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Lecture 2
Nucleation and Growth
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Importance - Nucleation and Growth
Important controlling Parameters:
Selection of Precursors
Purity of precursors
Precursor Concentration
Mixing Sequence
Reaction Temperature
Reaction Time
Precursor
Solution/Vapor
Ostwald
Ripening
Formation of Nuclei
Growth of
Nuclei
Agglomeration
Precipitation
Homogenous Heterogeneous
1. Variation of Particle Size Distribution
2. Hard agglomeration
3. Nano size change to micron
SIZE is an important phenomenon for NANOMATERIALS
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Monosize and Wide Size Distribution Nanoparticles!!
20nm
T. Hyeon et al, Nature Materials, 3, 2004, 891-95D. Sarkar et al, J American Ceramic Society, 92 [12], 2877 2882, 2009
Iron Oxide
5, 9, 12,16 and 22nm
Titanium Carbide
84nm
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Particle Size Distribution
Particle Size (nm)
VolumeFraction(
%)
monosize
narrow size
wide size
= Shape , b = Scale,g = location,x = particle size and f(x) = Cumulative undersize
Effect of Nucleation and Growth on particle size distribution
D. Sarkar et al, J American Ceramic Society, 92 [12] 2877 2882, 2009
Cumulative undersize mass distribution
gives the amount of particle smaller than
the defined size.
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Nucleation
Nucleation, the first step
First process is for microscopic clusters (nuclei) ofatoms 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 bepresent
Homogeneous - random accumulation of mother molecules
Heterogeneous -small particles present in the solution act as nuclei
Two Type:
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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
In the classic case of a spherical cluster
that liberates -Gv J/cc during formation, but
which must pay the positive cost of J/cm2
of surface interfacing with the surrounding
Homogenous Nucleation
Free Energy needed to form a cluster of radius r is ;
J/cc)
First term shows the energy gain of creating a new volume
Second term shows the energy loss due to surface tension of the new interface
Solid particle
G1 G2
DG = 4/3r3Gv + 4r2
orDG = 4/3r3Gv 4r2
G2 G1 = - DG
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Free energy to add molecules to this cluster, until the radius reaches Critical
radius
Homogenous NucleationContd
Addition of new molecules to clusters larger than this critical radius is no longer
l imited by nucleat ion, but perhaps by diffusion (i.e. the supply of molecules) or
cont inuous growth of nuc lei
The free energy needed to form this critical radius can be found by
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
r* =2
GvdGdr = 0Where,
DG* =163
3(Gv)2
Surface energy related to Gibbs free energy during nucleation !!
For Ti, DGv = 50J/cm3 and = 50mJ/m2
r* = 2nm (5-10times larger than single unit cell)
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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)
f(q) = + Cosq Cos3q
Energy needed for heterogeneous nucleation is reduced
Wetting angle determines the ease of nucleation byreducing the energy needed.
Hetero
Homo
DGr
r*
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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
Nucleation and Growth Process
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Strong overlap of growth and
nucleation rates
Nucleation rate is high
Growth rate is highBoth are high at the same
temperature
No overlap of growth and
nucleation rates Nucleation rate is small
Growth rate is small
At any one temperature one of the
two is zero
RateTemperature
Nucleation Rate
Growth Rate
Nucleation Rate
Growth Rate
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Nucleation
(critical size)
Agglomeration
Clusters
Crystallites
Primary particlesParticles
Growth
Typical precipitation reaction:
Reactant 1 + Reactant 2 Product + By-productT, t
Stabilizer
Nucleation & Growth
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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.
Homogeneousnucleation occurs when the nuclei is formed
from component ions of the precipitate; if foreign particles are
the nuclei, heterogeneousnucleation occurs.
Precipitation
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(b) Crystal growth: crystals form by the deposition of the precipitate
constituent ions onto nuclei.
Crystal growth rate can be expressed as:
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 givenparticle 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 asthe reaction rate at the crystal surface are rate limiting, n 1
dC
dtkS C C
n= ( *)
Precipitation.Contd
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(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 ripeningmayalso take place whereby the crystal size of the precipitate
increases.
Precipitation.Contd
Growth of Protein Crystal
Day 6 Day 10 Day 13 Day 16
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Ostwald ripeningOstwald 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 largercrystals are more energetically favored than smaller crystals.
While the formation of many small crystals is kineticallyfavored, (i.e. theynucleate more easily) large crystals are thermodynamicallyfavored.
Thus, from a standpoint of kinetics, it is easier to nucleate many smallcrystals. However, small crystals have a larger surface area to volume ratiothan 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 alower energy state.
Thus, many small crystals will attain a lower energy state if transformedinto large crystals and this is what we see in Ostwald ripening.
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Conclusion
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)
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 thesolution to their surface until the final size is attained