Equilibrium structures and flows of polar and nonpolar liquids and their mixtures in carbon nanotubes with rectangular cross sections
EEquilibrium structures and flows of polar and nonpolar liquids and their mixtures incarbon nanotubes with rectangular cross sections
L.V. Mirantsev ∗ and A. K. Abramyan Institute for Problems of Mechanical Engineering, Russian Academy of Sciences,199178, Bolshoi 61, V. O., St. Petersburg, Russia (Dated: December 27, 2018)Molecular dynamics (MD) simulations of equilibrium structures and flows of polar water, nonpolarargon and methane, and mixtures of water and methane confined by single - walled carbon nanotubes(SWCNTs) with different rectangular cross sections have been performed. The results of thesesimulations show that equilibrium structures and flows of all confined liquids significantly dependnot only on the shape of the SWCNT’s rectangular cross sections but also on the types of liquidsinside SWCNTs.
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I. INTRODUCTION
During the last 20 - 30 years, carbon nanotubes(CNTs) attract a huge interest of the international re-search community. There are two types of CNTs, namely,single - walled carbon nanotubes (SWCNTs) and multi- walled carbon nanotubes (MWCNTs) which are struc-tures composed of several concentric SWCNTs with dif-ferent chiral angles [1–5]. Diameters of CNTs range froma few to tens nanometers, and their lengths may reachseveral microns[6–9]. Both SWCNTs and MWCNTs ex-hibit superior mechanical, electronic, thermal and trans-port properties [10–22]. For example, experimental in-vestigations and computer simulations revealed that apressure drop driven fluid flows through CNTs could bemuch faster relative to predictions of classic hydrody-namics. Due to their unique properties, CNTs are widelyused in many electronic, medical, space, and military ap-plications [23–25]. Furthermore, membranes and filmsformed dy CNTs can be also used for a water purifica-tion and separation of various organic contaminants fromthe water [26–28].The most studies of CNTs have been dedicated to con-trolling their length [29, 30], wall number [31, 32], diame-ter [33–38], and chirality [39, 40], and in all these studiesCNTs had circular cross sections. However, in [41], apossibility of formation of CNTs with rectangular crosssections was discussed, and, recently, it was reported in[42] that such carbon nanotubes can be really formed. Inour previous paper [43], using molecular dynamics (MD)simulations, we performed study of equilibrium behav-iors and flows of polar and nonpolar liquids confined bySWCNTs with square cross section and showed that equi-librium fluid structures inside such SWCNTs and fluidflows through them can be strongly different from thosein SWCNTs with circular cross sections. Nevertheless,rectangular cross sections of CNTs could have shapes dif- ∗ author to whom correspondence should be addressed. Email ad-dress:[email protected]; ferent from the square one, and equilibrium and dynamicbehaviors of various fluids inside such CNTs could be sig-nificantly different from those in CNTs with square crosssections.In the present paper, using MD simulations, we studyequilibrium structures and flows of polar and nonpolarliquids and their mixtures confined by SWCNTs havingrectangular cross sections with different ratios betweentheir sides, namely, 1 : 1 (square cross section), 1 : 2,and 1 : 4. All these nanotubes have the same lengthand the cross section area. In our simulations, the polarfluid is the water, and the nonpolar ones are methaneand argon in their liquid phase. It has been found thatequilibrium structures of above mentioned confined liq-uids depend strongly on the shapes of cross sections ofSWCNTs. In addition, these equilibrium structures andflows of such fluids are very sensitive to a polarity of thefluid and interactions between their atoms (molecules)and interactions between liquid particles and boundarywall carbon atoms. II. SIMULATION DETAILS
As said above, using MD simulations, we investigatestatic and dynamic behavior of polar and nonpolar modelfluids and their mixtures confined by carbon nanotubeswith rectangular cross sections. As in [43, 44], we use avery simple model of polar fluid in which molecules areassumed to be point-like particles possessing a perma-nent dipole moment. These particles interact with eachother via the short-range Lennard-Jones (LJ) pairwisepotential U LJ ( r ij ) = 4 (cid:15) ij (cid:100) ( σ ij /r ij ) − ( σ ij /r ij ) (cid:101) , (1)where (cid:15) ij and σ ij are the strength and characteristiclength, respectively, for the LJ interaction between i -thand j -th molecules, and r ij is the distance between them, and the dipole-dipole interaction potential U dd ( r ij ) = ( (cid:126)d i · (cid:126)d j ) /r ij − (cid:126)d i · (cid:126)r ij )( (cid:126)d j · (cid:126)r ij ) /r ij , (2) a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec where (cid:126)d i is the dipole moment of the i -th molecule.Since the polar molecules are considered as the wa-ter ones, the LJ interaction constants (cid:15) ij and σ ij inEq. (1) should be similar to those for the LJ interac-tions between oxygen atoms of the i -th and j -th wa-ter molecules. These constants, (cid:15) H O and σ H O , aretaken from the well known model for water molecules[45, 46], and they are equal to (cid:15) H O = 1 . × − erg and σ H O = 3 . d w = 1 . × − g / cm / s − . An interaction betweenwater molecules and carbon atoms of SWCNTs is de-scribed by the LJ potential similar to Eq.(1)in whichthe interaction constants (cid:15) H O and σ H O are replacedby (cid:15) CH O and σ CH O , respectively. All these constantsare taken from [45]. As for nonpolar argon and methanemolecules, they are described in framework of the unitedatom model of methane [17] and the well known modelof argon atoms [47]. According to this models, the argonatoms Ar and the methane molecules CH (cid:15) CH , σ CH , (cid:15) CCH , and σ CCH are taken from [17], and the constants (cid:15) Ar , σ Ar for argonatoms are taken from [47]. As for interactions betweenargon and carbons atoms, the corresponding constants (cid:15) CAr , and σ CAr are determined by means of the Lorentz- Berthelot rules. The total force (cid:126)F i and the total torque (cid:126)τ i acting on the i-th fluid molecule due to interactionswith other molecules, and equations of motion of thismolecule are given by Eqs. (3) - (8) in [44]. All sim-ulations are performed in the N V T ensembles, and, ateach time step, the equations of motion of fluid moleculesare solved numerically by the standard method describedin [48]. The temperatures of the systems under consid-eration are kept constant ( T = 300 K for water and itsmixture with methane, T = 108 K for liquid phase ofmethane, and T = 85 K for liquid phase of argon) byemployment of the Berendsen thermostat [49]. III. SIMULATION RESULTS AND DISCUSSION
