Post on 15-Nov-2021
Chapter 4. Formation and Growth of Nanoparticles - Theories
4.1 Introduction
- Breakdown (Disintegration) : by crushing, milling, spraying
*"Wall of 3μm"
Bulk fine particles
- Buildup (Growth) : nucleation and growth (condensation, coagulation)
Atoms or molecules Nuclei Nanoparticles
*
4.2 Birth of Nanoparticles: Nucleation
Formation of nuclei: new-born particles
(1) Homogeneous nucleation
- Gibbs free energy change for a droplet from monomer molecules
where
vm: molecular mass
- Saturation ratio
For solution-to-particle:
For vapor-to-droplet or particle:
where c0, p0: equilibrium solubility, vapor pressure
v
p
p Gd
dG ∆−=∆6
3πσπ
volume energyinterfacial energy
0c
cS =
Gibbs free energy change for water droplet formation
Sv
kTG
m
v ln=∆
0p
pS =
- Critical nuclei
From
2)3(
16*
vGG
∆=∆ πσ
0=∆
pdd
Gd
0
ln
44*
PPkT
mv
vG
pdσσ =
∆−= Critical nuclei diameter
Kelvin equation
Energy barrier for nucleation
- Rate of nucleation
J(nuclei/(s)(cm3))=(frequency of collision) (species concentration) (probability)
∆−=kT
Gmnpd
mkT
pJ
*exp)2*(
2/1)2(π
π
∆−=kT
Gmn
dkTJ
*exp
3 µπ
nm: number concentration of the species
Solution-to-particle
Vapor-to-droplet or particle
- Critical saturation, Scrit*: S at J=1 Nucleation rate of
water droplets at
300K
*4
2)cos1)(cos2(** GGfhet
G ∆−+=∆=∆ θθ
(2) Heterogeneous nucleation
Correction term f vs. the contact angle in radian
* Used for formation of artificial rain (seed: AgCl, AgI)
Core-shell particles
Monodisperse particles
Films
* Critical supersaturation ratio (Scrit)
If S > Scrit ⇒ Homogeneous nucleation
S < Scrit ⇒ Heterogeneous nucleation
* Particles or Films?
High P/P0
or S > Scrit
Low P/P0
or S < Scrit
Homogeneous nucleation
If foreign nuclei exist
If free surface exists
Thin film Whiskers Crystals
Vapor molecules
Heterogeneous nucleation
Nuclei
Nuclei Particles
Growth
2/1
21
)2(
)(2
mkT
ppv
dt
ddmp
π−
=
kTd
ppDv
dt
dd
p
mp )(4 21 −=p
mp
d
nnDv
dt
dd )(4 21 −=
2/1
1
)2(
2
mkT
pvk
dt
ddmrp
πα
= 12 nvkdt
ddmr
p =
rMd
dt
dd R
i
iip
pp
= ∑
=13νρ
Droplet phase reaction limited
Surface reaction limited
Diffusion limited
Liquid phaseGas phase
gaspd λ<<
gaspd λ>>
- Rate of condensation
If p1<p2, evaporation instead of condensation
http://www.eng.uc.edu/~gbeaucag/Classes/Nanopowders/
4.3 Condensational Growth of Nanoparticles
Growth of particles by collision followed by sticking
2
0
2
pDp dtkd += )(8 21 nnDvk mD −=
ppp ddd δ+>=< 000 ppp ddd δ+>=<
00
2
0
2 22 pppDppp dddtkddd δδ ><+><+=><+><
p
pp
pd
ddd
00δδ =
2
0
2 ><+=>< pDp dtkd
2
0
00
pD
pp
p
dtk
ddd
+=
δδ
Substituting (**)
For diffusion-controlled condensation
Integration yields
Considering a variation in size
(*) where
and
Substituting these into (*), omitting square term which is small
Since (**)
Then
The variation in size decreases as growth proceeds!
0ppp dtkd +=12 nvkk mrp =
0pp dd δδ =∴
For surface reaction-controlled condensation
whereIntegration
The variation in size is kept constant but the relative size variation decrease as
growth proceeds!
4.4 Coagulation: Coalescence and Aggregation
Growth as a result of collision and subsequent coagulation with other particles.
- Sources of collision
Brownian motion, external force fields, particle-particle interaction
(polar, coulombic)
- Coagulation efficiency
decreaseincreaseTotal particle mass
concentration
decreaseNo changeParticle number
concentration
increaseincreasesize
coagulationcondensation
Comparison of condensation and coagulation
(1) Shapes of particles coagulated (secondary particles)
Spherical/near-spherical growth
: Liquid-phase or liquid-like growth
- Liquid particles grow by "coalescence"..
