nucleation and growth...
TRANSCRIPT
Introduction to Materials
Science and Engineering
21089201
Chedtha PuncreobutrDepartment of Metallurgical Engineering
Chulalongkorn University
Nucleation and Growth Kinetics
Phase transformation
2
Nucleation - The physical process by which a new phase is produced in a material. It is the initial process in crystallization.
Most phase transformations begin with the formation of numerous small particles of the new phase that increase in size until the transformation is complete.
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Nucleation
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• Nucleation is the process of forming a nucleus
• It is the process in which ions, atoms, or
molecules arrange themselves in a pattern
characteristic of a crystalline solid, forming a
site in which additional particles deposit as
the crystal grows
• Some examples of phases that may form by way
of nucleation in liquids are gaseous bubbles,
crystals, or glassy regions.
• Creation of liquid droplets in saturated vapour is
also characterized by nucleation
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Nucleation mechanisms
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• Homogeneous nucleation
Formation of a critically sized solid from the liquid by
the clustering together of a large number of atoms at
a high undercooling (without an external interface).
• Heterogeneous nucleation
Formation of a critically sized solid from the liquid on
an impurity surface.
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Driving force
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The change in free energy DG (function of the internal energy and enthalpy of the system) must be negative for a transformation to occur.
𝑻 > 𝑻𝒎
𝑻 < 𝑻𝒎
liquid
solid
∆𝐺 = ∆𝐻 − 𝑇∆𝑆
∆𝐺 = 0 = ∆𝐻 − 𝑇𝑚∆𝑆𝑎𝑡 𝑇𝑚Undercooling:
∆𝑆 =∆𝐻
𝑇𝑚=
∆𝐻𝑓
𝑇𝑚entropy of fusion
Temperature
Mo
lar
free
en
ergy
(D
G)
𝐺𝑆
𝐺𝐿
∆𝐺
𝑇𝑚𝑇
∆𝑇
for small ∆𝑇 ∆𝐺 = ∆𝐻𝑓 − 𝑇∆𝐻𝑓
𝑇𝑚∆𝐺 ≅
∆𝐻𝑓𝑇𝑚
∆𝑇
∆𝑇 = 𝑇𝑚 − 𝑇
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Homogeneous Nucleation
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• Assume that nuclei of the solid phase form spontaneously in the interior of the liquid as atoms cluster together-similar to the packing in the solid phase.
• Also, each nucleus is spherical and has a radius r
• Free energy changes as a result of a transformation:
1) the difference between the solid and liquid phases (volume free energy, DGV)
2) the solid-liquid phase boundary (surface free energy, DGS).
∆𝐺 = −𝑉𝑠∆𝐺𝑣 + 𝐴𝑠𝑙𝛾𝑠𝑙
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Homogeneous Nucleation & Energy Effects
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DGT = Total Free Energy
= DGS + DGV
Surface Free Energy - destabilizes
the nuclei (it takes energy to make
an interface)
D 24 rGS
= surface tension
Volume (Bulk) Free Energy –
stabilizes the nuclei (releases energy)
DD GrGV3
3
4
∆𝐺𝑉 = −4
3𝜋𝑟3
∆𝐻𝑓𝑇𝑚
∆𝑇
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Homogeneous Nucleation & Critical radius
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∆𝐺 = −4
3𝜋𝑟3
∆𝐻𝑓𝑇𝑚
∆𝑇 + 4𝜋𝑟2𝛾𝑆𝐿
𝑑∆𝐺
𝑑𝑟= 0at 𝑟∗
∆𝐺∗ =16𝜋𝛾𝑆𝐿
3𝑇𝑚
2
3∆𝐻𝑓2∆𝑇2
𝑟 < 𝑟∗
𝑟 > 𝑟∗
embryos - melt back
nuclei - grow
(∆𝐺ℎ𝑜𝑚𝑜)
𝑟∗ =2𝛾𝑆𝐿𝑇𝑚∆𝐻𝑓∆𝑇
surface free energy α 𝑟2
volume free energy
α 𝑟3∆𝑇
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Homogeneous Nucleation & Critical radius
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∆𝐺∗ =16𝜋𝛾𝑆𝐿
3𝑇𝑚
2
3∆𝐻𝑓2∆𝑇2
𝑟∗ =2𝛾𝑆𝐿𝑇𝑚∆𝐻𝑓∆𝑇
r* decreases as DT increases
For typical DT r* ~ 10 nm
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Critical Radius for the Solidification of Copper
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Calculate the size of the critical radius and the number of atoms in the critical nucleus when solid copper forms by homogeneous nucleation.
