Coalescence agenda
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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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Coalescence - Agenda

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Coalescence agenda

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


Coalescence agenda

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


Coalescence agenda

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


Coalescence agenda

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 underpredictprimary

  • 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)).


Coalescence agenda

Further motivation

  • 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.


Coalescence agenda

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


Coalescence agenda

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.


Coalescence agenda

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

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


Coalescence agenda

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)

  • May result in phase and transport

  • behavior different from P = 1 atm.


Coalescence agenda

-E

PV

-

ö

æ

D

D

exp

a

=

d

ç

÷

kT

o

ø

è

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

for diffusion,

cm3 molecule -1

[5]

  • For typical crystalline materials, increasing pressure

  • leads to decreasing diffusivity. Va is positive, ~ equal

  • to volume of diffusing species.


Coalescence agenda

  • 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 increasesas pressure increases.

  • Va in Eq. 5 is negative! Vaestimates 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).


Coalescence agenda

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.


Coalescence agenda

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


Coalescence agenda

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 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)


Coalescence agenda

Enhanced coalescence rate for particles

in initial stages of growth


Coalescence agenda

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.


Coalescence agenda

  • Particle growth for various coalescence times

Atomistic, with effect of pressure

Atomistic,

No effect of pressure

Viscous flow


Coalescence agenda

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 importantfor silica (and possibly other materials)

  • and should be considered when estimating material

  • properties of nanoparticles.


Discussion outlook

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


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