of the ammonia molecules
dipole moment. As can be seen in the top panel of Fig. 3, the higher the
temperature the higher variation of dipole values around means, speaking in
favor of increasing fluctuations in the system.

In the bottom panel of Fig. 3 we show the normal component of dipole moment of
individual ammonia molecules $\mu_{\rm norm}=\overline{|\mu_{\perp}|}/\mu$ and
the total dipole moment of ammonia chain tangential to the CNT axis $\mu_{\rm
tang}=\overline{\mu^{\rm tot}_{\parallel}}/(N\mu)$; here $\overline{(\ldots)}$
denotes the mean value of $(\ldots)$. Clearly, $\mu_{\rm tang}$ and $\mu_{\rm
norm}$ reach maximum and minimum, correspondingly, in vicinity of $T=40\ldots
50$ K as expected for the intermediate quasiphase with dipole moments aligned
along the CNT axis.

X. Ma et al. Ma2017  observed similar temperature profiles for confined water
and pointed out three types of water arrangement: 1) hydrogen-bonding over the
whole chain, when dipole moments of water molecules are tilted by 31∘ to the
CNT axis (quasiphase 1), 2) dipole moments tend to align along the CNT axis in
one direction (quasiphase 2), and 3) collective arrangement is completely
destroyed (quasiphase 3). On the basis of similar results reported in Fig. 3,
we may assume that the quasiphase 2 is achieved for ammonia in vicinity of
$T=40\ldots 50$ K, while the quasiphases 1 and 3 are located at lower and
higher temperatures, respectively.

Figure 4: Temperature dependence of mean distances between nearest nitrogen
atoms within ammonia chain.

In addition to MD analysis of the dipole moment magnitudes and orientations,
mean distances between nearest ammonia molecules, as they follow from MD
study, are also an important reference. We monitored nitrogen-nitrogen
distances during the simulation and then averaged them; such an average is
reported in Fig. 4. Interesting to note, that both mean N-N distance and its
variation increase once the system passes the temperature range of quasiphase
2.

We end up the section with a remark about ammonia molecule model based on the
Mulliken charges. The Mulliken case demonstrates a similar behavior as in
Figs. 2, 3, and 4 (cf. Figs. 7, 8, and 9 in Appendix B), i.e., predicts three
quasiphases and two quasiphase transitions between them, however, the
temperature range for the intermediate quasiphase 2 is now $T=90\ldots 100$ K.

##  III Lattice model

###  III.1 Lattice model formulation

We pass to statistical mechanics arguments for explaining behavior of ammonia
molecules forming a single-file chain in CNT and emergence of the intermediate
quasiphase Druchok2023 . We bear in mind that the system at hand is one-
dimensional and consists of finite (not very large) number of molecules $N$.
We have to account for short-range nearest-neighbor interactions leading to
hydrogen-bonded chains and long-range dipole-dipole interactions as well as
rotations with limitations imposed by the geometry of CNT having a very small
diameter that can only be filled with one ammonia molecule after the other
Yang2020 . Then, an interplay of these ingredients produces a finite range of
temperatures in which the states, in which ammonia molecule dipole moments are
directed along the CNT axis, dominate in the state space Druchok2023 .

More specifically, we consider a lattice model with $3$ states at each lattice
site subjected to certain restrictions (hydrogen-bonded chain consists at
least of two molecules) which effectively decrease the number of states per
site. Thus, we deal with $N$ rigid bodies each with the moment of inertia $I$,
which carry coplanar dipoles $\vec{\mu}_{j}$, $j=1,\ldots,N$; they are
rendered on a single straight line so that the distance between the
neighboring sites $j$ and $j+1$ is $a_{j,j+1}$. Each site $j$ may be in one of
the following 3 states $\xi_{j}$:

  * •

The state $\xi_{j}=1$, when $\mu_{\parallel,j}=\mu\cos\alpha_{1}$,
$|\mu_{\perp,j}|=\mu\sin\alpha_{1}$ and the extension of the occupied site is
$a_{1}$. The site being in such a state belongs to a hydrogen-bonded chain and
must have at least one neighboring site in the same state $\xi=1$.
Furthermore, the set of $\mu_{\perp,j}$ for the hydrogen-bonded chain forms a
staggered pattern.