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Rate of formation of homogeneous nuclei
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rNumber of stable nuclei
𝑛∗ = 𝑛𝐿𝑒𝑥𝑝 −∆𝐺∗
𝑘𝐵𝑇
= Boltzmann’s constant𝑘𝐵
𝑣𝑑 atomic vibration frequency
𝑝𝑐 probability of capturing an atom at the surface
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Rate of formation of homogeneous nuclei
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ሶ𝑁ℎ𝑜𝑚 = 𝑣𝑑𝑝𝑐𝑛𝐿𝑒𝑥𝑝 −16𝜋
3
𝛾𝑆𝐿3𝑇𝑚
2
𝑘𝐵𝑇∆𝐻𝑓2∆𝑇2
• At high temperatures, the thermodynamic
driving force for nucleation becomes less
due to the lower undercooling and the
number of critical clusters will be limited.
• At low temperatures, the thermodynamic
driving force for the nucleation is large due
to the large undercooling, but the rate of
diffusion of atoms to the nucleation site is
reduced
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ሶ𝑁ℎ𝑜𝑚
Heterogeneous nucleation
13Chedtha Puncreobutr
Heterogeneous nucleation
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When solidification is initiated from a foreign surface, i.e. solid particles suspended
in liquid, oxide layers, or surface contact with a crucible wall, it is said to nucleate
heterogeneously
Assuming a spherical cap
shape of nucleus formed
on a surface of foreign
substrate
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Heterogeneous nucleation
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The positive interfacial energy term can be reduced by
a factor
∆𝐺ℎ𝑒𝑡𝑒𝑟 =16𝜋𝛾𝑆𝐿
3𝑇𝑚
2
3∆𝐻𝑓2∆𝑇2
𝑓(𝜃)
∆𝐺ℎ𝑒𝑡𝑒𝑟 = ∆𝐺ℎ𝑜𝑚𝑜∙ 𝑓 𝜃
𝑓(𝜃)
= wetting angle
𝑓(𝜃)geometric factor given by ratio of the volumes of spherical cap and a full sphere of identical radius
Since the heterogeneous nucleation barrier is lower than that of homogeneousnucleation, it is much easier for heterogeneous nucleation to occur during solidification.
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Heterogeneous nucleation
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∆𝐺ℎ𝑒𝑡𝑒𝑟 =16𝜋𝛾𝑆𝐿
3𝑇𝑚
2
3∆𝐻𝑓2∆𝑇2
𝑓(𝜃)
=16𝜋𝛾𝑆𝐿
3𝑇𝑚
2
3∆𝐻𝑓2∆𝑇2
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Heterogeneous nucleation
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• Lower free energy for heterogeneous
nucleation means a smaller energy to
overcome during nucleation process,
therefore, heterogeneous nucleation
occurs more readily
• Much smaller degree of undercooling
is required for heterogeneous
nucleation.
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Growth
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ሶ𝐺 = 𝐶 𝑒𝑥𝑝 −𝑄
𝑘𝐵𝑇
• Growth rate is determined by the rate of diffusion, and its temperature dependence is the same as for the diffusion coefficient
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• Growth step in a phase transformation begins once an embryo has exceeded the critical size and becomes a stable nucleus
• Particle growth occurs by long-range atomic diffusion
Growth
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• Transformation occurs near melting point, low nucleation but high growth rate. Thus, resulting microstructure will consist few and relatively large particles (e.g. coarse grains)
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• Nucleation will continue to occur simultaneously with growth of the new phase particles. At a specific temperature, the overall transformation rate is equal to some product of nucleation and growth rates
• The size of the product phase particles will depend on transformation temperature
• Conversely, for transformations at lower temperatures, nucleation rates are high and growth rates low, which results in many small particles (e.g., fine grains).