Simple model of skeletal matter composed of magnetized electrically conducting thin rods

Transcription

1 Simple model of skeletal matte composed of magnetized electically conducting thin ods A.B. Kukushkin, K.V. Cheepanov NFI RRC Kuchatov Institute, Moscow, 1318, Russia A simple electodynamic model fo descibing the behavio of a skeletal matte composed of magnetized, electically conducting thin ods (1D magnetic dipoles) is poposed. It is aimed at modeling the self-assembling of a skeletal matte fom cabon nanotubes (o simila nanodust), as suggested in [1] fo intepeting the expeimental data on the long-lived filamentay stuctues in the high-cuent electic dischages. Hee the capability of the model is illustated with the example of how a staight tubula skeleton, which is composed of ~300 dipoles and cay cicula electic cuent in its wall, may be wapped up by a distant pulsed electic cuent to make a tooid-like stuctue. 1. Intoduction The self-simila skeletal stuctues [1,], composed of tubula blocks (sometimes with the catwheel in the butt-end of the tubule) which epeat themselves successively at vaious length scales to give, coespondingly, a factal of paticula topology of constituent blocks, wee called the Univesal Skeletal Stuctues (USS) [3]. The phenomenon of USS was suggested in [1] fo intepeting the expeimental data on the long-lived filamentay stuctues in the high-cuent electic dischages. The USS phenomenon was pedicted and taced in the vey wide ange of length scales, 10-5 cm cm, in the data fom vaious laboatoy expeiments and obsevations of sevee weathe phenomena and astophysical objects [] (fo the cuent status of the USS poject and its popula desciption see, espectively, [4,5] and [3]). The smallest block of USS was suggested [1] to be the widely known object, namely cabon nanotube, o simila nanostuctues with paticipation of othe chemical elements. The pediction [1] was based on appealing to exceptional electodynamic popeties of thei hypothetical building blocks -- fist of all, the ability of these blocks to facilitate the electic beakdown in laboatoy dischages and to assemble the mico- and macoskeletons. The selfassembling of skeletons was suggested to be based dominantly on magnetic phenomena. The indications on plausibility of the anomalous magnetism and, in paticula, on the ability of CNTs, and/o thei assemblies, to tap and almost dissipationlessly hold magnetic flux, with the specific magnetization high enough to stick the CNTs togethe, come fom obsevations of supeconducto-like diamagnetism in the assemblies of CNTs at high enough tempeatues. Such evidences ae obtained fo the self-assemblies of CNTs (which contain, in paticula, the ing-shaped stuctues of few tens of micons in diamete) inside non-pocessed fagments of cathode deposits, at oom tempeatues, [6] and fo the atificial assemblies, at 400 K [7]. The evidences and aguments fo the oom-tempeatue supeconductivity in individual CNT, and in atificial and natual assemblies of CNTs, ae summaized in [8]. The ecent suvey of expeimental evidences fo, and theoetical models of, the unexpected magnetism of cabon foams and heteostuctued nanotubes is given in [9]. Despite the above expeimental evidences and theoetical models need much stonge tests and confimations, they justify explicit demonstation of the capability of magnetized

2 nanotubula blocks to self-assemble a tubule of highe geneation [1(B,C)] and sustain the integity of the assembled skeleton. Similaly to development of, e.g., the plasma theoy, now it is woth to stat with analysing the stability of skeletal matte within the fame of as simple model as possible. This implies an analysis of the capability of nanotubes to sustain the integity of the hypothetically fomed tubula skeleton, which is composed of magnetized nanotubes (i.e. analyse the viability of the tubule of the -nd geneation). In the pesent pape, we (i) fomulate a simple model fo descibing the behavio of a skeletal matte composed of magnetized, electically conducting thin ods which behave as the 1D magnetic dipoles, and (ii) illustate the capability of the model on the example of how a staight tubula skeleton, which is composed of ~300 dipoles and cay cicula electic cuent in its wall, may be wapped up by a distant pulsed electic cuent to make a tooidlike stuctue.. Simple model of skeletal matte, composed of 1D magnetic dipoles We teat the poblem in as simple pictue as possible. Thus, we assume the elementay block of the skeletons to possess the following electodynamic popeties: the 1D static magnetic dipole (such a dipole may be epesented as a couple of magnetic monopoles located on the tips of the igid-body dipole; this appoximation seems to be good fo the tubules and/o od with the lage length-to-diamete atio), static positive electic chage, which is located in the cente of the od and is sceened by the ambient electons at some Debye adius (electic chaging is due to inevitable field emission, at least themal one, by the nanotubes), static electical conductivity, which is high enough to enable the tubula skeleton to tap, without dissipation, the magnetic flux inside the tubule (i.e. sustain cicula electic cuents in the tubule s wall). The above chaacteistics enable us to descibe the following inteactions of elementay blocks: mutual magnetic attaction and epulsion of the dipoles (i.e. inteaction of cicula electic cuent in the wall of one elementay block with simila cuent in anothe elementay block), action of extenal magnetic field on the magnetic dipole (i.e. inteaction of cicula electic cuents in the walls of the elementay block with the extenal electic cuent poducing the magnetic field), sceened electic epulsion of elementay blocks, action of magnetic field, poduced by the longitudinal electic cuent in all the magnetic dipoles, on the given dipole (i.e. inteaction of cicula electic cuent in the walls of the elementay block with the longitudinal electic cuent in the walls of othe blocks), inteaction of longitudinal electic cuent in the walls of the blocks. To simplify the desciption of dynamics of solid bodies we conside each dipole to be a couple of the point objects (coodinates i, masses m i, i=1,) which ae linked togethe with a igid-body massless bond and subjected to the action of the extenal foces applied to these objects, F 1 and F, and to the cente of mass of the system (i.e. to the massless inteconnecting bond), F cm. The exact system of equations fo such a system is descibed by the conventional

