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Methane Behavior in Carbon Nanotube as a Function of Pore Filling and Temperature Studied by Molecular Dynamics Simulations Katarzyna Bartu#, and Aleksander Brodka J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12855 • Publication Date (Web): 06 Feb 2017 Downloaded from http://pubs.acs.org on February 8, 2017

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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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The Journal of Physical Chemistry

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Methane Behavior in Carbon Nanotube as a Function of Pore Filling and Temperature Studied by Molecular Dynamics Simulations

Katarzyna Bartuś1 and Aleksander Bródka*,1,2 1

A. Chełkowski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice,

Poland 2

Silesian Center of Education and Interdisciplinary Research, 75 Pułku Piechoty St., 40-500

Chorzów, Poland

ABSTRACT: We report structural and dynamic properties of methane inside the (15,15) carbon nanotube (CNT) obtained from molecular dynamics simulations of flexible methane molecules, and intramolecular interactions are introduced by the reactive empirical bond order potential. The calculations are performed for wide range of temperature and loading, that corresponds to states from the dense gas phase to the liquid state. The properties of flexible and rigid models of methane molecules are compared. The diffusivity of molecular translations along the CNT and rotations increase with temperature and they decrease with pressure. Temperature dependences of the diffusion coefficients for flexible molecules are predicted by the Arrhenius equation. Internal motions of the CH4 atoms diminish the activation energies of the translational diffusion, and increase the energies of the rotational diffusion, especially for higher pressures. The results mean that possibility of changes of molecular bond lengths and valence angles in methane molecules causes a reduction of hindrances of their translations and at the same time it leads to the increase of rotational motion interruptions.


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1. INTRODUCTION Adsorption of different molecules on open surfaces1-4 or in porous systems3,5-39 has been the subject matter of intense investigations due to increasing importance of the storage of clean energy substances2,3,6,7,8,11-37 and utilization of toxic gases1,8 as well as greenhouse gases.5,7,8,10,16 The search for a suitable material for the alternative fuels storage is currently an active area of research. Fundamental studies of atomic and molecular processes related to transport and adsorption are needed for understanding the behavior of fluids and gases in systems with restricted geometry. Moreover, the studies are important also for predicting conditions and the type of materials for efficient adsorption storage. Methane is a possible source of clean and secure energy, and methane adsorbed in carbon nanotubes (CNTs) may be an alternative to compressed natural gas. Knowledge of structure and dynamics of methane inside CNTs may have practical benefits because it should be helpful in understanding of methane preservation and storing. Behavior of methane in CNTs was studied using computer simulations.15-35 Grand canonical Monte Carlo (GCMC) method was used to study adsorption and structure of methane in CNTs,15-20 and molecular dynamics (MD) simulations were conducted to investigate structural as well as dynamic properties of methane.17-35 In the MD simulation, the GCMC method was sometimes applied to generate initial configuration of the methane molecules in CNTs.18-21 Molecular transport through CNTs may be studied using modified MD such as non-equilibrium MD,33 a gravity-driven steady-state flow22 or dual control volume grand canonical molecular dynamics. 19 Symmetric structure of the CH4 molecule suggests the use of a 1-site model, i.e. spherical super-atom with a single Lennard-Jones (LJ) interaction center.16-29 In a 5-site model a methane molecule is represented by a rigid set of five LJ interaction centers at atom positions.17,28-30 In more realistic model, i.e. a five-site flexible model, intramolecular forces between atoms of


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methane molecule are included,31-35 and they are described by the reactive bond order (REBO) potential40 or harmonic potentials offered by the DLPOLY molecular dynamics simulation package.41 The structural properties of the 1-site and 5-site models of methane molecule have been compared by Nicholson and coworkers who studied adsorption of methane in CNTs,16 and in silica nanopores.36 Comparison of the two models of methane molecule have been done also by Do and Do3 who used the GCMC simulations to study adsorption of methane in graphitic slit pores. The studies show comparable pore densities for the two models, however, adsorption of the 1-site CH4 was overestimated compared to the 5-site model, that is attributed to enhanced 1site CH4-adsorbent potential energy. Similar comparison of methane structure and dynamics in a (15,15) CNT at temperature range from 173 K to 293 K has been done by us.28 Translational and rotational diffusion coefficients for the two models have similar values, however, their pressure dependencies vary with temperature slightly, and changes of the coefficients with temperature are described by the Arrhenius equation. Relatively free rotations of the 5-site molecules reduce the activation energies of translational diffusion compared to the energies for the 1-site molecules. The atomic model of the molecules increases density of methane in the vicinity of the nanotube wall, that may be observed in ref 16. The CNT flexibility, introduced by the REBO potential for interactions between carbon atoms of the nanotube, has weak impact on diffusivity of methane molecules, for densities considered in ref 27. However, the inclusion of the CNT flexibility increases slightly the activation energies of the translational diffusion, and diminishes the energies of the rotational diffusion for higher pressures.28 The CNT flexibility has a crucial influence on self-diffusion at low loadings, and neglecting the CNT flexibility can lead to overestimation of the diffusion coefficient of 1-site molecules by at least an order of magnitude,20-24 however, the overestimation factor for transport diffusivity of 5-site flexible molecules is about four for low pressure gradient using the DCV-GCMC method.34 Therefore, in


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the present study we switch on internal motion of atoms in methane molecules, and we investigate the influence of molecule flexibility on structure and dynamics of methane in the (15,15) CNTs. The earlier simulations of the 5-site flexible methane molecules were carried out usually at room temperature.31-34 Hence, in our MD simulations we change not only loading of methane molecules in the CNT but also temperature. The thermodynamic states are the same as those considered in ref 28, that allows us to compare properties of the 5-site flexible molecules inside the CNT with those obtained for the 5-site rigid molecules.


