Coalescence - Agenda What if particles are liquid, or are solid but temperatures are high enough, solid state diffusion can occur? Koch and Friedlander, coalescence limited approach Effect of partice internal pressure on coalescence rate

How about finite coalescence rate? Important for particle growth in steep T gradients, e.g. flames sintering complete chemical reaction particles grow by between collisions fast compared to collision/sintering particle formation times sintering incomplete between collisions particles are necked important characteristics: characteristic times: • primary particle size • time between particle • extent of agglomeration collisions • time required for particle coalescence

t Characteristic times and particle morphology time residence time coalesce collision unagglomerated necked agglomerated Characteristic times depend on concentration of particles and on material properties Desired degree of agglomeration depends on application

Motivations Models of nanoparticle growth important: reactor/process design understanding/predicting formation of unwanted byproducts of combustion. Models of particle growth for silica overpredict primary particle size if instantaneous coalescence is assumed (see for example Ulrich G.D., Milnes, B.A., and Subramanian, N.S., Combustion Sci. Technol. 14, 243 (1976)). Models of particle growth for silica underpredict primary particle size of finite coalescence times based upon bulk viscosity are used (see Xiong, Y. Akhtar M.K., and Pratsinis S.E., J. Aerosol Sci., 24, 301, (1993), Ehrman, S.H., Friedlander S.K., and Zachariah, M.R., J. Aerosol Sci., 29, 687 (1998)).

Further motivation Goal Because of high surface area to volume ratio, pressure inside nanoparticles may be very high. For materials which coalesce by viscous flow, rate is dominated by viscosity, an extremely temperature and pressure sensitive variable. Unlike typical crystalline materials, diffusivity of O2- and Si4+ ions in liquid silica increases with increasing pressure, resulting in a decrease in mobility (viscosity) with increasing pressure. Goal Incorporate this information into a traditional collision/ sintering model of aerosol growth.

a = surface area of aerosol assumptions no barrier to nucleation Collision/sintering see Koch and Friedlander, 1990; Friedlander and Wu, 1994; Lehtinen et al., 1996 flame generated silica particles a = surface area of aerosol assumptions no barrier to nucleation coalescence is rate-limiting gives solution for particle size after long residence times initial rate of growth important TEM - S.H. Ehrman

Characteristic coalescence time for viscous flow tc = dp [2] Frenkel (1945) J.Phys. 9,385. s h = viscosity dp = particle diameter s = surface tension What does this mean, viscosity in a nanoparticle? Especially a rapidly colliding and coalescing nanoparticle. Chemical bonds rapidly forming and breaking.

 = kT [4] Dl Coalescence as atomistic process: Coalescence via solid state diffusion mechanism [3] Friedlander and Wu, Phys. Rev. B, 49, 3622 (1994) vp = particle volume s = surface tension D = solid state diffusivity vo = volume of diffusing species As evidence of atomistic behavior in silica: viscosity related to diffusivity, D through Stokes-Einstein relationship:  = kT [4] Dl has been observed experimentally for mixed silicates by Shimizu and Kushiro (1984) Geochim. Cosmochim. Acta. 48, 1295. l = volume of oxygen anion

Pressure inside nanoparticles Laplace Equation Pi Pa Pi - Pa = 4s [1] dp s = surface tension dp = particle diameter Pi = internal pressure Pa = ambient pressure Pi for 3 nm diameter silica particle ~ 2000 atmospheres! (~ 0.2 gigaPascals) May result in phase and transport behavior different from P = 1 atm.

Effect of P on diffusivity For crystalline systems, diffusivity has exponential dependance on pressure as well as temperature: Ed = activation energy for diffusion, J molecule -1 Va = activation volume cm3 molecule -1 D = - æ è ç ö ø ÷ exp -E PV kT o d a [5] For typical crystalline materials, increasing pressure leads to decreasing diffusivity. Va is positive, ~ equal to volume of diffusing species.