First of all, we studied equilibrium static structuresof the polar water molecules, nonpolar argon atomsand methane molecules, and the mixture of water andmethane molecules in SWCNTs mentioned in Introduc-tion. The lateral projections and cross sections of theseSWCNTs are shown in figures 1 (a) - 1(c).All these carbon nanotubes have the same length L =3 . nm and the cross section area S = 1 . nm , and theirbounding walls have the graphene crystalline structure.Since the carbon atoms in CNTs are connected to eachother with very strong covalent bonds [50] with the in-teraction constants much larger than the LJ interactionconstants (cid:15) CH O , (cid:15) CAr and (cid:15)
CCH , these atoms are con-sidered to be fixed at their equilibrium sites. This ap-proximation is supported by estimations of thermal vi- FIG. 1: The lateral projections and cross sections of SWC-NTs used in our MD simulations. a - SWCNT with squarecross section; b - SWCNT with rectangular cross section andwith the ratio between its sides 1 : 2; c - SWCNT with rect-angular cross section and with the ratio between its sides 1 :4. brations of SWCNTs performed in [51]. According tothese estimations, average amplitudes of such vibrationsare of the order of ∼ .
01 nm that is significantly smallerthan typical molecular sizes. In order to obtain equilib-rium static structures of the water, argon, methane, andthe mixture of the water and methane molecules insidethe above mentioned SWCNTs, we performed MD simu-lations of free (without an external pressure drop) perme-ations of water and methane molecules, and argon atomsinto these SWCNTs. These simulations started from theinitial configuration schematically depicted in figure 2(a).This configuration is made as follows: initially we havetwo reservoirs of water or methane molecules, and ar-gon atoms separated from each other by the wall con-sisted of carbon atoms. Then, we make a channel inthis wall and insert the corresponding SWCNT into thechannel. Now, this SWCNT connects two reservoirs witheach other and liquid atoms (molecules) can freely per-meate into the SWCNT without any external action. Thesame procedure is performed for the mixture of the wa-ter and methane molecules. In this case, both reservoirscontain initially equal numbers of the water and methanemolecules. After running during about 100000 time steps(one time step was equal to 0.001 ps), one can reach theequilibrium configuration schematically shown in figure2(b).During simulation processes, the system under consid-eration is placed within the parallelepiped simulation boxof 6 . × . × . nm in size, and the periodicboundary conditions [48] were imposed on the system in x y , and z directions.Figures 3a - 3l demonstrate cross sections of equilib-rium configurations of water molecules (figures 3a - 3c),argon atoms (figures 3d - 3f), methane molecules (figures3g - 3i), and mixture of water and methane molecules(figures 3j - 3l) inside SWCNTs with the described above FIG. 2: The lateral projection of initial and final equilibriumconfigurations considered in our MD simulations. a - initialconfiguration; b - final equilibrium configuration. Blue circlesare liquid atoms (molecules), and red ones are carbon atoms. rectangular cross sections. The corresponding densityprofiles along z (vertical) and y (lateral) directions per-pendicular to the tube x axes are shown in figures 4 - 7.If we look at figures 3a - 3l, it is clearly seen that, forall types of liquid atoms (molecules), the main featuresof the corresponding equilibrium structures are similar,and these features are defined by the shapes of SWCNTcross sections. For example, all liquid atoms (molecules)inside SWCNTs with square cross sections form struc-tures with square - like cross sections which are reducedreplicas of the SWCNTs ones (see figures 3a, 3d, 3g, 3j).All liquid atoms (molecules) inside SWCNTs with rect-angular cross stctions having the ratio 1 : 2 between theirsides form equilibrium structures consisting of two planesparallel to the vertical bounding walls (see figures 3b, 3e,3h, 3k). Finally, all liquid atoms (molecules)inside SWC-NTs with rectangular cross sections having the ratios 1: 4 between their sides form equilibrium 2D structuresin a form of the plane parallel to the vertical boundingwalls (see figures 3c, 3f, 3i, 3l). Thus, the shape of therectangular SWCNT cross sections plays a dominant rolein a formation of equilibrium structures of liquid atoms(molecules) inside SWCNTs.However, it is simultaneously seen that the equilib-rium structures depicted in figures 3a - 3l demonstratecertain additional features depending on characteristicsof concrete liquid atoms (molecules) under considera-tion. It is easily seen that, inside all SWCNTs, liq-uid structures formed by argon atoms are most ordered.For example, argon atoms inside SWCNT with squarecross section form 9 well ordered chains parallel to thetube axis (see figure 3d). One of these chains coin-cides with the tube axis, and other 8 chains are dis-posed on bounding surfaces of imaginable parallelepipedinside this SWCNT. Simultaneously, methane moleculesinside the same SWCNT form similar structure but withmore smeared chains (see figure 3g). This fact is also re-flected in figures 5a and 6a, which demonstrate the den-sity profiles along z axis for argon atoms and methanemolecules inside the same SWCNT with square crosssection. One