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