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?
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
How about finite coalescence rate?
Important for particle growth in steep T gradients, e.g. flames
particles grow by
fast compared to
particle formation times
particles are necked
primary particle size
time between particle
extent of agglomeration
time required for particle
Characteristic times and particle morphology
times depend on
particles and on
depends on application
see Koch and Friedlander, 1990; Friedlander and Wu, 1994; Lehtinen et al., 1996
flame generated silica particles
TEM - S.H. Ehrman
Characteristic coalescence time
for viscous flow
tc = dp Frenkel (1945) J.Phys. 9,385.
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.
As evidence of atomistic behavior in silica: viscosity
related to diffusivity, D through Stokes-Einstein relationship:
= kT 
has been observed experimentally
for mixed silicates by Shimizu and
Kushiro(1984) Geochim. Cosmochim.
Acta. 48, 1295.
l = volume of oxygen anion
Coalescence as atomistic process:
Coalescence via solid state diffusion mechanism
 Friedlander and Wu, Phys.
Rev. B, 49, 3622 (1994)
vp = particle volume s = surface tension
D = solid state diffusivity vo = volume of diffusing species
Pressure inside nanoparticles
Pi - Pa = 4s
s = surface tension
dp = particle diameter
Pi = internal pressure
Pa = ambient pressure
Pi for 3 nm diameter silica
particle ~ 2000 atmospheres!
(~ 0.2 gigaPascals)
Effect of P on diffusivity
Ed = activation energy
J molecule -1
Va = activation volume
cm3 molecule -1
Pressure Facilitated Diffusion
(c) After decompression,
rearranged, and diffusion
has taken place.
(a) Silicon ( ) in tetra-
hedral coordination, pressure = 1 atm.
(b) As pressure increases, up
to Pcritical, areas of higher
coordinated silicon form locally.
Method proposed by Tsuneyuki and Matsui (1995) Phys. Rev. Let. 74, 3197.
Effect of P on D, for silica
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
tc (dp, T)
from equation 
dependance of diffusivity
tc (dp, P,T)
, , and 
to include effect
of internal pressure on D
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
-1 (a - as)
Model Results, Improvements!
Collision/sintering model for final primary particle size:
 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,
 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  Ehrman et al. (1998)
J. Aerosol Sci. 29, 687.
Atomistic, with effect of pressure
No effect of pressure
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