#  Stabilization of molecular hydrogen-bonded chains by carbon nanotubes

Alexander V. Savin  Semenov Institute of Chemical Physics, Russian Academy of
Sciences, Moscow 119991, Russia  Plekhanov Russian University of Economics,
Moscow 117997, Russia<EMAIL_ADDRESS>Yuri S. Kivshar  Nonlinear Physics
Center, Research School of Physics, Australian National University, Canberra
ACT 2601, Australia<EMAIL_ADDRESS>
###### Abstract

We study numerically nonlinear dynamics of several types of molecular systems
composed of hydrogen-bonded chains placed inside carbon nanotubes with open
edges. We demonstrate that carbon nanotubes provide a stabilization mechanism
for quasi-one-dimensional molecular chains via the formation of their
secondary structures. In particular, a polypeptide chain (Gly)N placed inside
a carbon nanotube can form of a stable helical chain ($3_{10}$, $\alpha$,
$\pi$ and $\beta$-helix) with parallel chains of hydrogen-bonded peptide
groups. A chain of hydrogen fluoride molecules can form hydrogen-bonded zigzag
chain. We reveal that in such geometries the hydrogen-bonded chains may remain
stable even at $T=500$ K. Thus, our results suggest that the use of carbon
nanotubes with encapsulated hydrogen fluoride molecules may support high
proton conductivity operating at high temperatures.

hydrogen bonded chains, graphene nanotubes, protein helices, chain of hydrogen
fluoride molecules

##  I Introduction

One of the research activities of David Campbell has been devoted to the study
of nonlinear waves and solitons in discrete systems david0 ; david00 ,
including the pioneering results on nonlinear dynamics of polyacetylene
$[(CH)_{x}]$ david1 . Many of the discrete nonlinear models are inspired by
the physics of molecular systems. Realistic molecular systems have a complex
multi-component structure, consisting of nonlinearly interacting subsystems
(electronic, excitonic, phononic, ionic, proton). In such systems, stable
propagation of nonlinear waves (such as polarons, solitons, kinks, etc.) can
be observed when nonlinear effects compensate for linear dispersion. As a
special example, highly efficient energy and charge transfer takes place in
molecular chains with hydrogen bonds.

Molecular systems with hydrogen bonds are characterised by a relatively low
stabilisation energy (0.12 - 0.40 eV per hydrogen bond), which ensures their
high dynamic mobility. Such structures include $\alpha$-helices and
$\beta$-sheets of proteins Pauling51 ; Pauling51a , DNA double helices,
organic crystals of acetamides (e.g, acetanilide (CH3CONHC6H5)x Careri84 ;
Eilbeck84 ), crystals of aramides (Kevlar crystals Chowdhury18 ), proton
transport channels in cell membranes Nagle78 ; Nagle83 ; Kreuer96 ; Nagamani11
, crystals of alcohols, hydrogen halides Jansen87 ; Sprinborg88 , and some
other systems.

However, many of the molecular systems studied theoretically are restricted by
the assumption of low dimensionality, so they become unstable being place into
three-dimensional space. Also, molecular complexes with hydrogen-bonded chains
become unstable at high temperatures (at $T>100^{\circ}$C all hydrogen bonds
are destroyed). In this paper, we study numerically the dynamics of quasi one-
dimensional molecular systems with hydrogen-bonded chains that can be
stabilized being placed inside open carbon nanotubes. In particular, we
demonstrate that a polypeptide chain (Gly)N may take the form of a stable
helical chain ($3_{10}$, $\alpha$, $\pi$ and $\beta$-helix) with parallel
chains of hydrogen-bonded peptide groups. A chain of hydrogen fluoride
molecules placed inside a nanotube of small radius can form hydrogen-bonded
zigzag chain $\cdots$FH$\cdots$FH$\cdots$FH supporting proton transport. In
such molecular systems, the hydrogen-bonded chains retain their structure and
the interaction with the nanotube lead to their additional stabilization. For
some molecular structures the hydrogen-bonded chains will remain stable even
at $T=500$ K.

##  II Nonlinear dynamics of molecular hydrogen-bonded chains

The presence of hydrogen-bonded chains of hydroxyl groups in the molecular
system

${\cdots}{\rm O{-}H}{\cdots}{\rm O{-}H}{\cdots}{\rm O{-}H}{\cdots}{\rm
O{-}H}{\cdots}{\rm O{-}H}$ (1)

ensures its high proton conductivity in the direction of these chains Zundel00
. Proton transport across cell membranes takes place through protein proton
channels and passes through hydrogen bond chains (1) formed by amino acid
residues containing the OH hydroxyl group (serine, threonine, tyrosine)
Nagle78 . In the bacteriorhodopsin molecule, the hydrogen bond chain is formed
by tyrosine residues contained in the seven transmembrane $\alpha$-helical
segments of the molecule Merz81 . Hydrogen-bonding chains (1) act as proton
”wire” and provide an efficient pathway for rapid proton transfer Fillaux02 .

Figure 1: Milestones of the nonlinear theory of hydrogen-bonded molecular
chains and soliton-based proton transport.

The idea of proton transport through the hydrogen bond chains of water
molecules was proposed by Theodore von Grotthuss in 1804 Marx06 . According to
modern ideas, proton transport in water and ice takes place along the hydrogen
bonding lattice and is divided into two phases: the passage of the ionic
defect (H+) (proton transport between water molecules involving the exchange
of a covalent bond between O and H with a hydrogen bond) and the passage of
the orientational defect (Bjerrum defect), which returns the hydrogen bonding
chain to its initial state Bjerrum52 . Proton transport along the zigzag chain
of hydrogen bonds occurs in the form of migration of two types of defects:
ionic and orientational Nagle83 ; Onsager69 . Defect migration can be
described as the motion of topological solitons Antonchenko83 ; Pnevmatikos91
; Karpan04 ; Pang11 ; Ndjike21 . The creation and development of the theory of
proton soliton transport in the literature is shown schematically in Fig. 1.

