The expression for number of layers of multilayered nanotube, which have been generated during the time of its growth in the plasma discharge, is derived.Отримано вираз для кількості прошарків багатошарової нанотрубки, що встигає сформуватися за час її росту в плазмовому розряді.Получено выражение для количества слоев многослойной нанотрубки, которое успевает сформироваться за время ее роста в плазменном разряде

Azarenkov, N.A.

Maslov, V.I.

Onishchenko, I.N.

Tretyakov, V.N.

Yegorov, A.M.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

NANOTUBE FORMATION ON CATALYTIC SURFACE IN PLASMA DISCHARGE V.I.Maslov, V.N.Tretyakov*, N.A.Azarenkov*, A.M.Egorov, I.N.OnishchenkoNSC Kharkov Institute of Physics & Technology, Kharkov, Ukrain,e-mail: vmaslov@kipt.kharkov.ua;*V.N. Karazin Kharkov National University, Kharkov, UkraineThe expression for number of layers of multilayered nanotube, which have been generated during the time of its growth in the plasma discharge, is derived. PACS Ref: 52.27.LwINTRODUCTIONControllable  formation  of  nanotubes,  superlong nanotubes, nanotubes with the necessary properties are the very important and now intensively investigated problems (see, for example, [1, 2]). In this paper the formation of carbon nanotubes in the plasma discharge is considered, when  on  catalytic  surface  or  surface  with  defects  the nanotubes under  effect  of  the  flow  of  carbon  plasma, bombarding  the  surface,  are  formed.  The  universal expression  for  number  of  layers  of  multilayered nanotubes, which have been generated during the time of its growth in the plasma discharge, is derived. THE DESCRIPTION OF NANOTUBE GROWTH Let us consider  the nanotube formation,  built  in the lattice  ordered  in  an  electric  field  of  the  nanotubes, perpendicular to the catalytic surface. Atoms of carbon, colliding with a lateral  surface of a tube and getting in area of a shadow concerning bombarding plasma stream, are quickly cooled,  cease to  join to nanotubes and "are blown"  from  a  wood  of  the  nanotubes  as  a  result  of bombardment by the following atoms of their stream. That is in the region of a shadow, closer to  the basis of the nanotube, layers do not grow on of the nanotubes. In other words, layers of nanotube grow on some distance from its head. We name this interval area outside of a shadow and equal  to  the  length  of  free  run  ℓfr of  carbon  atom concerning  collisions  with  nanotube  walls.  As probabilities of connection of atoms to the top and lateral side are practically identical, speeds of lengthening of a lateral  side  and  length  of  nanotube  are  identical.  Then during lengthening of nanotube on ℓfr its lateral side also to be extended on ℓfr. Then the amount of nanotube layers is  equal  Nmax=ℓfr/2πR  at  spiral  nanotube  which  cross-section,  perpendicular  to  tube  axes,  represents  an untwisted spiral. Here R is the average radius of nanotube. For  first  layer  R it  is  approximately equal  to  fullerene radius.  Thus  ℓfr≈Vz(ℓnan/Vthc ).  Here  ℓnan is  the  average distance  between  nanotubes,  Vz is  the  longitudinal lengthways of the nanotube velocity of carbon ions and atoms,  Vthc is  the  thermal  velocity  of  carbon  atoms.  If Vz≈Vthc,  then  ℓfr≈ℓnan.  Thus ℓnan≈(nfs)-1/2 .  Here nfs is  the superficial density of the nanotubes. Nmax≈1, if ℓnan≈2πR. The surface density of the nanotubes  σ  is derived from that  during  lengthening  with  speed  V nz  of  the nanotubes on ℓfr the density of the particle flow nc V z  fall on the surface, which with probability ωz  engenders nanotubes. Thus, σ=ℓ frV nzncV z ωz = nc V z2ωzV nz V thc 2/3  (1)One mechanism of the growth of the nanotube side is its bombardment by carbon atoms and ions of a plasma flow. The second mechanism of the nanotube growth is the following one. Fallen on the nanotube and sorbed on its  surface  carbon  atom diffuse  on  its  surface  up  to  a growing side,  that  is  growth of  a  side is  reduced to  its lengthening with some speed V nz . Now  we  show,  that  can  arise  as  spiral,  and azimuthally symmetric nanotubes. Also we find quantity of layers of the azimuthally symmetric nanotube. For last carbon atom getting between two next carbon atoms and to form closed cluster, which is the basis of the azimuthally  symmetric  nanotube,  it  should  overcome  a power  barrier  εз on  some  tens  percents  higher  than  to overcome a  barrier  εпр for  carbon  atom to  join  border open-ended spiral  cluster.  