1. 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

2. 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

3. 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

4. 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)).

5. 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.

6. 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

7. 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.

8.  = 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

9. 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.

10. 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.

11. 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).

12. 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.

13. 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

14. 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 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)

15. Enhanced coalescence rate for particles in initial stages of growth

16. 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 x x in cm [10] Ehrman et al. (1998)
J. Aerosol Sci. 29, 687.

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

18. 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.

19. 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