Reactive self-heating model of aluminum spherical nanoparticles
RReactive self-heating model of aluminumspherical nanoparticles
Karen S Martirosyan †∗ , Maxim Zyskin ‡ , † Department of Physics and Astronomy, University of Texas, Brownsville,80 Fort Brown, Brownsville, TX, USA, 78520; ‡ Rutgers University, 126 Frelinghuysen Road, Piscataway, NJ 08854-8019.
Abstract
Aluminum-oxygen reaction is important in many highly energetic, high pres-sure generating systems. Recent experiments with nanostructured thermitessuggest that oxidation of aluminum nanoparticles occurs in a few microsec-onds. Such rapid reaction cannot be explained by a conventional diffusion-based mechanism. We present a rapid oxidation model of a spherical alu-minum nanoparticle, using Cabrera-Mott moving boundary mechanism, andtaking self-heating into account. In our model, electric potential solvesthe nonlinear Poisson equation. In contrast with the Coulomb potential,a ”double-layer” type solution for the potential and self-heating leads to en-hanced oxidation rates. At maximal reaction temperature of 2000 o C, ourmodel predicts overall oxidation time scale in microseconds range, in agree-ment with experimental evidence. ∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec any previous studies have been directed to understand the mechanismand kinetics of aluminum particle oxidation. [1] -[9]. Aluminum oxidationexhibits high enthalpy and has been extensively used for propulsion, py-rotechnics and explosion reactions [10] [11]. Nanoenergetic materials (NM)based on aluminum thermites may store two times more energy per vol-ume than conventional monomolecular energetic materials [12]. The sizereduction of reactant powders such as aluminum from micro- to nano-sizeincreases the reaction front propagation velocity in some systems by two tothree orders of magnitude [12] [13]. The development of novel NM, theirdesign, synthesis and fabrication procedures are critical for national secu-rity and it was recognized to be significant to support advanced weaponsplatforms. Among numerous thermodynamically feasible Metastable Inter-molecular Composites mixtures the most widely investigated are Al/Fe O ,Al/MoO , Al/WO , Al/CuO, Al/Bi O and Al/I O nano systems [12]-[20].The main distinguishing features of these reactive systems are their signifi-cant enthalpy release and tunable rate of energy discharge, which gives riseto a wide range of combustion rates, energy release and ignition sensitivity.There are several advantages of using Al/Bi O and Al/I O nanocom-posites: (i) reduced ignition and reaction times; (ii) superior heat transferrates; (iii) tunability of novel energetic fuel/propellants with desirable phys-ical properties; (iv) enhanced density impulse; (v) incorporating nanoener-getic materials into the MEMs and NEMs systems [12]. Our recent experi-ments suggest that oxidation of nanoparticles of aluminum with Bi O andI O occurs in a few microseconds [17]-[19]. Rapid reaction in the nanostruc-tured thermites cannot be explained by a conventional mechanism based on2he diffusion of Al and O atoms in oxides.In this report, we present a rapid oxidation model of spherical aluminumnanoparticles surrounded by oxygen, using Cabrera-Mott oxidation model[7]-[9] with a self-consistent potential, and taking self-heating into account.Using nonlinear self-consistent potential in Cabrera-Mott model, as opposedto the Coulomb potential, is more accurate, and yields higher oxidation rates.For nanosized particles, nonlinear model gives ”double-layer” type solutionfor the potential, since potential changes rapidly near the metal-oxide inter-face. As a result, the electric field at the metal- oxide interface, determiningthe oxidation rate, is less sensitive to oxide thickness, and stays nearly con-stant at intermediate stages of oxidation (while for the Coulomb potential,electric field decreases, and the oxidation rate quickly drops). This nonlineareffect for the potential, combined with self-heating, leads to rapid tempera-ture increase, resulting in a dramatic (orders of magnitude) increase of oxi-dation rates for nano-sized particles. Our model can predict oxidation timesin microseconds range, in a good agreement with experimental data [12].To estimate the reaction times, we assume that aluminum sphere (radius25 nm, with thin oxide layer of 3 nm) is surrounded by oxygen. The sphereis rapidly heated to ignition temperature T , sufficient to initiate oxidationreaction, further boosted by self-heating as a result of oxidation. We assumethe spherical symmetry of the problem.In the Cabrera-Mott model of metal oxidation [1]-[3], [7] aluminum ionsare helped to escape aluminum boundary (overcoming ionization potential W ) by a self-consistent electric potential V. The electric field is induced inthe oxide layer by imbalance between excess positive aluminum ions and3lectrons in the oxide layer, due to difference in chemical potentials of themetal and oxidizer. Excess concentration of electrons and ions in the oxidelayer depends on the electric field potential V via appropriate Gibbs factors,leading to a self-consistent version of Poisson equation for V, which in thespherically-symmetric case is given by ∇ V ≡ r ddr (cid:18) r ddr V (cid:19) = 8 πk e N sinh (cid:18) e Vk b T (cid:19) , r ≤ r ≤ r ,V ( r ) = V ,V ( r ) = 0 . (1)Here r is the metal particle radius, ( r − r ) is oxide layer thickness, k =8 .