can see that the density profile for ar-gon atoms exhibits three peaks having almost the sameheight and width equal to about 0.6 σ Ar . The analo- gous profile for methane molecules has also three peaks,but the central one is noticeably lower than two oth-ers, and the width of these peaks is about 0.6 σ CH that is about 10 percent larger than that of the anal-ogous peaks in the density profile for argon atoms. If welook at figure 3a which exhibits the cross section of theequilibrium water structure inside the SWCNT with thesquare cross section and at figure 4a demonstrating thecorresponding density profile along z axis, then we findthat this structure is much more disordered relative tothose depicted in figures 3d and 3g for argon atoms andmethane molecules, respectively. Perhaps, it is due to theCoulomb - like dipole - dipole interactions between polarwater molecules which do not occur in ensembles of non-polar argon atoms and methane molecules. Accordingto the Earnshaw theorem [52], an ensemble of particlesinteracting via Coulomb - like forces cannot be main-tained in a stable stationary equilibrium configurations.Thus, the well ordered structures of water molecules in-side SWCNTs cannot exist for a sufficiently long times.If we look at figure 3j, which demonstrates the cross sec-tion of the equilibrium structure formed by the mixtureof water and methane molecules inside SWCNT withsquare cross section, we can find that this structure re-sembles the above mentioned structure formed by thepure methane inside the same SWCNT. This fact is alsoconfirmed by the density profiles for methane and wa-ter molecules depicted in figure 7a. The density pro-file for methane molecules (curve 2) exhibits three welldeveloped peaks of almost similar height, whereas theanalogous profile for water molecules (curve 1) demon-strates two well developed peaks at the edges and thestrongly smeared central one. This fact can be explainedby that the interaction constants (cid:15) CH and (cid:15) CCH for in-teractions between methane molecules and those betweenmethane molecules and boundary wall carbon atoms, re-spectively, are significantly larger than analogous inter-action constants (cid:15) H O and (cid:15) CH O for water molecules.So, if we look at figures 3b, 3c, 3e, 3f, 3h, 3i, 3k, 3l,which demonstrate equilibrium structures formed by wa-ter molecules, argon atoms, methane molecules, and themixture of water and methane molecules inside SWCNTswith rectangular cross sections having the ratios betweentheir sides 1 : 2 and 1 : 4, and at figures 4b - 4e, 5b - 5e, 6b- 6e, 7b - 7e demonstrating the corresponding equilibriumdensity profiles, we can conclude that the said above forequilibrium structures inside SWCNT with square crosssection is valid for equilibrium structures in all SWC-NTs with rectangular cross sections. The most orderedstructures are exhibited by argon atoms, whereas watermolecules form most disordered structures, and the posi-tional order of structures formed by methane moleculesand the mixtures H O + CH (cid:15) and σ playvery important role not only in equilibrium structuresand average liquid densities inside SWCNTs but alsoin a compositions of the mixture of water and methanemolecules inside different SWCNTs. For example, in-side SWCNT with square cross section, the ratio be-tween numbers of water and methane molecules is equalto 43 : 59, whereas inside SWCNTs with rectangularcross sections these molecules occur in almost equal pro-portions, namely, 52:49 for SWCNT with ratio betweenits sides 1 : 2 and 43 : 46 for SWCNT with the sideratio 1 : 4. This result can be qualitatively explainedas follows. In the case of SWCNT with square cross sec-tion, the distance between bounding walls is large enoughfor both water and methane molecules. Therefore, thestrengths of interactions between liquid molecules andbounding walls, (cid:15) CH O and (cid:15) CCH , play a main role intheir penetration into SWCNTs, and the effective sizesof these molecules, σ H O and σ CH , play a minor role.Since the interaction constant (cid:15) CCH for the interactionbetween methane molecules and bounding wall carbonatoms is noticeably larger than the analogous interac-tion constant (cid:15) CH O for the water molecules, then thenumber of methane molecules penetrating into SWCNTwith square cross section is larger than that of watermolecules. For SWCNTs with rectangular cross sectionshaving ratios between their sides equal to 1 : 2 and 1: 4, respectively, the distances between lateral boundingwalls along y direction become sufficiently small for largermethane molecules and and not so small for smaller wa-ter molecules (effective diameter σ CH of the methanemolecule is about 10 percent larger than the analogousdiameter σ H O of the water molecule). Thus, for theseSWCNTs, a competition between interactions of differ-ent liquid molecules with bounding wall atoms and theirmolecular sizes occurs, and these competition equalizesconcentrations of water and methane molecules insidesuch SWCNTs.Now, let us turn to results of MD simulations of flowsof the polar water, nonpolar methane and argon, andthe mixture of the water and methane through the abovementioned SWCNTs with rectangular cross sections un-der action of the external pressure drops across thesenanotubes. To simulate these flows, in the equilibriumconfigurations depicted in figure 2(b), we remove the liq-uid reservoirs and apply periodic boundary conditions toedges of SWCNTs. The external pressure drop acrossnatotubes is mimiced by the external force f x acting oneach liquid particle inside SWCNTs. In order to calcu-late velocity profiles, we devide the space inside carbonnanotubes into very thin sublayers parallel to the topand bottom bounding walls and calculate average molec-ular velocities inside these sublayers as a function of z coordinates of their centers. The fluid flow velocity pro-files obtained from our simulations are shown in figures8(a) - 8(c) for the polar water, the nonpolar argon, andthe mixture of the water and methane, respectively. Theprofiles for the water and argon flows are obtained forthe external force equal to f x = 0 .
