ISSN 10637842, Technical Physics, 2010, Vol. 55, No. 4, pp. 509–513. © Pleiades Publishing, Ltd., 2010. Original Russian Text © N.B. Volkov, A.E. Mayer, V.S. Sedoi, E.L. Fen’ko, A.P. Yalovets, 2010, published in Zhurnal Tekhnicheskoі Fiziki, 2010, Vol. 80, No. 4, pp. 77–80.
SOLID STATE
Mechanisms of Metallic Nanoparticle Generation during an Electric Explosion of Conductors N. B. Volkova, A. E. Mayerb, V. S. Sedoic, E. L. Fen’koa, and A. P. Yalovetsb a
Institute of Electrophysics, Ural Branch, Russian Academy of Sciences, ul. Amundsena 106, Yekaterinburg, 620016 Russia email:
[email protected] b Southern Ural State University, Leninskii pr. 76, Chelyabinsk, 454080 Russia c Institute of HighCurrent Electronics, Siberian Branch, Russian Academy of Sciences, Akademicheskii pr. 4, Tomsk, 634055 Russia Received July 1, 2009
Abstract—The mechanisms of conductor fracture and the generation of metallic nanoparticles during an electric explosion are discussed. The fracture of polycrystalline conductors with a crystallite size smaller than 100 nm during rapid introduction of energy is shown to occur due to its localization at grain boundaries. The dynamics of the explosion products (droplets, vapor) flying into a buffer gas is numerically simulated in terms of the mechanics of heterogeneous media with allowance for condensation and evaporation. The calculated size distributions of particles agree with the experimental distributions both qualitatively and quantitatively. DOI: 10.1134/S1063784210040122
An electric explosion of conductors (EEC) is one of the most promising methods for producing nanop owders [1, 2]. However, the following questions are still open. (i) Do nanoparticles form as a result of vapor condensation when explosion products fly into a buffer gas or as a result of the fragmentation of a liq uid metallic conductor during an explosion? (ii) How does the microstructure of the initial wire affect the nanoparticle size at EEC? This effect is indicated by the correlation between the crystallite size in the initial wire and the metallic nanoparticle size found in [2]. Therefore, the purpose of this work is to answer these questions and to simulate the EEC product spread dynamics in a buffer (inert) gas using a heterogeneous medium model with allowance for the processes that occur during the interaction of phases and phase tran sitions (evaporation, condensation, coalescence). We first determine the conditions under which EEC takes place. Let a cylindrical wire be homoge 2
neous over its cross section s = π r w (rw is the wire radius) and length L. The time of introduction of an 2
energy that is equal to sublimation energy Ws ≈ ρ C s /2 (where ρ and Cs are the density and sound velocity, respectively) into a conductor is th = Wsσ/j2 =
be met. From the equality th = ts, we can write the fol lowing lower estimate of the critical current density: j = * 3
where q∗ = ρ C s /2 is the energy flux characteristic of the given metal. To produce nanoparticles during EEC, it is better to use thin conductors. The minimum radius of such a conductor is determined by size effects and the appearance of a shunt discharge, which decreases the energy introduction efficiency along the conductor surface. The electrical resistivity of a cylindrical con ductor with allowance for size effects can be found –1 –1 –1 from the formula σ–1 = σ 0 (1 + lp r w ) = σ 0 (1 + –1
σ0 σ ∞ ) [3], where σ0 = e2nlp/pF = σ'lp; σ∞ = σ'rw; e, n, and lp are the electron charge, concentration, and free path length, respectively; and pF is the Fermi momen tum. We put the electric field strength at the conductor surface equal to the breakdown strength in the Pas chen curve Fb at a given buffer gas pressure P and obtain the lower estimate of the minimum conductor radius, 2 2 q* ⎛ 8σ 0 E b⎞ r wm = 2 ⎜ 1 + 1 + ⎟ . q σ' ⎠ 4σ 0 E b ⎝ *
2
ρσ C s /(2j2) (where σ is the electrical conductivity). The time of acoustic unloading of the conductor is ts = 2rw/Cs. For an EEC to occur, the condition th ≤ ts must
q σ/ ( 2r w ), *
(1)
In the absence of a shunt discharge along the con ductor surface, we have to meet the condition rw > rwm.
509
510
VOLKOV et al.