99 10 N m C , e = 1 .
60 10 − C , N = ( n e n i ) / , and n i and n e are concen-trations of respectively aluminum ions and of excess electrons in the oxidelayer, n e = 2 (cid:0) πm e k b T /h (cid:1) / exp (cid:18) − e φk b T (cid:19) ,n i = N i exp (cid:18) − e W i k b T (cid:19) , (2)where m e is the mass of electron, k b Boltzman constant, h Plank constant, N i is the concentration of sites available for hopping metal ions. Thus N = ( n e n i ) / = N (cid:16) TT (cid:17) exp (cid:16) − e ( φ + W i )2 k b T (cid:17) ,N = (2 N i ) (2 πm e k b T /h ) / ∼ . m − at T ∼ K. The physical meaning of W i is the difference of chemical potentials for metalions in the metal and the oxide; φ is the potential difference for electrons inthe conduction bands of aluminum metal and the oxide (a semi-conductor);4nd the value of V is determined from the condition that metal ion con-centration at the interface with the oxide equals n i . In Cabrera-Mott model,those ionization potentials may be considered as model parameters [7]-[9].Oxidation of metal occurs via tunneling of aluminum ions into the ox-ide layer, overcoming ionization potential of maximum height
W > W i , andassisted by the self-consistent electric potential. Electric field provides poten-tial energy decrease for ion hopping from the bottom to the top of ionizationpotential, at a distance of α ∼ . . This leads to an equation for themetal-oxide interface radius r : the normal velocity of the metal boundary u n is determined by the electric field ( −∇ V ) at the boundary, appearing ina Gibbs factor, u n ≡ dr dt = − Ω ν n exp (cid:18) − eWkT (cid:19) exp (cid:18) qeα | V (cid:48) ( r ) | k b T (cid:19) ; r (0) = r . (3)Here r is the initial metal sphere radius, which we assume to be 22 nm , Ω ≈ . is the volume of oxide per aluminum ion, n ∼
10 nm − isthe number of metal ions per unit surface area, ν ∼ s − is frequency oftunneling attempts, q = 3 is aluminum valency.In the Cabrera-Mott model, it is assumed that escaped metal ions migrateto the outer boundary of the oxide where they react with the oxygen, whilelocal Gibbs distribution of excess densities of electrons and metal ions insidethe oxide is essentially unaffected. Due to the spherical symmetry, the radiusof the oxide-oxidizer interface r changes uniformly, and can be found fromconservation of the number of metal ions, taking into account difference in5olumes per metal ion in the metal and the oxide, ( r − r ) = − κ ( r − r ) ,κ ≈ . . Thus r ≡ r ( r ) = ( r + κ ( r − r )) , κ ≈ . . (4)Here r = r + δ, where δ ∼ self-heating into ac-count, due to heat released by exothermic aluminum oxidation, resulting intemperature increase of the remaining metal and oxide layer. For nano-sizedparticles, temperature can be assumed to be uniform.Thus temperature canbe computed based on reaction heat release and specific heats of reagents.Assuming constant specific heats, T ≡ T ( r ) = T + σH Al ρ Al ( r − r ) c Al ρ Al r + c Al O ρ Al O ( r − r ) . (5)Here ρ Al , ρ Al O are densities of the aluminum and oxide, c Al , c Al O arespecifics heats per unit mass, H Al is the oxidation reaction enthalpy per unitmass of aluminum, and σ is the proportion of released heat which is used upfor self-heating. We take reaction initiation temperature T = 750 K. Since r can be found from r using (4), the temperature T in (5) is determined by r . Due to high enthalpy release in aluminum oxidation ( ∼
24 kJ per gramof aluminum ), adiabatic assumption σ = 1 yields unrealistically high maxi-mum temperature, even when melting and vaporization heats are taken intoaccount. We assume that σ = 0 .
