Oxidation of Metallic Nanoparticles - COMSOL.com

Excerpt from the Proceedings of the COMSOL Conference 2009 MilanOxidation of Metallic NanoparticlesA. Auge, 1,* A. Weddemann, B. Vogel, F. Wittbracht, and A. Hütten1 Bielefeld University, Department of Physics, Thin Films and Physics of Nanostructures*Corresponding author: Bielefeld University, Universitätsstr. 25, 33615 Bielefeld,email: aauge@physik.uni-bielefeld.deAbstract: The oxidation behavior of metallicnanoparticles is investigated in respect tomaterial parameters like Mott potential, defectson the microstructure and oxide volume increaseper ionic defect. An emphasis is laid on magneticnanoparticles where the degree of oxidation canbe measured via the reduction of the magneticmoment.Keywords: Nanoparticle, Oxidation, Level SetMethod1. IntroductionMetallic nanoparticles have becomeimportant in many fields like biotechnology,chemistry and physics. 1 One important examplein this area is magnetic nanoparticles. 2 For manyapplications it is mandatory that magneticproperties of the nanoparticles are stable for along time. 3 The reduction of the magneticmoment of the particle is due to oxidation of themagnetic material like Fe or Co. The oxidationprocess in nanoparticles can be described by amodel introduced by Cabrera and Mott 4 that wasfurther improved by Fromhold. 5In this work the simulation of oxidation isrealized via a Level set approach modeling theevolution of the metal-oxide interface. Thisoffers the possibility to investigate the growth ofthe oxide layer in dependence of themicrostructure that is introduced by grain orcore-shell structures.model was developed by Fromhold et al. 5According to the latter model the oxide growthcan be divided into the following steps:1. Dissociative adsorption of oxygen from thegas phase to the surface of the particle2. Oxidation of surface metal atoms3. Ionization of adsorbed oxygen atoms andmetal ions at the metal-oxide interface4. Build up of a self generated electric field5. Incorporation of metal ions into the oxidelayer at the metal-oxide interface6. Diffusion of metal ions within the oxidelayer7. Reaction of metal ions with oxygen ions toform metal oxideFor geometries below 200 nm the electric field isthe driving force of the oxidation, and hence itsorigin and strength is of major significance. Themechanism of ionization depends on the ambienttemperature, the work-function of the oxidizingmetal and the oxide thickness. For lowtemperatures the ionization occurs via electrontunneling from the metal-oxide interface to theadsorbed oxygen. Considering the energy leveldiagram in Fig. 1, the driving force of thetunneling can be easily understood.Vacuum potential2. Theoretical modelIn general the growth of oxide films iscontrolled by diffusion of ions and electronsunder the influence of gradients of theirconcentration and self generated electric fields.In the case of nanoparticles, the electric field isvery strong and hence its influence is significant.A generic model describing the kinetics of oxidegrowth under the effect of this electric field wasintroduced by Cabrera and Mott. 4 An improvedmetal oxide oxygenFIG. 1: Energy level diagram of the metal-oxideoxygensystem. The tunneling process stops as soon asthe potentials and reach equilibrium

Tunneling occurs due to the potentialdifference of the metal and the oxygenuntil the electric field of the ions is as strongas the potential difference(1)where the so called Mott potential and L theoxide thickness. For higher temperatures thermalemission of electrons is also important whichhowever is not considered here. The tunnelcurrent is given by the following expression 5 (2)with the Planck constant and the electronmass. The ionic diffusion current in the absenceof space charge effects is given in the steadystate approximation by 5 (3)FIG. 2 Nanoparticle with different oxide volumeincrease rates. Grains are colored in orange. The shelland core are shown as red and blue regions,respectively.where denotes the oxide volume increase pertransported ionic defect.3. Simulation modelThe parameters and are the bulkdefect concentrations of the diffusing ionicspecies at the oxide-oxygen and at the metaloxide interface, respectively. is the surfacecharge field in the oxide, the elementarycharge, the effective charge of thetransported ionic species, a the lattice constant, kthe Boltzmann constant, the temperature andthe thermal activation energy for ionicmotion. The assumption that the steady statecurrents are equal in magnitude but opposite insign leads to the following equation 5 (4)For this simulation model the nanoparticle isapproximated by a sphere of radius R and oxidelayer thickness L. The growth of the oxide layeris implemented via a Level set approach. 7 Thismethod is capable of describing the motion ofsurfaces and often used in multiphase flows.Here it is used to track the motion of the metaloxideinterface.The basic idea of the Level set method is toexpress a surface in an implicit form, as the zerolevel set or isophote of a higher dimensionalfunction and then trace the deformation ofthe surface by means of deformation of thisembedding function. Generally, for a givendomain with smooth boundary, we assume theexistence of an implicit function whichsatisfiesThis condition is used to determine the selfgenerated electric field in the oxide layer. Thegrowth of the oxide layer is then given by 5 (5)