The following two limiting cases follow from Eq. (1): (1) 2 lp Ⰶ rw when r wm = q∗(2σ0 B b )–1 and lp Ⰷ rw when (2)
–1
r wm = (q∗/2σ'))1/2 E b . As follows from Eq. (1), an increase in Eb with the buffer gas pressure leads to a decrease in the minimum wire radius at a retained energy input efficiency. We now consider the most important mechanisms of conductor fragmentation by an electric current in order to produce metallic nanoparticles. The magnetic induction at the surface of a cylindri cal conductor is Be = μ0I/(2πrw), where I is the electric current and μ0 is the magnetic constant. When energy is slowly introduced into a conductor, its heating is assumed to be uniform over the cross section. In this case, destruction via the fragmentation (dispersion) of the quasihomogeneous liquid metallic conductor that forms during melting into transverse strata is pos sible as a result of constriction (sausage) magnetohy drodynamic (MHD) instability [4, 5]. According to [4], the constriction instability devel opment time is tin = rw/(vAΩmax), where vA = Be/ μ 0 ρ is the Alfven velocity and Ωmax = 2 is the maximum increment corresponding to the development of a sur face perturbation with magnetosound velocity vsA = 2
2
C s + v A at vA = Cs. It follows from the condition tin ≤ th that, for a conductor with constriction MHD insta bility to be fragmented by an electric current, we must meet the condition r ≥ r * = 2ν /C , where ν = w
w
m
s
m
(μ0σ)–1 is the magnetic diffusion coefficient. The MHD perturbation wavelength on the conductor sur face is determined by a balance between the capillary forces and the magnetic pressure; therefore, the size of the forming particles is smaller than the conductor radius. When producing nanoparticles by EEC, it is more interesting to analyze the stratification of an exploding conductor that was observed in experiments on a nanosecond electric explosion of wires several microns in diameter in vacuum [5], where the electric current density was well above that in the experiments in [4]. The increase in the surface perturbation growth rate induced by constriction instability is also favored by the removal of the material from a constriction region and its effective cooling due to radiation [6], which was not taken into account in the linear MHD instability theory in [4, 5]. Moreover, the linear MHD instability theory does not take into account the con tribution of convection, which also takes place in experiments, to the development of MHD instability. As shown in [7–10], taking into account the con vective terms in the equations of motion and magnetic diffusion results in the development of convective MHD instability even in an incompressible liquid. This instability appears when the convective contribu
tion of a magnetic field to a conductor becomes equal to the diffusion contribution or greater than it. The nonuniformity of the electric current distribution over the cross section of the conductor is a substantial fac tor promoting the development of convective MHD instability, since its increment increases with the elec trical conductivity of the conductor. (Note that, in the general case of a timevariable current, its distribution over the cross section of the conductor is always non uniform; therefore, heating of the conductor is also nonuniform.) The perturbation wavelength corre sponding to the maximum increment in the convec tive MHD instability is λ = 2.32rw. Convective instability in a liquid metallic conduc tor, carrying a current, results in hydrodynamic and current vortex structures [7–10]. Surface perturba tions in the form of narrow ring slots appear at the cen ters of vortices between two hydrodynamic vortex rings with oppositely directed particle motion; in a finite time, they divide the conductor into particles whose diameter is on the order of the conductor diam eter [8, 9]. The voltage drop across a slot growing deep into the conductor behaves as U ~ (t∗ – t)–1/2, and the current in the conductor behaves as I ~ (t∗ – t)1/2 (where t∗ is the time it takes for the ringlike slot to reach the axis) [8, 9]. The power consumed for the for mation of a ringlike slot is P = UI ~ (t∗ – t)0 = const. Therefore, the splitting of a conductor into particles on the order of the conductor diameter in size is ener getically favorable as compared to the situation where it remains uniform along the length during an explo sion. Moreover, hot points with a temperature exceed ing the average temperature of the conductor volume by an order of magnitude form in the conductor axis between two vortex rings with oppositely directed par ticle motion at the centers of the vortices as a result of localized Joule heat sources [7, 10]. These hot points favor further fragmentation of the particles. Thus, even in the case of a quasihomogeneous liq uid metallic conductor carrying a current, there exist nonlinear mechanisms leading to its fragmentation. Real wires used to produce nanoparticles during EEC are polycrystalline wires with randomly oriented grains, whose average size is d = 20–80 nm for various materials depending on treatment [2]. The boundaries of such grains (highangle grain boundaries) are known to be saturated by defects, mainly dislocation pileups, and contain disclinations at triple junctions [11, 12]. These defects change the mechanical strength of the metal and serve as the centers of scat tering and localization of conduction electrons [13], significantly changing the electron transport proper ties of the metal. The authors of [14] obtained an expression for the electrical resistivity of a polycrystal line film, the grain boundaries in which, were simu TECHNICAL PHYSICS