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COMPUTATIONAL DETAILS We consider the single-walled (15,15) CNT formed by NC=2700 atoms, and its diameter

and length are d=2.034 nm and L=11.068 nm, respectively. To diminish methane density the CNT is elongated twice and four times. The systems analyzed here are analogous to those studied ealier,28 and they are listed in Table 1.

Table 1. Systems under consideration and lengths of the MD runs. Molecule number/CNT length MD run length [ps] Period of the molecular configuration acquisition [ps] Number of simulation runs

240/1L 180/1L 120/1L 60/1L 60/2L 60/4L 40/4L 20/4L 50

50

50

50

300

300

300

400

0.025

0.025

0.025

0.025

0.15

0.15

0.15

0.2

4

4

4

4

4

4

4

7

Interactions of carbon atoms in the nanotube and atoms in a methane molecule are modelled by the reactive empirical bond order (REBO) potential,40 and the binding energy of atoms is expressed as follows:

Eb =

∑∑ f (rij )[U R (rij ) − bijU A (rij )], i

(1)

j >i

where rij is an interatomic distance between the atoms i and j. The sum in Eq. (1) is carried out over the nearest neighbors, that is ensured by the switching function f(r) whose value decreases from one to zero at a distance range characteristic for interacting atoms.40 UR and UA are the repulsive and attractive potentials, respectively. The attractive part of the inter-atomic interaction is modified by the bond order function bij, which depends on the local coordination and bond angles for the atoms i and j. The REBO potential was originally parameterized to examine the growth of diamond thin films by chemical vapor deposition, and it was also successfully used to reproduce properties of the others solid carbons and hydrocarbon molecules.40 Particularly, the


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REBO potential has been applied to study the mechanical properties of CNTs37,42-44 as well as structure of nanodiamonds and its graphitization.45,46 It has been also used to reproduce results of X-ray and neutron diffractions of CNTs,47 nanodiamonds,48 activated carbons,49 and carbon nanohorns.50 A methane molecule interacts with another molecule and a carbon atom of the nanotube via the spherically truncated and force-shifted Lennard-Jones (LJ) potentials. The spherical truncation as well as the minimum image convention are applied to the centers of molecular masses. The LJ potential parameters and cut-off radii are the same as for the 5-site rigid methane molecule,28 and they are collected in Table 2. For methane atoms the parameters were proposed by Severin and Tildesley,4 and for a nanotube carbon atom the parameter values were the same as for carbon atoms in graphite.51 For unlike interacting centers the parameters were calculated using the Lorentz–Berthelot mixing rules.52

TABLE 2. Parameters of the intermolecular LJ interactions. Interaction

ε/kB [K]

σ [nm]

rc [nm]

Cmol-Cmol

51.198

0.335

1.06

Cmol-Hmol

23.798

0.299

Hmol-Hmol

4.87

0.261

Cmol-CNT

47.68

0.330

Hmol-CNT

17.00

0.298

1.04

The initial configurations of the molecules considered here, i.e. positions and velocities of the molecule atoms are taken from the final configurations of the 5-site rigid molecules.28 The MD simulations are performed with one dimensional periodic boundary conditions (PBCs), i.e. PBCs in the direction of the CNT axis (the z direction) are applied to interaction centers. Therefore, the PBCs are used to calculate interatomic interactions and forces acting on the centers as well as


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contributions to their positions and velocities arising from equations of motion, and the position contributions are used to upgrade the z components with and without PBCs. The second values of the z position components are employed to calculate a function of the mean square displacement along the CNT, defined by Eq. (11). That approach is especially important for a low density systems, when molecules can move a distance longer than the CNT length. The calculations are carried out at temperatures 173, 193, 213, 233, 253, 273 and 293 K, and the constrain method,52 proposed by Evans, Hoover and coworkers,54,55 is used to keep the temperature constant. For αth atom of the ith methane molecules the equation of motion is following 5

N

..

miα r iα = Fiα −

.

∑∑ F j

β

j =1 β =1 N 5 .

⋅ r jβ .

∑∑ r j ⋅ r j β

.

r iα ,

(2)

β

j =1 β =1

.

where r iα means velocity of the atom with mass miα , and Fiα is force acting on the atom originating from intramolecular interactions as well as interactions of the atom with atoms of other molecules and the CNT. Similar method is used to maintain the constant temperature of the CNT, and translation of the αth nanotube atom is described by the following equation NC

..

mC r α = Fα −

∑

.

Fβ ⋅ rβ

β =1 NC .

.

.

rα ,

(3)

r β ⋅ rβ ∑ β =1

where the force Fα is result of interactions of the αth atom with the others atoms of the CNT and atoms of the methane molecules. To eliminate possible temperature drift due to accumulation of numerical errors the velocity scaling is applied to the CNT atoms as well as atoms of the methane molecules. Moreover, to control energy equipartition between translations and rotations


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of the methane molecules we scale separately their translational and angular velocities. At each time step we calculate position of the mass center for the ith molecule, ri, and its translational .

velocity, r i , defined as follows 5

ri =

∑

5

miα riα

α =1 5

.

ri =

,

∑ mi

∑

.

miα r iα

α =1 5

.

