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The Influence of the Rigidity of Carbon Nanotube on the

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									Journal of the Physical Society of Japan
Vol. 79, No. 6, June, 2010, 064608
#2010 The Physical Society of Japan



                                  The Influence of the Rigidity of a Carbon Nanotube
                                 on the Structure and Dynamics of Confined Methanol
                   Vitaly V. C HABAN, Oleg N. K ALUGINÃ, Bradley F. H ABENICHT1 , and Oleg V. PREZHDO1 y
                         Department of Inorganic Chemistry, V. N. Karazin, Kharkiv National University, Kharkiv, Ukraine
                                 1
                                   Department of Chemistry, University of Washington, Seattle, WA 98195, U.S.A.
                                    (Received February 7, 2010; accepted April 13, 2010; published June 10, 2010)
                    In this paper, we compare the behavior of liquid methanol confined by an open-ended single-walled
                  nanotube (SWCNT) under four different simulation conditions by using the molecular dynamics (MD)
                  simulations technique. The first model is a rigid and fixed SWCNT with all its carbon atoms fixed at their
                  initial positions; the second is a flexible and fixed SWCNT with its centre-of-mass fixed at the center of
                  the MD box and with the carbon–carbon bond potential applied; the third is a rigid and floating SWCNT,
                  and the fourth is the most realistic flexible and floating SWCNT model — without fixed atoms and with
                  bond potential. The microscopic structure and transport properties of bulk methanol confined by the four
                  different SWCNTs were analyzed. No changes in the radial distribution functions of the hydrogen bond
                  between MeOH molecules are found, and the self-diffusion constant and microscopic dipole relaxation
                  time are essentially unaffected by the confinements. In spite of the flexible/rigid or fixed/floating ð15; 15Þ
                  SWCNT model used, the structure and transport properties of confined MeOH are found to be very close
                  in all the simulated cases. We conclude that using the approximation of rigid or/and fixed SWCNT does
                  not lead to any systematic errors in properties of the confined liquid. The results show that simulations
                  using rigid carbon nanotubes provide a reliable description of molecular diffusion and other solvent
                  properties in a variety of applications, such electro-chemical devices, membranes and sensors that rely on
                  these properties.

               KEYWORDS: confined fluid, carbon nanotube, molecular dynamics simulation, methanol, intramolecular poten-
                            tials, diffusion coefficient, hydrogen bond
               DOI: 10.1143/JPSJ.79.064608



                                                                            less than a factor of two. Jakobtorweihen, et al.12) confirmed
1.   Introduction                                                           the above results also for CH4 at low loadings and showed
   Confined liquids are now of exceptional interest because                  how the influence of flexible walls can be taken into account
of their unusual properties1–5) and high importance for                     by means of a Lowe–Andersen thermostat.13,14)
industrial applications.6–11) Molecular dynamics simulations                   Other authors dealing with transport diffusion of argon19)
of confined liquids have already become a common                             and with water structure20) found almost no noticeable
practice,1,3,12–14) as their experimental investigation is a                influence both on molecular structure and transport. Again,
difficult problem. Open-ended single-walled carbon nano-                      neglect of the velocities of the SWCNT sites during MD
tubes (SWCNT) with an ideal geometry (zigzag or armchair)                   simulation can be a substantial approximation, possibly
provide an excellent model to study the properties of the                   distorting dynamical constants of the molecules absorbed
confined liquids at the microscopic level.                                   inside the SWCNT. We analyze this effect as well.
   Our present study was inspired by a number of questions                     Hence, in the present paper, we make a comparison
arising from earlier reported molecular dynamics (MD)                       among four differing models of the ð15; 15Þ SWCNT,
simulations of confined non-aqueous liquids.15–17) These                     representing the most common simplifications applied in
investigations involved SWCNTs with fixed atomic coor-                       practical simulations: rigidity instead of flexibility and
dinates and caused concern about how rigidity and fixation                   centre-of-mass fixation instead of free nanotube motion
of the carbon nanotube can influence the properties of the                   across the MD cell. Liquid methanol (MeOH) at 298 K was
molecules inside the confinements.                                           chosen as a test media because of its specific structure in
   The above mentioned problem, namely the effect of                         the bulk phase and the novel interest to ‘‘SWCNT–MeOH’’
rigidity/flexibility was considered in a few recent pa-                      systems in connection with green chemistry direct methanol
pers12,18–20) devoted to the MD simulations of gases and                    fuel cells.21)
water confined by SWCNTs. However, different results were                        Below, we compare the structure and transport properties
obtained for confined gases and liquids. Chen et al.18)                      of MeOH confined inside the SWCNT: rigid fixed [the
investigated the influence of nanotube flexibility on the                     center of ð15; 15Þ SWCNT coincides with a center of the
transport diffusion of methane (CH4 ) in ð15; 0Þ and ð20; 0Þ                 box], fixed flexible, rigid floating (initially placed at the
tubes. According to their simulations, the diffusion of CH4 is               center of the box and then allowed to float as an ordinary
hugely reduced by SWCNT flexibility at pressure values                       particle) and, eventually, flexible floating (the most realis-
close to zero but at pressures around 1 bar the corresponding               tic), denoted as systems I, II, III, and IV, respectively.
values for rigid and flexible SWCNTs differ on average by
                                                                            2.   MD Simulations Details
Ã
 E-mail: Oleg.N.Kalugin@univer.kharkov.ua                                      The four ‘‘SWCNT–MeOH’’ systems (Table I) consisted
y
 E-mail: prezhdo@u.washington.edu                                           of the non-capped ð15; 15Þ armchair SWCNT with diameter,
                                                                     064608-1
J. Phys. Soc. Jpn., Vol. 79, No. 6                                                                                    V. V. C HABAN et al.