  * •

The state $\xi_{j}=2$, when $\mu_{\parallel,j}=\mu\cos\alpha_{2}$,
$|\mu_{\perp,j}|=\mu\sin\alpha_{2}$, $0^{\circ}\leq\alpha_{2}<\alpha_{1}$, and
the extension of the occupied site is $a_{2}=a_{1}(1+\varepsilon_{2})>a_{1}$.

  * •

The state $\xi_{j}=3$, $\mu_{\parallel,j}=\mu\cos\alpha_{3}$,
$|\mu_{\perp,j}|=\mu\sin\alpha_{3}$, $\alpha_{1}<\alpha_{3}\leq 90^{\circ}$,
and the extension of the occupied site is
$a_{3}=a_{1}(1+\varepsilon_{3})>a_{2}$. This state represents a completely
independent ammonia molecule with a random orientation of $\vec{\mu}_{j}$.

The introduced rules reduce the number of states $W_{N}$ for the lattice of
$N$ sites, which is now $W_{N}\approx 2.52^{N}<3^{N}$ Druchok2023  (i.e., the
lattice model has $\approx 2.52<3$ states per each site).

From the mathematical point of view, $W_{N}$ is equal to the number of words
of length $N$ over the three-letter alphabet $\\{1,2,3\\}$ without words
having isolated 1’s Birmajer2016 ; Sloane1964 . Moreover, $W_{N}$ is defined
recursively according to the recurrence relation
$W_{N}=3W_{N-1}-2W_{N-2}+2W_{N-3}$, $W_{0}=1$, $W_{1}=2$, $W_{2}=5$ and can be
obtained from the generating function $g(z)=\sum_{N=0}^{\infty}z^{N}W_{N}$,
which is given by $g(z)=(1-z+z^{2})/(1-3z+2z^{2}-2z^{3})$, see Ref. Sloane1964
. Furthermore Hall1967 , inserting $W_{N}\propto a^{N}$ for $N\to\infty$ into
the recurrence relation, one gets a cubic equation for $a$,
$a^{3}-3a^{2}+2a-2=0$, the largest root of which
$(27+3\sqrt{78})^{1/3}/3+(27+3\sqrt{78})^{-1/3}+1\approx 2.521\,379\,706\,8$
settles $W_{N}\approx 2.52^{N}$ as $N\to\infty$.

Now, the thermodynamic average is defined as follows:

$\displaystyle\langle(\ldots)\rangle=\frac{1}{Z}{\sum_{\xi_{1}\ldots\xi_{N}}}^{\prime}\sum_{\rm
rot}\left(\exp\left[-\frac{E(\xi_{1}\ldots\xi_{N})}{k_{\rm
B}T}\right](\ldots)\right),$ $\displaystyle
Z={\sum_{\xi_{1}\ldots\xi_{N}}}^{\prime}\sum_{\rm
rot}\exp\left[-\frac{E(\xi_{1}\ldots\xi_{N})}{k_{\rm B}T}\right].$ (1)

Here the prime near the first sum indicates the discussed above restriction on
the set of values $\xi_{1}\ldots\xi_{N}$ (no words with isolated 1’s) and the
second sum denotes the summation over rotational degrees of freedom for given
(allowed) set $\xi_{1}\ldots\xi_{N}$. Moreover, $E(\xi_{1}\ldots\xi_{N})$
stands for the sum of the rotation energy and the interaction energy which
contribute to the rotation part $K$ and the interaction part $Q$ of the
partition function $Z=Z(T,N)$.

We take into account the short-range nearest-neighbor interactions by treating
the molecules at sites $j$ and $j+1$ with $a_{j,j+1}=a_{1}$ as linked (i.e.,
rigidly connected) through a hydrogen bond. More generally, we have to assume
in addition that the energy of the hydrogen-bonded chain also decreases
because of the bonding, see below. The long-range interactions between all
molecules $U_{\xi_{1}\ldots\xi_{N}}$ is the sum over all $N(N-1)/2$ pairs of
the dipole-dipole interaction $u_{ij}$, $i<j$, $i=1,\ldots,N-1$,
$j=2,\ldots,N$. Moreover,

$\displaystyle
u_{ij}=k\frac{\mu_{\perp,i}\mu_{\perp,j}-2\mu_{\parallel,i}\mu_{\parallel,j}}{a^{3}_{ij}},$
(2)