Based on the helical structure of protein Pauling51 ; Pauling51a  and the
theory of wave excitations (excitons) of molecular crystals Davydov51 ;
Moffitt56 ; Davydov68 ; Davydov69  A.S. Davydov and N.I. Kislukha in 1973
Davydov73  (see also Davydov76 ; Davydov79 ; Davydov81 ), proposed the theory
of soliton energy transport along the alpha-helix of protein molecules. The
alpha-helix of a protein is stabilised by three parallel chains of hydrogen-
bonded HNCO peptide groups (PGs):

${\rm H{-}N{-}N{-}C{=}O}{\cdots}{\rm H{-}N{-}N{-}C{=}O}{\cdots}{\rm
H{-}N{-}N{-}C{=}O}{\cdots},$ (2)

where the lines represent valence bonds and the dots represent hydrogen bonds.
According to Davydov’s theory, these chains can transport energy released
during the hydrolysis of the adenosine triphosphate (ATP) molecule in the form
of self-localised states of vibrations of amide-I double valence bonds C=O,
which are part of PG. The self-localisation of the amid-I vibration is caused
by its non-linear interaction with longitudinal deformations of the hydrogen
bond chains (2). Using this model, it was shown that in $\alpha$-helical
protein molecules nonlinear collective self-localised excitations can
propagate without energy loss and shape change, which were later called
Davydov solitons Hyman81 ; Scott82 ; Lomdahl85 ; Scott85 ; Scott92 .

In the first papers by Davydov et al. the continuum approximation was used and
the $\alpha$-helix was treated as a continuous medium. The discrete model was
used to numerically model the dynamics of Davydov solitons in the work of
Scott et al. Hyman81 ; Scott82 . These works have led to an increased interest
in the Davydov model Lomdahl85a ; Kuprievich85 ; Brizhik88 ; Savin93 ;
Brizhik93 ; Zolotaryuk95 ; Brizhik04 . The Davydov model is still actively
used today, see Refs. Georgiev20 ; Georgiev22 ; Cruzeiro16 ; Cruzeiro20 ;
Cruzeiro22 .

The presence of hydrogen bonding chains (2) also allows for the possibility of
external electron transfer in $\alpha$-helical proteins in the form of an
electrosoliton (a bound state of a localised electron with the region of helix
deformation resulting from the interaction of the electron with the PG chains)
Davydov79a ; Davydov84 ; Davydov91 .

Practically all molecular complexes with hydrogen bonding chains are no longer
stable at high temperatures (at $T>100^{\circ}$C all hydrogen bonding chains
are destroyed). This complicates the use of such molecular systems in
nanotechnology. In this work it will be shown that the use of complexes of
such molecular systems with carbon nanotubes can significantly increase the
stability of the hydrogen bond chains. For this purpose, quasi one-dimensional
molecular systems ($\alpha$-helix of protein, zigzag chain of hydrogen
fluoride molecules) will be placed in carbon nanotubes of a certain radius. In
this case, the hydrogen bonding chains will retain their structure and the
interaction with the nanotube will lead to their additional stabilisation,
which will allow them to be used for energy and charge transfer over a wider
temperature range. In such composite systems the hydrogen bonding chains will
remain stable even at $T=500$ K.

Figure 2: Helical structures of the peptide chain (Gly)52: (a) $3_{10}$-helix
(stationary structure energy $E=16.26$ eV, hydrogen bond energy $E_{\rm
hb}=0.175$ eV, length $L=9.66$ nm); (b) $\alpha$ helix ($E=12.00$, $E_{\rm
hb}=0.258$, $L=7.64$); (c) $\pi$ helix ($E=11.32$, $E_{\rm hb}=0.268$,
$L=6.01$); (d) $\beta$ helix ($E=10.03$, $E_{\rm hb}=0.268$, $L=4.67$). Carbon
atoms are shown in green, nitrogen atoms in blue, oxygen atoms in red and
hydrogen atoms in white. Hydrogen bonds are shown as thin lines. Hydrogen
atoms of $\alpha$-carbon atoms are not shown.

##  III Helices of proteins

To analyse helical structures, consider a polypeptide chain (Gly)N consisting
of $N$ peptide groups. The chemical formula of such a chain is

${\rm C}_{\alpha}{\rm H}_{3}{\rm-HNCO{-(}C}_{\alpha}{\rm
H}_{2}{\rm{-HNCO-)}}_{N-2}{\rm C}_{\alpha}{\rm H}_{2}{\rm-H}_{2}{\rm NCO},$
(3)

where two hydrogen atoms are attached to each $\alpha$-carbon atom (in order
not to leave free valences, three hydrogen atoms are attached to the first
$\alpha$-carbon atom and one hydrogen atom is attached to the nitrogen atom of
the last peptide group).

When modelling a polyglycine chain (3), it is convenient to consider
$\alpha$-carbon atoms with attached hydrogen atoms as one united atom
$C_{\alpha}$. In this case each chain link will consist of 5 atoms Cα, C, O,
N, H with coordinates $\\{{\bf u}_{n,i}\\}_{i=1}^{5}$, where $n$ is the link
number of the polypeptide chain and $i$ is the atom number. The atomic masses
are $M_{n,1}=14m_{p}$, $M_{n,2}=12m_{p}$, $M_{n,3}=16m_{p}$,
$M_{n,4}=14m_{p}$, $M_{n,5}=m_{p}$ (for end links $M_{1,1}=15m_{p}$,
$M_{N,4}=15m_{p}$), where $m_{p}=1.6603\times 10^{-27}$kg – the mass of a
proton. To describe the interatomic interaction, we will use the AMBER general
force field for organic molecules (version 2.1, April 2016) Amber .

The Hamiltonian of a polypeptide chain has the form

$H=\sum_{n=1}^{N}\sum_{i=1}^{5}\left[\frac{1}{2}(M_{n,i}\dot{\bf
u}_{n,i},\dot{\bf u}_{n,i})+P(\\{{\bf u}_{n,i}\\}_{n=1,i=1}^{N,~{}5})\right],$
(4)

where the first sum defines the kinetic energy (the three-dimensional vector
${\bf u}_{n,i}=(x_{n,i},y_{n,i},z_{n,i})$ defines the coordinates of the
$i$-th atom of the $n$-th chain link), and the last sum – the potential energy
of the chain found using the AMBER force field.