11 of the heat released contributes to selfheating, while the rest is lost due to radiation, heat conduction, and convec-6ion. For such value of σ , the maximum reaction temperature, correspondingto r = 0 , r = r (0) in (4), (5), will be T M ∼ o C, which agrees withexperiments [12].We use experimentally determined maximum temperature to control theeffect of the heat transfer process on oxidation rates. Detailed modeling ofthe heat transfer is outside the scope of this Letter. However, we note thatan assumption of fairly large heat loss is sensible. For a nanoscale particle,the convection loss mechanism is important. Indeed, in the steady statelimit, heat loss of a small sphere due to convection can be modeled by theNewton’s law of cooling, with the heat transfer coefficient η, [ W m − K − ],given by η = kr [21], where k, [ W m − K − ], is air’s thermal conductivity,and r is Al sphere radius (this limit is applicable for small Grashof numbers,which is the case for nano-spheres). Thus, the convective loss rate η (4 πr ) δT is proportional to r , while the total heat released due to chemical reaction isproportional to r . As a result, convective heat loss becomes significant forsmall spheres. An estimate shows that for an aluminum particle of radius25 nm, all the heat released by oxidation may be lost to steady convectionin a time scale of 10 − s, which is comparable to the time scale of the finalrapid phase of the oxidation process in our model. Radiation loss, whichis proportional to r T , contributes less than convection for the values ofparameters in our model.We note that Eq. (3) for the metal boundary velocity is similar to themodel of metal sphere oxidation considered in [1], however we use a differentpotential (a solution of the nonlinear Poisson equation (1) rather than theCoulomb potential), and we also take the self-heating effect into account. As7 result, oxidation rates are dramatically increased.We have found the self-consistent potential V, solving numerically theboundary value problem for the Poisson equation (1) for various radii r (wehave used the Newton’s method to solve a discrete version of the problem).Our results for the potential are illustrated in Figure 1. Our values of ion-ization parameters V = 0 .
65 V, φ + W i = 1 . W = 1 . a = 0 . r = 22nm for the initial metal radius, and r = 25 nm for the initial metal+oxideradius. When the remaining metal radius is r ≤ r , the radius of the metal+oxide r can be found from Eq. (4), and the temperature, with the selfheating taken into account, is determined by (5). We use T = 750 K as theinitial temperature.We note that for the case of cylindrical symmetry, the potential equationis equivalent to Painleve 3 (with particular values of parameters).Solution of the nonlinear Poisson equation (1) enables us to compute thegradient of the potential on the metal surface, and corresponding potentialdecrease due to electric field near metal-oxide interface, for various radii r . In Figure 2, we show electric potential decrease at a distance a = 0 . nm from the metal-oxide interface for the solution of nonlinear Poisson equa-tion (solid line), and compare it with a potential decrease computed usingCoulomb potential (broken line). Potential decrease determines oxidationspeed, via an exponential Gibbs factor in Eq. (3). Our results demonstratethat the nonlinear model for the potential, with the self-heating effect, yieldsconsiderably higher oxidation rates.Using results for the gradient of the potential on metal-oxide boundary,8e can find the radius of the metal sphere as a function of time, by solvingthe moving boundary Eq. (3). This equation is separable, since it followsfrom Eqs. (4),(5) that the right hand side is a function of r only. Thus, asolution can be found by a numerical integration. The solution is shown inFigure 3. The overall oxidation time scale is in microseconds, much fasterthan for macroscopic particles. We further observe that in our model mostof oxidation occurs very quickly towards the end of oxidation process, sincereaction rate dramatically increases with the temperature rise due to theself-heating.Our model yields oxidation time of 32 µs for a spherical aluminum nanopar-ticle of radius 25 nm at initial temperature of 750 K. By using exactly thesame ionization