$k=1/(4\pi\epsilon_{0})$, $\epsilon_{0}$ is the vacuum permittivity (SI
units), if the both sites $i$ and $j$ belong to the same hydrogen-bonded
chain. However,

$\displaystyle
u_{ij}=k\frac{-2\mu_{\parallel,i}\mu_{\parallel,j}}{a^{3}_{ij}},$ (3)

if the sites $i$ and $j$ belong to different hydrogen-bonded chains or at
least one of these sites is in the state 2 or 3. In other words, the
$\mu_{\perp}$ on-site components contribute to the intersite interaction
$u_{ij}$ only if the both sites rotate as a whole, but do not contribute to
the intersite interaction if they rotate independently. In contrast, the
$\mu_{\parallel}$ on-site components always contribute to the intersite
interaction $u_{ij}$.

Limited (because of CNT geometry) rotations are accounted as follows. A
hydrogen-bonded chain consisting of $n$ molecules has the moment of inertia
$nI$ and rotates along one axis only, which coincides with the nanotube axis.
Its energy is given by $E_{m}=\hbar^{2}m^{2}/(2nI)$ with $m=0,\pm 1,\pm
2,\ldots$ and hence each energy level $E_{m}$ except the one with $m=0$ is
two-fold degenerate Galitski1981 . The rotational partition function of the
hydrogen-bonded chain reads:

$\displaystyle
K_{n}^{(1)}=\sum_{m=-\infty}^{\infty}\exp\left(-\frac{m^{2}}{n\tau}\right),$
(4)

$\tau=T/T_{\rm rot}$, $k_{\rm B}T_{\rm rot}=\hbar^{2}/(2I)$. Furthermore, we
assume that a site being in the state 2 corresponds to a molecule which
rotates similarly to the hydrogen-bonded chain, i.e., contributes
$K_{1}^{(1)}$ to the rotation part of the partition function. Moreover, a site
being in the state 3 corresponds to a molecule which rotates along three axes;
its energy is given by $E_{J}=\hbar^{2}J(J+1)/(2I)$ with $J=0,1,2,\ldots$ and
the degeneracy of the energy level $E_{J}$ is $(2J+1)^{2}$. The partition
function of such a rotor reads:

$\displaystyle
K_{1}^{(3)}=\sum_{J=0}^{\infty}\left(\left(2J+1\right)^{2}\exp\left[-\frac{J(J+1)}{\tau}\right]\right).$
(5)

Having the partition function $Z$ (III.1), we immediately get the Helmholtz
free energy $F=-k_{\rm B}T\ln Z$ and hence the entropy $S=-\partial F/\partial
T$, the internal energy $E=F+TS$, and the specific heat $C=T\partial
S/\partial T$. According to Eq. (III.1) we also get the average dipole moment
(per site) or, more precisely, the following quantities:

$\displaystyle\mu_{\parallel}=\frac{1}{N}\sum_{j=1}^{N}\langle\mu_{\parallel,j}\rangle,\;\;\;|\mu_{\perp}|=\frac{1}{N}\sum_{j=1}^{N}\langle|\mu_{\perp,j}|\rangle.$
(6)

Obviously, $\mu_{\rm tang}=\mu_{\parallel}/\mu$ and $\mu_{\rm
norm}=|\mu_{\perp}|/\mu$, see Sec. II.2 and Fig. 3. Finally, we can calculate
the average length of the chain $L$ and the coefficient of linear thermal
expansion $\alpha_{L}=(1/L)(\partial L/\partial T)$:

$\displaystyle L=\sum_{j=1}^{N-1}\langle
a_{j,j+1}\rangle,\;\;\;\alpha_{L}=\frac{1}{L}\frac{{\rm d}L}{{\rm d}T}.$ (7)

The length of the chain per site $L/N$ might be related to the mean distance
between the nearest nitrogen atoms, see Fig. 4.