To find the stationary state of the chain, we need to numerically solve the
minimum potential energy problem

$P\rightarrow\min:\\{{\bf u}_{n,i}\\}_{n=1,i=1}^{N,~{}5}.$ (5)

The numerical solution of the problem (5) showed that the polypeptide chain
(3) has four stationary states as a helix chain: $3_{10}$-helix,
$\alpha$-helix, $\pi$-helix and $\beta$-helix – see Fig. 2. Let us assume that
the peptide groups $n$ and $k$ ($n<k$) form a hydrogen bond if the non-valent
interaction energy of the corresponding chain links is

$E_{n,k}=U(\\{{\bf u}_{n,i}\\}_{i=1}^{5},\\{{\bf
u}_{k,j}\\}_{j=1}^{5})<-0.16~{}{\rm eV}.$

Each helix is characterised by the steady state energy $E$, the number of
hydrogen bond chains $N_{\rm sp}$, the number of hydrogen bonds $N_{\rm hb}$,
the average bond energy

$E_{\rm hb}=-\sum_{l=1}^{N_{\rm hb}}E_{n_{l},k_{l}}/N_{\rm hb}$

and the length of the helix (the distance between the end alpha carbon atoms
$L=|{\bf u}_{N,1}-{\bf u}_{1,1}|$).

Table 1:  Energy $E$, number of hydrogen bond chains $N_{\rm sp}$, number of hydrogen bonds $N_{\rm hb}$, average hydrogen bond energy $E_{\rm hb}$ and chain end distance $L$ for different stationary states (conformations) of the polypeptide chain (Gly)73.  conformation |  $E$ (eV) |  $N_{\rm sp}$ |  $N_{\rm hb}$ |  $E_{\rm hb}$ (eV) |  $L$ (nm)
---|---|---|---|---|---
$3_{10}$-helix | 22.56 | 2 | 71 | 0.1704 | 13.59
$\alpha$-helix | 16.22 | 3 | 70 | 0.2583 | 10.79
$\pi$-helix | 14.93 | 4 | 69 | 0.2695 | 8.46
$\beta$-helix | 14.00 | 5 | 70 | 0.2560 | 5.94
globule | 13.33 | - | 60 | 0.2341 | 0.57

The chain of $N=73$ peptide groups was used to analyse the helical structures.
The values of $E$, $N_{\rm sp}$, $N_{\rm hb}$, $E_{\rm hb}$, and $L$ for
different stationary states (conformations) of the polypeptide chain (Cly)73
are given in Table 1. As can be seen from the table, the most energetically
unfavourable structure is the $3_{10}$ helix. Such a helix is highly
stretched, it has two chains of weakened hydrogen bonds – see Fig. 2. Further
more favourable in energy are the $\alpha$, $\pi$, $\beta$ helix and globule
structures.

Let us consider the dynamics of helical structures. To do this, we place the
polypeptide chain in a Langevin thermostat of temperature $T$ and simulate the
dynamics of the system numerically during the time $t=100$ ns. To do this, we
numerically integrate the system of Langevin equations

$\displaystyle M_{n,i}\ddot{\bf u}_{n,i}=-\frac{\partial H}{\partial{\bf
u}_{n,i}}-\Gamma M_{n,i}\dot{\bf u}_{n,i}-\Xi_{n,i},$ (6)
$\displaystyle~{}n=1,...,N,~{}i=1,...,5,$

where $\Gamma=1/t_{r}$ is the friction coefficient,
$\Xi_{n,i}=\\{\xi_{n,i,k}\\}_{k=1}^{3}$ is 3-dimensional vector of normally
distributed random Langevin forces with the following correlations:

$\langle\xi_{n_{1},i,k}(t_{1})\xi_{n_{2},j,l}(t_{2})\rangle=2M_{n_{1},i}k_{B}T\Gamma\delta_{n_{1},n_{2}}\delta_{ij}\delta_{kl}\delta(t_{1}-t_{2}).$

Here $M_{n,i}$ is mass of $i$-th atom of $n$-th segment of polypeptide chain,
$k_{B}$ is Boltzmann constant, $T$ is temperature of the Langevin thermostat,
numbers $n_{1},n_{2}=1,...,N$, $i,j=1,...,5$, $k,l=1,2,3$. The parameter
$t_{r}=10$ ps characterizes the intensity of energy exchange between molecular
system and the thermostat.

The equations of motion Eq. (6) are solved numerically using the velocity
Verlet method Swope82 . A time step of 1 fs is used in the simulations, since
further reduction of the time step has no appreciable effect on the results.

Let us numerically integrate the system of equations of motion (6) with the
initial condition corresponding to the stationary state of the polypeptide
chain. After the dynamics of the molecular system reaches the steady state, we
will find the time averages of the system energy $\bar{E}(T)$, the number of
hydrogen bonds $\bar{N}_{\rm hb}(T)$ and chain length $\bar{L}(T)$.

The state of the system can be conveniently characterized by its dimensionless
heat capacity

$c=\frac{1}{15Nk_{B}}\frac{d\bar{E}(T)}{dT},$ (7)

the normalized number of hydrogen bonds $n_{\rm hb}=\bar{N}_{\rm hb}(T)/N$ and
chain length $\bar{L}(T)$. The dependence of these quantities on temperature
is shown in Fig. 3.

Figure 3: Dependence of (a) the normalised number of hydrogen bonds $n_{\rm
hb}$, (b) the dimensionless heat capacity $c$ and (c) the chain end spacing
$\bar{L}$ for a polypeptide chain (Gly)73 originally shaped as a
$\alpha$-helix. The vertical lines indicate the temperature values of 95, 235
and 285K.

The $3_{10}$-helix is the most unstable. At temperature $T=10$ K it changes to
$\pi$-helix. The alpha helix of the chain remains stable at $T<T_{1}=95$ K, at
$T_{1}<T<T_{2}=235$ K the $\alpha$-helix changes into $\pi$-helix, and at
$T_{2}<T<T_{3}=285$ K into $\beta$-helix. At $T\geq T_{3}$, the chain
transforms into a globule shape – see Fig. 3. The $\pi$-helix remains stable
at $T<T_{2}$, at $T=T_{2}$ it changes into a $\beta$-helix. The beta helix is
stable at $T<T_{3}$, at $T>T_{3}$ the chain adopts the globule form. The
transitions between the forms of the helix occur without changing the heat
capacity of the molecular system, i.e. these are second order phase
transitions. The increase in heat capacity occurs only when the chain
transitions to the globular form. The temperature $T=T_{3}$ can be considered
as the melting point of the $\beta$-helix. The transition of a helix into a
globule is first order phase transition, here the dimensionless heat capacity
increases from $c=1$ to $c=1.2$.