05 (in reduced MDunits [48]), and for the flow of the mixture of the waterand methane f x = 0 . FIG. 3: Cross sections of equilibrium structures of watermolecules (3a - 3c), argon atoms (3d - 3f), methane molecules(3g - 3f), and the mixture of the water and methane molecules(3j - 3l) inside SWCNTs with rectangular cross sections de-picted in figures 1a - 1c. In all figures brown circules de-note the bounding wall carbon atoms. In figures 3a - 3i,blue circules denote liquid atoms (molecules). In figures 3j- 3l, blue and red circules denote the water and methanemolecules, respectively. The equilibrium ratios between thewater and methane molecules are 43 : 59, 52 : 49, and 43 :46 for SWCNTs with rectangular cross sections with ratiousbetween their sides equal to 1 : 1, 1 : 2, and 1 : 4, respectively. through all rectangular SWCNTs under consideration atthese values of f x is absent. This case will be discussedbelow.One can see from figures 8a - 8c that fluid flows throughSWCNTs with rectangular cross sections depend stronglyon both the type of the fluid inside the tube and theshape of its cross section. For example, it is easilyseen that, for the polar water, the average flow velocity v averx should have a maximum value for SWCNT hav-ing rectangular cross section with the ratio between itsside 1 : 4 ( v averx = 1 .
15 in MD units). The interme-diate value v averx = 0 .
97 corresponds to the water flowthrough SWCNT having rectangular cross section withthe ratio between its side 1 : 2, and the minimum value v averx = 0 .
28 exhibits the water flow through SWCNTwith square cross section. For nonpolar argon, the av-erage fluid flow velocity has also the minimum value v averx = 0 .
08 in the case of SWCNT with square crosssection but the results for two SWCNTs with other rect-angular cross sections change places: the maximum av-erage flow velocity v averx = 1 . v averx = 0 .
7. Finally, for the mix-ture of polar water and nonpolar methane, we have themaximum value v averx = 0 . v averx = 0 . -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,50,00,51,01,52,02,53,03,54,04,55,05,5 (a) z/ H2O -3 -2 -1 0 1 2 30,00,51,01,52,02,53,03,54,0 (b) z/ H2O -1,5 -1,0 -0,5 0,0 0,5 1,0 1,50123456789 (c) y/ H2O -5 -4 -3 -2 -1 0 1 2 3 4 50,00,51,01,52,02,53,0 (d) z/ H2O -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5024681012 (e) y/ H2O
FIG. 4: Equilibrium density profiles for the water inside SWCNTs with different rectangular cross sections. a - The densityprofile along z axis for SWCNT with square cross section. For symmetry reasons the analogous profile along y axis should besimilar. ρ = 0 . y axis for the same SWCNT. ρ = 0 . z axis for SWCNT with rectangular ( cross section having the ratio between its sides1 : 4. e - The density profile along y axis for the same SWCNT. ρ = 0 . fluid flow velocity v averx = 0 . f x , which is a given constant, and by certain retarding forces dueto the interactions between liquid atoms (molecules) andbounding wall carbon atoms. It is also clear that thestronger these interactions the stronger retarding forces,and, hence, the slower the fluid flow. In our simulations,the interactions between liquid atoms (molecules) andthe bounding wall carbon atoms are modelled by means -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,00,00,51,01,52,02,53,03,54,0 (a) z/ Ar -3 -2 -1 0 1 2 30,00,51,01,52,02,53,03,54,04,5 (b) z/ Ar -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5012345678 (c) y/ Ar -4 -3 -2 -1 0 1 2 3 40,00,51,01,52,02,53,0 (d) z/ Ar -1,5 -1,0 -0,5 0,0 0,5 1,0 1,50246810 (e) y/ Ar FIG. 5: Equilibrium density profiles for argon atoms inside SWCNTs with different rectangular cross sections. a - The densityprofile along z axis for SWCNT with square cross section. ρ = 0 . y axis for the same SWCNT. ρ = 0 . z axis for SWCNT with rectangular cross section having the ratio between its sides 1 : 4. e - Thedensity profile along y axis for the same SWCNT. ρ = 0 . of the LJ pairwise potentials which are characterizedby the above mentioned interaction constants (cid:15) CH O , (cid:15) CCH , (cid:15) CAr , and characteristic lengths σ CH O , σ CCH , σ CAr . It is also well known that these LJ potentials havethe minimum disposed at the distance r ∗ to a given car-bon atom equal to r ∗ = 2 / σ , and, at this minimum,the force acting on the liquid particle is equal to zero.When the distance between the liquid particle and the carbon atom is less than r ∗ this force is repulsive, and,for distances larger than r ∗ it is attractive. When westudy the flow of the same liquid particles through dif-ferent SWCNTs, the interaction constant (cid:15) for LJ interac-tions between liquid particles and bounding wall carbonatoms is the same for all SWCNTs, and only distances be-tween wall atoms and liquid particles define a differencein their flows. Let us consider the water flows through -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0012345 (a) z/ CH4 -3 -2 -1 0 1 2 30,00,51,01,52,02,53,03,54,0 (b) z/ CH4 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,50123456 (c) y/ CH4 -4 -3 -2 -1 0 1 2 3 40,00,51,01,52,02,53,03,5 (d) z/ CH4 -1,0 -0,5 0,0 0,5 1,002468101214 (e) y/ CH4
