PHYSICAL REVIEW B, VOLUME 65, 235410 Structure and properties of silica nanoclusters at high temperatures I. V. Schweigert,* K. E. J. Lehtinen,† M. J. Carrier,‡ and M. R. Zachariah§ Departments of Mechanical Engineering and Chemistry, University of Minnesota and the Minnesota Supercomputer Institute, Minneapolis, Minnesota 55455 ͑Received 29 January 2001; revised manuscript received 13 December 2001; published 28 May 2002͒ The properties of silica clusters at temperatures from 1500 to 2800 K are investigated using classical molecular dynamics simulations for particles containing up to 1152 atoms. We found that atoms in the cluster were arranged in a shell-like structure at low temperatures, and that the radial density profile peaked near the outer edge of the particle. Smaller clusters have much higher pressures, with the magnitudes corresponding quite well to those obtained from the Laplace–Young equation, when evaluated in a self-consistent manner using our derived surface tension. Our computed surface tension did not show any significant size-dependent behavior in contrast to the decreasing surface tension observed for Lennard–Jones liquid clusters. Finally our computed diffusion coefficients in the liquid state are seen to be larger than bulk computed diffusivities. A discussion regarding the relevance of these computations on the growth of silica nanoparticles is presented. DOI: 10.1103/PhysRevB.65.235410 PACS number͑s͒: 61.46.ϩw, 61.20.Ja I. INTRODUCTION ␶ ϭ ␩ ␴ ͑ ͒ f 2 R p / , 2 ␩ 11 where is the viscosity, and R p is the radius of the par- One of the most import methods of synthesizing nanopar- 7 ticles is through vapor phase nucleation and growth. Several ticle. Ehrman et al. discussed the possible causes of the en- methods are used, including combustion, plasmas, thermal hanced rate of silica particle sintering observed in an experi- reactors, and evaporation-condensation.1–3 The evolution of ment, including the presence of impurities such as alkali an aerosol undergoing a gas-to-particle growth process is de- metals which could result in a lower viscosity of silica. On scribed by a master equation, the ‘‘aerosol general dynamic the other hand, in theoretical considerations kinetic coeffi- equation’’ ͑GDE͒.4 The GDE is a population balance equa- cients such as the diffusion coefficient and the viscosity, as tion for the behavior in time and space of the particle size well as the surface tension, were taken from the properties of distribution function, which can include all the driving forces bulk material. However, we know that what makes the nano- for particle growth ͑nucleation, surface growth, coagulation/ particles so interesting is that their fundamental properties coalescence, transport, etc.͒. Less conventionally, but of in- are intimately related to the physical size of the component.6,12 For an example, a 10-nm-diameter iron par- terest for nanoparticle synthesis research, the GDE can be 13 extended to track the degree of coalescence of particle ticle has almost 20% of its atoms on the surface. This high 5 fraction of surface atoms alters properties such as the melting aggregates. One of the biggest challenges in the implemen- 14 tation of the GDE is knowledge of the rates of these indi- point and presumably the surface tension and accounts for vidual processes. For nanoparticle synthesis, one of the most the high reactivity of nanoparticles both toward each other fundamental and important kinetic properties of interest in and also toward other materials and gases. nanoparticle growth is the sintering or coalescence rate be- One of the interesting aspects of nanoparticles properties tween particles during vapor-phase growth. A knowledge of we investigate in this paper is the variation of the internal these rates and their dependence on process parameters ulti- pressure with particle size, as discussed in Ref. 15. Laplace’s mately impacts the ability to control primary particle and equation agglomerate growth, which are of critical importance to ⌬Pϭ2␴/R ͑3͒ nanoscale particles whose properties depend strongly on p size, morphology, and crystal structure.6 tells us that the pressure difference between a particle and the Previous experimental and theoretical studies of sintering surrounding environment ⌬P goes as the surface tension ␴, in flame reactors demonstrated good agreement for the case divided by the particle radius. This implies that as the par- of titania particle growth,7–9 based on a characteristic coales- ticle becomes smaller, the pressure will approach infinity as cence time obtained from a solid-state diffusion model, in the radius becomes vanishingly small. Of course, this as- which the characteristic time is written as sumes that the surface tension is size independent, but in ␶ ϭ ␲ ␴ ͑ ͒ fact, at some point, the surface tension will tend toward zero. f 3kBTv p/64 D v0 , 1 Nevertheless, we should expect that the pressure of a particle where T is the gas temperature, v p is the particle