Table I. The potential parameters for the interacting sites of the MeOH   to manage the existing constraints. The interatomic poten-
  and SWCNT.                                                              tials for carbons in the flexible SWCNT (systems II and IV)
                                  ii          "ij              zi        were taken for sp2 carbon centres from25) developed by
   Molecule         Site
                                 (nm)      (kJ molÀ1 )         (e)        fitting experimental lattice parameters, elastic constants and
   MeOH             H           0.0         0.0              þ0:431       phonon frequencies for graphite: Morse bond stretches
                    O           0.3083      0.731169         À0:728       (Rb ¼ 1:4114, kb ¼ 720, Db ¼ 133:0), cosine angle bends
                    Me          0.3861      0.757859         þ0:297       (a ¼ 120, k ¼ 196:13, kr ¼ À72:41, krr ¼ 68) and a
                                                                          twofold torsion (Vt ¼ 21:28), where all distances are in
   SWCNT            C           0.34        0.41               0
                                                                          angstroms, all angles in degrees, all energies in kcal molÀ1 ,
                                                                          and all force constants in kcal molÀ1 , angstroms and radian
                                                                          units. The density and dielectric constant of MeOH was set
                                                                          equal to the experimental values at 298 K: 786.37 kg/m3 and
                                                                          32.66, respectively.26)
                                                                             The simulations were performed using the proprietary
                                                                          package MDCNT17) and GROMACS27) MD engine. For
                                                                          each system, the equilibration process during 200 ps was
                                                                          followed by the five consecutive collection runs of 500 ps.
                                                                          To eliminate the boundary effect, only the molecules at the
                                                                          current moment located more than 1.0 nm from the SWCNT
                                                                          edge were taken into account when calculating the properties
                                                                          of the MeOH confined inside.
                                                                          3.    Results and Discussion
                                                                             The comparison of the fixed and floating versions of
                                                                          the ‘‘SWCNT–MeOH’’ systems (Fig. 1) indicates that the
                                                                          SWCNT can undergo substantial translational and rotational
                                                                          motions on a fairly short time-scale. The left and right
                                                                          panels in Fig. 1 are separated by 2.7 ns. Within this time
                                                                          interval the tube has rotated and moved. The tube mass is
                                                                          29,520 a.m.u., which is about 923 times larger than the mass
                                                                          of a single methanol molecule. One could expect that such a
                                                                          heavy object would not move much within a nanosecond;
                                                                          however, this is not so. An estimated diffusion coefficient for
                                                                          the SWCNT center of mass is about 4 Â 10À10 m2 /s, which
                                                                          is only 6 times smaller than that of bulk methanol. A floating
Fig. 1. (Color online) Sketch of the simulated MD cell: 4 000 MeOH        CNT tends to orient diagonally in the simulation cell. These
  molecules and SWCNT ð15; 15Þ.                                           results show that a priori one may expect a notable influence
                                                                          of the SWCNT motion on the properties of the confined
                                                                          solvent.
dSWCNT ¼ 2:035 nm, and length, LSWCNT ¼ 9:968 nm. The                        The short-range structure of liquid methanol confined
SWCNT was filled with liquid methanol and simulated at                     within the different ð15; 15Þ SWCNT models has been
298 K in the NVT ensemble with Berendsen temperature                      characterised by a set of radial distribution functions
coupling every 0.1 ps and a timestep of 0.001 ps. In each                 (RDFs), g ðrÞ, as well as by the running co-ordination
system, the ð15; 15Þ SWCNT was placed in a periodic MD                    numbers (r.c.n.) defined as
cell with box side lengths equal to 4.63791, 4.63791, and                                                Zr
13.75626 nm. The centered SWCNT was additionally                                         n ðrÞ ¼ 4     g ðr 0 Þr 02 dr 0 ;   ð1Þ
surrounded by several layers of MeOH molecules in all                                                    0