(4)

∑ mi

α

α

α =1

α =1

Moreover, for the molecule angular momentum, Li, and the inertia tensor with respect to the molecular center of mass, Ιi, are calculated, 5

Li =

∑ (ri

α

α =1

.  . − ri × miα  r iα − r i  ,  

)

5

(Ii )βγ = ∑ mi α =1

α

(

(5)

)(

δ r − r 2 − β − β γ − γ iα i iα i  βγ iα i

),

β, γ = x, y, z ,

(6)

which allow us to obtain an angular velocity in the laboratory frame, ωi, given by

ωi = I i−1Li .

(7)

Using the quantities defined by Eqs. (4-7) the translational and rotational temperatures are calculated. When absolute value of difference between the translational temperature and the required one is greater than 3 K we calculate the internal velocity as follows .

.

(

.

)

ρiα = r iα − r i − ωi × riα − ri .

(8) .

After that the translational velocity of mass center, r i , and angular velocity, ωi, are scaled for all .

molecules, and the scaled velocities together with ρiα , riα , and ri are used in Eq. (8) to calculate .

the real atomic velocity r iα .


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The equations of motion are integrated with time step 0.125 fs using six-value predictorcorrector algorithm.53 Because of hydrogen atoms in methane molecules the time step is four times shorter than that used previously for rigid molecules.27,28 Moreover, such a reduction of time step decreases frequency of the velocity scaling described above, that is illustrated in Figure S1 of the Supporting Information. The MD run lengths presented in Table 1 give approximately zero value of the velocity and angular velocity correlation functions after half of the simulation run length. For each system the first run is treated as an equilibration run, and the remaining runs constitute the main part of the MD, in which the system configuration, i.e. positions and velocities of all methane atoms are recorded (see Table 1). To quantify the pressure of methane in the CNT we estimate an average value of the diagonal elements of the pressure tensor52

Pγγ = ρk B T −

1 3V

1 ∂U (riα jβ ) rij ⋅ riα jβ . ∂riα jβ α β

∑∑∑∑ r i j >i i j i j α

β

(9)

In the above equation U ( riα jβ ) is the LJ interaction energy between atoms iα and jβ of the molecules i and j, respectively, kB means the Boltzmann constant, ρ=N/V is the number density, and V is the internal volume of the CNT with smooth walls and effective diameter def = d−σCNT, where σCNT is the collision parameter of the LJ potential of a nanotube carbon atom.51 To characterize structure of methane in the CNT we calculate distribution of the molecule mass centers across the CNT, ρR(r), which is given by

R

ρ (r ) =

N R (r , r + ∆r ) NdS

,

(10)

In the above equation NR(r,r+∆r) is number of molecules in a cylindrical shell with radius r and thickness ∆r=0.01 nm, and dS is the area of the cylindrical shell base. Angular brackets in Eqs. (9) and (10) indicate averaging over time and simulation runs.


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Using molecular configurations stored during the production run we calculate the mean square displacements (MSDs) along, ∆r||2 (t ) , and across the CNT, ∆r⊥2 (t ) , which are defined as follows ∆r||2 (t ) = [z (t ) − z (0)]2 , ∆r⊥2 (t ) =

1 [x(t ) − x(0)]2 + [ y (t ) − y (0)]2 , 2

(11)

(12)

where x, y and z are Cartesian coordinates of the center of molecular mass given by the first formula of Eq. (4). From the MSD ∆r||2 (t ) the translational diffusion coefficients DT for motion parallel to the CNT axis is calculated applying the Einstein formula52 DT = lim

1

t →∞ 2t

∆r||2 (t ) .

(13)

Rotational motion of the methane molecules is characterize by the rotational diffusion coefficient obtained from the angular velocity correlation function (AVCF)

DR =

1 3

∞

∫ ω(t ) ⋅ ω(0) dt ,

(14)

0

where the angular velocity, ω, is calculated according to Eq. (7). Angular brackets in Eqs. (11), (12) and (14) mean averaging over molecules, initial time and simulation runs. Temperature dependences of the translational and rotational diffusion coefficients are analyzed in terms of an activated diffusion model described by the Arrhenius equation D = D0 exp ( − Ea /k BT) ,

(15)

where D0 is the pre-exponential factor with units of diffusivity, and Ea is the activation energy of diffusion process.