Probabilities to  join to open-ended circular  wпр or  to  close  wз it  are  proportional  to wпр~exp(-εпр/T),  wз~exp(-εз/T).  Hence, the probability to close  wз of  open-ended  circular  cluster  is  less  than probability  to  join  wпр to  open-ended  circular  cluster wз<wпр. Thus, the probability of origin of spiral cluster is more than probability of origin of closed circular cluster. However,  the  spiral  nanotube  (Figure  a)  is  less favourable from the power point of view, because there are many nonsaturated connections. Hence, if during the nanotube growth the bombardment is intensive the part of the spiral nanotubes decreases due to "heating". Probability  to  arise  new  layer  -  cluster  around azimuthally  symmetric  nanotube  is  small.  It  is proportional to nanotube radius. The factor of diffusion of carbon atoms on the nanotube surface is also proportional to  nanotube  radius.  Thus,  at  formation  of  multilayered azimuthally symmetric nanotube the distribution on radius r  lengths  of  layers  along  z-axis  has  qualitatively  kind submitted in Figure b. The number of carbon atoms getting in time unit on the  external  surface  of  the  nanotube  we  estimate  as follows:188                  Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 188-190dNcmax/dt=npvp2πRmaxHHere H is the height of the nanotube, np, Vp are the density and  velocity  of  carbon  ions  in  the  flow  (plasma) accordingly, 2πRmax is the length of "circle" of an external layer of the nanotube. a ba) Origin spiral nanotube; b)distribution on radius r of  lengths of nanotube layers The  nanotube  growth  rate  upwards  due  to  carbon atom  connection,  which  are  placed  on  the  nanotube surface,  to  a  face  edge  is  determined  by  the  following expression:V||(1)=dH(1)/dt=(aw0 dN||/dt)/Na  (2)Here  a  is  the  size  of  internuclear  distance  in  nanotube structure,  w0 is  the  atom,  diffused  from  the  nanotube surface,  probability of connection to the edge, dN||/dt is the number of carbon atoms getting in time unit on a face edge, Na is the number of carbon atoms on a face edge. Taking into account that dN||/dt=w||dNcmax/dt  (3)and Na=L/a  (4)we derive the following expression:dH(1)/dt=(a2w0w||dNcmax/dt)/L  (5)Here w|| is the probability of that the carbon atom, which is  placed  on  the  nanotube  surface,  diffuse  on  the  face edge, L is  the length of the lateral face edge. With the account (1) the expression (5) has the following kind:dH(1)/dt=(a2w0w||npvp2πRmaxH)/L  (6)Now we take into account direct hits of carbon ions from the  discharge  plasma flow on  a  face  edge  of  the nanotube:V||(2)=dH(2)/dt=(aw00 dNc/dt)/Na  (7)Here w00 is the probability of connection of a carbon ion to a face edge at direct hit from the plasma flow, dNc/dt is the number of carbon ions getting from the plasma flow on  the  end  face  in  time  unit.  It  can  be  estimated  as follows:dNc/dt=npvpLΔR  (8)Here  R  is  the  nanotube   "thickness".  Then,  with  the account (4) and (8) the equation (7) can be presented as:dH(2)/dt=a2 w00npvpΔR  (9)Then the full growth rate of the face edge is equal V||=dH/dt=dH(1)/dt+dH(2)/dt==a2npvp(w00ΔR+w0w||2πRmaxH /L)  (10)Let  us  estimate  the  growth  rate  of  the  nanotube surface sideways:Vθ=dL/dt=(aw0dNθ/dt)/Naθ  (11)Here  dNθ/dt  is  the number  of  atoms,  diffused from the nanotube surface on the lateral edge in time unit, Naθ is the number of carbon atoms on the lateral edge. In our casedNθ/dt=wθdNcmax/dt=wθnpvp2πRmaxH  (12)and Naθ=H/a  (13)Then with the account (12) and (13) the equation (11) we present in the following kind:dL/dt=a2w0wθnpvp2πRmax  (14)Here wθ is the probability of that the carbon atom, which is  placed  on  the  nanotube  surface,  is  diffused  on  the lateral edge.As a result we have derived the following system of the  equations  describing  the  nanotube  growth  upwards and sideways:dH/dt=a2npvp(w00ΔR+w0w||2πRmaxH /L)  (15)dL/dt=a2npvpw0wθ2πRmax  (16)It is necessary to take into account, that in system of the equations (15) - (16 three values, namely w||, Rmax and  wθ are the functions of variables H and L.The  length  of  an  external  lateral  surface  of  the nanotube can be found from the assumption that it can be presented as Arhimed’s spiral. Then L=δRφ2/4π  (17)Thus, we derive 2πRmax=L(φ)–L(φ-2π)=δR(φ2–(φ-2π)2)/4π==δR(φ-2π)/2  (18)Here δR is the distance between layers of the nanotube, φ is  the  angular  coordinate.  