There are only a few parameters which are used as the input for the lattice
model described above. We begin with the energy scale which is determined by
$T_{\rm rot}$. Using the values $I_{A}=I_{B}=2.8\times 10^{-47}$ or
$I_{C}=4.4\times 10^{-47}$ in units of SI Badger1929 ; David1996  we obtain
for $T_{\rm rot}$ the values about $14$ or $9$ K. In our calculations we set
$T_{\rm rot}=10$ K. The energy scales which is determined by $T_{\rm
dip}\propto(k\mu^{2}/a^{3})/k_{\rm B}$ depends on the values of the dipole
moment $\mu$ and the characteristic length $a$. The quantum-chemical
computations and MD simulations illustrated in Sec. II allow us to choose
these parameters. Using data for $\mu=1.35\ldots 1.28$ D and $a=3.1\ldots 4.6$
Å from Figs. 3 and 4 (Löwdin charges), we may set $R=0.07$, that is, $T_{\rm
dip}=T_{\rm rot}/R\approx 143$ K. Note that for the Mulliken charges, Figs. 8
and 9, one has to decrease $R$ slightly. Next, we fix the angles as follows:
$\alpha_{1}=56^{\circ}$, $\alpha_{2}=20^{\circ}$, $\alpha_{3}=80^{\circ}$, cf.
Fig. 2. Finally, the length scale is determined by $a_{1}$, $a_{2}$, and
$a_{3}$: It might be reasonable to set $a_{1}=3.10$ Å, $\varepsilon=0.25$,
$a_{2}=(1+\varepsilon)a_{1}\approx 3.88$ Å, and
$a_{3}=(1+2\varepsilon)a_{1}=4.65$ Å in view of the MD results reported in
Fig. 4. The values $0^{\circ}\leq\alpha_{2}<\alpha_{1}<\alpha_{3}\leq
90^{\circ}$ and $a_{1}<a_{2}<a_{3}$ are chosen to be in line with whole
temperature dependencies shown in the bottom panel of Fig. 3 and in Fig. 4.

Importantly, the value of $\varepsilon$ must exceed a certain threshold value
in order to have as the ground state the hydrogen-bonded chain $111\ldots$
rather than the state $22\ldots 2$. More realistic description mentioned above
would imply including an energy of the hydrogen-bonded chain which diminishes
the energy of the state $111\ldots$ . In the present study, we do not take
into account the bonding energy which may modify quantitatively the lattice-
model outcome.

We emphasize here that our aim is not to reproduce MD simulations by the
lattice model analysis, but only to demonstrate a reasonable agreement between
the conclusions yielded by these approaches, i.e., MD simulations and
statistical mechanics calculations.

###  III.2 Lattice model results

We perform all lattice-model calculations described in previous subsection for
$N=8,9,10,11,12$ using the Maple software package implemented on a personal
computer. Our findings are reported in Figs. 5 and 6 and are discussed below.

To illustrate how the intermediate phase shows up, we single out in the
partition function $Z$ (III.1) the contributions of hydrogen-bonded chains of
various lengths. Namely, of the length $N$,
${\cal{Z}}_{N}=K_{N}^{(1)}{\cal{Q}}_{N}$, of the length $N-1$,
${\cal{Z}}_{N-1}=K_{N-1}^{(1)}{\cal{Q}}_{N-1}$, …, of the length 2,
${\cal{Z}}_{2}=K_{2}^{(1)}{\cal{Q}}_{2}$, and the contributions without the
hydrogen-bonded chains ${\cal{Z}}_{0}$, i.e.,
$Z={\cal{Z}}_{N}+{\cal{Z}}_{N-1}+\ldots+{\cal{Z}}_{2}+{\cal{Z}}_{0}$, where
$K_{n}^{(1)}$ is given in Eq. (4) and ${\cal{Q}}_{n}$ denotes the remaining
part of ${\cal{Z}}_{n}$, $n=N,\ldots,2$, see Ref. Druchok2023 . Moreover, we
introduce the probabilities $p_{N}={\cal{Z}}_{N}/Z$,
$p_{N-1}={\cal{Z}}_{N-1}/Z$, …, $p_{2}={\cal{Z}}_{2}/Z$, and
$p_{0}={\cal{Z}}_{0}/Z$; obviously, $p_{N}+p_{N-1}+\ldots+p_{2}+p_{0}=1$.

The temperature-dependent probabilities $p_{N}$, $p_{N-1}$, …, $p_{2}$, and
$p_{0}$ control the role of the configurations with different length of
hydrogen-bonded chains in thermodynamics. In the zero-temperature limit $T\to
0$, when the lowest-energy ground state dominates, $Z\to{\cal{Z}}_{N}$ and
$p_{N}\to 1$. In the high-temperature limit $T\to\infty$, when the dipole-
dipole interactions become irrelevant $Z\to{\cal{Z}}_{0}\to(K_{1}^{(3)})^{N}$
and $p_{0}\to 1$.