Thus, in an isolated polyglycine peptide chain, regular chains of hydrogen
bonds (2) can only exist at low temperatures $T<285$ K.

##  IV Conformations of a peptide chain inside a carbon nanotube

Polypeptide molecules tend to self-insertion into carbon nanotubes with open
ends. Inside the nanotube, polypeptide atoms are able to fully realise non-
valent interactions with its surface Liu05 ; Xiu13 ; Zhang14 . This is
particularly significant for hydrophobic polypeptides such as (Gly)N, (Ala)N,
(Val)N, (Ile)N. Let us determine what conformations the polypeptide chain
(Gly)N inside a single-walled carbon nanotube with chirality index $(n,m)$
(CNT(n,m)) can take.

To describe dynamics of CNT we present the nanotube Hamiltonian in the form

$H_{\rm c}=\sum_{n=1}^{N_{\rm c}}\left[\frac{1}{2}M_{n,0}(\dot{\bf
v}_{n},\dot{\bf v}_{n})+Q_{n}\right],$ (8)

where $N_{\rm c}$ is the number of atoms in the CNT, $M_{n,0}$ is the mass of
$n$-th carbon atom, ${\bf v}_{n}=\\{v_{n,i}(t)\\}_{i=1}^{3}$ is the radius
vector of $n$-th atom at the time $t$. The term $Q_{n}$ describes the
interaction energy of the atom with index $n$ with neighboring atoms.

To describe the carbon-carbon valence interactions, let us use a standard set
molecular dynamics potentials Savin10 ; Savin17 . We consider a hydrogen-
terminated nanotube with open ends, where edge atoms correspond to the
molecular group CH. We will consider such a group as a single effective
particle at the location of the carbon atom. Therefore, in our model of open
CNT we take the mass of atoms inside the nanotube as $M_{n,0}=12m_{p}$, and
for the edge atoms as $M_{n,0}=13m_{p}$.

The valence bond between two neighboring carbon atoms $n$ and $k$ can be
described by the Morse potential

$U_{1}({\bf v}_{n},{\bf
v}_{k})=\epsilon_{1}\\{\exp[-\alpha(r_{nk}-r_{1})]-1\\}^{2},$ (9)

where $r_{nk}=|{\bf v}_{n}-{\bf v}_{k}|$, $\epsilon_{1}=4.9632$ eV is the
valence bond energy, and $r_{1}=1.418$ Å is the equilibrium valence bond
length.

The valence angle deformation energy between three adjacent carbon atoms $n$,
$k$, and $l$ can be described by the potential

$U_{2}({\bf v}_{n},{\bf v}_{k},{\bf
v}_{l})=\epsilon_{2}(\cos\varphi-\cos\varphi_{0})^{2},$ (10)

where $\cos\varphi=({\bf v}_{n}-{\bf v}_{k},{\bf v}_{l}-{\bf
v}_{k})/r_{nk}r_{kl}$, and $\varphi_{0}=2\pi/3$ is the equilibrium valent
angle. Parameters $\alpha=1.7889$ Å-1 and $\epsilon_{2}=1.3143$ eV can be
found from the small-amplitude oscillations spectrum of the graphene sheet
Savin08 . Valence bonds between four adjacent carbon atoms $n$, $m$, $k$, and
$l$ constitute torsion angles, whose potential energy can be defined as

$U_{3}(\phi)=\epsilon_{3}(1-\cos\phi),$ (11)

where $\phi$ is the corresponding torsion angle ($\phi=0$ is the equilibrium
value of the angle) and $\epsilon_{3}=0.499$ eV. A detailed discussion about
the choice of the interatomic potential parameters can be found in Savin10 .

We describe the interaction of polypeptide chain atoms with nanotube atoms
using Lennard-Jones potentials with parameters from the AMBER force field.

Table 2:  Values of the nanotube radius $R$, the helix type of the polypeptide chain, the energy $E$ of the stationary state of the complex (Gly)${}_{73}{\in}$CNT(n,m), the interaction energy $E_{\rm i}$ of the polypeptide chain with the nanotube, and the average hydrogen bonding energy $E_{\rm hb}$ for different nanotubes with chirality index $(n,m)$ (nanotube length $L_{\rm cnt}=16.1$ nm).  $(n,m)$ |  $R$ (Å) |  helix |  $E$ (eV) |  $-E_{\rm i}$ (eV) |  $E_{\rm hb}$ (eV)
---|---|---|---|---|---
(12,0) | 4.74 | $3_{10}$ | 221.53 | 39.16 | 0.2756
(7,7) | 4.76 | $3_{10}$ | 219.25 | 39.66 | 0.2744
(13,0) | 5.13 | $3_{10}$ | 204.16 | 37.33 | 0.2005
$\alpha$ | 203.88 | 33.39 | 0.2711
(8,8) | 5.43 | $\alpha$ | 188.93 | 34.06 | 0.2618
(14,0) | 5.52 | $\alpha$ | 186.57 | 33.65 | 0.2584
(15,0) | 5.91 | $\alpha$ | 178.93 | 28.74 | 0.2314
$\pi$ | 175.29 | 30.87 | 0.2705
(9,9) | 6.11 | $\alpha$ | 175.68 | 25.53 | 0.2481
$\pi$ | 170.79 | 29.12 | 0.2641
$\beta$ | 171.71 | 30.33 | 0.2588
(16,0) | 6.29 | $\alpha$ | 172.18 | 24.25 | 0.2558
$\pi$ | 167.92 | 27.03 | 0.2614
$\beta$ | 167.64 | 28.63 | 0.2532
(17,0) | 6.68 | $\alpha$ | 164.70 | 21.74 | 0.2568
$\pi$ | 161.99 | 22.92 | 0.2652
$\beta$ | 157.05 | 27.68 | 0.2708
(10,10) | 6.78 | $\alpha$ | 162.77 | 21.06 | 0.2555
$\pi$ | 160.16 | 22.15 | 0.2658
$\beta$ | 155.55 | 26.46 | 0.2700
(18,0) | 7.07 | $\alpha$ | 157.64 | 19.89 | 0.2587
$\pi$ | 155.23 | 20.83 | 0.2659
$\beta$ | 152.06 | 23.31 | 0.2653
(11,11) | 7.46 | $\alpha$ | 150.77 | 18.48 | 0.2570
$\pi$ | 148.85 | 19.33 | 0.2660
$\beta$ | 146.56 | 20.24 | 0.2636
Figure 4: State of the polypeptide chain (Gly)73 located in the nanotube
(13,0) with open edges: (a) $3_{10}$-helix ($T=10$ K), (b) $\alpha$-helix at
$T=10$ K, and (c) $T=300$K (the peptide chain is $3_{10}$-helix). For ease of
review, only one half of the nanotube atoms is shown (nanotube atoms are shown
in grey). The length of the nanotube is $L_{\rm cnt}=16.14$ nm, the number of
carbon atoms is $N_{\rm c}=1976$.