volume, D can be many hundreds of atmospheres, and may impact the is the diffusion coefficient usually reported as an Arrhenious chemical, phase and crystalline structure of particles. For an 10 function of the temperature, ␴ is the surface tension, and example, the pressure inside a 10-nm-diameter TiO2 particle ͓␴ϭ 2 ͑ ͔͒ v0 is the molecular volume for diffusion. In the case of sili- 0.5 J/m Ref. 16 is on the order of 2000 atmo- con dioxide ͑silica͒, however, the primary particle size is spheres. Such elevated pressures may have a significant typically underpredicted based on a viscous flow coalescence impact on properties such as diffusion coefficients17 time written as and the viscosity, which in turn affect the rates of particle 0163-1829/2002/65͑23͒/235410͑9͒/$20.0065 235410-1 ©2002 The American Physical Society SCHWEIGERT, LEHTINEN, CARRIER, AND ZACHARIAH PHYSICAL REVIEW B 65 235410 sintering.10,11 Given these constraints, data obtained at large grain sizes when extrapolated to smaller dimensions may lead to considerable errors. Experimental studies are cer- tainly desirable for investigating these characteristics; how- ever, experiments aimed at determining the fundamental transport and thermodynamic properties of fine aerosols are not easily obtained, because of the ensemble nature of most studies. In this paper we study the nanoscale silica clusters of different sizes (Nϭ72, 288, 576, and 1152͒ and at different temperatures ͑1500–2800 K͒, using a classical molecular- dynamics ͑MD͒ simulation to determine the structural prop- erties, internal pressure, diffusion coefficients, and surface tension. We are particularly interested in the particle size dependence of these properties, and its influence on coalescence/sintering modeling. FIG. 1. The radial density distribution for clusters with 288 II. MODEL SYSTEM atoms ͑dashed line͒, 576 atoms ͑solid line͒ and 1156 atoms ͑dotted line͒ at Tϭ2080 K. The sketch of surface structure is shown in the The clusters consist of N classical atoms which interact inset. through a pair model potential developed by Tsuneyuki et al.,18 which takes into account the steric repulsion due to equilibration procedure for each prepared cluster, we gradu- atomic sizes, screened Coulomb interaction from charge ally decreased the temperature from 4000 to 1500 K, fol- transfer, charge-dipole interactions due to the large electronic lowed by a reversal of this procedure to the desired target polarizability of the anions, and dipole-dipole interactions to temperature. A comparison of cluster properties such as the mimic the short- and intermediate-order interaction. The in- radial density distribution, obtained by cooling and then re- teraction potential has an analytical form of the Born- verse heating, shows good agreement. Huggins-Mayer type V ϭq q /r ϩa exp(Ϫb r )Ϫc /r6 , ij i j ij ij ij ij ij ij The total angular momentum of cluster motion is always where qi is the charge of the i-type atom, rij is the inter- reset to zero by the transformation of velocities in order to atomic distance, and aij , bij , and cij are the parameters evaluate transport properties such as diffusion coefficients. taken from Ref. 18. For higher temperature the potential was We did not analyze the cluster behavior at higher tempera- modified in Ref. 19 by adding a Lennard-Jones term, in order tures TϾ3000 K, where a significant dissociation of frag- to avoid the occasional overriding of the O-O repulsion bar- ments takes place during the time interval of the calculation rier at a shorter interatomic distance at a higher temperature. (tϷ0.15 ns). This is because this interatomic potential can- The modified interatomic interaction is referred to as the 18 not realistically describe the vaporization processes of *ϭ ϩ ⑀ ͓ ␴ 22 TTAMm potential with the form Vij Vij 4 ij ( ij /rij) silica. For instance, in our calculations we observe the va- Ϫ ␴ 6 ⑀ ␴ ( ij /rij) ͔, where ij and ij are the usual Lennard–Jones 23 19 porization of SiO2 molecules, while it is well known that parameters specified in. silica vaporizes by decomposition to SiO and O . The equations of motion for each atom was solved using 2 the standard velocity version of the Verlet algorithm.20,21 In our calculations we constructed a silica cluster with an ap- III. RESULTS propriate number of atoms, and placed it in the center of a spherical cavity of radius R*, which comprised the calcula- A. Particle density tional cell. The center of the calculation cell (rϭ0) coin- At lower temperature Tϭ1500 K in a solid ‘‘glass’’ state, cides with the center of mass of the cluster. The radius of the all clusters exhibit an oscillating density distribution. With cell was kept at 2 nm for all clusters. In order to conserve increasing temperature there seems to be a tendency to mass at higher temperatures, we force any cluster fragment dampen out the radial density gradient, though some peaks which escapes the cluster due to evaporation, and reaches the still remain.