FIG. 6: Equilibrium density profiles for methane molecules inside SWCNTs with different rectangular cross sections. a - Thedensity profile along z axis for SWCNT with square cross section. ρ = 0 . y axis for the same SWCNT. ρ = 0 . z axis for SWCNT with rectangular cross section having the ratio between its sides1 : 4. e - The density profile along y axis for the same SWCNT. ρ = 0 . different rectangular SWCNTs. It iseasy to calculate av-erage minimum distances d avermin from water molecules tothe bounding wall carbon atoms corresponding to equi-librium structures of water molecules inside these SWC-NTs depicted in figures 3a - 3c. These distances areequal to 1 . σ H O for SWCNT with square cross sec-tion, 1 . σ H O for SWCNT with rectangular cross sec-tionwith the ratio between its sides 1 : 2, and 1 . σ H O for SWCNT with other rectangular cross section withanalogous ratio equal to 1 : 4. The distance r ∗ for LJinteractions between water molecules and bounding wallcarbon atoms is equal to r ∗ = 1 . σ H O . One can seethat, for all these SWCNTs, d avermin is smaller than r ∗ ,and, therefore, inside these nanotubes, the forces actingon water molecules from bounding wall carbon atoms arerepulsive. In addition, the larger difference between r ∗ -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,50,00,51,01,52,02,5 (a) z/ H2O -3 -2 -1 0 1 2 30,00,51,01,52,02,5 (b) z/ H2O -1,5 -1,0 -0,5 0,0 0,5 1,0 1,50,00,51,01,52,02,53,03,54,04,5 (c) y/ H2O -4 -3 -2 -1 0 1 2 3 40,00,20,40,60,81,01,21,41,61,8 (d) z/ H2O -1,0 -0,5 0,0 0,5 1,00,00,51,01,52,02,53,03,54,04,55,05,5 (e) y/ H2O
FIG. 7: Equilibrium density profiles for mixtures of water and methane molecules inside SWCNTs with different rectangularcross sections 1 - the water profiles, 2 - the methane profiles. a - The density profile along z axis for SWCNT with squarecross section. ρ = 0 . y axis for the same SWCNT. ρ = 0 . z axis forSWCNT with rectangular cross section having the ratio between its sides 1 : 4. e - The density profile along y axis for thesame SWCNT. ρ = 0 . and d avermin the stronger these forces. Hence, these forcesare strongest for SWCNT with square cross section, theyare weakest for SWCNT with rectangular cross sectionhaving the ratio between its sides 1 : 4, and, for otherSWCNT with rectangular cross section we have an inter-mediate value for the force between water molecules andbounding wall carbon atoms. So, one can conclude that the water flow through SWCNT with rectangular crosssection having the ratio between its sides 1 : 4 should befastest, for water flow through SWCNT with square crosssection should be slowest, and the water flow throughSWCNT with other rectangular cross section should haveintermediate average liquid flow velocity. These specu-lations are in a qualitative agreement with the velocityprofiles depicted in figure 8a. One can repeat such qual-itative analysis for the flow of argon atoms through theabove mentioned SWCNTs with rectangular cross sec-tions. For equilibrium structures of argon atoms insidethese nanotubes depicted in figures 3d - 3f, we obtain thevalues of d avermin equal to 0 . σ Ar , 0 . σ Ar , and 0 . σ Ar for SWCNTs having rectangular cross sections with theratios between their sides equal to 1 : 1, 1 : 2, 1 : 4,respectively, and r ∗ for LJ interactions between argonand carbon atoms equal to r ∗ = 1 . σ Ar . Therefore,inside all these SWCNTs, argon atoms are subjected torepulsive forces from the bounding wall carbon atoms,and these forces are strongest for SWCNT with squarecross section, weakest for SWCNT with rectangular crosssection with the ratio between its sides 1 : 2, and theyhave an intermediate value for other SWCNT with rect-angular cross section. Then, the argon flow should befastest for SWCNT having the rectangular cross sectionwith the ratio between its sides 1 : 2, the average fluidflow velocity should be lowest for SWCNT with squarecross section, and the argon flow through SWCNT hav-ing rectangular cross section with the ratio between itssides 1 : 4 should have an intermediate value of the av-erage fluid flow velocity. The results of this analysis isalso in a qualitative agreement with the fluid flow profilesdepicted in figure 8b.As said above, for the external forces equal to f x =0 .