The Hamiltonian of the molecular system (Gly)${}_{N}{\in}$CNT will have the
form

$\displaystyle H=\sum_{n=1}^{N_{\rm c}}\frac{1}{2}(M_{n,0}\dot{\bf
v}_{n},\dot{\bf
v}_{n})+\sum_{n=1}^{N}\sum_{i=1}^{5}\frac{1}{2}(M_{n,i}\dot{\bf
u}_{n,i},\dot{\bf u}_{n,i})$ $\displaystyle+E(\\{{\bf v}_{n}\\}_{n=1}^{N_{\rm
c}},\\{{\bf u}_{n,i}\\}_{n=1,i=1}^{N,~{}5}),~{}~{}~{}$ (12)

where the first summand defines the kinetic energy of the carbon nanotube
($N_{\rm c}$ is the number of atoms of the nanotube), the second – the kinetic
energy of the polypeptide chain, the third – the interaction energy of the
atoms of the system.

To find the stationary state of the two-component system (Gly)${}_{N}{\in}$CNT
it is necessary to solve numerically the problem on the minimum of the
interaction energy

$E\rightarrow\min:\\{{\bf v}_{n}\\}_{n=1}^{N_{\rm c}},\\{{\bf
u}_{n,i}\\}_{n=1,i=1}^{N,~{}5}.$ (13)

Let us take the polypeptide chain (Gly)73 and place it inside a nanotube of
length $L_{\rm cnt}=16.1$ nm. The numerical solution of the problem (13) shows
that the polypeptide chain inside the nanotube takes the form of a helix, the
type of which depends on the radius of the nanotube $R$ – see Table. 2. Thus
at $R<5$ Å , the chain inside the nanotube can take only one stationary state
– helix 310. Two stationary states of the chain helix $3_{10}$ and
$\alpha$-helix are possible inside the nanotube (13,0) – see Fig. 4 (a),(b).
The $\alpha$-helix can exist in all nanotubes with radius $R>5.1$ Å,
$\pi$-helix – in nanotubes with $R>5.9$ Å, $\beta$-helix – in nanotubes with
$R>6.1$ Å. Note that in nanotubes with $R<5$ Å , hydrogen bonds become strong
for 310 helix, the strongest interaction between the polypeptide chain and the
nanotube occurs here (the smaller the radius of the nanotube, the greater the
interaction energy).

Let us consider the dynamics of the (Gly)${}_{73}{\in}$CNT complex. To do
this, we place the nanotube in a Langevin thermostat of temperature $T$ (the
thermalisation of the polypeptide chain inside the CNT will occur through its
interaction with the nanotube) and numerically simulate the dynamics of the
molecular complex during the time $t=20$ ns. To do this, we numerically
integrate the system of Langevin equations

$\displaystyle M_{n,0}\ddot{\bf v}_{n}=-\frac{\partial H}{\partial{\bf
v}_{n}}-\Gamma M_{n,0}\dot{\bf v}_{n}-\Theta_{n},$ (14) $\displaystyle
n=1,...,N_{\rm c}$ $\displaystyle M_{n,i}\ddot{\bf u}_{n,i}=-\frac{\partial
H}{\partial{\bf u}_{n,i}},$ (15) $\displaystyle n=1,...,N,~{}i=1,...,5,$

where $\Theta_{n}=\\{\eta_{n,k}\\}_{k=1}^{3}$ is 3-dimensional vector of
normally distributed random Langevin forces with the following correlations:

$\langle\eta_{n_{1},k}(t_{1})\eta_{n_{2},l}(t_{2})\rangle=2M_{n_{1},0}k_{B}T\Gamma\delta_{n_{1},n_{2}}\delta_{kl}\delta(t_{1}-t_{2}).$

We numerically integrate the system of equations of motion (14), (15) with the
initial condition corresponding to the solution of the minimum problem (13).
After the dynamics of the molecular complex reaches the steady state, we will
find the time averages of the complex energy $\bar{E}(T)$, the number of
hydrogen bonds $\bar{N}_{\rm hb}(T)$ and peptide chain length $\bar{L}(T)$.

The state of the molecular complex can be characterized by its dimensionless
heat capacity

$c=\frac{1}{3(N_{\rm c}+5N)k_{B}}\frac{d\bar{E}(T)}{dT},$ (16)

the normalized number of hydrogen bonds $n_{\rm hb}=\bar{N}_{\rm hb}(T)/N$ and
peptide chain length $\bar{L}(T)$. The dependence of this quantities on
temperature for different nanotubes is shown in Fig. 5 and 6.