directions (Fig. 1) to reproduce a correct interacting                    where  is the number density of species . As an example,
potential between MeOH and the ð15; 15Þ SWCNT. The                        the intermolecular O···H RDFs, gOH ðrÞ, and corresponding
intermolecular interactions were represented as a sum of                  r.c.n.’s, are displayed in Fig. 2, and the positions/heights
Coulomb and Lennard-Jones (LJ) ð12; 6Þ potentials. The                    of the first maxima and minima in the gOH ðrÞ are listed in
techniques of reaction field and shifted force were applied to             Table II.
Coulomb and LJ ð12; 6Þ interactions, respectively. The force                 The sharp first maximum in the O···H RDF is located at
field of the carbon nanotube was adopted to be purely LJ                   0.183 –185 nm. It is followed by a deep minimum at 0.257 –
ð12; 6Þ with  ¼ 0:34 nm and " ¼ 0:41 kJ/mol parameter-                   263 nm and a well pronounced plateau in nOH ðrÞ % 1. The
ized for a graphene sheet. The potential parameters for the               RDF clearly shows formation of two hydrogen bonds by each
interacting sites of the MeOH (force field model H122)                     MeOH molecule in the liquid inside the ð15; 15Þ SWCNT as
successfully used in our previous simulations16,23) and                   well as in bulk. Table II and Fig. 2 indicate that all five
carbon atoms of the ð15; 15Þ SWCNT24) are listed in Table I.              SWCNT models agree with each other. A minor difference in
Mixed LJ ð12; 6Þ parameters were calculated according to                  the position and height (Table II) of the RDF maxima for
the Lorentz–Berthelot rule. SHAKE algorithm was applied                   bulk and confined methanol is caused by the non-spherical
                                                                     064608-2
J. Phys. Soc. Jpn., Vol. 79, No. 6                                                                                                  V. V. C HABAN et al.




                                                                                                                                                (d)




Fig. 2. Radial distribution functions, gOH ðrÞ, and running co-ordination
  numbers, nOH ðrÞ, inside the ð15; 15Þ SWCNT (solid, dash-dot-dotted,
  dashed, dash-dotted, and long-dashed curves stand for the systems I, II,
  III, IV, and V respectively) derived from the MD simulations.                                                                                 (c)



Table II. The parameters of the first maxima of the RDFs for O–H pair
  (also see Fig. 2), self-diffusion coefficients, DMeOH , and microscopic
  dipole relaxation times,  , of the confined systems (I–IV) and bulk
  (system V) methanol molecules from MD simulations.

