Lecture 2

Nucleation and Growth

 Importance - Nucleation and Growth
Precursor
Solution/Vapor
Formation of Nuclei

Homogenous Heterogeneous

Important controlling Parameters: Growth of

• Selection of Precursors Nuclei

• Purity of precursors Ostwald Agglomeration

Ripening
• Precursor Concentration
• Mixing Sequence Precipitation
• Reaction Temperature 1. Variation of Particle Size Distribution
2. Hard agglomeration
• Reaction Time 3. Nano size change to micron

SIZE is an important phenomenon for NANOMATERIALS

 Monosize and Wide Size Distribution Nanoparticles!!

20nm

Titanium Carbide
84nm

Iron Oxide
5, 9, 12,16 and 22nm
T. Hyeon et al, Nature Materials, 3, 2004, 891-95
D. Sarkar et al, J American Ceramic Society, 92 [12], 2877 – 2882, 2009
Effect of Nucleation and Growth on particle size distribution
Particle Size Distribution
monosize
Volume Fraction (%)

narrow size

wide size

Particle Size (nm)

 = Shape , b = Scale, g = location,

x = particle size and f(x) = Cumulative undersize

D. Sarkar et al, J American Ceramic Society, 92 [12] 2877 – 2882, 2009

 When Nucleation Start?

Liquid
A
Temperature
Consider cooling a
Te B liquid into a solid
C through a eutectic point
Solid D

Composition

at point A: solid is not stable so will not form

at point B: solid and liquid are both stable so no driving force to solid
at point C: liquid is unstable - will form solid
at point D: liquid is unstable - will form solid
Beyond equilibrium => Need driving force to form solid
Requires:
Transformation to solid phase 1. Nucleation of new phase
2. Growth of new phase
 Nucleation
• Nucleation, the first step…
• First process is for microscopic clusters (nuclei) of
atoms or ions to form
– Nuclei possess the beginnings of the structure of the crystal
– Only limited diffusion is necessary
– Thermodynamic driving force for crystallization must be
present

Two Type:
Homogeneous - random accumulation of mother molecules
Heterogeneous -small particles present in the solution act as nuclei
 Homogenous Nucleation
G1 G2 G2 – G1 = - DG

Solid particle
The change in free energy is balanced by
the energy gain of creating a new volume,
and the energy cost due to creation of a
new interface.

When the overall change in free energy,

DG is negative, nucleation is favored
J/cc)
In the classic case of a spherical cluster
that liberates -Gv J/cc during formation, but
which must pay the positive cost of σ J/cm2
of surface interfacing with the surrounding

Free Energy needed to form a cluster of radius r is ;

– DG = – 4/3r3Gv + 4r2
or DG = 4/3r3Gv – 4r2
 Homogenous Nucleation…Contd
Free energy to add molecules to this cluster, until the radius reaches Critical
radius
2 dG For Ti, DGv = 50J/cm3 and  = 50mJ/m2
r* = Where, =0
Gv dr r* = 2nm (5-10times larger than single unit cell)

Addition of new molecules to clusters larger than this critical radius is no longer
limited by nucleation, but perhaps by diffusion (i.e. the supply of molecules) or
continuous growth of nuclei

The free energy needed to form this critical radius can be found by
163
DG* =
3(Gv)2
which occurs at the maximum DG where dG / dr = 0
As the phase transformation becomes more and more favorable, the
formation of a given volume of nucleus frees enough energy to form an
increasingly large surface
Surface energy related to Gibbs free energy during nucleation !!
 Heterogeneous Nucleation
Heterogeneous nucleation occurs much more often than homogeneous nucleation

It forms at preferential sites such as phase boundaries or impurities like dust and
requires less energy than homogeneous nucleation.

At such preferential sites, the effective surface energy is lower, thus diminished the
free energy barrier and facilitating nucleation.