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3. RESULTS AND DISCUSSION Examples of thermodynamic data obtained from the MD simulation are presented in Supporting Information. Changes of the translational and rotational temperatures as well as intermolecular energy with time are presented in Figure S2 of the Supporting Information. In the figure there are depicted results for the lowest density system, when one observes the largest fluctuations of the quantities. Translational and rotational temperatures show good stability of their average values, and their RMSs are about 2 K and 3 K, respectively. The increase of methane density decrease slightly the temperature fluctuations. Changes of the intermolecular potential energy are more pronounced. When molecular density increases from the lowest value to the highest one the average energy value diminishes from -7.75 to -9.02 kJ/mol for the lowest temperature, and from -6.94 to -7.47 kJ/mol for the highest temperature, and its fluctuation decreases about three times. In Figure S3 of the Supporting Information we present distributions of bond lengths and angles between bonds in methane molecules for the systems 20/4L. The maximum value of the C-H bond length distributions appears at 0.11 nm, and distribution of shorter distances is higher than that for longer ones. The temperature increase causes broadening of the distribution, and similar changes of the distribution are observed with increasing density. Distribution of angles between the C-H bonds shows that the tetrahedral angle is the most probable value, and the distribution is temperature and density independent. Changes of the bond length distribution with temperature and molecular density may suggest energy transfers between methane molecules and the CNT. One must note that fluctuations of the CNT temperature are an order of magnitude lower than those for translational and rotational motions of methane molecules. This observation is not surprising because interaction between carbon atoms in the nanotube is two orders of magnitude stronger than intermolecular interaction and interaction between a methane molecule and the CNT. The internal energy per carbon atom of the nanotube changes slightly with


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temperature, and it increase from -7.355 eV for 173 K to -7.340 eV for 293 K. Those values are independent of number of methane molecules in the CNT, and their uncertainty is 0.018 eV. The energies when the CNT contains rigid methane molecules28 are very similar, and their values are -7.356 and -7.338 eV, respectively, but their uncertainty is and order of magnitude smaller than that obtained here. It must be noted that in this paper we report results of equilibrium MD simulations, and different fluctuations of the internal nanotube energies observed for flexible and rigid molecules may indicate energy transfers between substrate and adsorbate, however, decisive approval about the transfer requires further studies. Snapshots of final positions of methane molecules in the CNT for two systems under consideration at the lowest temperature and the highest one are presented in Figure 1. The results are visualized with the program VMD.56 A monolayer of methane in the vicinity of the CNT wall is observed for low densities up to 60/1L. For higher densities some molecules from the contact layer are pushed out to the central part of the CNT that is clearly visible for the largest density. The layered structure of methane in CNTs is also shown by probability distribution functions presented in Figure 1. For the smallest density a contact layer is represented by a single peak which shifts towards lower distances as well as becomes wider and more asymmetric when temperature increases. Similar temperature dependence of the peak connected with the contact layer is observed for higher densities of methane molecules. In that case a second wide peak appears in the distribution function which is related to molecules in the inner part of the CNT. As it can be seen in Figure 1 temperature behavior of the second peak is different to that of the contact peak. When temperature increases the inner peak becomes higher whereas the other decreases, that indicates a movement of methane molecules from the contact layer to the central part of the CNT. The distribution functions presented here are almost identical to the distributions when a methane molecule is modelled by a rigid set of five interaction centers.28 Two layers of methane were observed for 5-site model in the (15,15) CNT30 and for the 1-site


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methane molecules in the (16,16) CNT,22 and multilayer structure of methane in the (30,30) CNT was reported for the 1-site model.19 Examples of the MSD functions for translations along the CNT, ∆r||2 (t ) , are presented in Figure 2. In the figure there is also shown time dependence of the slope of a function ln[∆r||2(t)] with respect to the ln(t). Values of the slope start from 2 for very short times, and diminish to 1 for longer times. In other words at the beginning the MSD functions show quadratic dependence on time, that indicates ballistic motion of the molecules, and at longer times the functions increase linearly with time, that means normal mode diffusion. The normal mode diffusion was observed for methane in CNTs using 1-site model19,24 and 5-site flexible model in the CNTs of diameters from 0.72 nm to 1.75 nm.31-33 Superdiffusive behavior of the 5-site rigid methane molecules in the (8,8) CNT was reported by Bhide and Yashonath,29 however, the MSD functions were calculated up to 50 ps, and in that time range the functions behave similarly to those presented in Figure 2 for the lowest density. The slope of the linear part of the MSD function shown in Figure 2 gives information about the translational diffusion coefficient, and the Einstein formula (13) is used to calculate the coefficient omitting 20÷30 % of the initial data. The MSD functions shown in Figure 2 illustrate also impact of temperature and density on molecular diffusion. The increase of temperature from the lowest value to the highest one rises diffusion coefficient by a factor about 2, whereas the increase of density in the range considered here reduces the diffusion coefficient about two orders of magnitude. The MSD function for motion perpendicular to the CNT axis, ∆r⊥2 (t ) , is calculated up to 75 ps when the function approaches its maximum value, and examples of the MSD function are shown in Figure 3. For short times, up to 1÷2 ps, one observes changes of the MSD typical for ballistic motion. For longer times the MSD increases with rate depending on molecular density, and then rises to a maximum value. It must be noted that for a small density all molecules are in the contact layer and motion of each molecule is restricted in a similar manner. When molecular


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density increases, contributions to the MSD function originate from the methane molecules located in the contact layer as well as relatively freely moving molecules from the central part of the CNT. An exponential rise of the MSD ∆r⊥2 (t ) to a maximum value is predicted by solution of the diffusion equation for a point particle moving on the cylinder surface38 as well as for the particle inside a cylinder,39 but because of ballistic behavior of the displacement of real molecule at short times the theoretical formulas cannot be fitted satisfactory to the simulation results. However, considering a particle on the surface of a cylinder of radius RL38 and a particle inside a cylinder of radius RCTR,39 the maximum value of the MSDs may be described as follows