Figure 5: Dependence of (a) the normalised number of hydrogen bonds $n_{\rm
hb}$, (b) the dimensionless heat capacity $c$ and (c) the distance between the
ends of the peptide chain $\bar{L}$ for the molecular complex
(Gly)${}_{73}{\in}$CNT(13,0). Curves 1, 2, 3 show dependencies for the
$3_{10}$ helix chain, curves 4, 5, 6 – for the $\alpha$ helix chain. The
length of the CNT is $L_{\rm cnt}=16.14$ nm, the number of atoms $N_{\rm
c}=1976$, at $T=75$K there is a transition from $\alpha$-helix to
$3_{10}$-helix.  Figure 6: Dependence of (a) the normalized number of hydrogen
bonds $n_{\rm hb}$, (b) the dimensionless heat capacity $c$ and (c) the
distance between the ends of the peptide chain $\bar{L}$ for the molecular
complex (Gly)${}_{73}{\in}$CNT for nanotube (8,8) (curves 1, 4, 7), (9,9)
(curves 2, 5, 8) and (10,10) (curves 3, 6, 9). In nanotube (8,8), the peptide
chain is $\alpha$-helix, in (9,9) is $\pi$-helix, in (10,10) is $\beta$-helix.
Figure 7: Configuration of peptide chain (Gly)73 in (a) carbon nanotube (8,8)
($L_{\rm cnt}=11$ nm), (b) nanotube (9,9) ($L_{\rm cnt}=8.8$) and (c) nanotube
(10,10) ($L_{\rm cnt}=7.3$). The temperature is $T=300$K. In nanotube (8,8)
the chain is in the form of a $\alpha$-helix, in nanotube (9,9) in the form of
a $\pi$-helix, and in (10,10) in the form of a $\beta$-helix. For convenience,
some carbon atoms of the nanotubes are not shown.

Numerical integration of the system of equations of motion (14), (15) showed
that at $T<500$ K there is only one basic helix type for each nanotube. For
CNT(13,0) the $3_{10}$ helix will be the basic helix. Here, at $T<75$ K, two
forms of helix can exist: $3_{10}$ and $\alpha$ helix, and at $T>75$ K, only
the $3_{10}$ helix can exist (thermal fluctuations lead to the transition of
the $\alpha$ helix to the $3_{10}$ helix) – see Fig. 4 and Fig. 5. The
$3_{10}$ helix exists at all temperatures $T<500$ K. As the temperature
increases, only partial breaking of the edge hydrogen bonds occurs, leading to
a decrease in $n_{\rm hb}$ and an increase in the peptide chain length
$\bar{L}$. The dimensionless heat capacity of the (Gly)${}_{73}{\in}$CNT(13,0)
complex remains constant at $c=1$.

Note that for an isolated peptide chain, the $3_{10}$ helix form is the most
unstable. Even at $T=10$ K, thermal fluctuations lead to a rapid transition to
the $\pi$-helix, and at $T=285$ K the chain melts (the chain changes from
helical to molten globule form).

In nanotube (8,8) the ground state of the peptide chain will always be
$\alpha$-helix, in CNT (9,9) – $\pi$-helix and in CNT (10,10) – $\beta$-helix.
The appearance of these helical states of the peptide chain in nanotubes with
chirality indices (8,8), (9,9) and (10,10) is shown in Fig. 7. All these
helixes are stable to thermal fluctuations at $T<500$ K. With increasing
temperature there is only a small decrease in the number of hydrogen bonds –
see Fig. 6 (a). At the same time, the dimensionless heat capacity always
remains equal to one and the length of the helix is almost unchanged – see
Fig. 6 (b) and (c).

Thus, placing a polypeptide chain in a nanotube of small radius will stabilise
its helical shape. For the simplest polypeptide chain (Gly)N, the most stable
to thermal fluctuations will be the $\alpha$-helix lying inside the nanotube
(9,9). Here the $\alpha$-helix is conserved at all temperatures $T<500$ K.
Note that such stabilisation of the helical shape is impossible for an
isolated polypeptide chain. Note that the optimal nanotube radius for helix
stabilisation depends on the size of the amino acid residues. For larger Ala,
Val, Ile residues, the best stabilisation will occur in nanotubes of larger
radius.

##  V Hydrogen-bonded chains of hydrogen fluoride molecules

Experimental and theoretical studies of water-filled carbon nanotubes have
shown that water molecules can enter the interior of the open nanotubes and
form extended hydrogen bond chains Hummer01 ; Dellago03 ; Kofinger11 ;
Mendonca19 ; Druchok23 . Small-diameter nanotubes with a chain of water
molecules inside can act as proton-conducting channels Chen13 ; Ma20 .

Carbon nanotubes are hydrophobic and the presence of water molecules inside
them is not energetically favourable. Inside the nanotube, water molecules
cannot organise a three-dimensional lattice of hydrogen bonds. For example,
inside a CNT(6,6) (diameter $D=0.80$ nm) only one chain of water molecules can
be found, in which each molecule participates in the formation of two hydrogen
bonds (in the bulk phase each molecule participates in the formation of four
bonds) Hanasaki08 .

Hydrogen fluoride molecules FH can also form zigzag chains of hydrogen bonds
(FH)∞:

${\cdots}{\rm F{-}H}{\cdots}{\rm F{-}H}{\cdots}{\rm F{-}H}{\cdots}{\rm
F{-}H}{\cdots}{\rm F{-}H},$ (17)

which are similar to zigzag chains of hydroxyl groups (1). At temperature
$T<189.6$ K, hydrogen fluoride has a crystal structure formed by parallel flat
hydrogen bond chains Atoji54 . Note that a separate zigzag chain of hydrogen
bonds (17) becomes unstable in three-dimensional space. The chain breaks down
into cyclic chains of small size, the most stable being chains of 6 molecules
in the shape of a regular hexagon Orabi20 . Hydrogen bond chains (17) can be
stabilised by placing them inside the CNT, i.e. forming a
(FH)${}_{\infty}{\in}$CNT structure.

In each chain (17), one hydrogen fluoride molecule participates in the
formation of two hydrogen bonds. Analysis of the interaction of the
F–H${\cdots}$F–H dimer with the CNT shows that it is energetically more
favourable for the hydrogen fluoride dimer to be inside the nanotube than
outside Roztoczynska16 . Therefore, (FH)N chains can easily be placed inside
open CNTs to form stable (FH)${}_{N}{\in}$CNT structures ($N$ – number of
hydrogen fluoride molecules). Keeping hydrogen fluoride inside nanotubes may
help to avoid problems associated with its high toxicity Gtari18 .

Let us simulate molecular complexes (FH)${}_{N}{\in}$CNT. We describe the
valence bond deformation of the two-atom molecule FH by a harmonic potential

$V(\rho)=\frac{1}{2}K(\rho-\rho_{0})^{2},$ (18)

where $\rho$ and $\rho_{0}=0.929$ Å are the current and equilibrium bond
lengths, $K=444.3$ N/m is the bond stiffness.