- Solid particles grow by "aggregation (agglomeration)" and
followed by rapid sintering
Random (fractal) growth
: Solid-phase growth
- Fractal representation
where V: solid volume
l: characteristic length of the aggregate
D: fractal dimension(≠1, 2, 3)
DlV ~Aggregate of TiO2 nanoparticles
Directional (anisotropic) growth
With interparticle forces - electrical or magnetic dipoles
e.g. Smoke particles, iron nanoparticles
Iron particles
Fast sintering Slow sintering Dipole aggregation
Primary particles Secondary particles
Fraction growth
- Coalescence or Aggregation?
-Depends on the characteristic times of collision and coalescence
* Solid particle growth
(2) Mathematical expression for coagulation
Ni,j : Number of collisions occurring per unit time per unit volume between the
particles having diameters (volumes) di(vi) and dj(vj) respectively.
ni, nj: number concentration of colliding particles vi, vj, respectively.(no./cm3)
i, j: number of basic units making particles (e.g. i-mer and j-mer)
In terms of continuous size distribution
where bi,j: collision frequency function
"coagulation coefficient"
Increase in size
jijiji nnbN ,, =
( ) 3/133
, pjpinewp ddd +=
jijijiji dvdvvnvnvvbvvN )()(),(),( =
jinew vvv +=
- Change in number concentration of the particles, v
- For polydisperse particles
When di > dj, bi, j> bi,i or bj,j
- Different-size coagulation occurs faster than that between similar-size coagulation.
Monodisperse particles becomes polydisperse…
Polydisperse particles becomes monodisperse…
∴Self preserving model: distribution converges to
+
+
= 3
1
3
1
3/13/1'
'
11
3
2)',( vv
vv
kTvvb
µ
( )pjpi
pjpi
dddd
kTvvb +
+
=
11
3
2)',(
µ
4.1=gσ
or
∫∫∞
−−−=00
')'()',()(')'()'()','(2
1)(dvvnvvbvndvvvnvnvvvb
dt
vdnv
Input by coagulation Output by coagulation
∑ ∑∑ ∑=+
∞
==+
∞
=
−=−=kji i
ikikjiji
kji i
kijik nbnnnbNN
dt
dn
0
,,
0
,,2
1
2
1
For continuum regime
- For Brownian coagulation
or
Solving for total number concentration of particles, N and
average diameter of the particles, d
0
KtN
NtN
0
0
1)(
+=
3/1
0
0 )(
)(
=
tN
N
d
tdand
smCKkT
b c /100.33
8 316−×===µ
∑ ∑=+
∞
=
−=kji i
ikjik nKnnn
K
dt
dn
02
- For monodisperse particles
Under standard conditions
Saturation ratio, S
Time
Induction
period
Nucleation
Condensation,
Coagulation
Aging ⇒
Critical supersaturation ratio
1.0
Scrit
Saturation ratio, S
Time
Induction
period
Nucleation
Condensation,
Coagulation
Aging ⇒
Critical supersaturation ratio
1.0
Scrit
4.5 Some Comments on Particle Growth
(1) Overall growth
(2) Growth mechanisms and particle size distribution
* Nucleation
- increase of particle number concentration
- may cause accelerating the rate of coagulation
- gives delta function in particle size distribution in given condition
* Condensation
- No effect of particle number concentration
- results in monodisperse size distribution.
* Coagulation
- Decreases in particle number concentration
- gives polydisperse size distribution in growth process
- Matijevic’s method
(3) Preparation of monodisperse particles
- Maintain low rate of nucleation
- using low supersaturation
- inducing heterogeneous nucleation
- Allow the same growth time for all the particles by shortening the time of
nucleation
- Suppress coagulation
- Using electrostatic repulsion (electrical double layer)
- Using adsorption of surfactants and macromolecules
- Rapid cooling and dilution
(4) Composite particles
* Mixed –phase particles from mixtures of precursors:
Since reaction (or evaporation), critical supersaturation, saturation:
different for each component,
- Different history of nucleation
- Possible occurrence of heterogeneous nucleation
- Different condensation rate
- uniform distribution of components within each particle: unexpected
“Composition gradient”
- Confined growth: preferred
e.g. spray-pyrolysis, emulsion, in-pore and in-layer growth
* Core-shell composite particles
coating of existing particles
start with heterogeneous nucleation and subsequent growth