parameters and the self-heating mechanism, but taking theCoulomb potential rather than a solution of the self-consistent equation (1),oxidation time is calculated to be 2.5 ms, that is almost 100 times slower.As for the diffusion limit, it is known to correspond to the linear in theCoulomb potential V (cid:48) approximation of the Eq. (3). It is clear from Eq. (3)that the oxidation rate in the linearized model will much slower than for theMott model. Using the same model parameters and the self heating as inour model, oxidation time in the diffusion limit would be about 40 ms; andwithout the self-heating it is estimated to be 14 seconds at T = 750 K . Exper-imentally, an explosive oxidation reaction of nanosized particles is initiatedat a temperature of about 750 K (i.e below aluminum melting temperature),with the reaction time scale estimated to be in microseconds [12]. Thiscomparison strongly suggests the Cabrera-Mott model with a self-consistentpotential as the most likely reaction mechanism.9igure 4 illustrates the oxidation rate of change of mass of aluminum ina metal-oxide sphere with initial radius 25 nm , as a function of aluminummass and temperature T . Such a rate of change of reacting mass appears e.g.in Semenov explosion model, with m dmdt usually given by the Arrhenius factorexp( − ET ) with a constant E. However in Cabrera-Mott oxidation model, E depends on the remaining mass and temperature (since electric potentialdrop in the self-consistent Cabrera-Mott model depends on the metal radiusand temperature).Nonlinearity in equation for the self-consistent electric potential, togetherwith the self-heating effect, lead to significant increase of oxidation rate fornano-sized aluminum particles, compared to the model with Coulomb po-tential and without self-heating. Results of our modeling suggests oxida-tion time scales in microseconds range for nano-sized aluminum particles, inagreement with the experimental evidence [12]. Results of computation ofoxidation time scale are very sensitive to values of ionization potentials, sincethey appear as Gibbs factors in the equation for boundary velocity. There-fore we expect that oxidation rate will be very sensitive to defects, and todeviation from spherical symmetry.After this work was completed, we have learned that in a recent paper[4] a different mechanism of oxidation initiation, based on high temperaturerun-off and the Coulomb potential, was proposed for a smaller, 10 nm radius,particle with very thin initial oxide layer. In our model with the nonlinearequation for the potential, rapid full oxidation of nanosized particle can occurat lower, experiment-matched, maximal temperature, due to nonlinear effectsfor the potential near metal-oxidizer interface.10e acknowledge the financial support of this research by the NationalScience Foundation grant 0933140. References [1] A. Ermoline , E. L. Dreizin , Equations for the Cabrera-Mott kineticsof oxidation for spherical nanoparticles, Chemical Physics Letters ,4750, (2011).[2] V.P. Zhdanov, B Kasemo, Chem. Phys. Lett. , 285 (2008).[3] V.P. Zhdanov, B Kasemo, Appl. Phys. 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Figures
Figure 1:
Self-consistent potential V for r = 2 , , , , ,
22 nm. Here r = 22nm, r = 25 nm, V = 0 .
65 V, φ + W i = 1 . W = 1 . T = 750 K. r , nm0.20.40.60.8 ∆ V a (cid:144) V o (cid:56)(cid:60) Figure 2:
Potential drop a distance a = 0 . etal-oxide interface, shown as a proportion of the total Cabrera-Mott potential V = 0 .
65 V, as a function of the remaining metal radius, and taking self-heatinginto account. Solid red curve: nonlinear self-consistent model for the potential;black dashed curve: Coulomb potential. Here r = 22 nm, r = 25 nm, φ + W i =1 . W = 1 . T = 750 K , T M = 2273 K. a Figure 3:
Oxidation time scales: (a) ratio of remaining aluminum metal mass toinitial aluminum mass, as a function of time. Here r = 22 nm, r = 25 nm, φ + W i = 1 . W = 1 . T = 750 K , T M = 2273 K, a = 0 . igure 4: Oxidation rate of change of mass of metal aluminum in a metal-oxidesphere of initial radius 25 nm, as a function of metal mass m and temperature T (K), in log scale. Metal mass is shown as a ration to the initial mass of theparticle m . Here initial radius is 25 nm, φ + W i = 1 . W = 1 . T = 750K, T M = 2273 K, a = 0 .4 nm.