∆r⊥2 (max) =

N − N CTR 2 NCTR 2 RL + RCTR , N 2N

(16)

where NCTR means number of molecules in the central part of the CNT, i.e. in a cylinder of radius RCTR. Using the distribution functions for the smallest density (see Figure 1) one may calculate average radius of the contact layer, RL, and it is 0.659 and 0.637 nm for the lowest temperature and the highest one, respectively. In this case all molecules are in the contact layer and NCTR=0, that leads to the maximum values of the MSDs given by RL2 , and represented by black circles in Figure 3. Assuming that the inner part of the CNT is formed by a cylinder with radius RCTR=0.5 nm, and using the probability distribution functions presented in Figure 1 we are able to estimate a number of the central molecules, NCTR. The number strongly depends on temperature that is suggested by distribution functions, and its average value for the system 240/1L increases from 8.4 to 41 when temperature is raising from 173 K to 293 K. Using the above data in Eq. (16) the maximum values of the MSDs are denoted by red triangles in Figure 3. A small difference between the maxima values of the MSDs for the systems 20/4L and 240/L at 173 K is result of small number of the central molecules in the second system. The un-normalized AVCF is used to calculate rotational diffusion coefficient Eq. (14). The normalized AVCFs are presented in Figure 4 for three methane densities as well as the lowest


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and highest temperatures. Initial decay of the angular velocity correlation function is relatively fast, and at times longer than 0.5 ps one observes slow decay of the function to zero value. The increase of methane density leads to faster decay of the AVCF, however, the changes are weak at low densities, when the molecules are moving within the contact layer, and they become stronger for higher densities, when motion of a molecule is hindered by interactions with other molecules. The temperature increase slows down initial decay of the AVCFs, but for long time behavior of the function is almost temperature independent. It is interesting to note a monotonic decay of the AVCFs, characteristic for weakly hindered rotations. The rotational diffusion coefficient is calculated integrating the un-normalized AVCF up to 15 ps. To check accuracy of the upper limit of integration we increase it to 25 ps, and this modification does not change the coefficient values. The MSD functions presented in Figures 2 and 3 as well as the AVCFs shown in Figure 4 are very similar to those obtained from the MD simulations of the 5-site rigid molecules. Therefore, the correlation functions for the rigid model are not presented here, however, they are compared quantitatively with results for the flexible molecules in Figures 5 and 6. In Figure 5a there are presented translational diffusion coefficients of methane in the CNT. The simulation data are shown as a function of pressure for three temperatures, and a similar plot for the rotational diffusion coefficient is presented in Figure 5b. The translational and rotational diffusion coefficients decrease with pressure in different manner, and to analyze the data we assume linear dependence of logD on logp for translational motion, whereas for rotational motion such a relation is described by a quadratic function. The diffusion coefficients for the two models of the methane molecule have similar values. However, the translational diffusion coefficients for the flexible molecule change with pressure slightly faster than those for the 5-site rigid model, whereas rotational diffusion coefficients for flexible model are slightly higher than those for rigid molecule.


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Applying the functions fitted to the simulation data one can find temperature dependence of the diffusion coefficients for constant pressure. The diffusion coefficients as functions of reciprocal temperature are shown in insets in Figure 5, and linearity of the functions indicate Arrhenius behavior of the diffusion coefficients for methane molecules in the CNT. Eq. (15) gives possibility to estimate activation energies, whose pressure dependence is shown in Figure 6. In the figure there are also presented activation energies for rigid molecules. The errors depicted in Figure 6 are result of linear regression analysis of data presented in the insets in Figure 5. Similar uncertainties have been estimated for activation energies of the rigid molecules and for clarity of Figure 6 they are not shown. Values of the activation energy for translational diffusion of the flexible molecules are lower than those for the rigid molecules. The difference decreases from about 0.6 kJ/mol for the lowest pressure, when behavior of the molecules is determined by their interactions with the CNT atoms, to 0.3 kJ/mol for the highest pressure, when intermolecular interaction is dominated. The above results indicate that intramolecular motions of atoms make easier translations of methane molecules. For rotational motion the activation energy behaves differently. At low pressures, up to 20 bar, the activation energy of rotational diffusion is almost the same for both models of the methane molecule, and for higher pressures the energy for the flexible molecules is elevated by about 0.2 kJ/mol with respect to the value for the rigid molecule. When intermolecular distances are large and in consequence interactions are weak, the molecule flexibility has weak impact on the energy value, but the increase of pressure makes the intermolecular interactions stronger that leads to fast increase of the activation energy for flexible molecules. The obtained results suggest that thermal fluctuations of the methane atoms make roughness of the methane molecule more evident, and in consequence hindrances of molecular rotation become stronger.