The interaction of FH molecules is conveniently described by the potential
12-6-1 Cournoyer84 ; Orabi20

$U=\sum_{i=1}^{3}\sum_{j=1}^{3}\kappa
q_{i}q_{j}/r_{ij}+4\epsilon[(\sigma/r)^{12}-(\sigma/r)^{6}],$ (19)

where the first sum defines the Coulomb interaction between systems of point
charges approximating the charge distribution in a pair of interacting FH
molecules ($r_{ij}$ – the distance between the charge $q_{i}$ of the first
molecule and the charge $q_{j}$ of the second molecule), $r$ – the distance
between the centres of the fluorine atoms, the coefficient $\kappa=14.400611$
eVÅ/$e^{2}$. Two positive charges $q_{2}$, $q_{3}$ are on atoms F and H, the
negative charge $q_{1}=-(q_{2}+q_{3})$ is at a distance $r_{1}$ from atom F on
the interval [FH]. If for the potential (19) does not fix the position of the
two charges on the atoms of the molecule, the potential will be given by seven
parameters: two charges, three distances $\\{r_{i}\\}_{i=1}^{3}$, specifying
the position of the charges on the line connecting atoms F and H, and two
parameters $\epsilon$ and $\sigma$ of the Lenard-Jones potential.

Figure 8: Stationary state of the hydrogen bond chain (FH)N inside an open
single-walled CNT with chirality index $(n,m)$: (a) (FH)${}_{32}{\in}$CNT(5,5)
(nanotube chemical formula C600H20); (b) (FH)${}_{35}{\in}$CNT(9,0) (C648H18);
(c) (FH)${}_{38}{\in}$CNT(10,0) (C720H20); (d) (FH)${}_{36}{\in}$CNT(6,6)
(C720H24); (e) (FH)${}_{44}{\in}$CNT(11,0) (C792H22); (f)
(FH)${}_{106}{\in}$CNT(7,7) (C840H28). For ease of review, some atoms of the
nanotubes are not shown.

The relative simplicity and a sufficiently large number of free parameters of
the 12-6-1 potential make it very convenient for modelling the dynamics of
multi-molecular complexes of two-atom FH molecules. The FH molecule usually
participates in the formation of two strong hydrogen bonds, polarising
significantly in the process. We will assume that when the valence bond length
$r=|{\rm FH}|$ changes, the charge arrangement distances from the centre of
the fluorine atom also change proportionally. To model the dynamics of the
hydrogen bond chain (17) it is best to use the potential (19) with the
parameters

$\displaystyle q_{1}=-0.6397e,~{}~{}q_{2}=0.6159e,~{}~{}q_{3}=0.0238e,$
$\displaystyle
r_{1}=0.2577\rho/\rho_{0},~{}~{}r_{2}=0.9356\rho/\rho_{0},~{}~{}r_{3}=-1.6237\rho/\rho_{0},$
$\displaystyle\epsilon=0.0079778~{}{\rm eV},~{}~{}\sigma=2.837109~{}{\rm\AA},$

where $\rho$ and $\rho_{0}$ are the actual and equilibrium lengths of the F-H
valence bond, $e$ is the electron charge (dimension $[r_{i}]=$Å). With this
set of parameters, the FH molecule will have the correct values for the dipole
and quadrupole moments

$\mu=\sum_{i=1}^{3}q_{i}r_{i},~{}~{}Q=\sum_{i=1}^{3}q_{i}(r_{i}-r_{m})^{2}$

($r_{m}$ – the distance from the centre of gravity of the molecule to the
centre of the fluorine atom). The energy and configuration of ground-state of
the dimer (FH)2 will best agree with the experimental data Dyke72  and the
results of the quantum calculations Nemuchin92 .

Consider open armchair and zigzag nanotubes with hydrogen atoms attached to
the edge carbon atoms – see Fig. 8. To find the ground state
(FH)${}_{N}{\in}$CNT it is necessary to solve numerically the problem on the
minimum of potential energy of the system (13). By choosing the initial
position of the molecules, we can obtain all possible stationary states of the
two-component system. At the position of the chain (FH)N along the outer
surface of the nanotube we obtain a flat zigzag chain of hydrogen bonds with
pitch $a=|{\rm FF}|=2.49$ Å and zigzag angle $\alpha=\angle{\rm
FFF}=117^{\circ}$. Torsion angle of the chain (dihedral angle formed by four
consecutive fluorine atoms) $\phi=\angle{\rm FFFF}=180^{\circ}$.

Table 3:  Values of nanotube diameter $D$, pitch $a$ and angle $\alpha$ of zigzag and angle of torsion $\phi$ of hydrogen bonding chain (FH)∞ inside CNT(n,m).  $(n,m)$ |  $D$ (Å) |  $a$ (Å) |  $\alpha$ (∘) |  $\phi$ (∘)
---|---|---|---|---
(5,5) | 6.8 | 2.48 | 138 | 180
(9,0) | 7.1 | 2.48 | 132 | 180
(10,0) | 7.8 | 2.47 | 114 | 180
(6,6) | 8.1 | 2.47 | 113 | 180
(11,0) | 8.6 | 2.47 | 108 | 120
(7,7) | 9.5 | 2.60 | 88 | 93

The numerical solution of the minimum energy problem (13) showed that the
hydrogen bond chain (17) retains a flat zigzag shape when placed inside
nanotubes of diameter $D<0.85$ nm – see Fig. 8 and Tab. 3. The chain pitch $a$
is weakly dependent on the nanotube diameter, the chain zigzag angle $\alpha$
decreases monotonically with increasing diameter, the chain torsion angle
$\phi=180^{\circ}$ at $D<0.85$ nm. In the nanotube (11,0) ($D=0.86$ nm) the
chain takes the form of a three-dimensional helix (torsion angle
$\phi<180^{\circ}$) running along the nanotube surface.

Figure 9: Stationary state of the hydrogen bond chain (FH)N inside CNT(n,m)
with narrowed open edges: (a) (FH)36@CNT(10,0) (nanotube chemical formula
C430H12), ground state; (b) a chain with a positive orientational defect; (c)
a chain with a negative orientational defect; (d) (FH)38@CNT(6,6) (C402H20),
ground state; (e) a chain with a positive orientational defect; (f) a chain
with a negative orientational defect (defects are located in the centre of the
chain). For simplicity, some atoms of the nanotubes are not shown.