Vol. 55
No. 4
2010
MECHANISMS OF METALLIC NANOPARTICLE GENERATION
lated by potential barriers with an arbitrary reflection coefficient, ρ0 2 3 3 1 = 1 – α + 3α – 3α ln ⎛ 1 + ⎞ , ⎝ 2 α⎠ ρg
(2)
where α = l0R/dD; l0 is the electron free path length without regard for scattering by a grain boundary; R is the coefficient of electron reflection by a potential barrier; D = 1 – R is the transmission coefficient of the potential barrier; d is the crystallite size; and ρ0 and ρg are the electrical resistivity without and with allow ance for electron scattering by crystallite boundaries, respectively. According to [12], Eq. (2) with R = 0.23 well describes the electrical resistivity of a polycrystalline conductor with a crystallite size ranging from 300 mm to 13 μm. In turn, the reflection coefficient depends substantially on the grain size and temperature, for example, R = 0.24 for coarsegrained copper and R = 0.468 at T = 100 K and R = 0.506 at T = 275 K for nanostructured copper [15]. In the case of α Ⰷ 1 (which corresponds to the pre dominant contribution of crystallite boundaries to the electrical resistivity of the conductor), the asymptotics ρg = 4αρ0/3 holds true [14]. In this case, the tempera ture leveling inside crystallites is controlled by phonon thermal conductivity, which is inversely proportional to temperature, rather than electron thermal conduc tivity. Figure 1 shows the ratio of the time of temperature leveling in a crystallite as a result of phonon thermal conductivity to the time it takes for a conduction elec tron to travel across a crystallite at various tempera tures. When analyzing the behavior of the curves in Fig. 1, we arrive at the conclusion that a polycrystal line metal with a crystallite size smaller than 100 nm should be considered as a heterogeneous medium with a nonuniform temperature distribution. When an electric current passes through such a medium, the substance in intercrystallite regions is first heated; as a result, the conductor transforms into a mixture of vapor and droplets (clusters) with sizes close to the 1
crystallite size in the initial wire. When metallic nanopowders are produced by EEC, the operating conditions of experimental setups 1 Let
us make two notes. The authors of [16] annealed a rather thick wire and grew a granular structure with boundaries normal to its axis. They assumed that the wire melted in intercrystallite regions and constriction instability developed there as a result of energy release at grain boundaries. The initiation of partial elec tric arcs in intercrystallite regions was thought to cause a con trolled number of steps in the time dependence of voltage. The experimental results obtained in [16] did not contradict these assumptions. In review [17], Lebedev hypothesized that an exploding conduc tor can transform into a sol on the order of the free path length in size when energy is rapidly introduced into it. Our analysis made above not only agrees with this hypothesis but also sup ports it theoretically. TECHNICAL PHYSICS
Vol. 55
No. 4
2010
511
tχ/tF
1
8000 2 4000 3 0 10
30
50
70
90 d, nm
Fig. 1. Ratio of the time of temperature equalization over a crystallite to the travel time of conduction electrons through a crystallite vs. the crystallite size. T = (1) 1396, (2) 800, and (3) 300 K.
are chosen so that the entire energy accumulated in a source was released in a conductor by the time of no current condition. Thereafter, the EEC products (a mixture of vapor and droplets (clusters of various sizes)) expand into an inert gas. The expansion of the EEC products was simulated in a onedimensional approximation of the heterogeneous medium model proposed by us in [18], which includes balance equa tions for the mass, momentum, and energy of each component with allowance for the interaction of phases and phase transitions (evaporation, condensa tion, coalescence). As the initial state of the explosion products, we chose the following two limiting cases. The first case corresponds to the expansion of a vapor cylinder into an inert gas. In this case, nanoparticles form as a result of vapor condensation in the zone where vapor mixes with a buffer gas. Our calculations demonstrate that the particle size in this case is Dp ≤ 10 nm. The second limiting case is the expansion of a vapor–droplet mixture into an inert gas. The initial droplet sizes in this mixture were assumed to be the same and close to the average crystallite size in the ini tial wire. The mass fraction of the vapor component in the mixture, which forms due to boiling of intercrys tallite regions, is substantially lower than the mass fraction of the droplet component in the mixture. Fig ures 2 and 3 show the results of a numerical solution of the dynamic equations for the expansion of the vapor–droplet mixture into a buffer gas. As follows from Fig. 2, the calculated dependence of the average nanoparticle size agrees with the exper imental dependence both qualitatively and quantita tively. The calculated size distribution of particles (Fig. 3) is narrower than the experimental distribution (see [2]), which is mainly caused by the initial mono
512
VOLKOV et al. Dp, nm
N/N0 1
70 Cu 2
0.4
Al 50 Al Al
0.2
30
0 20
Ti 10 20
40
60
40
60
80
80 d, nm
Fig. 2. Average particle size vs. the average crystallite size in the initial wire: (1) calculation and (2) experiment. The experimental results were borrowed from [2].
100 Dp, nm
Fig. 3. Calculated size distribution of Al particles, d = 73 nm.