4. CONCLUSIONS


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The MD simulations of flexible methane molecules in the (15,15) CNT are performed, when interatomic interactions in CH4 molecules as well in the CNT are described by the REBO potential, whereas intermolecular interactions are represented by the LJ potential. We use procedure of velocity scaling, which gives the average kinetic energy per degree of freedom in the translational motion of a molecule equal that of its rotational motions for molecules with internal degrees of freedom. In other words the procedure leads to alike temperatures connected with translational and rotational motions of flexible molecules. The structural and dynamic properties of the flexible methane molecules are compared with those obtained from simulations for rigid molecules. Radial distributions of methane across the CNT illustrate contact layer near the CNT surface as well as methane molecules in the central part of the CNT whose number increases with molecular density and temperature. The number gives possibility to estimate maximum values of the MSDs across the CNT. The MSDs along the CNT indicate ballistic motion at short times, which transforms into normal mode diffusion at longer times. The initial ballistic range of time is almost temperature independent, but according to our expectation the density increase strongly reduces the time range. Flexibility of molecules does not change radial distribution of methane in the CNT. The translational diffusivity of molecules along the carbon nanotube and rotational diffusivity increase with temperature, and they decrease with pressure. Translational and rotational diffusions are activated processes, and temperature dependences of the diffusion coefficients are described by the Arrhenius equation. The methane molecule flexibility has weak impact on diffusivity of methane molecules, however, intramolecular motions of the methane atoms diminish the activation energy values of the translational diffusion, and increase slightly the energies of the rotational diffusion for higher pressures. The results suggest that thermal fluctuations of the bond lengths and valence angles in methane molecules make their translations easier whereas they are source of additional restrictions of rotational motion.


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ASSOCIATED CONTENT Supporting Information Frequency of velocity scaling. Time dependencies of translational and rotational temperatures, and intermolecular interaction energy. Distribution of the bond lengths and angles between bonds in methane molecules. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS The authors acknowledge Prof. D.W. Brenner for making available the code for MD simulation with the REBO potential, which has been modified for methane molecules in a carbon nanotube.

REFERENCES (1) Kuoa, J-K.; Huang, P-H.; Huang, T-H.; Luo, J.-Y. Adsorption Behaviors of Hydrogen Sulphide on Au(110) Nanoslit Array Surfaces Using Molecular Dynamics Simulations, Mol. Sim. 2016, 42, 1429-1436.

(2) Okamoto, Y.; Miyamoto, Y. Ab Initio Investigation of Physisorption of Molecular Hydrogen on Planar and Curved Graphenes. J Phys Chem B 2001, 105, 3470–3474. (3) Do, D. D.; Do, H. D. Evaluation of 1-Site and 5-Site Models of Methane on Its Adsorption on Graphite and in Graphitic Slit Pores. J. Phys. Chem. B 2005, 109, 19288-19295.


18


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 33

(4) Severin, E. S.; Tildesley, D. J. A Methane Molecule Adsorbed on a Graphite Surface, Mol. Phys. 1980, 41, 1401-1418.

(5) Matranga, C.; Chen, L.; Smith, M.; Bittner, E.; Johnson, J.K.; Bockrath, B. Trapped CO2 in Carbon Nanotube Bundles, J. Phys. Chem. B 2003, 107, 12930-12941. (6) Kowalczyk, P.; Bhatia, S.K. Optimization of Slitlike Carbon Nanopores for Storage of hythane Fuel at Ambient Temperatures, J. Phys. Chem. B 2006, 110, 23770-23776. (7) Bienfait, M.; Zeppenfeld, P.; Dupont-Pavlovsky, N.; Muris, M.; Johnson, M.R.; Wilson, T.; DePies, M.; Vilches, O.E. Thermodynamics and Structure of Hydrogen, Methane, Argon, Oxygen, and Carbon Dioxide Adsorbed on Single-Wall Carbon Nanotube Bundles, Phys. Rev. B 2004, 70, 035410.

(8) Zhao, J.; Buldum, A.; Han, J.; Lu, J.P. Gas Molecule Adsorption in Carbon Nanotubes and Nanotube Bundles, Nanotechnology 2002, 13, 195–200. (9) Huang P.-H.; Hung S.-C.; Huang M.-Y. Molecular Dynamics Investigations of Liquid–Vapor Interaction and Adsorption of Formaldehyde, Oxocarbons, and Water in Graphitic Slit Pores, Phys. Chem. Chem. Phys. 2014, 16, 15289- 15298. (10) Huang P-H.; Chen, S-H. Effect of Moisture Content, System Pressure, and Temperature on the Adsorption of Carbon Dioxide in Carbon Nanotube and Graphite Composite Structures Using Molecular Dynamics Simulations, Journal of Nanoscience and Nanotechnology 2016, 16, 8654-8661.

(11) Beerdsen, E.; Dubbeldam, D.; Smit, B. Understanding Diffusion in Nanoporous Materials, Phys. Rev. Lett. 2006, 96, 044501. (12) Zeitler, T.R.; Allendorf, M.D.; Greathouse, J.A. Grand Canonical Monte Carlo Simulation of Low-Pressure Methane Adsorption in Nanoporous Framework Materials for Sensing Applications, J. Phys. Chem. C 2012, 116, 3492–3502.