                             rmax1             rmin1   DMeOH 109                       (a)                      (b)
System        SWCNT                gðrmax1 Þ
                             (nm)              (nm)     (m2 sÀ1 )   (ps)
    I    Fixed+rigid16Þ       3.5      0.183   0.263 1:25 Æ 0:06 24 Æ 2      Fig. 3. (Color online) An instantaneous configuration of methanol mole-
                                                                               cules confined within the rigid ð15; 15Þ SWCNTs taken from MD
   II    Fixed+flexible        3.4      0.183   0.261 1:31 Æ 0:06 22 Æ 2
                                                                               simulation on system (I). (a) ‘‘Ball-and-stick’’ and (b) ‘‘stick’’ represen-
  III    Floating+rigid       3.5      0.185   0.259 1:30 Æ 0:06 23 Æ 2        tations. The H-bonds between MeOH molecules are indicated by dashed
  IV     Floating+flexible     3.5      0.185   0.257 1:22 Æ 0:06 24 Æ 2        lines. Methyl groups, oxygens and hydrogens are depicted in green, red
   V     Bulk MeOH16Þ         3.6      0.185   0.261 2:16 Æ 0:02 12 Æ 2        and white, respectively. An example of the helix-like H-bonded cluster
                                                                               formed by eight methanol molecules is shown on the right in a ‘‘spacefill’’
                                                                               representation. (c) Projection as inside the CNT. (d) Projection rotated by
                                                                               90 along the CNT z-axis towards the observer. The H-bonds in this
area in which these RDFs are calculated. In principle, such                    cluster are shown in blue in parts (a) and (b).
a geometrical effect can be corrected by the analysis of
excluded volume effects.28) The parts of MeOH molecules
that are located near the inner SWCNT wall has no                               The corresponding spectral densities of the hindered
neighbouring solvent molecules at the distance of the first                   translational motion of the MeOH molecules were calculated
maximum. The equality of the RDFs parameters for all                         by the cosine Fourier transformation,
ð15; 15Þ SWCNT models brings us to the conclusion that                                                Z1
there is no observable impact of the fixation of the SWCNT                                   Svv ð!Þ ¼     Cvv ðtÞ cosð!tÞ dt:       ð3Þ
geometry and centre-of-mass on the confined liquid structure.                                                  0

   Further insights into the structure of the confined                           The time evolution of Cvv ðtÞ and Svv ð!Þ are shown in
methanol are gained from Fig. 3, which shows an instanta-                    Fig. 4. The self-diffusion coefficients of the centre-of-mass
neous configuration of methanol molecules confined within                      of the MeOH molecules were calculated according to the
the rigid ð15; 15Þ SWCNTs taken from MD simulation                           Green-Kubo relation, by integrating the velocity autocorre-
on system (I). Molecular helix-like chains formed by                         lation function,
the hydrogen-bonded MeOH molecules can be seen. An                                                      Z
                                                                                                      1 1
example of a helix-like H-bonded cluster formed by eight                                      D ¼ lim      hvð0ÞvðtÞi dt:            ð4Þ
                                                                                                  t!1 3 0
methanol molecules is shown explicitly in Fig. 3. Similar
chains can be found in the simulation of the flexible SWCNT                   The resulting DMeOH values for all the simulated systems are
systems. This finding is in agreement with the results of Liu                 summarised in Table II. Table III shows that the diffusion
et al.,29) who found the same layered distribution inside                    constant does not depend on the distance from the SWCNT
carbon nanotubes for water by simulating rigid and non-                      wall. One may have expected a variation in the solvent
moving SWCNTs without an outside solvent layer.                              mobility near the tube wall due to the difference in the
   The dynamics of the MeOH molecules were examined in                       solvent–solvent and solvent–tube interaction. This is not
terms of the normalized velocity autocorrelation functions                   observed in our simulations, since the space confinement
(VACF) of the translational motion of the centre-of-mass of                  induced by the SWCNT reinforces solvent–solvent inter-
the methanol molecules                                                       actions. A well-defined hydrogen-bonding molecular net-
                                      hvð0ÞvðtÞi                             work, confirmed by the cylindrical distribution func-
                          Cvv ðtÞ ¼              :                    ð2Þ    tion,15–17) controls the movement of the confined methanol
                                      hvð0Þvð0Þi
                                                                             system as a whole.
                                                                      064608-3
J. Phys. Soc. Jpn., Vol. 79, No. 6                                                                                                   V. V. C HABAN et al.




                                                                                Fig. 5. Natural logarithm of the autocorrelation function of dipole vector,
                                                                                  C ðtÞ, inside the ð15; 15Þ SWCNT (solid, dash-dot-dotted, dashed, dash-
                                                                                  dotted, long-dashed curves stand for the systems I, II, III, IV, and V,
                                                                                  respectively) derived from the MD simulations (also see  in Table II).