REFERENCES dispersity of the vapor–droplet mixture and a small amount of evaporated substance in grain boundaries. Thus, our consideration demonstrates that it is necessary (1) to take into account the effect of the microstructure of an exploding conductor on both its fragmentation during an electric explosion and the formation of metallic nanoparticles when the explo sion products fly into an inert gas, (2) to pass to higher energy introduction rates and smaller conductor diameters, and (3) to increase the buffer gas pressure. The latter requirement simultaneously solves the following problems: it decreases the shunt discharge probability along the conductor surface, increases the energy input efficiency, and increases the nanoparticle coalescence probability in the zone of mixing of the explosion products and a buffer gas. ACKNOWLEDGMENTS This work was supported by the Ministry of Educa tion and Science of the Russian Federation (grant no. 2.513.11.3127), INTAS (project no. 061000013 8949), the Russian Foundation for Basic Research (project no. 060800355a), the Ural–Russian Foun dation for Basic Research (project no. 070896032), the Presidium of the Ural Branch of the Russian Academy of Sciences in terms of the program for sup porting integration projects performed in cooperation with the Siberian Branch and Far East Branch of the Russian Academy of Sciences, and the basic research program Thermophysics and Mechanics of Extreme Energy Actions and the Physics of Strongly Com pressed Matter of the Presidium of the Russian Acad emy of Sciences.
1. Y. A. Kotov, J. Nanopart. Res. 5, 539 (2003). 2. V. S. Sedoi and Y. F. Ivanov, Nanotechnology 19, 154710 (2008). 3. I. M. Lifshits, M. Ya. Azbel’, and M. I. Kaganov, Elec tronic Theory of Metals (Nauka, Moscow, 1971) [in Russian]. 4. K. B. Abramova, V. P. Valitskii, Yu. V. Vandakurov, et al., Dokl. Akad. Nauk SSSR 167, 778 (1966) [Sov. Phys. Dokl. 11, 301 (1966)]. 5. G. V. Ivanenkov, S. A. Pikuz, T. A. Shelkovenko, et al., in Encyclopedia of LowTemperature Plasma, Ser. B: Appendices: Databases and Data Banks, Vol. IX3: Radiation Plasma Dynamics, Ed. by V. A. Gribkov (YanusK, Moscow, 2007), p. 288 [in Russian]. 6. N. A. Bobrova and V. S. Imshennik, in Encyclopedia of LowTemperature Plasma, Ser. B: Appendices: Data bases and Data Banks, Vol. IX2: HighEnergy Plasma Dynamics, Ed. by A. S. Kingsep (YanusK, Moscow, 2007), p. 104. 7. N. B. Volkov and A. M. Iskoldsky, J. Phys. A 26, 6635 (1993). 8. N. B. Volkov and A. M. Iskol’dskii, Pis’ma Zh. Tekh. Fiz. 20 (24), 71 (1994) [Tech. Phys. Lett. 20, 1004 (1994)]. 9. N. B. Volkov and A. M. Iskoldsky, J. Phys. A 28, 1789 (1995). 10. A. M. Iskoldsky, N. B. Volkov, and O. V. Zubareva, Physica D 91, 182 (1996). 11. V. N. Chuvil’deev, Nonequilibrium Grain Boundaries in Metals: Theory and Applications (Fizmatlit, Moscow, 2004) [in Russian]. 12. R. Z. Valiev and I. Z. Aleksandrov, Bulk Nanostruc tured Metallic Materials: Preparation, Structure and TECHNICAL PHYSICS
Vol. 55
No. 4
2010
MECHANISMS OF METALLIC NANOPARTICLE GENERATION
13. 14. 15. 16.
Properties (IKTs Akademkniga, Moscow, 2007) [in Russian]. V. F. Gantmakher, Electrons in Disordered Media (Fiz matlit, Moscow, 2003) [in Russian]. A. F. Mayadas and M. Shatzkes, Phys. Rev. B 1, 1382 (1970). A. I. Gusev, Nanomaterials, Nanostructures, Nanotech nologies (Fizmatlit, Moscow, 2007) [in Russian]. V. L. Budovich, O. A. Zakstel’skaya, I. S. Kotova, and I. P. Kuzhekin, Zh. Tekh. Fiz. 48, 1219 (1978) [Sov. Phys. Tech. Phys. 23, 681 (1978)].
TECHNICAL PHYSICS
Vol. 55
No. 4
2010
513
17. S. V. Lebedev and A. I. Savvatimskii, Usp. Fiz. Nauk 144, 215 (1984) [Sov. Phys. Usp. 27, 749 (1984)]. 18. E.L. Fenko, N. B. Volkov, and A. P. Yalovets, in Pro ceedings of the 9th International Conference on Modifica tion of Materials with Particle Beams and Plasma Flows, Tomsk, 2008, Ed. by N. Koval and A. Ryabchikov (IAO SB RAS, Tomsk, 2008), p. 701.
Translated by K. Shakhlevich