The modelling shows that the hydrogen bond chain closest in structure to the
hydrogen bond chain of the hydroxyl groups (1) is formed by hydrogen fluoride
molecules inside nanotubes with chirality indices (6,6) and (10,0). We will
therefore consider these complexes further.

To avoid the possibility of hydrogen fluoride molecules escaping from the
nanotubes, it is better to narrow their open edges as shown in Fig. 9. The
small diameter of the edge openings ($D=0.47$ and 0.42 nm) will prevent a
large fluorine atom from leaving, but will preserve the possibility of protons
leaving and entering. So we get a chain of hydrogen bonds encapsulated in a
carbon nanotube: (FH)N@CNT. Note that in 2016 endohedral fullerene HF@C60 was
synthesised containing Buckminsterfullerene inside a molecule of hydrogen
fluoride Krachmalnicoff16 .

Proton transport along the zigzag chain of hydrogen bonds proceeds by
migration of two types of defects: ionic and orientational Nagle83 ; Onsager69
. In the first stage of transport, the proton moves as an ionic defect. There
is a sequential movement of hydrogen atoms along the hydrogen bond lines from
one fluorine atom to the neighbouring one, transferring the chain from the
$\cdots$FH$\cdots$FH$\cdots$FH$\cdots$FH$\cdots$ to the
$\cdots$HF$\cdots$HF$\cdots$HF$\cdots$HF$\cdots$. In the second stage, an
orientation defect (sequential rotation of the FH chain molecules by the
zigzag angle $\alpha$) must pass through the chain to bring the chain back to
its original orientation.

Note that the used force field (18), (19) allows to model the formation and
motion of orientational defects.

The chain of hydrogen bonds inside the nanotube is bistable, all molecules of
the chain can be orientated either from left to right or from right to left.
An orientation defect occurs when the molecules of the chain are directed in
different directions in the first and second parts of the chain. When the
molecules face each other, an excess density of protons appears in the defect
localisation region, so such a defect is called a positive defect – see Fig. 9
(b) and (e). When the molecules are directed away from each other in the
defect localisation region, the proton density decreases, so such a defect is
called negative – see Fig. 9 (c) and (f).

The stationary orientational defects of the hydrogen bonding chain (FH)N
inside CNT (6,6) and (10,0) are shown in Fig. 9 (b, c, e, f). As can be seen
from the figure, the orientational defects are localised at 3-4 chain links.
In nanotube (6,6), the energy of positive defect (the difference of the energy
of the chain with defect from the energy of the ground state of the chain) is
$\Delta E_{+}=0.56$ eV, the energy of negative defect is $\Delta E_{-}=0.51$
eV. In the (10,0) nanotube, the orientational defect energies of the chain
$\Delta E_{+}=0.06$ eV, $\Delta E_{-}=0.91$ eV.

To test the stability of a chain of hydrogen bonds (FH)N placed inside a CNT,
a molecular dynamics simulation was carried out on a chain of $N=177$ FH
molecules placed inside a CNT(6,6) with narrowed open edges. With a length of
$L=38.7$ nm, such a nanotube is presented by the formula C3620H20. A zigzag
chain of hydrogen bonds (FH)177 completely fills the internal cavity of the
nanotube. Such a nanotube of length $L=8.7$ nm is shown in Fig. 9 (d).

Numerical modeling of the system of equations of motion (14), (15) showed that
the chain of hydrogen bonds inside the nanotube, complex (FH)177@CNT(6,6), at
all considered temperatures $T<500$ K remains stable – it always remains
inside and maintains a zigzag shape. At $T<400$ K, the chain of hydrogen bonds
always remains in the initial ground state, and at $T>400$ K a positive
orientational defect Savin20  can already appear at the right end of the chain
and then move along it.

Simulation of the dynamics of a complex with an orientational defect in the
internal chain of hydrogen bonds showed that the movement of the defects is
thermally activated (at low temperatures the defects remain pinned) and occurs
in the form of Brownian motion of the particle along the chain. The higher the
temperature, the higher the mobility of defects. A positive orientation defect
has higher mobility than a negative defect. A defect disappears when it
reaches the end of the chain and the chain goes into a homogeneous ground
state.

Thus, thermal vibrations do not lead to breaks in the chain of hydrogen bonds
(FH)∞ inside the carbon nanotube, so the chain can always be used for proton
transport. When a proton is transferred along a chain of hydrogen bonds, the
longest time is required for its reorientation after passing through the ionic
defect. This reorientation, the transition of the chain from the
HF$\cdots$HF$\cdots$HF$\cdots$ state to the FH$\cdots$FH$\cdots$FH$\cdots$
state, occurs due to thermally activated passage along the chain of positive
orientation defect. Therefore, thermal vibrations not only do not interfere
with the transport of protons along the chain of hydrogen bonds inside the
nanotube, but they are its necessary condition.

##  VI Conclusion

Being inspired by the pioneering studies of David Campbell on nonlinear
dynamics in molecular structures, here we have studied numerically the
structure and dynamics of several types of hydrogen-bonded molecular chains
placed inside caped or open carbon nanotubes. We have demonstrated that a
polypeptide chain is stabilized by a carbon nanotube, and it may take the form
of a helical structure with parallel chains of hydrogen-bonded peptide groups.
A chain of hydrogen fluoride molecules placed inside a nanotube of small
radius can form hydrogen-bonded zigzag chain being highly suitable for the
proton transport. In such molecular systems, the hydrogen-bonded chains retain
their structure, being stabilized by the interaction with the nanotube. For
some molecular structures the hydrogen-bonded chains will remain stable even
at $T=500$ K. Thus, the results of our numerical modeling suggest that the use
of carbon nanotubes with encapsulated hydrogen fluoride molecules would allow
to construct anhydrous molecular systems supporting high proton conductivity
being capable of operating at high temperatures.

###### Acknowledgements.

This work was supported by the Australian Research Council (Grants Nos.
DP200101168 and DP210101292). Computational facilities were provided by the
Interdepartmental Supercomputer Center of the Russian Academy of Sciences.

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