3 equations fo the motion of a solid body specified fo the above paticula case. The equations fo the momentum and angula momentum, espectively, of the solid body ae as follows: m t m+ m = F+ F+ F cm. (1) ( ) 1 1 1, m F F F 1, 1, 1, t + t 1 1 = [ ] + [ ] + [ cm cm ], () whee the squae backets denote the vecto multiplication, and the adius vecto of the cente of mass of the system is equal to m + cm m m + m ( 11 ) ( 1 ) /. (3) One may easily check that the solution to the system of Eqs. (1) and () may be found as a solution to the following system of equations: m m A F t m m F = cm + 1. (4) m m A F t m m F = cm. (5) 1+ (6) ( 1 1 ) The fist tem in the ight-hand side of Eqs. (4) and (5) descibes the action of the igid-body bond between the point objects 1 and. One can find the value of A fom the condition of igidity of a solid body, ( v v ) ( v, ) 0 1, = whee and v ae the velocities of the point objects. This gives v 1, (7) F A = µ ν m F m 1,, (8) whee µ 1 is the educed mass of the system of two point masses. Majo dimensionless vaiables of the outlined above poblem ae as follows. The space coodinates, time and velocity ae taken in the units of dipole s length L, t 0 and v 0, espectively:

4 = L 0, t ml Z e M 3 0 =, v Z M e = ml 0, Φ0 Z M 4π e =, (9) whee m=m 1 =m, Z M is the modulus of magnetic chage of the monopole taken in the units of electon chage e, Φ 0 is magnetic flux tapped in the dipole. Electic chage Z will be taken in the units of magnetic chage. All the foces ae expessed in the units of magnetic inteaction attaction at the distance L. The pai inteaction of longitudinal electic cuents of the value J o though the dipole is taken in the units of F 0JJ, and the inteaction of the dipoles with extenal cuent J ext - in the units of F Jext : J L 0 F = 0 JJ, czm e F J cz ext M L e Jext = (10) The electodynamic foces ae assumed to lagely exceed the gavity of the dipoles. To descibe sticking of the dipoles we allow the magnetic monopoles to move feely in an isotopic potential well which is fomed by (a) magnetic attaction of monopoles of the opposite sign and (b) thei epulsion due to elasticity of the tips of the tubules/ods of finite diamete. The fom of the potential and the espective foce ae shown in Figue 1. This potential povides smooth tansition fom the Coulomb potential fo >* to epulsion potential at small adii. Also, in the egion <* we intoduced the following fiction foce: whee v1 F bake = kb v 1 v 1, (11) is the elative velocity, and the coefficient kb is taken in the units m/l. The above stong simplification of the oiginal pictue of the motion of solid ods is acceptable if the spatial density of the ods is athe small and, espectively, the sticking and collisions of the ods ae govened mostly by the inteaction of stong magnetic monopoles on the tips of these ods. 3. Dynamics of tubula skeleton, composed of 1D magnetic dipoles Hee we illustate the capability of the model, outlined in the pevious Section, to descibe the integity of skeleton unde the action of extenal foces. Fist, we constuct the ideal tubula skeleton accoding to the ules suggested in [1(B,C)]: namely, the skeleton is composed of hexagons assembled fom the dipoles. The stuctue of the tubula staight skeleton, whose wall is assembled fom hexagons and whose coss section has also a hexagonal stuctue, is shown in Figs. and 3, fo the total numbe of the dipoles N dip = 94. The coesponding magnetic theading of such a netwok will be ideal if the skeleton is composed of the dipoles of magnetic chages Z M which diffe by the facto of. In geneal case, it is possible to compose a skeleton fom abitay polygons povided the magnetic chages on the tips of the blocks suppot the espective magnetic theading.