From  (17)  we  obtain dependence φ=φ(L):φ=2(πL/δR)1/2  (19)From here, using (17) and (19), we find2πRmax=δR((πL/δR)1/2-π)  (20)Taking into account, that L>>δR, we have from (20) the following expression:2πRmax=δR(πL/δR)1/2=(πLδR)1/2  (21)Then the system of the equations (15) - (16) becomes:dH/dt=a2npvp(w00ΔR+w0w||(πδR/L)1/2H )  (22)dL/dt=a2npvpw0wθ(πδRL)1/2  (23)QUANTITY OF LAYERS IN GROWING IN THE DISCHARGE MULTILAYERED NANOTUBEUnder a shadow we believe that part of the nanotube surface on which bombarding carbon ions and atoms get only after  collision  with  another  nanotube.  We believe that  on  the  nanotube  surface,  located  outside  of  the shadow there appears fast (induced) surface diffusion. We believe that  the atom, which is  placed on the nanotube surface, diffuses with equal probability in any direction. Therefore on the end face of the nanotube which length is equal 2π R N , the following part of atoms comes 189zrωup=2π R N2  z sh2π R N .On the lateral side of the growing nanotube, which length is equal z sh , the following part comes ω¿=z shz sh2π R NHere N˙  is the number of carbon atoms, settling on the lateral surface of the nanotube outside of a shadow in time unit:N˙=2πRmax zsh noV th ωsR  is  the average radius of the multilayered nanotube, N  is  the quantity of  layers in multilayered  nanotube, Rmax  is the radius of the multilayered nanotube. Rmax  is possible to express through N  and distance between layers in multilayered nanotubeRmax=Nr ℓ . The quasi-stationary density of sorbed carbon atoms ncs  is  determined  by  balance  of  sorption  with probability ωs  and a leaving with probability ωds   ωs no V thс=ωds ncs .Here  no  is the plasma density. It is necessary to note, that  ωds  depends on induced effective temperature of sorbed atoms. Speed of lengthening of growing nanotube is equalV II=N˙ aωo ωup2π R N /aHere ωo  is the probability of carbon atom connection to a growing side or the end face of the nanotube. The growth rate of the lateral side of the nanotube is equalV ¿=N˙ aωo ω¿z sh /aV II  and V ¿   are connected between themselves by the following ratioV IIV ¿=zsh2π R NFrom here we find quantity of layers in multilayered of the nanotubeN= z sh2πr ℓ V ¿V IIThe nanotube length, located outside of a shadow, is equalz sh=RnV IIeV thHere V IIe  is the component of the carbon atom velocity, directed  perpendicularly  to  the  catalytic  surface; Rn=n sn−1 /2 is the distance between nanotubes;  nsn  is the surface density of the nanotubes. N=Rn2πr ℓ 2π R Nz sh V IIeV th ω¿ωup =¿Rnπr ℓ V IIeV th =1πr ℓnsn V IIeV thSince  π R N≈ z sh ,  the  area  of  cross-section  of multilayered  nanotube  is  equal  Sup=πRmax2 =4π R2 . The area of its lateral surface, on which the carbon atoms are sorbed and fastly diffuse due to surface diffusion, is equal  Sc=2π R NπR min .  Taking  into  account  that R=Nr ℓ /2 , we derive Sc=4π2 R2 Rminr ℓ. Thus, we obtain  that  Sup  is  not  much  less  Sc ,  i.e. S cS up=πRminr ℓ3 . In the case of azimuthally symmetric nanotubeN=ωln  zsh2πr ℓ V ¿V IIωln  is the probability of origin of a new layer.REFERENCES1. G.-H.Jeong, R.Hatakeyama, T.Hirata et al. // Proc. of  XXV ICPIG. Nagoya, 2001, v.2, p.155. 2. R.Hatakeyama, Y.Abe, H.Ishida et al. // Proc. of XXV ICPIG. Nagoya, 2001, v.2, p.159. ФОРМИРОВАНИЕ НАНОТРУБОК НА КАТАЛИТИЧЕСКОЙ ПОВЕРХНОСТИ В ПЛАЗМЕННОМ РАЗРЯДЕ В.И. Маслов, В.Н. Третьяков, Н.А. Азаренков, А.М. Егоров, И.Н. ОнищенкоПолучено выражение для количества слоев многослойной нанотрубки, которое успевает сформироваться за время ее роста в плазменном разряде. ФОРМУВАННЯ НАНОТРУБОК НА КАТАЛІТИЧНІЙ ПОВЕРХНІ В ПЛАЗМОВОМУ РОЗРЯДІ В.І. Маслов, В.М. Третьяков, М.О. Азаренков, О.М. Єгоров, І.М. ОнищенкоОтримано вираз для кількості прошарків багатошарової нанотрубки, що встигає сформуватися за час її росту в плазмовому розряді. 190

Nanotube formation on catalytic surface in plasma discharge

http://dspace.nbuv.gov.ua/bitstream/handle/123456789/79796/127-Maslov.pdf?sequence=1

Nanotube formation on catalytic surface in plasma discharge

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)