05 and f x = 0 .
1, the flow of methane moleculesthrough all SWCNTs under consideration is absent.Therefore, we increased little by little the external force f x and found that there are certain threshold or criticalvalues of f x , f cx , above which methane molecules canflow through SWCNTs with rectangular cross sections.These critical values, which can be considered as certainstrengths of breakaway, depend strongly on the shape ofSWCNT cross sections. We found that, for SWCNT withthe square cross section, f cx = 0 .
275 (in reduced MDunits), for SWCNT with the rectangular cross sectionwith the ratio between its sides 1 : 2, f cx = 0 .
15, and,for SWCNT having the rectangular cross section with theratio between its sides 1 : 4, f cx = 0 .
8. The followingquestions arise: i) Why the liquid methane flows throughSWCNTs with rectangular cross sections demonstrate anexistence of strengths of breakaway that is absolutely notinherent to flows of ordinary liquids?; ii) Why we do notobserve such strengths of breakaway for the water andargon flows through the same SWCNTs?; iii) How canwe explain the above mentioned dependence of f cx onthe shape of the SWCNT cross sections?. The answer tothe first question seems to be sufficiently obvious. If welook at figures 3a - 3i, which exhibit equilibrium struc-tures of argon atoms and water and methane moleculesinside SWCNTs under consideration, we can see an oc-currence of different types of positional order which is notinherent to an ordinary liquid phase. Thus, fluid atoms(molecules) inside our SWCNTs form solid -like struc-tures, and, as is well known, the strength of breakawayis a typical phenomenon for sliding a solid along a solid surface.The answer to the second question is also simpleenough. The interaction constant (cid:15) CCH , which de-fines a strength of interaction between methane moleculesand bounding wall carbon atoms, is considerably largerthan analogous constants (cid:15) CH O and (cid:15) CAr which de-fine strengths of interactions between bounding wall car-bon atoms and water molecules and argon atoms, re-spectively. In addition, the effective size of methanemolecules, σ CH , is larger than effective sizes of argonatoms σ Ar and water molecules σ H O . Therefore, the in-teraction between methane molecules and bounding wallcarbon atoms is significantly stronger than analogous in-teractions of water molecules and argon atoms. Perhaps,their flows through SWCNTs under consideration couldalso exhibit certain strengths of breakaway, but thesestrengths are much lower than the force f x used to driveargon atoms and water molecules.In order to answer to the third question, we should,as we made above, calculate average minimum distances d avermin between methane molecules and bounding wall car-bon atoms for equilibrium structures formed by methanemolecules inside SWCNTs under consideration. Our cal-culations give d avermin = 0 . σ CH for SWCNT with squarecross section, d avermin = 0 . σ CH for SWCNT having rect-angular cross section with the ratio between its sides 1 :2, and d avermin = 0 . σ CH for SWCNT with other rectan-gular cross section. We also obtain r ∗ = 1 . σ CH forLJ interactions between methane molecules and bound-ing wall carbon atoms. Repeating the above reason-ing about relationship between difference r ∗ − d avermin andthe strength of interactions between liquid particles andbounding wall carbon atoms, one can conclude thatsuch interaction between methane molecules and carbonatoms should be strongest for SWCNT having rectangu-lar cross section with the ratio between its sides 1 : 4,weakest for SWCNT with other rectangular cross section,and intermediate for SWCNT with square cross section.Thus, one can explain why the strength of breakawayshould be highest for SWCNT with rectangular cross sec-tion with the ratio between its sides 1 : 4, lowest forSWCNT with other rectangular cross section, and inter-mediate for SWCNT with square cross section.The qualitative explanation of velocity profiles for theflows of the mixture H2O + CH4 through SWCNTs withdifferent rectangular cross sections depicted in figure 8ccan be obtained by means of similar analysis. Sincein this mixture, methane molecules are characterizedby largest constant (cid:15) CCH for LJ interactions of thesemolecules with bounding wall carbon atoms, they aresubjected to strongest retarding forces from boundingwalls. The stronger these forces the slower the fluid ofwater and methane molecules through SWCNT and viceversa. Therefore, we should calculate d avermin for methanemolecules in equilibrium structures formed by the mix-ture H2O + CH4 inside SWCNTs with different rect-angular cross sections (see figures 3j - 3l) and comparethese values with r ∗ for LJ interactions between methane0molecules and bounding wall carbon atoms. Such calcu-lations give d avermin = 1 . σ H O for SWCNT with thesquare cross section, d avermin = 1 . σ H O for SWCNTwith the rectangular cross section with the ratio betweenits sides 1 : 2, and d avermin = 1 . σ H O for SWCNT withother rectangular cross sections. r ∗ for LJ interactionsbetween methane molecules and carbon atoms is equal to r ∗ = 1 . σ H O . Then, one can see that the value d avermin for SWCNT with the square cross section is closest to r ∗ , d avermin for SWCNT with rectangular cross section withthe ratio between sides is most different from r ∗ , and d avermin for SWCNT with other rectangular cross sectionhas an intermediate value between two above mentionedones. Then, one can conclude that the flow of the mix-ture H2O + CH4 through SWCNT with the square crosssection should be fastest, the flow of this mixture throughSWCNT with the rectangular cross section with the ratiobetween its sides 1 : 2 should be slowest, and the flowof such mixture through SWCNT with other rectangularcross section should have an intermediate average flowvelocity. It is easily seen than these conclusions are in aqualitative agreement with the velocity profiles depictedin figure 8c.As said above, for external driving force f x = 0 . v averx . It means that, sincethe external force f x is switched on, liquid particles be-gin to move along the tube axis with a certain acceler-ations untill the average fluid flow velocity achieves thesteady value v averx . This is a quite expected behavior offluid flows through SWCNTs. However, when the ex-ternal driving force is two times larger, f x = 0 .