Temperature dependencies of $p_{N}$, $p_{N-1}$, …$p_{2}$, and $p_{0}$ for
$N=10$ and $N=12$ are shown in Fig. 5. As can be seen from this figure, there
is a wide temperature range of $55\ldots 75$ K where the largest probability
$p_{2}$ exceeds 40% for both values of $N$. More detailed analysis of ${\cal
Q}_{2}$ shows that the main contribution to $p_{2}$ comes from the subset of
configurations in which the remaining $N-2$ molecules are in the state 2
Druchok2023 .

Figure 5: Probabilities of various configurations versus temperature for the
chains of $N=10$ (top) and $N=12$ (bottom) sites. There is a temperature
region $50\ldots 75$ K where the probability $p_{2}$ (blue curves) dominates.

In summary, the considered probabilities $p_{N},\ldots,p_{0}$ for
$N=8,\,9,\,10,\,11,\,12$ provide evidence that the states, which contain
ammonia molecules in the on-site states 1 and 2, are the most relevant ones in
the temperature range $50\ldots 75$ K and result in the emergence of an
intermediate quasiphase. Since the ammonia molecules in the state 2 and in the
short hydrogen-bonded chains contribute to $\mu_{\parallel}$ but not to
$\mu_{\perp}$, the tangential/normal component of total dipole moment should
increase/decrease in this temperature interval. Further evidence for that
follows from analysis of temperature dependencies of observable quantities to
be discussed below.

The intermediate quasiphase is stable in a rather wide temperature region
$T_{1}\ldots T_{2}$. An estimate for the temperatures of quasiphase
transitions $T_{1}<T_{2}$ can be drawn by equating the corresponding
probabilities, that is, $p_{3}(T_{1})=p_{2}(T_{1})$ and
$p_{2}(T_{2})=p_{0}(T_{2})$. For the chosen set of parameters we get
$T_{1}\approx 39,\,41,\,44,\,46,\,49$ K and $T_{2}\approx
69,\,72,\,75,\,77,\,79$ K as $N=8,\,9,\,10,\,11,\,12$.

Figure 6: Temperature dependencies of (from top to bottom)
$\mu_{\parallel}/\mu$ (blue) and $|\mu_{\perp}|/\mu$ (orange), see Eq. (6),
specific heat $c(T)=C/(k_{\rm B}N)$, average length (coefficient of linear
thermal expansion is shown the inset), see Eq. (7), and correlators
$\langle\mu_{\parallel,1}\mu_{\parallel,i}\rangle/\mu^{2}$, $i=1,\ldots,N$ for
the chains of $N{=}8,\ldots,12$ sites. MD simulations ($N=35$ ammonia
molecules in the CNT of length $\approx 170$ Å, filled circles) are shown in
the panels with $\mu_{\parallel}/\mu$, $|\mu_{\perp}|/\mu$ and the average
length [we set $L(0)/(N-1)=3.10$ Å].

In Fig. 6 we show temperature dependencies for various quantities of interest
for the lattice model of $N=8,9,10,11,12$ sites. The introduced model predicts
an increase of $\mu_{\parallel}$ (6) and decrease of $|\mu_{\perp}|$ (6) in
the temperature range $30\ldots 80$ K in agreement with MD simulations, see
blue and orange curves in the top panel of Fig. 6. The specific heat per site
in the interval $30\ldots 70$ K has minimum with the minimal value which
noticeably depends on $N$. At high temperatures the specific heat approaches
$3k_{\rm B}/2$ as it should for independent three-axes rotators, see Eq. (5).
The average length $L$ increases with the temperature growth, however,
differently at different temperatures. This can be also seen from the
temperature dependence of $\alpha_{L}$ from Eq. (7) shown in the inset. We
plot also the results of MD simulations shown previously in Fig. 4 after
assuming $L(0)/(N-1)=3.10$ Å. Again both results, lines and filled circles,
agree qualitatively.

