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?
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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
necked
unagglomerated
agglomerated
t
time
coalesce
t
collision
residence time
Characteristic times and particle morphology
Characteristic
times depend on
concentration of
particles and on
material properties
Desired degree
of agglomeration
depends on application
Motivations
Further motivation
Collision/sintering
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[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.
As evidence of atomistic behavior in silica: viscosity
related to diffusivity, D through StokesEinstein 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
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
Pressure inside nanoparticles
Laplace Equation
Pi
Pi  Pa = 4s[1]
dp
s = surface tension
dp = particle diameter
Pi = internal pressure
Pa = ambient pressure
Pa
Pi for 3 nm diameter silica
particle ~ 2000 atmospheres!
(~ 0.2 gigaPascals)
E
PV

ö
æ
D
D
exp
a
=
d
ç
÷
kT
o
ø
è
Effect of P on diffusivity
Ed = activation energy
for diffusion,
J molecule 1
Va = activation volume
for diffusion,
cm3 molecule 1
[5]
Why?
Pressure Facilitated Diffusion
(c) After decompression,
tetrahedral framework
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
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
dp
kT
3
E
t
ö
exp
æ
=
d
÷
lsvo
ç
c
128D
kT
ø
è
o
dp
kT
3
t
=
lsvo
c
128D
o
tc as function of T and P
tc (dp, T)
from equation [3]
incorporating T
dependance of diffusivity
[6]
tc (dp, P,T)
ù
è
s
é
4
æ
P
+
ú
ç
ê
ç
ç
E
V
a
ç
d
+
ú
combining eqn’s
[1], [3], and [5]
to include effect
of internal pressure on D
ê
[7]
exp
æ
p
è
ú
d
a
ê
ú
ê
kT
ë
û
Ed = 5.44 x 1019 J molecule1 (328 kJ/mole) RodriguezViejo et al. (1993) Appl. Phys. Lett. 63, 1906.
Do = 1.1 x 102 cm2 sec1 , ibid.
vo = 6.9cm3 (based upon diameter of oxygen ion, 2.8 A)
Pa = 1 atm (1.013 bars)
s = 0.3 J m2 Kingery et al. (1976) Introduction to Ceramics
Va =19.2 cm3 mole1, Aziz et al., Nature, 390, 596 (1997)
Enhanced coalescence rate for particles
in initial stages of growth
da
dt
1 (a  as)
=
tc
dvp
dt
0.31 vp
tc
=
Model Results, Improvements!
Collision/sintering model for final primary particle size:
[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,
[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.
Atomistic, with effect of pressure
Atomistic,
No effect of pressure
Viscous flow
Summary/Conclusions
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