1, thesituation changes radically, and one can observe two dras-tically different behaviors that depend on types of fluidparticles and the shapes of rectangular sections of SWC-NTs. In the case of the water flows through SWCNTswith different rectangular cross sections, one can observeagain the flows with steady average flow velocities v averx ,which are higher than those for f x = 0 .
05, but remain fi-nite. For argon atom flows through SWCNTs with squarecross section and rectangular cross section with the ratiobetween its sides 1 : 2, one can observe the similar fluidflows with steady and finite flow velocities (see curves1 and 2 in figure 9a). This figure exhibits time depen-dences of v averx averaged over subsequent time intervalswith a duration equal to 100 MD time units (symbolson these curves correspond to central points of such timeintervals). One can see that, for argon flows throughSWCNTs with such rectangular cross sections, the fluidflow velocities averaged over subsequent time intervalsfirst grow with time, and then reach saturation at certainsteady and finite values. However, for argon flow throughSWCNT with the rectangular cross section with the ratiobetween its sides 1 : 4, the fluid flow velocity averagedover above mentioned subsequent time intervals exhibitan unlimited growth with no signs of saturation. More-over, if we then switch off the external force ( f x = 0, the average flow velocity remains constant with no signs ofdecay (curve 4 in figure 9a). In order to understand suchextraordinary behavior of argon flows through SWCNTswith different rectangular cross sections, we must an-alyze a time dependence of all forces acting on argonatoms during their flows through SWCNTs. Each atom(molecule) inside SWCNT is subjected to two forces di-rected along the tube axis, namely, the external drivingforce f x and retarding force f rx due to interactions be-tween a given atom (molecule) and bounding wall carbonatoms. The external driving force f x is constant, andtypical time dependence of instant value of f rx is shownin figure 9b. It is easily seen that this time dependencehas a stochastic - like character, and we must performtime averaging of this force over the above mentionedsubsequent time intervals. The results of such time av-eraging for argon flows through SWCNTs with differentrectangular cross sections are shown in figure 9c, whichexhibits time dependences of the ratio | f rx | /f x , where | f rx | is the absolute value of the time averaged retardingforce f rx (if f x is positive f rx after time averaging isalways negative) for argon flows through SWCNTs withdifferent rectangular cross sections. One can see fromthis figure that for argon flows through SWCNTs withsquare cross section and rectangular cross section withthe ratio between its sides 1 : 2, the ratios | f rx | /f x firstgrow with time and then reach saturation at the steadyvalue equal to nealy 1 (curves 1 and 2). It means thatthe absolute value of the time averaged retarding force f rx becomes equal to the external driving force f x , butit has an opposite sign. As a result, the total force act-ing on carbon atoms during their flows through SWCNTsvanishes, and they move with certain constant time aver-aged velocities. One can also see from this figure (curve3) that, for argon flow through SWCNT with rectangu-lar cross section having the ratio between its sides 1 : 4,the ratio | f rx | /f x decays with time to nearly zero, andargon atoms begin to move along tube axis in a ballis-tic regime. This fact can explain the unlimited growthof the average fluid velocity during argon flow throughSWCNT with such rectangular cross section under ac-tion of the external driving force f x = 0 .
1. It shouldbe noted that qualitatively similar phenomenon, namely,ballistic frictionless gas flow through two - dimensionalchannels made from graphene or boron nitride has beenexperimentally observed [53].At first glance, this phenomenon seems to be somewhatsimilar to the superfluidity that occurs, for example, inhelium - 4 near the absolute zero [54]. However, thereare several principal differences between well known reg-ular classic superfluidity and our results on argon flowthrough SWCNT with one of above mentioned rectangu-lar cross sections. First of all, the regular classic superflu-idity is the macroscopic quantum phenomenon whereasour MD simulations are based on the usual classic me-chanics. Secondly, our ”pseudo superfluidity” dependson the shape of the cross section of SWCNT, whereasthe ”true” superfluidity is independent of shapes of chan-1 -5 -4 -3 -2 -1 0 1 2 3 4 50,00,20,40,60,81,01,21,4 (a) x z/ H2O -4 -3 -2 -1 0 1 2 3 40,00,51,01,52,02,53,0 (b) x z/ Ar -5 -4 -3 -2 -1 0 1 2 3 4 50,00,10,20,30,40,50,60,70,80,9 (c) x z/ H2O
FIG. 8: The fluid flow velocity profiles for flows of polar water molecules, nonpolar argon atoms and mixtures of waterand methane molecules through SWCNTs with different rectangular cross sections. 8a - fluid flow velocity profiles for watermolecules, 8b and 8c - analogous profiles for argon atoms and mixtures of water and methane molecules, respectively. f x = 0 . f x = 0 . nels. Thirdly, when the external driving force is equal to f x = 0 .