In the lowest panel of Fig. 6 we report the lattice-model predictions for the
dipole correlators $\langle\mu_{\parallel,1}\mu_{\parallel,i}\rangle$,
$i=1,\ldots,8$ ($N=8$), which also indicate a correlated state of ammonia
molecules in CNT at low temperatures. From this figure we conclude that all
correlators practically coincide, i.e., are independent on distance between
the sites, up to 15 K having the value about $\cos^{2}56^{\circ}$ (quasiphase
1 when molecules form a hydrogen-bonded chain), and with further temperature
growth up to 55 K the correlators with $i=2,\ldots,6$ remain almost
indistinguishable passing the maximal value about 0.5 (quasiphase 2). Only for
higher temperatures, the dipole correlations decrease with increase of the
distance between sites and gradually fall down to $\cos^{2}80^{\circ}$
(quasiphase 3).

In the end of subsection we remark that the estimates of the temperature
interval for the intermediate quasiphase, as they follow from inspection of
various quantities, are slightly different and depend on the quantity under
analysis. This is yet another indication that we face a gradual emergence of
the intermediate quasiphase but not a strict phase transition.

##  IV Discussion and summary

In conclusion, motivated by the experiments reported in Ref. Ma2017  and their
theoretical support Ma2017 ; Druchok2023 , which demonstrated a temperature-
driven water dipole quasiordering in one dimension, we address a question
whether such phenomenon can be observed for other molecules and focus on the
ammonia molecules NH3. Our theoretical study, which includes quantum chemistry
calculations for the model of ammonia molecule and MD simulations for ammonia
molecules encapsulated in (6,5) CNT as well as statistical mechanics arguments
on the basis of a lattice model, gives an affirmative answer to this question:
Ammonia dipoles show quasiorder below 100 K becoming parallel to the CNT axis.
Even though details of quasiordering depend on the input characteristics of
the ammonia molecule inside CNT (Löwdin or Mulliken charges) the quasiordering
cannot be questioned.

Both the water molecules and the ammonia molecules show similar behavior.
Namely, atomic charges in CNT are reduced and rotations within CNT becomes
restricted, the temperature dependence of the probability distribution density
of the angle $\theta$ between the molecule dipole moment and the CNT axis
indicate three phases and two quasiphase transitions between them, an
interplay of the interaction contribution and the entropy contribution
(rotations within restricted one-dimensional geometry) produce a highly
ordered structures in a temperature range $T_{1}\ldots T_{2}$. In this
temperature range, as it follows from the lattice-model analysis, the states
with dipole moments oriented along the CNT axis dominate the partition
function that can be seen in the temperature dependencies of
$\mu_{\parallel}$, $|\mu_{\perp}|$, the specific heat or dipole correlators.
Clearly, we deal with a finite number of sites one-dimensional lattice model
and any true phase transitions cannot be expected, however, a gradual
replacement of one quasiphase by another is possible and several calculated
quantities do indicate this. The quasiordering at intermediate temperatures is
a classical phenomenon: It occurs at temperatures at which the thermal de
Broglie wavelength of ammonia molecule is much smaller than, e.g.,
intermolecular distances.

Most evident difference between the water molecules and the ammonia molecules
is the angle of molecule dipoles relative to the nanotube axis in the
hydrogen-bonded quasiphase, which is about two times larger for ammonia case
in comparison the water case (recall, $\alpha_{1}=31^{\circ}$ for water
Druchok2023  versus $\alpha_{1}=56^{\circ}$ for ammonia).

The most interesting question which naturally shows up, is whether it is
possible to encapsulate ammonia molecules inside a narrow CNT and then examine
extensively ammonia-filled and empty CNTs for comparison. Unfortunately, we
are not aware about this. We hope that the presented theoretical study may be
of use for corresponding experiments in the future.

## Acknowledgments

The authors thank Danylo Dobushovskyi for bringing Refs. Sloane1964 ;
Birmajer2016  to their attention.

## Author declarations

Conflict of interest:

The authors have no conflicts to disclose.

Author contributions:

O. D. conceived the study; M. D. performed molecular dynamics simulations; V.
K. performed quantum chemical calculations; T. K. performed calculations for
the lattice model. All authors discussed the results and commented on the
manuscript.

## Data availability

The data that support the findings of this study are available within the
article.