Surfaces promote nucleation because of wetting – contact angles greater than zero
between phases encourage particles to nucleate

The free energy needed for heterogeneous nucleation is equal to the product of
homogeneous nucleation and a function of the contact angle :

DGheterogeneous = DGhomogenous x f(q) Homo

f(q) = ½ + ¾ Cosq – ¼ Cos3q Hetero

DG
r
Energy needed for heterogeneous nucleation is reduced r*

Wetting angle determines the ease of nucleation by

reducing the energy needed.
 Nucleation and Growth Process
Formation of Nuclei
• formation favor:
– high initial concentration or supersaturation
– low viscosity
– low critical energy barrier
• uniform nanoparticle size:
– same time formation
– abruptly high supersaturation
Growth of Nuclei
• Growth processes then enlarge existing nuclei
• Smallest nuclei often redissolve
• Larger nuclei can get larger through diffusion and adsorption
• Thermodynamics favors the formation of larger nuclei
 Strong overlap of growth and
Growth Rate nucleation rates
•Nucleation rate is high
•Growth rate is high
Nucleation Rate •Both are high at the same
temperature

Temperature
Rate
Growth Rate
No overlap of growth and
nucleation rates
• Nucleation rate is small
• Growth rate is small
At any one temperature one of the Nucleation Rate
two is zero
 Nucleation & Growth

Typical precipitation reaction:

Reactant 1 + Reactant 2 T, t
Product + By-product
Stabilizer

Nucleation Agglomeration
(critical size) Primary particles
Particles
Growth

Crystallites
Clusters
 Precipitation
Precipitation has generally been shown to occur in four steps:

(a) nucleation
(b) crystal growth
(c) agglomeration and
(d) ripening of the solids

(a) Nucleation: a nucleus is a fine particle on which the

spontaneous formation or precipitation of a solid phase can
take place in a supersaturated solution.

Homogeneous nucleation occurs when the nuclei is formed

from component ions of the precipitate; if foreign particles are
the nuclei, heterogeneous nucleation occurs.
 Precipitation….Contd
(b) Crystal growth: crystals form by the deposition of the precipitate
constituent ions onto nuclei.
Crystal growth rate can be expressed as:

dC
=  kS (C  C*) n
dt
where
C* = saturation concentration (mole/L)
C = actual concentration of limiting ion (mole/L)
k = rate constant (Ln / time mg)
S = surface area available for precipitation (mg/L of a given
particle size)
n = constant
When the diffusion rate of ions to the surface of the crystal controls
the crystal growth rate, exponent n = 1; when other processes such as
the reaction rate at the crystal surface are rate limiting, n ≠ 1
 Precipitation….Contd
(c) Agglomeration : conversion of small particles into larger
particles is enhance by agglomeration of particles to form
larger particles, which is the continual growth until equilibrium
is reached. The changes in crystal structure that take place
over time are often called aging.

(d) Ostwald Ripening : A phenomenon called ripening may

also take place whereby the crystal size of the precipitate
increases. Growth of Protein Crystal
Day 6 – Day 10 – Day 13 – Day 16
 Ostwald ripening
Ostwald ripening is an observed phenomenon in solid (or liquid) solutions
which describes the change of an inhomogeneous structure over time. The
phenomenon was first described by Wilhelm Ostwald in 1896
• It is a spontaneous process that occurs because larger
crystals are more energetically favored than smaller crystals.

• While the formation of many small crystals is kinetically favored, (i.e. they
nucleate more easily) large crystals are thermodynamically favored.

• Thus, from a standpoint of kinetics, it is easier to nucleate many small

crystals. However, small crystals have a larger surface area to volume ratio
than large crystals.

• Molecules on the surface are energetically less stable than the ones
already well ordered and packed in the interior.

• Large crystals, with their greater volume to surface area ratio, represent a
lower energy state.

• Thus, many small crystals will attain a lower energy state if transformed
into large crystals and this is what we see in Ostwald ripening.
 Key Points
In a homogenous precipitation, a short single burst of nucleation occurs when the
concentration of constituent species reaches critical supersaturation
The nuclei so obtained are allowed to grow uniformly by diffusion of solutes from the
solution to their surface until the final size is attained

To achieve monodispersity, these two stages must be separated and nucleation

should be avoided during the period of growth (Curve I)
Curve III represents self-sharpening growth process, i.e Ostwald ripening
Uniform particles can be obtained due to aggregation of much smaller subunits
rather than continuous growth by diffusion (Curve II)