5 The dynamics of the skeleton in Figs.,3 is tested against the petubation intoduced by the distant extenal electic cuent fo the following conditions,: magnetic chages Z M =, fo ed thick blocks, and Z M =, fo all the othes, electic chages Z =1 fo all the blocks, sceening (Debye) adius D =1, bake coefficient k b =100, cuent-cuent foce coefficient F 0JJ =, cuent-extenal-cuent foce coefficient F Jext =50, extenal electic cuent flows along X-diection, the line of cuent is located in the point {Y=-15, Z=15} and acts fom time t=0 to t=1. The esults of numeical modeling ae shown in Figues 4-7 fo vaious time moments. The futue dynamics of the skeleton -- collision of the tips of the skeleton, which follows the closue of the loop, as is seen in Figue 7 -- may not be descibed by the model of Sec. because we neglected mechanical collision of the ods along entie length of the blocks. The esults of Figs. 4-7 may be intepeted as an illustation of the possibility of skeletons -- if fomed in the high-cuent electic dischages o simila conditions - to fom the toodal-like and catwheel-like stuctues (cf. lase-induced poduction of lage cabon-based tooids epoted in [10], see the Q-shaped tooids in Fig. 3 of this pape). Acknowledgments One of the authos (A.B.K.) highly appeciates his long-tem collaboation with V.A. Rantsev-Katinov in thei eseach of skeletal stuctues [1-5]. The pesent wok is suppoted by the Russian Foundation fo Basic Reseach (poject No ).

6 Refeences 1. A. B. Kukushkin and V. A. Rantsev-Katinov. (A) Fusion Enegy 1998 (Poc. 17th IAEA Conf., Yokohama, 1998) IAEA, Vienna, 1999, Vol. 3, pp ( (B) Poc. 6-th Eu. Phys. Soc. conf. on Plasma Phys. and Cont. Fusion (Maasticht, Nethelands, June 1999), pp ( (C) Long-living filamentation and netwoking of electic cuent in laboatoy and cosmic plasmas: fom micoscopic mechanism to self-similaity of stuctuing, in Cuent Tends in Intenational Fusion Reseach: Review and Assessment (Poc. 3 d Symposium, Washington D. C., 1999), edited by E. Panaella, NRC Reseach Pess, Ottawa, Canada, 00, pp A. B. Kukushkin and V. A. Rantsev-Katinov. Phys. Lett. A 306, (00). 3. A. B. Kukushkin and V. A. Rantsev-Katinov. Univesal skeletal stuctues: in lab and in... space. Science in Russia, 004, # 1, pp A. B. Kukushkin and V. A. Rantsev-Katinov. Evidences fo and the models of self-simila skeletal stuctues in fusion devices, sevee weathe phenomena and space. Rep. 6-th Symposium «Cuent Tends in Intenational Fusion Reseach: A Review», Washington, D.C., USA, Mach 7-11, 005 ( see also the pevious suveys fo the simila symposia: and 5. A. B. Kukushkin and V. A. Rantsev-Katinov. Advances in Plasma Phys. Reseach, 00, Vol. (Ed. F. Gead, Nova Science Publishes, New Yok), p V. I. Tsebo and O. E. Omel'yanovskii, Phys. Usp. 43, 847 (000). 7. G. Zhao and Y. S. Wang, Pepint cond-mat/ (001) at 8. G. Zhao, Pepints cond-mat/ (003) and cond-mat/04138 (004) at 9. D. Tomanek. J. Phys.: Condens. Matte 17, R413 R459 (005). 10. M. E. Lyn, J. He, B. Koplitz. Appl. Suf. Sci., 46, (005).

7 Figue 1. Radial dependence of the effective model potential (taken in the units of (Z M e) /L, and multiplied by 10) and the espective foce (in the units (Z M e/l) ) fo the inteaction of two attacting magnetic monopoles. Hee, tansition adius is * = 0.06.

8 Figue. Tubula skeletal stuctue composed of 94 magnetic dipoles. Magnetic chage of the dipoles shown as ed thick ods is twice of that fo thin ods.

9 Figue 3. Magnified image of the pat of tubula skeleton in Fig.. The cosses on the dipoles indicate noth pole pat of the dipole.

10 Figue 4. The image of the esults of numeical modeling of the behavio of tubula skeletal stuctue of Fig., unde conditions listed in Sec. 3, at dimensionless time moment t = 1.0.

11 Figue 5. The pictue simila to Fig.4, fo time t =.0.

12 Figue 6. The pictue simila to Fig.4, fo time t =.6.

13 Figue 7. The pictue simila to Fig.4, fo time t=.8.