                                                                                                                   huð0Þ Á uðtÞi
                                                                                                       C ðtÞ ¼                 :                     ð5Þ
                                                                                                                   huð0Þ Á uð0Þi
                                                                                C ðtÞ as functions of time are displayed in Fig. 5 for the
                                                                                MeOH molecules in the simulated systems. Clearly at the
                                                                                long time, these correlation functions can be reasonably
Fig. 4. Normalized autocorrelation functions [Cvv ðtÞ] (at the top) of the      described by exponential decay. The corresponding re-
  translational motion of the centre-of-mass of the methanol molecules          orientational relaxation times of the methanol dipole,  ,
  inside the ð15; 15Þ SWCNT (dash-dotted, dashed, dash-dot-dotted, solid,       calculated from the slopes of ln C ðtÞ using the least-
  and long-dashed curves stand for the systems I, II, III, IV, and V,           squares-fit at long times (4 t 10 ps) are summarized in
  respectively). Svv ð!Þ (at the bottom) are cosine Fourier transforms of the
  corresponding ACFs (on the left), and represent the spectral densities of     Table II. The re-orientational relaxation times are twice
  the hindered translations of the MeOH molecules.                              larger for methanol inside the ð15; 15Þ SWCNT systems than
                                                                                bulk, compare 22 – 24 ps with 12 ps. The re-orientational
                                                                                relaxation times obtained with the different SWCNT models
Table III. Methanol self-diffusion coefficient, DMeOH 109 (m2 sÀ1 ), as a          for the confined solvent agree with each other (Table II).
  function of the distance from the SWCNT wall, Rwall .
                                                                                4.   Conclusions
  Rwall
  (nm)
              Fixed+rigid Fixed+flexible Floating+rigid Floating+flexible            The molecular dynamics simulations of liquid methanol
                                                                                confined by differing ð15; 15Þ SWCNT models in the NVT
 0.0 – 0.4       1.25          1.29            1.30              1.21
                                                                                ensemble at 298 K were performed in order to determine
0.40 – 0.75      1.26          1.32            1.30              1.23
                                                                                whether the rigidity and fixation of the SWCNT can affect
0.75 –1.05       1.24          1.28            1.29              1.20
                                                                                the properties of the confined fluid. For all systems, neither
                                                                                structural properties nor dynamics of the MeOH were found
                                                                                to differ more than one standard deviation. The diffusion
   The Cvv ðtÞ functions of the confined MeOH (Fig. 4) show                      constant of MeOH, DMeOH , inside the ð15; 15Þ SWCNT is
some changes in comparison with the bulk phase. The                             two times lower than in the bulk and appeared very close
minima in the linear VACF for the confined liquid are                            in spite of the differences between the ð15; 15Þ SWCNT
deeper and more pronounced than for bulk. However, the                          simulation conditions used.
VACF of the confined liquid depends neither on the ð15; 15Þ                         The lack of dependence of the solvent properties on the
SWCNT flexibility/rigidity nor on its fixation/flotation.                          SWCNT flexibility and mobility can be rationalized by the
One can also see that in comparison with bulk MeOH, an                          weak solvent-SWCNT interactions, compared to the solvent–
insignificant blue-shift is observed in Svv ð!Þ upon confine-                     solvent interactions. Liquid methanol forms a strong hydro-
ment. The blue-shift indicates the increased dynamical                          gen-bonding network both in bulk phase and inside the
correlation among MeOH molecules for all SWCNT force                            SWCNT. In the latter case, the network is reinforced and
fields used in the present study. The diffusion coefficients                        reorganized into helical structures near SWCNT walls due
of the confined MeOH molecules are about twice lower                             to spatial confinement. The hydrogen bonding network is
compared to bulk,16) 1.2 –1.3 vs 2:2 Â 10À9 m2 ÁsÀ1 . The                       sufficiently flexible and dynamic in order to respond to small
deviations in the diffusion coefficients obtained for the                          changes in the SWCNT curvature. The low energy acoustic
different SWCNT simulation conditions are within the range                       modes, that are active in SWCNTs at room temperature, are
of the statistical errors (Table II).                                           characterized by long vibrational periods and small displace-
   To further examine the re-orientational dynamics of the                      ment of individual atoms. Methanol molecules are able to
MeOH molecules inside the ð15; 15Þ SWCNT, we evaluated                          respond to these motions of the SWCNT walls quickly and
the re-orientational ACFs of the unit vector u along the                        easily. Thus, the diffusion coefficients are not greatly influ-
molecular dipole ,                                                             enced by the motions of the SWCNT at room temperature.
                                                                         064608-4
J. Phys. Soc. Jpn., Vol. 79, No. 6                                                                                                  V. V. C HABAN et al.


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