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Introduction to Materials Science and Engineering 21089201 Chedtha Puncreobutr Department of Metallurgical Engineering Chulalongkorn University Nucleation and Growth Kinetics

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Page 1: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Introduction to Materials

Science and Engineering

21089201

Chedtha PuncreobutrDepartment of Metallurgical Engineering

Chulalongkorn University

Nucleation and Growth Kinetics

Page 2: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

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.

Chedtha Puncreobutr

Page 3: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Nucleation

3

• 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

Chedtha Puncreobutr

Page 4: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Nucleation mechanisms

4

• 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.

Chedtha Puncreobutr

Page 5: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Driving force

5

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

Chedtha Puncreobutr

Page 6: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Homogeneous Nucleation

6

• 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

Chedtha Puncreobutr

Page 7: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Homogeneous Nucleation & Energy Effects

7

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

Chedtha Puncreobutr

Page 8: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Homogeneous Nucleation & Critical radius

8

∆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

Chedtha Puncreobutr

Page 9: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Homogeneous Nucleation & Critical radius

9

∆𝐺∗ =16𝜋𝛾𝑆𝐿

3𝑇𝑚

2

3∆𝐻𝑓2∆𝑇2

𝑟∗ =2𝛾𝑆𝐿𝑇𝑚∆𝐻𝑓∆𝑇

r* decreases as DT increases

For typical DT r* ~ 10 nm

Chedtha Puncreobutr

Page 10: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Critical Radius for the Solidification of Copper

10

Calculate the size of the critical radius and the number of atoms in the critical nucleus when solid copper forms by homogeneous nucleation.

Chedtha Puncreobutr

Page 11: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Rate of formation of homogeneous nuclei

11

rNumber of stable nuclei

𝑛∗ = 𝑛𝐿𝑒𝑥𝑝 −∆𝐺∗

𝑘𝐵𝑇

= Boltzmann’s constant𝑘𝐵

𝑣𝑑 atomic vibration frequency

𝑝𝑐 probability of capturing an atom at the surface

Chedtha Puncreobutr

Page 12: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Rate of formation of homogeneous nuclei

12

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

Chedtha Puncreobutr

ሶ𝑁ℎ𝑜𝑚

Page 13: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Heterogeneous nucleation

13Chedtha Puncreobutr

Page 14: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Heterogeneous nucleation

14

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

Chedtha Puncreobutr

Page 15: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Heterogeneous nucleation

15

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.

Chedtha Puncreobutr

Page 16: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Heterogeneous nucleation

16

∆𝐺ℎ𝑒𝑡𝑒𝑟 =16𝜋𝛾𝑆𝐿

3𝑇𝑚

2

3∆𝐻𝑓2∆𝑇2

𝑓(𝜃)

=16𝜋𝛾𝑆𝐿

3𝑇𝑚

2

3∆𝐻𝑓2∆𝑇2

Chedtha Puncreobutr

Page 17: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Heterogeneous nucleation

17

• 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.

Chedtha Puncreobutr

Page 18: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Growth

18

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

Chedtha Puncreobutr

• 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

Page 19: Nucleation and Growth Kineticspioneer.netserv.chula.ac.th/~pchedtha/Lecture_nucleation-growth-kinetics_2189201_cp.pdfNucleation 3 • Nucleation is the process of forming a nucleus

Growth

19

• 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)

Chedtha Puncreobutr

• 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).