19

Page 27 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(13) Houndonougbo, Y. A.; Signer, C.; He, N.; Morris, W.; Furukawa, H; Ray, K. G.; Olmsted, D. L.; Asta, M.; Laird, B. B.; Brian, B,; Yaghi, O. M. A Combined ExperimentalComputational Investigation of Methane Adsorption and Selectivity in a Series of Isoreticular Zeolitic Imidazolate Frameworks, J. Phys. Chem. C 2013, 117, 10326-10335. (14) Tepper, H.L.; Hoogenboom, J. P.; van der Vegt, N. F. A.; Briels, W.J. Unidirectional Diffusion of Methane in AlPO4-5, J. Chem. Phys. 1999, 110, 11511-11516. (15) Cao, D.; Zhang, X.; Chen, J.; Wang, W.; Yun, J. Optimization of Single-Walled Carbon Nanotube Arrays for Methane Storage at Room Temperature. J. Phys. Chem. B 2003, 107, 13286-13292. (16) Liu, L.; Nicholson, D.; Bhatia, S. K. Adsorption of CH4 and CH4/CO2 Mixtures in Carbon Nanotubes and Disordered Carbons: A Molecular Simulation Study. Chem. Eng. Sci. 2015, 121, 268–278

(17) Skoulidas, A. I.; Ackerman, D. M.; Johnson, J. K.; Sholl, D. S. Rapid Transport of Gases in Carbon Nanotubes. Phys. Rev. Lett. 2002, 89, 185901. (18) Chen, H.; Sholl, D.S. Rapid Diffusion of CH4/H2 Mixtures in Single-Walled Carbon Nanotubes. J. Am. Chem. Soc. 2004, 126, 7778-7779. (19) Cao, D.; Wu, J. Self-Diffusion of Methane in Single-Walled Carbon Nanotubes at Sub- and Supercritical Conditions. Langmuir 2004, 20, 3759-3765. (20) Chen, H.; Johnson, J. K.; Sholl, D. S. Transport Diffusion of Gases Is Rapid in Flexible Carbon Nanotubes. J. Phys. Chem. B 2006, 110, 1971-1975. (21) Jakobtorweihen, S.; Lowe, C. P.; Keil, F. J.; Smit, B. Diffusion of Chain Molecules and Mixtures in Carbon Nanotubes: The Effect of Host Lattice Flexibility and Theory of Diffusion in the Knudsen Regime. J. Chem. Phys. 2007, 127, 024904. (22) Sokhan, V. P.; Nicholson, D.; Quirke, N. Fluid Flow in Nanopores: Accurate Boundary Conditions for Carbon Nanotubes. J. Chem. Phys. 2002, 117, 8531-8539.


20


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 33

(23) Jakobtorweihen, S.; Verbeek, M. G.; Lowe, C. P.; Keil, F. J.; Smit, B. Understanding the Loading Dependence of Self-Diffusion in Carbon Nanotubes. Phys. Rev. Lett. 2005, 95, 044501. (24) Jakobtorweihen, S.; Lowe, C. P.; Keil, F. J.; Smit, B. A Novel Algorithm to Model the Inﬂuence of Host Lattice Flexibility in Molecular Dynamics Simulations: Loading Dependence of Self-Diffusion in Carbon Nanotubes. J. Chem. Phys. 2006, 124, 154706. (25) Jakobtorweihen, S.; Keil, F. J.; Smit, B. Temperature and Size Effects on Diffusion in Carbon Nanotubes. J. Phys. Chem. B 2006, 110, 16332-16336. (26) Chopra, M.; Choudhury, N. Comparison of Structure and Dynamics of Polar and Nonpolar Fluids through Carbon Nanotubes. J. Phys. Chem. C 2013, 117, 18398−18405. (27) Bartuś, K.; Bródka, A. Methane in Carbon Nanotube: Molecular Dynamics Simulation. Mol. Phys. 2011, 109, 1691-1699.

(28) Bartuś, K.; Bródka, A. Temperature Study of Structure and Dynamics of Methane in Carbon Nanotubes, J. Phys. Chem. C 2014, 118, 12010−12016. (29) Bhide, S. Y.; Yashonath, S. Orientational Preference and Inﬂuence of Rotation on Methane Mobility in One-Dimensional Channels. J. Chem. Phys. 2002, 116, 2175-2183. (30) Zhu, X.; Zhao, Y.-P. Atomic Mechanisms and Equation of State of Methane Adsorption in Carbon Nanopores. J. Phys. Chem. C 2014, 118, 17737−17744. (31) Mao, Z.; Garg, A.; Sinnott, S. B. Molecular Dynamics Simulations of the Filling and Decorating of Carbon Nanotubules. Nanotechnology 1999, 10, 273-277. (32) Mao, Z.; Sinnott, S. B. A Computational Study of Molecular Diffusion and Dynamic Flow through Carbon Nanotubes. J. Phys. Chem. B 2000, 104, 4618-4624. (33) Lee, K.-.H.; Sinnott, S. B. Computational Studies of Non-Equilibrium Molecular Transport through Carbon Nanotubes. J. Phys. Chem. B 2004, 108, 9861-9870.