05, the average fluid argon flow velocity throughour SWCNT is finite, whereas disappearance of viscosityof helium - 4 depends only on its temperature and doesnot depend on the external driving forces. Thus, the re-sults of our simulations on the liquid argon flows throughSWCNTs with rectangular cross sections have nothingto do with the classic superfluidity. Perhaps, these re-sults are due a combination of several factors, namely,the equilibrium structure formed by argon atoms insideSWCNT with the rectangular cross section with the ra-tio between its sides 1 : 4, and the time averaging ofretarding forces acting on argon atoms from boundingwall carbon atoms. May be, the analysis of the time de-pendence of the instant retarding force f rx depicted infigure 9b will allow us to elucidate this challenge. IV. CONCLUSION
We performed MD simulations of equilibrium struc-tures and flows of polar water, nonpolar argon and methane, and mixtures of water and methane confinedby SWCNTs with square cross section and rectangularcross sections having the same area and the ratios be-tween their sides 1 : 2 and 1 : 4. The results of our sim-ulations show that equilibrium structures of all confinedliquids depend mainly on the shape of the SWCNT’s rect-angular cross sections, namely, the cross sections of thesestructures resemble replicas of those of SWCNTs. Never-theless, the types of liquids confined by above mentionedSWCNTs also have some influence on their equilibriumstructures. For example, the results of performed MDsimulations revealed that nonpolar argon atoms form in-side SWCNTs with rectangular cross sections the mostspatially ordered equilibrium structures, whereas, polarwater molecules form the least spatially ordered ones.The corresponding decrease in the spatial order is dueto the Coulomb - like dipole - dipole interactions be-tween polar water molecules. As for the external pressuredriven flows of all above mentioned liquids through SWC-NTs with different rectangular cross sections, these flowsdepend strongly on both the shapes of the rectangularcross sections and the type of the confined liquids. For2 (a)
43 21
Vxaver (MD units) t * (MD units) (b) r x t * (MD units) (c) r x0x t * (MD units) FIG. 9: Time dependences of average argon flow velocities v averx through SWCNTs with different rectangular cross sectionsand analogous dependeces for retarding forces acting on argon atoms from bounding wall carbon ones. 9a - time dependence of v averx averaged over subsequent time intervals with duration equal to 100 reduced MD units. 9b - typical time dependence ofinstant retarding force f rx acting on argon atoms from bounding wall carbon atoms during argon flow through SWCNT withrectangular cross section with the ratio between its sides 1 : 4. 9c - time dependences of the ratios | f rx | /f x averaged overabove subsequent time intervals during the argon atom flows through SWCNTs with different rectangular cross sections. In8a and 8c curves 1 correspond to SWCNT with square cross section; curves 2 and 3 correspond to SWCNTs with rectangularcross sections having the ratios between their sides 1 : 2 and 1 : 4, respectively. For all figures f x = 0 . example, our MD simulations revealed that, for nonpolarmethane inside above SWCNTs with different rectangu-lar cross sections, there are critical (threshold) values ofthe external driving force f x , that mimics the externalpressure drop through SWCNTs, below which the av-erage flow velocity is nearly zero, and above which theliquid methane flow occurs. Our simulations revealedalso that these critical values, which can be consideredas certain strengths of breakaway, depend strongly on theshape of rectangular cross sections of our SWCNTs. Per-haps, this phenomenon, which is absolutely not inherentto flows of ordinary liquids, is due to a certain spatial or-der formed by argon and methane atoms inside SWCNTswith rectangular cross sections. We show that observeddependence of the strengths of breakaway obtained fromour MD simulations on the shapes of rectangular crosssections of SWCNTs can be qualitatively explained interms of interactions between liquid particles and bound-ing wall carbon atoms. The stronger these interactionsthe stronger the strength of breakaway for SWCNTs with rectangular cross sections.Another interesting phenomenon was revealed fromour MD simulations of the liquid argon flows throughabove mentioned SWCNTs with rectangular cross sec-tions. For the external driving force f x = 0 .
05, theseflows are characterized by the average flow velocities v averx that are different for different shapes of SWCNT’srectangular cross sections but still remain finite. How-ever, for f x = 0 .
1, the liquid argon flows through SWC-NTs with square cross section and rectangular cross sec-tion with the ratio between its sides 1 : 2 also exhibit dif-ferent finite values of v averx , whereas the liquid argon flowthrough SWCNT with rectangular cross section with theratio between its sides 1 : 4 occurs in the ballistic regime,i. e., the average flow velocity v averx exhibits an unlimitedgrowth with time. It was revealed that, for the liquid ar-gon flows through two former SWCNTs with rectangularcross sections, the retarding force f rx acting on liquidparticles from the bounding wall carbon atoms averagedover a certain subsequent time intervals first grows with3time until it reaches a saturation at the value equal tothe external driving force f x = 0 . f rx decreases withtime to nearly zero. Therefore, for the liquid argon flowsthrough two former SWCNTs with rectangular cross sec-tions, the total steady force acting on argon atoms along the tube axis is equal to zero, whereas, for liquid ar-gon flow through SWCNT with rectangular cross sectionwith the ratio between its sides 1 : 4, this force is almostequal to the external driving force f x = 0 .
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