FIG. 3: Comparison between experimental data forthe oxidation of Co nanoparticles withand simulation results for different values ofthe Mott potential. All other parameters are given inTab.1.The dynamic evolution of the Level setfunction follows the convection diffusionequation(5)where the normal velocity of the movingboundary and D is the diffusion coefficient. Inthe investigated systems diffusion is onlyintroduced to maintain numerical stability.Equation (5) defines the motion of the zero levelset . As an initial condition, we set(6)This is a signed distance function, andtherefore, has the advantage thatfor most of the domain except at the centerwhere certain partial derivates are not defined.The normal vector of the implicit surface iscalculated via . The growth of the oxidelayer and thus the normal velocity is given byequation (5). It is implemented in the followingway(7)The ionic current is determined for each timestep using equation (4). For this the distance Lhas to be calculated which is done via linearregression.(8)FIG. 4: Oxidation in dependence of temperature of ananoparticle with R = 5 nm and an initial oxide layerof 1nm. All other parameters are given in Tab. 1.It is also possible to implement further detailslike structural defects, grain structures or coreshell nanoparticles by varying the oxide volumeby an increase of on the nanoparticle domain.This is shown in Fig. 2. In this example the shellhas a high oxide volume increase per ionicdefect. Grains within the material are presentedas orange subdomains inside the core.4. Simulation resultsTo verify the simulation model, experimentaldata are compared to calculated oxidation curves.For all simulations an initial oxide thickness of 1nm is assumed. The oxidation degree givenneglects this initial oxidation. This coincideswith experimental conditions, where themeasurements start after a certain oxidation time.In Fig. 3 experimental data of Co nanoparticleswith a mean radius ofandsimulation results are shown. The nanoparticlesoxidized at room temperature and the loss ofmagnetization is measured via an AlternatingGradient Magnetometer. To calculate the loss ofmagnetization the degree of oxidation is used.The simulation fits well to the experimental dataif a Mott potential ofand remaining parameters according to Tab. 1are chosen.Furthermore, the oxidation of nanoparticlesin dependence of temperature and the oxygenwork function are investigated. In Fig. 4oxidation curves for different temperatures isshown. All curves show a rapid increase ofoxidation followed by a slow linear increase. For

FIG. 5: Oxidation in dependence of the oxide workfunction of a nanoparticle with R = 5nm and an initialoxide layer of 1 nm. All other parameters are given inTab. 1.rising temperature, the initial fast oxidationincreases rapidly. The slow increase is alsoaffected by a change of slope. The change of theoxidation rate is due to the increase of the ioniccurrent, which is exponentially affected by thetemperature.In Fig. 5 the oxidation in dependence of theoxygen work function is shown. Forthe initial fast oxidation is visible. Forthe growth shows a linearbehavior. This can be attributed to growthkinetics limited by the electron current. In Fig. 6the effect of different growth regions, which ispossible with the Level set method, is shown.Deformations of the metal-oxide interface arevisible where the growth rate is enhanced.5. Summary and OutlookIn this work a Level set based method isintroduced to model the oxidation dynamics ofmetallic nanoparticles. A comparison betweenexperimental data from the oxidation of Conanoparticles shows good agreement with thesimulations. It has also been shown, that it ispossible with the proposed approach to modeldifferent regions of growth within thenanoparticle. For the future, further studies onthe influence of the latter are planned.FIG. 6: Regions with different growths rates ofnanoparticle as well as an oxide-metal interface (greenlayer) are shown. A distortion of the interface isvisible at the areas with increased oxide growth rate.6. AppendixSymbol Definition ValueMetal oxide work function 2(eV)work function (eV) 2.5Mott potential (eV) -0.5T Temperature (K) 3002a Ionic jump distance 4.25Ionic defect concentration atmetal oxide interface (Ionic defect concentration atoxide oxygen interface (Oxide volume increase pertransported ionic defectTable 1: Parameters used for simulationsReferences19.191 J. Blackman, Metallic Nanoparticles, Elsevier (2009)2 S. P. Gubin, Magnetic Nanoparticles, Wiley-VCH(2009)3 G. A. Niklasson, R. Karmhag, Surface Science 532–535, 324–327 (2003)4 N. Cabrera et al., Rept. Progr.Phys. 12, 163 (1949)5 A. T. Fromhold, E.A. Cook, Phys. Rev. 158, 600–612 (1967)6 S. Osher, R. Fedkiw, Level Set Methods and DynamicImplicit Surfaces, Springer (2002)