## Appendix A: Quantum chemistry calculation details

Table 1: Quantum chemistry calculations for the ammonia molecules restricted to one dimension (without CNT), see Sec. II.1. Semi-empirical methods AM1 and PM6. | AM1 | PM6
---|---|---
$q_{\rm N}$ ($e$) | $-0.3965$ | $-0.6127$
$q_{\rm H}$ ($e$) | 0.1322 | 0.2042
$\alpha_{\rm H-N-H}$ (∘) | 107.8 | 105.7
$r_{\rm N-N}$ (Å) | 2.95 | 2.97
$r_{\rm N-H}$ (Å) | 1.00 | 1.01
Table 2: Quantum chemistry calculations for the ammonia molecules inside the CNT, see Sec. II.1. Semi-empirical methods AM1 and PM6. | AM1 | PM6
---|---|---
$q_{\rm N}$ ($e$) | $-0.3913$ | $-0.6281$
$q_{\rm H}$ ($e$) | $0.1289$ | $0.2100$
$\alpha_{\rm H-N-H}$ (∘) | $107.6$ | $105.1$
$r_{\rm N-N}$ (Å) | $3.05$ | $2.94$
$r_{\rm N-H}$ (Å) | $1.00$ | $1.01$
Table 3: Quantum chemistry calculations for ammonia molecules restricted to one dimension (without CNT), see Sec. II.1. Hartree-Fock method results with the STO-2G and Huzinaga MINI basis sets. A hydrogen-bonded chain implies, first, a forming bond hydrogen charge $q_{\rm H}$ (the first number before slash in the third row) and two dangling hydrogens having charge $q_{\rm H}$ (the second number after slash in the third row) and, second, a shorter and a longer ${\rm N}$-${\rm H}$ bonds with $r_{\rm N-H}$ given by the first and the third numbers in the last row, respectively, as well as the dangling hydrogens with $r_{\rm N-H}$ given by the second number in the last row. The partial charges $q_{\rm N}$ and $q_{\rm H}$ are determined from the Löwdin population analysis (the upper rows in the second and third rows) or from the Mulliken population analysis (the lower rows in the second and third rows). | STO-2G | MINI
---|---|---
$q_{\rm N}$ ($e$) | $\begin{array}[]{c}-0.2352\\\ -0.3735\end{array}$ | $\begin{array}[]{c}-0.4272\\\ -0.6150\end{array}$
$q_{\rm H}$ ($e$) | $\begin{array}[]{c}0.1249/0.0553\\\ 0.1821/0.0958\end{array}$ | $\begin{array}[]{c}0.1662/0.1317\\\ 0.2350/0.1911\end{array}$
$\alpha_{\rm H-N-H}$ (∘) | 103.5/101.9 | 108.3/107.8
$r_{\rm N-N}$ (Å) | 2.83 | 3.29
$r_{\rm N-H}$ (Å) | 1.06/1.04 | 1.03/1.02
Table 4: Quantum chemistry calculations for the ammonia molecule restricted to one dimension (without CNT), see Sec. II.1. Density-functional-theory method (B3LYP) results, see explanations in the title of Table 3. | B3LYP
---|---
$q_{\rm N}$ ($e$) | $\begin{array}[]{c}-0.7814\\\ -0.9582\end{array}$
$q_{\rm H}$ ($e$) | $\begin{array}[]{c}0.2704/0.2540\\\ 0.3761/0.2918\end{array}$
$\alpha_{\rm H-N-H}$ (∘) | 105.5/104.9
$r_{\rm N-N}$ (Å) | 3.18
$r_{\rm N-H}$ (Å) | $1.03/1.02$

In this appendix, we report more results of quantum chemistry calculations for
ammonia molecules encapsulated in the $(6,5)$ CNT. First of all, we show some
characteristics of the ammonia molecule restricted to one dimension, but
without CNT (Table 1) and then inside the CNT (Table 2) as they follow from
semi-empirical methods Christensen2016 . Comparing numbers, we notice that the
presence of the CNT results only in small changes. This allows us to restrict
ourselves to less computer demanding case of the ammonia molecule without CNT
while performing quantum chemistry calculations at the Hartee-Fock level
(Table 3) or the density-functional-theory level (Table 4).

Similarly to the previous studies for water Ma2017 ; Druchok2023 , we observe
rather scattered outcomes for the charges of nitrogen and hydrogen atoms of
NH3, see the second and third rows in Tables 1 – 4. Most importantly and again
similarly to the previous studies for water Druchok2023 , not all sets of
charges $q_{\rm N}$ and $q_{\rm H}$ being plugged in MD simulations yield two
quasiphase transitions between three quasiphases. For example, $q_{\rm N}$ and
$q_{\rm H}=-q_{\rm N}/3$ optimized at the AM1 level (Tables 1 and 2) do not
reproduce a temperature-driven orientational quasiordering in MD simulations.