The special case of silica It has been observed experimentally for pure silica and for some mixed silicates (NaAlSi2O6, Na2Si4O9) and also in molecular dynamics simulations of pure silica - up to a certain pressure Pcritical , diffusivity of oxygen and silicon ions increases as pressure increases. Va in Eq. 5 is negative! Va estimates range from volume of oxygen ion to volume of SiO4 tetrahedra references: Shimizu and Kushiro Geochim. Cosmochim. Acta, 48, 1295 (1984). Tsuneyuki and Matsui Phys. Rev. Let. 74, 3198 (1995) . Poe et al. Science, 276, 1245 (1997). Aziz et al., Nature, 390, 596 (1997).

Why? Pressure Facilitated Diffusion (a) Silicon ( ) in tetra- hedral coordination, pressure = 1 atm. (b) As pressure increases, up to Pcritical, areas of higher coordinated silicon form locally. (c) After decompression, tetrahedral framework rearranged, and diffusion has taken place. Method proposed by Tsuneyuki and Matsui (1995) Phys. Rev. Let. 74, 3197.

Pcritical estimates range from 1 to 10 gPa Effect of P on D, for silica Diffusivity Pressure Pcritical P < Pcritical = activation energy for diffusion related to activation energy for forming higher coordinated silica. P > Pcritical = activation energy related to activation energy for formation of tetragonal silica. Pcritical estimates range from 1 to 10 gPa as reference point, for limiting case of 1 SiO4 tetrahedra, Pi = 0.3 GPa

tc as function of T and P t t lsvo s lsvo 128D dp kT = E 4 P + 128D dp tc (dp, T) 128D lsvo t c dp kT 3 o = from equation [3] incorporating T dependance of diffusivity E kT exp æ è ç d ö ø ÷ [6] tc (dp, P,T) é ù 4 s è æ P + 128D lsvo t c dp kT 3 o = ê ç ú ç ç E V a combining eqn’s [1], [3], and [5] to include effect of internal pressure on D exp ê + ç d ú æ [7] d a è p ê ú ú ê kT ë û Ed = 5.44 x 10-19 J molecule-1 (328 kJ/mole) Rodriguez-Viejo et al. (1993) Appl. Phys. Lett. 63, 1906. Do = 1.1 x 10-2 cm2 sec-1 , ibid. vo = 6.9cm3 (based upon diameter of oxygen ion, 2.8 A) Pa = 1 atm (1.013 bars) s = 0.3 J m-2 Kingery et al. (1976) Introduction to Ceramics Va =19.2 cm3 mole-1, Aziz et al., Nature, 390, 596 (1997)

Enhanced coalescence rate for particles in initial stages of growth

Model Results, Improvements! Collision/sintering model for final primary particle size: da dt -1 (a - as) tc = [8] Koch and Friedlander (1990) J. Colloid Interface Sci. 140,419. In terms of particle volume for the case of two particles coalescing at one time, dvp dt 0.31 vp tc = [9] Lehtinen et al. (1996) J. Colloid Interface Sci. 182,606. Linear temperature profile and plug flow velocity profile: T(x) = 1720 K - 106 x x in cm [10] Ehrman et al. (1998) J. Aerosol Sci. 29, 687.

Particle growth for various coalescence times Atomistic, with effect of pressure Atomistic, No effect of pressure Viscous flow

Summary/Conclusions Magnitude of the pressure dependence appears to be significant Including pressure dependence increases rate of growth in initial stages. Effect becomes stronger as temperature decreases. Predictions of particle size made with collision/sintering model are closer to experimental values when effect of pressure is included. Though still not in quantitative agreement with experimental values, results from this study suggest effect of internal pressure is important for silica (and possibly other materials) and should be considered when estimating material properties of nanoparticles.

Discussion - outlook You’d better know temperature. Coalescence very T sensitive. Recent developments from Zachariah group (2003) - energy from heat released by reduction of surface area, heats particle above background gas T, and leads to quicker coalescence. Surface tension as a function of T also may important Still need better estimates of surface tension as function of particle size Impurities may affect coalescence