21

Page 29 of 33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(34) Mutat, T.; Adler, J.; Sheintuch, M. Single species transport and self-diffusion in wide single-walled carbon nanotubes. J. Chem. Phys. 2012, 136, 234902. (35) Vela, S.; Huarte-Larranaga, F. A Molecular Dynamics Simulation of Methane Adsorption in Single Walled Carbon Nanotube Bundles. Carbon 2011, 49, 4544-4553. (36) Bhatia, S. K.; Nicholson, D. Adsorption and Diffusion of Methane in Silica Nanopores: A Comparison of Single-Site and Five-Site Models. J. Phys. Chem. C 2012, 116, 2344-2355. (37) Ni, B.; Sinnott, S.B. ; Mikulski, P.T.; Harrison, J.A. Compression of carbon nanotubes filled with C60, CH4, or Ne: predictions from molecular dynamics simulations, Phys. Rev. Lett. 2002, 88, 205505. (38) Bródka, A. Molecular dynamics study of cyclohexane in a cylindrical pore of amorphous silica. Mol. Phys. 1994, 83, 803-813. (39) Bródka, A. Diffusion in restricted volume. Mol. Phys. 1994, 82, 1075-1078. (40) Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, N.; Sinnott, S. B. A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression for Hydrocarbons. J. Phys.: Condes. Matter 2002, 14, 783-802. (41) Smith, W. DLPOLY – applications to molecular simulation. Mol. Simul. 2006, 32, 935-943. (42) Yakobson, B. I.; Brabec, C. J.; Bernholc, J. Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response. Phys. Rev. Lett. 1996, 76, 2511-2514. (43) Sinnott, S. B.; Shenderova, O. A.; White, C. T.; Brenner, D. W. Mechanical Properties of Nanotubule Fibers and Composites Determined from Theoretical Calculations and Simulations. Carbon 1998, 36, 1-9. (44) Garg, A.; Sinnott, S. B. Effect of chemical functionalization on the mechanical properties of carbon nanotubes. Chem. Phys. Lett., 1998, 295, 273-278. (45) Bródka, A.; Zerda, T.W.; Burian, A. Graphitization of small diamond cluster – Molecular dynamics simulation, 2006, Diamond Relat. Mater. 15, 1818–1821.


22


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 33

(46) Bródka, A.; Hawełek, Ł.; Burian, A.; Tomita, S.; Hönkimäki, V. Molecular Dynamics Study of Structure and Graphitization Process of Nanodiamonds. J. Mol. Struct. 2008, 887, 34-40. (47) Bródka, A.; Kołoczek, J.; Burian, A. Application of Molecular Dynamics Simulations for Structural Studies of Carbon Nanotubes. J. Nanosci. Nanotechnol. 2007, 7, 1505-1511. (48) Hawełek, Ł.; Bródka, A.; Dore, J.C.; Hönkimäki, V.; Tomita, S.; Burian, A. Structural studies of nanodiamond by high-energy X-ray diffraction, Diamond Relat. Mater. 2008, 17, 1186-1193. (49) Hawełek, Ł.; Bródka, A.; Dore, J.C.; Hönkimäki, V.; Burian, A. Fullerene-like structure of activated carbons, 2008, Diamond Relat. Mater. 17, 1633-1638. (50) Hawełek, Ł.; Bródka, A.; Dore, J. C.; Hannon, A. C.; Iijima, S.; Yudasaka, M.; Ohba, T.; Kaneko, K.; Burian, A. Structural Modeling of Dahlia-Type Single-Walled Carbon Nanohorn Aggregates by Molecular Dynamics, J. Phys. Chem. A 2013, 117, 9057−9061. (51) Girifalco, L. A.; Hodak, M.; Lee, R. S. Carbon Nanotubes, Buckyballs, Ropes, and a Universal Graphitic Potential. Phys. Rev. B 2000, 62, 13104-13110. (52) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1989. (53) Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; PrenticeHall: Englewood Cliffs, NJ, 1971. (54) Hoover, W. G.; Ladd, A. J. C.; Moran, B. High-Strain-Rate Plastic Flow Studied via Nonequilibrium Molecular Dynamics, Phys. Rev. Lett. 1982, 48, 1818–1820. (55) Evans, D. J.; Hoover, W. G.; Failor, B. H.; Moran, B.; Ladd, A. J. C. Nonequilibrium molecular dynamics via Gauss's principle of least constraint. Phys. Rev. A 1983, 28, 1016– 1020. (56) Humphrey, W.; Dalke, A.; Schulten, K. VMD: visual molecular dynamics, J. Mol. Graph. 1996, 14, 33-38.


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Figure captions Figure 1. Examples of probability distribution for methane molecules in the CNT as a function of distance from the CNT axis, ρR(r), for indicated molecular systems and temperatures. Snapshots along the CNT axis are shown for two systems and the lowest and highest temperatures considered here. Figure 2. Mean square displacement functions of methane translations along the CNT ∆r||2(t) (right scale) and slope of ln[∆r||2(t)] with respect to the ln(t) (left scale) as functions of time for the lowest and highest densities and temperatures considered here. Figure 3. Mean square displacement functions of methane translations across the CNT ∆r⊥2(t) for the lowest and highest densities and temperatures considered here. Circles and triangles mean values estimated using formula (16) and data given in the text. Figure 4. Examples of normalized correlation function of angular velocity for the lowest and highest temperatures. Figure 5. Diffusion coefficients for translational motion of the methane molecules, DT, (a) and rotational one, DR, (b) as functions of pressure p for indicated temperatures. The solid and open symbols mean simulation results for flexible and rigid model of the methane molecule, respectively. The lines represent linear function and quadratic function for translations and rotations, respectively, fitted to the simulation data represented by symbols. The solid and dashed lines mean results for flexible and rigid methane molecules, respectively. In the insets there are presented Arrhenius plots of the diffusion coefficients of the flexible methane molecules for indicated pressures.


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Figure 6. Activation energies of translational and rotational diffusions for the flexible model of the CH4 molecules (filled symbols and solid lines) and rigid molecules (open symbols and dashed lines). The solid lines are to guide the eye.


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