## Appendix B: Mulliken population analysis for MD model for ammonia

Figure 7: The same as in Fig. 2, however, for the ammonia molecule model with
Mulliken charges. $p(\theta)/\sin\theta$ has maximum at $\theta\approx
56^{\circ}$ for $T=10$ K.

Figure 8: The same as in Fig. 3, however, for the ammonia molecule model with
Mulliken charges. Solid curves in the lower panel correspond to the lattice-
model results, $N=10$ with $R=0.03$ for $\mu_{\parallel}$ (blue) and
$|\mu_{\perp}|$ (orange). Figure 9: The same as in Fig. 4, however, for the
ammonia molecule model with Mulliken charges. Solid curve corresponds to the
lattice-model result, $N=10$ with $R=0.03$ and $L(0)/(N-1)=2.90$ Å.

In this appendix, we report MD results for the ammonia molecule model with the
Mulliken charges. Namely, Figs. 7, 8, and 9 (Mulliken charges) correspond to
Figs. 2, 3, and 4 (Löwdin charges). All shown dependences are qualitatively
the same, however, the temperature range for the intermediate phase is shifted
to higher temperatures and is $90\ldots 100$ K. In Figs. 8 and 9 we also show
by solid curves the lattice-model predictions, see Sec. III, considering as an
example $N=10$. To catch the change in atomic charges we have decreased $R$
(in comparison with $R=0.07$ we used for the Löwdin charges) and set now
$R=0.03$ (solid curves). Moreover, to get the average length in angstroms
(Fig. 9), we set $L(0)/(N-1)=2.90$ Å. As can be seen from Figs. 7, 8, and 9,
overall, the lattice-model calculations again agree with the MD simulations,
although the minimum or maximum of $\mu_{\parallel}(T)$ (blue) and
$|\mu_{\perp}|(T)$ (orange) are sharper than their MD counterparts (compare to
green and red symbols, respectively).

## Appendix C: Reference data for water

Figure 10: Angles between CNT axis and dipole moments of water molecules, cf.
Figs. 2 and 7 for the ammonia case. $p(\theta)/\sin\theta$ has maximum at
$\theta\approx 35^{\circ}$ for $T=10$ K. Figure 11: $\mu_{\rm norm}$ (red) and
$\mu_{\rm tang}$ (green) for water molecule, cf. the bottom panels of Fig. 3
and Fig. 8. Figure 12: Temperature dependence of mean distances between
nearest oxygen atoms within water chain, cf. Figs. 4 and 9 for the ammonia
case.

Water molecules encapsulated in the $(6,5)$ CNT were examined in Ref.
Druchok2023 . However, the main focus of that paper was on the case $N=11$
water molecules encapsulated inside a CNT of $\approx 40$ Å. In this appendix
we show results for a longer chain of $N=35$ molecules H2O in correspondence
to the ammonia case. Thus, we considered the $(6,5)$ CNT of the length
$\approx 170$ Å which contains $N=35$ water molecules H2O, i.e., $4.86$ Å per
molecule (while $3.64$ Å per molecule in Ref. Druchok2023 ), used the SPC/E
water model with the charges $q_{\rm O}=-0.4348e$ and $q_{\rm H}=0.2174e$
optimized at the AM1 level Druchok2023 , and carried out MD simulations
arriving at results reported in Figs. 10, 11, and 12. Overall, the obtained
results are similar to the ones for a shorter chain Druchok2023  and to those
of Ref. Ma2017 . Note, however, that according to Ref. Ma2017 , the most
probable angle of water dipoles relative to the nanotube axis in the hydrogen-
bonded quasiphase is $31^{\circ}$, whereas we obtained almost $35^{\circ}$,
Fig. 10. This may be related to difference in the precise values of water
oxygen and hydrogen charges or to specific water molecule model used in
simulations, i.e., the TIP3P Ma2017  or SPC/E Druchok2023  water models. On
the other hand, comparing MD simulation data in Fig. 3 (and Fig. 8) to the
ones in Fig. 11, we conclude that $N=35$ ammonia and water molecules in the
$(6,5)$ CNT of the length $\approx 170$ Å show quite similar behavior: In both
cases a quasiphase with dipoles tending to aling along the CNT axis emerges at
intermediate temperatures.

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